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An Accurate Worst Case Timing Analysis for RISC Processors.

An accurate and safe estimation of a tasks worst case execution time (WCET) is crucial for reasoning about the timing properties of real-time systems. In RISC processors, the execution time of a program construct (e.g., a statement) is affected by various factors such as cache hits/misses and pipeline hazards, and these factors impose serious problems in analyzing the WCETs of tasks. To analyze the timing effects of RISCs pipelined execution and cache memory, we propose extensions to the original timing schema where the timing information associated with each program construct is a simple time-bound. In our approach, associated with each program construct is worst case timing abstraction, (WCTA), which contains detailed timing information of every execution path that might be the worst case execution path of the program construct. This extension leads to a revised timing schema that is similar to the original timing schema except that concatenation and pruning operations on WCTAs are newly defined to replace the add and max operations on time-bounds in the original timingschema. Our revised timing schema accurately accounts for the timing effects of pipelined execution and cache memory not only within but also across program constructs. This paper also reports on preliminary results of WCET analysis for a RISC processor. Our results show that tight WCET bounds (within a maximum of about 30% overestimation) can be obtained by using the revised timing schema approach.

INTRODUCTION In real-time computing systems, tasks have timing requirements (i.e., deadlines) that must be met for correct operation. Thus, it is of utmost importance to guarantee that tasks finish before their deadlines. Various scheduling techniques, both static and dynamic, have been proposed to ensure this guarantee. These scheduling algorithms generally require that the WCET (Worst Case Execution Time) of each task in the system be known a priori. Therefore, it is not surprising that considerable research has focused on the estimation of the WCETs of tasks. In a non-pipelined processor without cache memory, it is relatively easy to obtain a tight bound on the WCET of a sequence of instructions. One simply has to sum up their individual execution times that are usually given in a table. The WCET of a program can then be calculated by traversing the program's syntax tree bottom-up and applying formulas for calculating the WCETs of various language constructs. However, for RISC processors such a simple analysis may not be appropriate because of their pipelined execution and cache memory. In RISC processors, an instruction's execution time varies widely depending on many factors such as pipeline stalls due to hazards and cache hits/misses. One can still obtain a safe WCET bound by assuming the worst case execution scenario (e.g., each instruction suffers from every kind of hazard and every memory access results in a cache miss). However, such a pessimistic approach would yield an extremely loose WCET bound resulting in severe under-utilization of machine resources. Our goal is to predict tight and safe WCET bounds of tasks for RISC processors. Achieving this goal would permit RISC processors to be widely used in real-time systems. Our approach is based on an extension of the timing schema [1]. The timing schema is a set of formulas for computing execution time bounds of language constructs. In the original timing schema, the timing information associated with each program construct is a simple time-bound. This choice of timing information facilitates a simple and accurate timing analysis for processors with fixed execution times. However, for RISC processors, such timing information is not sufficient to accurately account for timing variations resulting from pipelined execution and cache memory. This paper proposes extensions to the original timing schema to rectify the above problem. We associate with each program construct what we call a (Worst Case Timing Abstraction). The of a program construct contains timing information of every execution path that might be the worst case execution path of the program construct. Each timing information includes information about the factors that may affect the timing of the succeeding program construct. It also includes the information that is needed to refine the execution time of the program construct when the timing information of the preceding program construct becomes available at a later stage of WCET analysis. This extension leads to a revised timing schema that accurately accounts for the timing variation which results from the history sensitive nature of pipelined execution and cache memory. We assume that each task is sequential and that some form of cache partitioning [2, 3] is used to prevent tasks from affecting each other's timing behavior. Without these assumptions, it would not be possible to eliminate the unpredictability due to task interaction. For example, consider a real-time system in which a preemptive scheduling policy is used and the cache is not partitioned. In such a system, a burst of cache misses usually occurs when a previously preempted task resumes execution. Increase of the task execution time resulting from such a burst of cache misses cannot be bounded by analyzing each task in isolation. This paper is organized as follows. In Section II, we survey the related work. Section III focuses on the problems associated with accurately estimating the WCETs of tasks in pipelined processors. We then present our method for solving these problems. In Section IV, we describe an accurate timing analysis technique for instruction cache memory and explain how this technique can be combined with the pipeline timing analysis technique given in Section III. Section V identifies the differences between the WCET analysis of instruction caches and that of data caches, and explains how we address the issues resulting from these differences. In Section VI, we report on preliminary results of WCET analyses for a RISC processor. Finally, the conclusion is given in Section VII. II RELATED WORK A timing prediction method for real-time systems should be able to give safe and accurate WCET bounds of tasks. Measurement-based and analytical techniques have been used to obtain such bounds. Measurement-based techniques are, in many cases, inadequate to produce a timing estimation for real-time systems since their predictions are usually not guaranteed, or enormous cost is needed. Due to these limitations, analytical approaches are becoming more popular [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Many of these analytical studies, however, consider a simple machine model, thus largely ignoring the timing effects of pipelined execution and cache memory [8, 12, 13, 15]. A. Timing Analysis of Pipelined Execution The timing effects of pipelined execution have been recently studied by Harmon, Baker, and Whalley [6], Harcourt, Mauney, and Cook [5], Narasimhan and Nilsen [11], and Choi, Lee, and Kang [4]. In these studies, the execution time of a sequence of instructions is estimated by modeling a pipelined processor as a set of resources and representing each instruction as a process that acquires and consumes a subset of the resources in time. In order to mechanize the process of calculating the execution time, they use various techniques: pattern matching [6], SCCS (Synchronous Calculus of Communicating Systems) [5], retargetable pipeline simulation [11], and ACSR (Algebra of Communicating Shared Resources) [4]. Although these approaches have the advantage of being formal and machine independent, their applications are currently limited to calculating the execution time of a sequence of instructions or a given sequence of basic blocks 1 . Therefore, they rely on ad hoc methods to calculate the WCETs of programs. The pipeline timing analysis technique by Zhang, Burns and Nicholson [16] can mechanically calculate the WCETs of programs for a pipelined processor. Their analysis technique is based on a mathematical model of the pipelined Intel 80C188 processor. This model takes into account the overlap between instruction execution and opcode prefetching in 80C188. In their approach, the WCET of each basic block in a program is individually calculated based on the mathematical model. The WCET of the program is then calculated using the WCETs of the constituent basic blocks and timing formulas for calculating the WCETs of various language constructs. Although this approach represents significant progress over the previous schemes that did not consider the timing effects of pipelined execution, it still suffers from two inefficiencies. First, the pipelining effects across basic blocks are not accurately accounted for. In general, due to data dependencies and resource conflicts within the execution pipeline, a basic block's execution time will differ depending on what the surrounding basic blocks are. However, since their approach requires that the WCET of each basic block be independently calculated, they make the worst case assumption on the preceding basic block (e.g., the last instruction of every basic block that can precede the basic block being analyzed has data memory access, which prevents the opcode prefetching of the first instruction of the basic block being analyzed). This assumption is reasonable for their target processor since its pipeline has only two stages. However, completely ignoring pipelining effects across basic blocks may yield a very loose WCET estimation for more deeply pipelined processors. Second, although their mathematical model is very effective for the Intel 80C188 processor, the model is not general enough to be applicable to other pipelined processors. This is due to the many machine specific assumptions made in their model that are difficult to generalize. 1 A basic block is a sequence of consecutive instructions in which flow of control enters at the beginning and leaves at the end without halt or possibility of branching except at the end [17]. B. Timing Analysis of Cache Memory Cache memories have been widely used to bridge the speed gap between processor and main memory. However, designers of hard real-time systems are wary of using caches in their systems since the performance of caches is considered to be unpredictable. This concern stems from the following two sources: inter-task interference and intra-task interference. Inter-task interference is caused by task preemption. When a task is preempted, most of its cache blocks 2 are displaced by the newly scheduled task and the tasks scheduled thereafter. When the preempted task resumes execution, it makes references to the previously displaced blocks and experiences a burst of cache misses. This type of cache miss cannot be avoided in real-time systems with preemptive scheduling of tasks. The result is a wide variation in task execution time. This execution time variation can be eliminated by partitioning the cache and dedicating one or more partitions to each real-time task [2, 3]. This cache partitioning approach eliminates the inter-task interference caused by task preemption. Intra-task interference in caches occurs when more than one memory block of the same task compete with each other for the same cache block. This interference results in two types of cache miss: capacity misses and conflict misses [19]. Capacity misses are due to finite cache size. Conflict misses, on the other hand, are caused by a limited set associativity. These types of cache miss cannot be avoided if the cache has a limited size and/or set associativity. Among the analytical WCET prediction schemes that we are aware of, only four schemes take into account the timing variation resulting from intra-task cache interference (three for instruction caches [10, 9, 7] and one for data caches [14]). The static cache simulation approach which statically predicts hits or misses of instruction references is due to Arnold, Mueller, Whalley and Harmon [10]. In this approach, instructions are classified into the following four categories based on a data flow analysis: ffl always-hit: The instruction is always in the cache. ffl always-miss: The instruction is never in the cache. ffl first-hit: The first reference to the instruction hits in the cache. However, all the subsequent references miss in the cache. A block is the minimum unit of information that can be either present or not present in the cache-main memory hierarchy [18]. . if (cond) elsek Fig. 1. Sample C program fragment ffl first-miss: The first reference to the instruction misses in the cache. However, all the subsequent references hit in the cache. This approach is simple but has a number of limitations. One limitation is that the analysis is too conservative. As an example, consider the program fragment given in Fig. 1. Assume that both of the instruction memory blocks corresponding to S i (i.e., are mapped to the same cache block and that no other instruction memory block is mapped to that cache block. Further assume that the execution time of S i is much longer than that of S j . Under these assumptions, the worst case execution scenario of this program fragment is to repeatedly execute S i within the loop. In this worst case scenario, only the first access to b i will miss in the cache and all the subsequent accesses within the loop will hit in the cache. However, by being classified as always-miss, all the references to b i are treated as cache misses in this approach, which leads to a loose estimation of the loop's WCET. Another limitation of this approach is that the approach does not address the issues regarding pipelined execution and the use of data caches, which are commonly found in most RISC processors. In [9], Niehaus et al. discuss the potential benefits of identifying instruction references corresponding to always-hit and first-miss in the static cache simulation approach. However, as stated in [10], their analysis is rather abstract and no general method for analyzing the worst case timing behavior of programs is given. In [7], Liu and Lee propose techniques to derive WCET bounds of a cached program based on a transition diagram of cached states. Their WCET analysis uses an exhaustive search technique through the state transition diagram which has an exponential time complexity. To reduce the time complexity of this approach, they propose a number of approximate analysis methods each of which makes a different trade-off between the analysis complexity and the tightness of the resultant WCET bounds. Although the paper mentions that the methods are equally applicable to the data cache, the main focus is on the instruction cache since the issues pertinent to the data cache such as handling of write references and references with unknown addresses (cf. Section V) are not considered. Also, it is not clear how one can incorporate the analysis of pipelined execution into the framework. Rawat performs a static analysis for data caches [14]. His approach is similar to the graph coloring approach to register allocation [20]. The analysis proceeds as follows. First, live ranges of variables and those of memory blocks are computed 3 . Second, an interference graph is constructed for each cache block. An edge in the interference graph connects two memory blocks if they are mapped to the same cache block and their live ranges overlap with each other. Third, live ranges of memory blocks are split until they do not overlap with each other. If a live range of a memory block does not overlap with that of any other memory block, the memory block never gets replaced from the cache during execution within the live range. Therefore, the number of cache misses due to a memory block can be calculated from the frequency counts of its live ranges (i.e., how many times the program control flows into the live ranges). Finally, the total number of data cache misses is estimated by summing up the frequencies of all the live ranges of all the memory blocks used in the program. Although this analysis method is a step forward from the analysis methods in which every data reference is treated as a cache miss, it still suffers from the following three limitations. First, the analysis does not allow function calls and global variables, which severely limits its applicability. Second, the analysis leads to an overestimation of data cache misses resulting from the assumption that every possible execution path can be the worst case execution path. This limitation is similar to the first limitation of the static cache simulation approach. The third limitation of this approach is that it does not address the issues of locating the worst case execution path and of calculating the WCET, again limiting its applicability. 3 A live range of a variable (memory block) is a set of basic blocks during whose execution the variable (memory potentially resides in the cache [14]. RD IF ALU MD mult $25, $24 nop lw $24, 16($22) nop lw $25, 16($23) Fig. 2. Sample MIPS assembly code and the corresponding reservation table III PIPELINING EFFECTS In pipelined processors, various execution steps of instructions are simultaneously overlapped. Due to this overlapped execution, an instruction's execution time will differ depending on what the surrounding instructions are. However, this timing variation could not be accurately accounted for in the original timing schema since the timing information associated with each program construct is a simple time-bound. In this section, we extend the timing schema to rectify this problem. In our extended timing schema, the timing information of each program construct is a set of reservation tables rather than a time-bound. The reservation table was originally proposed to describe and analyze the activities within a pipeline [21]. In a reservation table, the vertical dimension represents the stages in the pipeline and the horizontal dimension represents time. Fig. 2 shows a sample basic block in the MIPS assembly language [22] and the corresponding reservation table. In the figure, each x in the reservation table specifies the use of the corresponding stage for the indicated time slot. In the proposed approach, we analyze the timing interactions among instructions within a basic block by building its reservation table. In the reservation table, not only the conflicts in the use of pipeline stages but also data dependencies among instructions are considered. A program construct such as an if statement may have more than one execution path. Moreover, in pipelined processors, it is not always possible to determine which one of the execution paths is the worst case execution path by analyzing the program construct alone. As an example, suppose that an if statement has two execution paths corresponding to the two reservation tables shown in Fig. 3. The worst case execution path here depends on the instructions in the preceding program constructs. For example, if one of the instructions near the end of the preceding program construct uses the MD stage, the execution path corresponding to R 1 will become the worst case execution path. On the other hand, if there is an instruction using the DIV stage instead, the execution path corresponding to R 2 will become the worst case execution path. Therefore, we should keep both ALU RD IF MD Fig. 3. Two reservation tables with equal struct pipeline timing information f time t max; reservation table head[ffi head ]; reservation table tail[ffi tail ]; d d MD RD head tail Fig. 4. Reservation table data structure reservation tables until the timing information of the preceding program constructs is known. Fig. 4 shows the data structure for a reservation table used in our approach in both textual and graphical form. In the data structure, t max is the worst case execution time of the reservation table, which is determined by the number of columns in the reservation table. In implementation, not all the columns in the reservation table are maintained. Instead, we maintain only a first few (i.e., columns and a last few (i.e., ffi tail ) columns. The larger ffi head and ffi tail are, the tighter the resulting WCET estimation is since more execution overlap between program constructs can be modeled as we will see later. corresponds to the case where the full reservation table is maintained. As explained earlier, we associate with each program construct a set of reservation tables where each reservation table contains the timing information of an execution path that might be the worst case execution path of the program construct. We call this set the WCTA (Worst Case Timing Abstraction) of the program construct. This WCTA corresponds to the time-bound in the original timing schema and each element in the WCTA is denoted by (t With this framework, the timing schema can be extended so that the timing interactions across ALU RD MD IF ALU RD IF MD ALU RD IF MD Fig. 5. Example application of \Phi operation program constructs can be accurately accounted for. In the extended timing schema, the timing formula of a sequential statement S: are the WCTAs of S, S 1 and S 2 , respectively. The operation between two WCTAs is defined as are reservation tables and the \Phi operation concatenates two reservation tables resulting in another reservation table. This concatenation operation models the pipelined execution of a sequence of instructions followed by another sequence of instructions. The semantics of this operation for a target processor can be deduced from its data book. Fig. 5 shows an application of the \Phi operation. From the figure, one can note that as more columns are maintained in head and tail, more overlap between adjacent program constructs can be modeled and, therefore, a tighter WCET estimation can be obtained. The above timing formula for S: effectively enumerates all the possible candidates for the worst case execution path of S 1 However, during each instantiation of this timing formula, a check is made to see whether the resulting WCTA can be pruned. An element in a WCTA can be removed from the WCTA if we can guarantee that that element's WCET in the worst case scenario is shorter than the best case scenario WCET of some other element in the same WCTA. This pruning condition can be more formally specified as follows: A reservation table w in a WCTA W can be pruned without affecting the prediction for the worst case timing behavior of W if In this condition, w:t max is w's execution time when we assume the worst case scenario for w (i.e., when no part of w's head and tail is overlapped with the surrounding program constructs). On the other hand, w tail is the execution time of w 0 when we assume the best case scenario for w 0 (i.e., when its head is completely overlapped with the tail of the preceding program construct and its tail is completely overlapped with the head of the succeeding program construct). The timing formula of an if statement S: if (exp) then S 1 else S 2 is given by are the WCTAs of S, exp, S 1 and S 2 , respectively and S is the set union operation. As in the previous timing formula, pruning is performed during each instantiation of this timing formula. Function calls are processed like sequential statements. In our approach, functions are processed in a reverse topological order in the call graph 4 since the WCTA of a function should be calculated before the functions that call it are processed. 4 A call graph contains the information on how functions call each other [23]. For example, if f calls g, then an arc connects f 's vertex to that of g in their call graph. Finally, the timing formula of a loop statement S: while (exp) S 1 is given by where N is a loop bound that is provided by some external means (e.g., from user input). This timing formula effectively enumerates all the possible candidates for the worst case execution scenario of the loop statement. This approach is exact but is computationally intractable for a large N . In the following, we provide approximate methods for loop timing analysis. Approximate Loop Timing Analysis The problem of finding the worst case execution scenario for a loop statement with loop bound N can be formulated as a problem to find the longest weighted path (not necessarily simple) containing exactly N arcs in a weighted directed graph. Thus, the approximate loop timing analysis method is explained using a graph theoretic formulation. be a weighted directed graph where is the set of the execution paths in the loop body that might be the worst case execution path (i.e., those in associated with each arc is weight w ij which is the execution time of path its execution is immediately preceded by path p i . Define D ';i;j as the weight of the longest path (not necessarily simple) from p i to p j in G containing exactly ' arcs. With this definition, the t max of the loop's worst case execution scenario that starts with path p i and ends with path p j is given by p i :t max +DN \Gamma1;i;j where p i :t max is t max of path p i . The WCTA of this worst case execution scenario inherits p i 's head since it starts with p i . Likewise, it inherits p j 's tail. From these, the of the loop's worst case execution scenario that starts with path p i and ends with path p j , which is denoted by wcta(wp N ij ), is given by (p i :t Since the actual worst case execution scenario of the loop depends on the program constructs surrounding the loop statement, we do not know with which paths the actual worst case execution scenario starts and ends when we analyze the loop statement. Therefore, one has to consider all the possibilities. The corresponding WCTA of the loop statement is given by ( (exp). The only remaining problem is to determine DN \Gamma1;i;j . We determine the value by solving the following equations. pk 2P Computation of D ';i;j for all using dynamic programming takes O(N \ThetajP time. For a large N , this time complexity is still unacceptable. In the following, we describe a faster technique that gives a very tight upper bound for D ';i;j . This technique is based on the calculation of the maximum cycle mean of G. The maximum cycle mean of a weighted directed graph G is ranges over all directed cycles in G and m(c) is the mean weight of c. The maximum cycle mean can be calculated in O(jP j \Theta jAj) time, which is independent of N , using an algorithm due to Karp [24]. Let m be the maximum cycle mean of G, then D ';i;j can safely be approximated as We prove this in the following proposition. Proposition 1 If D ';i;j is the maximum weight of a path (not necessarily simple) from p i to p j containing exactly ' arcs in a complete weighted directed graph and m is the maximum cycle mean of G, then D ';i;j - D 0 Proof . Assume for the sake of contradiction that D ';i;j is greater than ' \Theta m+ (m \Gamma w ji ). Then we can construct a cycle containing by adding the arc from p j to p i to the path from which D ';i;j is calculated. The arc should exist since G is a complete graph. The resulting cycle has a mean weight greater than m since D ';i;j +w ji m. This implies an existence of a cycle in G whose mean weight is greater than m. This contradicts our hypothesis that m is the maximum cycle mean of G and thus D ';i;j - ' \Theta m Moreover, it has been shown that D 0 ';i;j \Gamma D ';i;j , which indicates the looseness of the approx- imation, is bounded above by 3 \Theta (m \Gamma wmin ) where wmin is the minimum weight of an arc in A [25]. We can expect this bound to be very tight since m ' wmin . (Remember that P consists of the paths in W (exp) that cannot be pruned by each other.) Interference Up to now, we have assumed that tasks execute without preemption. However, bb contents cache cache cache Fig. 6. Sample instruction block references from a program construct in real systems, tasks may be preempted for various reasons: preemptive scheduling, external inter- rupts, resource contention, and so on. For a task, these preemptions are interference that breaks in the task's execution flow. The problem regarding interference is that of adjusting the prediction made under the assumption of no interference such that the prediction is applicable in an environment with interference. Fortunately, the additional per-preemption delay introduced by pipelined execution is bounded by the maximum number of cycles for which an instruction remains in the pipeline (in MIPS R3000 it is 36 cycles in the case of the div instruction). Once this information is available, adjusting the predictions to reflect interference can be done using the techniques explained in [26]. IV INSTRUCTION CACHING EFFECTS For a processor with an instruction cache, the execution time of a program construct will differ depending on which execution path was taken prior to the program construct. This is a result of the history sensitive nature of the instruction cache. As an example, consider a program construct that accesses instruction blocks 5 (b 2 , b 3 , b 2 , b 4 ) in the sequence given (cf. Fig. 6). Assume that the instruction cache has only two blocks and is direct-mapped. In a direct-mapped cache, each instruction block can be placed exactly in one cache block whose index is given by instruction block number modulo number of blocks in the cache. In this example, the second reference to b 2 will always hit in the cache because the first reference to b 2 will bring b 2 into the cache and this cache block will not be replaced in the mean time. On 5 We regard a sequence of consecutive references to an instruction block as a single reference to the instruction block without any loss of accuracy in the analysis. struct pipeline cache timing information f time t max; reservation table head[ffi head ]; reservation table tail[ffi tail ]; block address first reference[n block ]; block address last reference[n block ]; Fig. 7. Structure of an element in a the other hand, the reference to b 4 will always miss in the cache even when b 4 was previously in the cache prior to this program construct because the first reference to b 2 will replace b 4 's copy in the cache. (Note that b 2 and b 4 are mapped to the same cache block in the assumed cache configuration.) Unlike the above two references whose hits or misses can be determined by local analysis, the hit or miss of the first reference to b 2 cannot be determined locally and is dependent on the cache contents immediately before executing this program construct. Similarly, the hit or miss of the reference to b 3 will depend on the previous cache contents. The hits or misses of these two references will affect the (worst case) execution time of this program construct. Moreover, the cache contents after executing this program construct will, in turn, affect the execution time of the succeeding program construct in a similar way. These timing variations, again, cannot be accurately represented by a simple time-bound of the original timing schema. This situation is similar to the case of pipelined execution discussed in the previous section and, therefore, we adopt the same strategy; we simply extend the timing information of elements in the WCTA leaving the timing formulas intact. Each element in the WCTA now has two sets of instruction block addresses in addition to t max , head, and tail used for the timing analysis of pipelined execution. Fig. 7 gives the data structure for an element in the WCTA in this new setting where n block denotes the number of blocks in the cache. In the given data structure, the first set of instruction block addresses (i.e., first reference) maintains the instruction block addresses of the references whose hits or misses depend on the cache contents prior to the program construct. In other words, this set maintains for each cache block the instruction block address of the first reference to the cache block. The second set (i.e., last reference) maintains the addresses of the instruction blocks that will remain in the cache after the execution of the program construct. In other words, this set maintains for each cache block head tail bb first-reference last-referenceX X ALU RD IF MD RD ALU MD Fig. 8. Contents of the element corresponding to the example in Fig. 6 the instruction block address of the last reference to the cache block. These are the cache contents that will determine the hits or misses of the instruction block references in the first reference of the succeeding program construct. In calculating t max , we accurately account for the hits and misses that can be locally determined such as the second reference to b 2 and the reference to b 4 in the previous example. However, the instruction block references whose hits or misses are not known (i.e., those in first reference) are conservatively assumed to miss in the cache in the initial estimate of t max . This initial estimate is later refined as the information on the hits or misses of those references becomes available at a later stage of the analysis. Fig. 8 shows the timing information maintained for the program construct given in the previous example. With this extension, the timing formula of S: This timing formula is structurally identical to the one given in the previous section for the sequential statement. The differences are in the structure of the elements in the WCTAs and in the semantics of the \Phi operation. The revised semantics of the \Phi operation is procedurally defined in Fig. 9. The function concatenate given in the figure concatenates two input elements puts the result into w 3 , thus implementing the \Phi operation. In lines 9-12 of function concatenate, first reference if the corresponding cache block is accessed in w 1 . If the cache block is not accessed in w 1 , the first reference to the cache block in w 1 \Phi w 2 is from w 2 . Therefore, struct pipeline cache timing information concatenate(struct pipeline cache timing information w 1 , 3 struct pipeline cache timing information w 2 ) struct pipeline cache timing information w 3 ; 8 for 9 if (w 1 .first reference[i] == NULL) .first reference[i]; else .first reference[i]; .last reference[i] == NULL) .last reference[i]; else .last .last reference[i]; .last reference[i] == w 2 .first reference[i]) .head 22 w 3 26 g Fig. 9. Semantics of the \Phi operation would inherit w 2 's first reference. Likewise, in lines 13-16, w 3 inherits w 2 's last reference if the corresponding cache block is accessed in w 2 or w 1 's last reference otherwise. By comparing first reference with w 1 's last reference, lines 17-18 determine how many of the memory references in w 2 's first reference will hit in the cache. These cache hits are used to refine w 3 's (Remember that all the memory references in w 2 's first reference were previously assumed to miss in the cache in the initial estimate of w 2 's t max .) In lines 20-21, w 3 inherits w 1 's head and taking into account the pipelined execution across w 1 and w 2 and the cache hits determined in lines 17-18. In this calculation, the \Phi pipeline operation is the \Phi operation defined in the previous section for the timing analysis of pipelined execution and miss penalty is the time needed to service a cache miss. As before, an element in a WCTA can safely be eliminated (i.e., pruned) from the WCTA if we can guarantee that the element's WCET is always shorter than that of some other element in the same regardless of what the surrounding program constructs are. This condition for pruning is procedurally specified in Fig. 10. The function prune given in the figure checks whether either one struct pipeline cache timing information prune(struct pipeline cache timing information w 1 , 3 struct pipeline cache timing information w 2 ) 7 for 8 if (w 1 .first reference[i] != w 2 .first reference[i]) 9 n diff ++; .last reference[i] != w 2 .last reference[i]) else else 19 return NULL; Fig. 10. Semantics of pruning operation of the two execution paths corresponding to the two input elements can be pruned and returns the pruned element if the pruning is successful and null if neither of them can be pruned. In the function prune, lines 6-12 determine how many entries in w 1 's first reference and last reference are different from the corresponding entries in w 2 's first reference and last reference. The difference bounds the cache memory related execution time variation between checks whether w 2 can be pruned by w 1 . Pruning of w 2 by w 1 can be made if w 2 's WCET assuming the worst case scenario for w 2 is shorter than w 1 's WCET assuming best case scenario. Likewise, line 16 checks whether w 1 can be pruned by w 2 . Again as before, the timing formula of S: if (exp) then S 1 else S 2 is given by As in the previous section, the problem of calculating W (S) for a loop statement S: while (exp) S 1 can be formulated as a graph theoretic problem. Here, wcta(wp N ij ) is given by :first reference; p j :last reference) After calculating wcta(wp N can be computed as follows: The loop timing analysis discussed in the previous section assumes that each loop iteration benefits only from the immediately preceding loop iteration. This is because in the calculation of w ij , we only consider the execution time reduction of p j due to the execution overlap with p i . This assumption holds in the case of pipelined execution since the execution time of an iteration's head is affected only by the tail of the immediately preceding iteration. In the case of cache memory, however, the assumption does not hold in general. For example, an instruction memory reference may hit to a cache block that was loaded into the cache in an iteration other than the immediately preceding one. Nevertheless, since the assumption is conservative, the resulting worst case timing analysis is safe in the sense that the result does not underestimate the WCET of the loop statement. The degradation of accuracy resulting from this conservative assumption can be reduced by analyzing a sequence of k (k ? 1) iterations at the same time rather than just one iteration [25]. In this case, each vertex represents an execution of a sequence of k iterations and w ij is the execution time of sequence j when its execution is immediately preceded by an execution of sequence i . This analysis corresponds to the analysis of the loop unrolled k times and trades increased analysis complexity for more accurate a) Set associative caches: Up to now we have considered only the simplest cache organization called the direct-mapped cache in which each instruction block can be placed exactly in one cache block. In a more general cache organization called the n-way set associative cache, each instruction block can be placed in any one of the n blocks in the mapped set 6 . Set associative caches need a policy that decides which block to replace among the blocks in the set to make room for a block fetched on a cache miss. The LRU (Least Recently Used) policy is typically used for that purpose. Once this replacement policy is given (assuming that it is not random), it is straightforward to implement \Phi and prune operations needed in our analysis method. 6 In a set associative cache, the index of the mapped set is given by instruction block number modulo number of sets in the cache. The timing analysis of data caches is analogous to that of instruction caches. However, the former differs from the latter in several important ways. First, unlike instruction references, the actual addresses of some data references are not known at compile-time. This complicates the timing analysis of data caches since the calculation of first reference and last reference, which is the most important aspect of our cache timing analysis, assumes that the actual address of every memory reference is known at compile-time. This complication, however, can be avoided completely if a simple hardware support in the form of one bit in each load/store instruction is available. This bit, called allocate bit, decides whether the memory block fetched on a miss will be loaded into the cache. For a data reference whose address cannot be determined at compile-time, the allocate bit is set to zero preventing the memory block fetched on a miss from being loaded into the cache. For other references, this bit is set to one allowing the fetched block to be loaded into the cache. With this hardware support, the worst case timing analysis of data caches can be performed very much like that of instruction caches, i.e. treating the references whose addresses are not known at compile-time as misses and completely ignoring them in the calculation of first reference and last reference. Even when such hardware support is not available, the worst case timing analysis of data caches is still possible by taking two cache miss penalties for each data reference whose address cannot be determined at compile-time, and then ignoring the reference in the analysis [27]. The one cache miss penalty is due to the fact that the reference may miss in the cache. The other is due to the fact that the reference may replace a cache block that contributes a cache hit in our analysis. The second difference stems from accesses to local variables. In general, data area for local variables of a function, called the activation record of the function, is pushed and popped on a runtime stack as the associated function is called and returned. In most implementations, a specially designated register, called sp (Stack Pointer), marks the top of the stack and each local variable is addressed by an offset relative to sp. The offsets of local variables are determined at compile- time. However, the sp value of a function differs depending on from where the function is called. However, the number of distinct sp values a function may have is bounded. Therefore, the of a function can be computed for each sp value the function may have. Such sp values can be calculated from the activation record sizes of functions and the call graph. The final difference is due to write accesses. Unlike instruction references, which are read-only, data references may both read from and write to memory. In data caches, either write-through or write-back policy is used to handle write accesses [18]. In the write-through policy, the effect of each write is reflected on both the block in the cache and the block in main memory. On the other hand, in the write-back policy, the effect is reflected only on the block in the cache and a dirty bit is set to indicate that the block has been modified. When a block whose dirty bit is set is replaced from the cache, the block's contents are written back to main memory. The timing analysis of data caches with the write-through policy is relatively simple. One simply has to add a delay to each write access to account for the accompanying write access to main memory. However, the timing analysis of data caches with the write-back policy is slightly more complicated. In a write-back cache, a sequence of write accesses to a cached memory block without a replacement in-between, which we call a write run, requires only one write-back to main memory. We attribute this write-back overhead (i.e., delay) to the last write in the write run, which we call the tail of the write run. With this setting, one has to determine whether a given write access can be a tail to accurately estimate the delay due to write-backs. In some cases, local analysis can determine whether a write access is a tail or not as in the case of hit/miss analysis for a memory reference. However, local analysis is not sufficient to determine whether a write access is a tail in every case. Hence, when this is not possible, we conservatively assume that the write access is a tail and add a write-back delay to t max . However, if later analysis over the program syntax tree reveals that the write access is not a tail, we subtract the incorrectly attributed write-back delay from t max . This global analysis can be performed by providing a few bits to each block in first reference and last reference and augmenting the \Phi and pruning operations [27]. VI EXPERIMENTAL RESULTS We tested whether our extended timing schema approach could produce useful WCET bounds by building a timing tool based on the approach and comparing the WCET bounds predicted by the timing tool to the measured times. Our timing tool consists of a compiler and a timing analyzer (cf. Fig. 11). The compiler is a modified version of an ANSI C compiler called lcc [28]. The modified compiler accepts a C source program and generates the assembly code along with the program syntax information and the call graph. The timing analyzer uses the assembly code and the program syntax WCET WCEP Modified Information User-provided Graph Call Information Program Code Assembly Program Analyzer Timing Fig. 11. Overview of the timing tool information along with user-provided information (e.g., loop bound) to compute the WCET of the program. We chose an IDT7RS383 board as the timing tool's target machine. The target machine's CPU is a 20 MHz R3000 processor which is a typical RISC processor. The R3000 processor has a five-stage integer pipeline and an interface for off-chip instruction and data caches. It also has an interface for an off-chip Floating-Point Unit (FPU). The IDT7RS383 board contains instruction and data caches of 16 Kbytes each. Both caches are direct-mapped and have block sizes of 4 bytes. The data cache uses the write-through policy and has a one-entry deep write buffer. The cache miss service times of both the instruction and data caches are 4 cycles. The FPU used in the board is a MIPS R3010. Although the board has a timer chip that provides user-programmable timers, their resolutions are too low for our measurement purposes. To facilitate the measurement of program execution times in machine cycles, we built a daughter board that consists of simple decoding circuits and counter chips, and provides one user-programmerable timer. The timer starts and stops by writing to specific memory locations and has a resolution of one machine cycle (50 ns). Three simple benchmark programs were chosen: Clock, Sort and MM. The Clock benchmark is a program used to implement a periodic timer. The program periodically checks 20 linked-listed timers and, if any of them expires, calls the corresponding handler function. The Sort benchmark sorts an array of 20 integer numbers and the MM program multiplies two 5 \Theta 5 floating-point matrices. Table 1 compares the WCET bounds predicted by the timing tool and the measured execution times for the three benchmark programs. In all three cases, the tool gives fairly tight WCET bounds (within a maximum of about 30% overestimation). A closer inspection of the results revealed that Clock Sort MM Predicted Measured 2768 11471 6346 (unit: machine cycles) Table 1. Predicted and measured execution times of the benchmark programs more than 90% of the overestimation is due to data references whose addresses are not known at compile-time. (Remember that we have to account for two cache miss penalties for each such data reference.) Program execution time is heavily dependent on the program execution path, and the logic of most programs severely limits the set of possible execution paths. However, we intentionally chose benchmark programs that do not suffer from overestimation due to infeasible paths. The rationale behind this selection is that predicting tighter WCET bounds by eliminating infeasible paths using dynamic path analysis is an issue orthogonal to our approach and that this analysis can be introduced into the existing timing tool without modifying the extended timing schema framework. In fact, a method for analyzing dynamic program behavior to eliminate infeasible paths of a program within the original timing schema framework is given in [29] and we feel that our timing tool will equally benefit from the proposed method. We view our experimental work reported here as an initial step toward validating our extended timing schema approach. Clearly, much experimental work, especially with programs used in real systems, need to follow to demonstrate that our approach is practical for realistic systems. VII CONCLUSION In this paper, we described a technique that aims at accurately estimating the WCETs of tasks for RISC processors. In the proposed technique, two kinds of timing information are associated with each program construct. The first type of information is about the factors that may affect the timing of the succeeding program construct. The second type of information is about the factors that are needed to refine the execution time of the program construct when the first type of timing information of the preceding program construct becomes available at a later stage of WCET analysis. We extended the existing timing schema using these two kinds of timing information so that we can accurately account for the timing variations resulting from the history sensitive nature of pipelined execution and cache memory. We also described an optimization that minimizes the overhead of the proposed technique by pruning the timing information associated with an execution path that cannot be part of the worst case execution path. We also built a timing analyzer based on the proposed technique and compared the WCET bounds of sample programs predicted by the timing analyzer to their measured execution times. The timing analyzer gave fairly tight predictions (within a maximum of about 30% overestimation) for the benchmark programs we used and the sources of the overestimation were identified. The proposed technique has the following advantages. First, the proposed technique makes possible an accurate analysis of combined timing effects of pipelined execution and cache memory, which, previously, was not possible. Second, the timing analysis using the proposed technique is more accurate than that of any other technique we are aware of. Third, the proposed technique is applicable to most RISC processors with in-order issue and single-level cache memory. Finally, the proposed technique is extensible in that its general rule may be used to model other machine features that have history sensitive timing behavior. For example, we used the underlying general rule to model the timing variation due to write buffers [27]. One direction for future research is to investigate whether or not the proposed technique applies to more advanced processors with out-of-order issue [30] and/or multi-level cache hierarchies [18]. Another research direction is in the development of theory and methods for the design of a retargetable timing analyzer. Our initial investigation on this issue was made in [31]. The results indicated that the machine-dependent components of our timing analyzer such as the routines that implement the concatenation and pruning operations of the extended timing schema can be automatically generated from an architecture description of the target processor. The details of the approach are not repeated here and interested readers are referred to [31]. --R "Reasoning About Time in Higher-Level Language Software," "SMART (Strategic Memory Allocation for Real-Time) Cache Design," "Software-Based Cache Partitioning for Real-time Applications," "Timing Analysis of Superscalar Processor Programs Using ACSR," "High-Level Timing Specification of Instruction-Level Parallel Processors," "A Retargetable Technique for Predicting Execution Time," "Deterministic Upperbounds of the Worst-Case Execution Times of Cached Programs," "Evaluating Tight Execution Time Bounds of Programs by Annotations," "Predictable Real-Time Caching in the Spring System," "Bounding Worst-Case Instruction Cache Performance," "Portable Execution Time Analysis for RISC Processors," "Experiments with a Program Timing Tool Based on Source-Level Timing Schema," "Calculating the MaximumExecution Time of Real-Time Programs," "Static Analysis of Cache Performance for Real-Time Programming," "Pipelined Processors and Worst-Case Execution Times," Computer Architecture: A Quantitative Approach. 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pipelined execution;cache memory;worst case execution time;real-time system;RISC processor

631174

Conversion of Units of Measurement.

Algorithms are presented for converting units of measurement from a given form to a desired form. The algorithms are fast, are able to convert any combination of units to any equivalent combination, and perform dimensional analysis to ensure that the conversion is legitimate. Algorithms are also presented for simplification of symbolic combinations of units. Application of these techniques to perform automatic unit conversion and unit checking in a programming language is described.

Introduction Although many programming languages are described as "strongly typed", in most languages the types of numeric quantities are described only in terms of the numeric representation (real, integer, etc.) but not in terms of the units of measurement (meters, feet, etc.) of the quantity represented by a variable. The assignment of a value represented in one unit to Computer equipment used in this research was furnished by Hewlett Packard and IBM. a variable that is assumed to be in a different unit is clearly an error, but such errors cannot be detected if the type system does not include units of measurement. Conversion of units must be done explicitly by the programmer; this can be both burdensome and error-prone, since the conversion factors used by the programmer might be entered incorrectly or might have limited accuracy. Failure to represent units explicitly within program code is a serious shortcoming in specification of the program, since later modification of the program might be performed under the assumption of the wrong units. Hundreds of units of measurement are in general use; entire books [2] [13] [25] [27] [29] are devoted to tables of unit conversions. This paper presents methods for symbolic representation of units of measurement. Efficient algorithms are presented that can convert any combination of units to any equivalent combination, while verifying the legitimacy of the conversion by dimensional analysis. Algorithms are also presented for simplification of combinations of symbolic units. Applications of these techniques in programming languages and in data conversion are discussed. Related Work 2.1 Units and Unit Conversion [18] and [14] describe the Syst'eme International or International System of Units, abbreviated SI; these are the definitive references on SI. [4] provides style guidelines for use of SI units and tables of conversion factors. Several books provide conversion factors and algorithms for use in unit conversion. The available books differ widely in the number of units covered, the accuracy of the conversion factors, and the algorithms that some books present for unit conversion. Although one might think that unit conversion is easy and "everyone knows how to do it", the number of books and the variety of methodologies and algorithms they present suggest otherwise. Horvath [13] has an especially complete coverage of different units, as well as an extensive bibliography. The tables in this book give conversion factors from a given unit to a single SI unit; this is similar to the approach taken in the present paper, although Horvath does not present conversion algorithms per se. Semioli and Schubert [27] present voluminous tables that combine multiplication of the conversion factor by the quantity of the source unit to be converted. They also present somewhat complex methods for obtaining additional accuracy and shifting the decimal place of the result. This book has the flavor of a book of logarithm tables, although it was published in 1974, when pocket calculators were available. presents a series of directed acyclic graphs; each node of a graph is a unit, and arcs between nodes are labeled with conversion factors. The nodes are ordered by size of the unit. In order to convert from one unit to another, the user traverses the graph from the source unit to the goal unit, multiplying together all the conversion factors encountered along the way (or dividing if moving against the directed graph arrows). Although this technique presents many conversions together in a compact structure, its use involves many steps and thus many opportunities for error and loss of accuracy. Karr and Loveman [15] outline a computational method for finding conversion factors. Their method involves writing dimensional quantities and conversion factors in terms of logarithms, making a matrix of the equations in logarithmic form, and solving the matrix by linear algebra. Since the size of the matrix is the number of units involved in the conversion multiplied by the number of units the system knows about, both the matrix and the time required to solve it could quickly become large. Schulz [26] describes COMET, an APL program for converting measurements from the English system used in the U.S. to the metric system. COMET focuses on conversion of machine part specifications that include allowable tolerances. Gruber and Olsen [9] describe an ontology for engineering mathematics, including representation of units of measurement as an Abelian group. Their system can convert units, presumably by a process of logical deduction that would be significantly slower than the methods we describe. 2.2 Units in Programming Languages Units of measurement are allowed in the ATLAS language [5], although ATLAS allows only a limited set of units and a limited language for constructing combinations of units. Cunis [8] describes Lisp programs for converting units. These programs combine units with numeric measurements at runtime and perform runtime conversion. While this is consistent with the Lisp tradition of runtime type checking, it does not allow detection of conversion errors at compile time. Gehani [10] argues in favor of compile-time checking. Hilfinger [12] describes methods for including units with numeric data using Ada packages and discusses modifications of Ada compilers that would be required to make use of these packages efficient and allow compile-time checking of correctness of conversions. Karr and Loveman [15] propose incorporation of units into programming languages; they discuss methods of unit conversion, dimensional analysis, and language syntax issues. We believe that the unit conversion algorithms described in the present paper are simpler: our methods require only one scalar operation per unit for conversion and one scalar operation per unit for checking, whereas the methods of [15] are based on manipulation of matrices that could be large. 2.3 Data Translation Reusing an existing procedure may require that data be translated into the form expected by that procedure; we describe in [21] some methods for semi-automatic data translation. If a procedure requires that its data be presented in particular units, then unit conversion may also be required. The unit conversion methods of this paper can be combined with the methods of [21] to accomplish this. Unit conversion may also be required in preparing data for transmission to a remote site over a network, or for use in a remote procedure call. IDL (Interface Description Language) [16] allows exchange of large structured data, possibly including structure sharing, between separately written components of a large software system such as a compiler. Use of IDL requires that the user write precise specifications of the source and target data structures. Herlihy and Liskov [11] describe a method for transmission of structured data over a network, with a possibly different data representation at the destination. Their method employs user-written procedures to encode and decode the data into transmissible representations. 3 Unit Representation We use the term simple unit to refer to any named unit for which an appropriate conversion factor and dimension (as described later) have been defined. Simple units include the base units of a system of measurement, such as meter and kilogram, named units such as horsepower that can be defined in terms of other units, and the SI prefixes such as nano that are used for scaling. Positive, nonzero numeric constants are also allowed as simple units. In addition, common abbreviations may be defined as synonyms for the actual units, e.g., kg is defined as a synonym for kilogram.A composite unit is a product or quotient of units, and a unit is either a simple unit or a composite unit. We represent units in Lisp syntax, so that composite units are written within parentheses, preceded by an operator that is * or /. Thus, the syntax of units is as follows: simple-unit ::= symbol j number unit ::= simple-unit j composite-unit We say that a unit is normalized if nested product and quotient terms have been removed as far as possible, so that the unit will be at most a quotient of two products. Clearly, any unit can be normalized; algorithms for simplification of units are described later. normalized-unit A single numeric conversion factor is associated with each simple unit. The conversion factor is the number by which a quantity expressed in that unit must be multiplied in order to be expressed in the equivalent unit in the standard system of units. We have chosen as our Treating abbreviations as synonyms, rather than giving them a conversion factor and dimension and treating them as units, avoids the possibility that slightly different numeric factors might be specified for the same unit under different names. standard the SI (Syst'eme International d'Unit'es) system of units [18] [14] [1]; a different standard system could be chosen without affecting the algorithms described here. Thus, the conversion factor for meter is 1.0, while the conversion factor for foot is 0.3048 since meter. The conversion factor for a numeric constant is just the constant itself. The conversion factor for a product or quotient of units is, respectively, the product or quotient of the factors for the component units. Based on these definitions, it is easy to define a recursive algorithm that computes the conversion factor for any unit, whether simple or composite. This algorithm is shown in pseudo-code form in Fig. 1. Assuming that the definitions of units are acyclic, the algorithm is guaranteed to terminate and requires time proportional to the size of the unit expression tree. function 1. if unit is a simple unit (number or symbol), (a) if unit is a number, return unit; (b) if unit is a synonym of unit 0 , return factor(unit 0 ); (c) if unit has a predefined conversion factor f , return f ; (d) else, error: unit is undefined. 2. otherwise, (a) if unit is a product, (* (b) if unit is a quotient, (/ u 1 (c) else, error: unit has improper form. Figure 1: Conversion Factor Algorithm Our system provides facilities for defining named simple units with specified numeric conversion factors and for defining units in terms of previously defined units. Examples of unit definitions are shown in Fig. 2. In the first example, each unit is defined by its name, numeric conversion factor, and a list of synonyms. In the second example, the conversion factor is specified as an expression in terms of previously defined units. We now consider conversion from a source unit, unit s , to a desired unit, unit d . Let q s be the numeric quantity expressed in the source unit, q SI be the equivalent quantity in the standard (SI) system, and q d be the quantity in the desired unit. Let f the conversion factor of the source unit and f be the conversion factor of the desired unit. Then we have the equations: f d (defsimpleunits 'length '((meter 1.0 (m meters)) (foot 0.3048 (ft feet)) (angstrom 1.0e-10 (a angstroms)) (defderivedunits 'force '((newton (/ (* kilogram meter) (* second second)) (nt newtons)) (pound-force (/ (* slug foot) (* second second)) (ounce-force (/ pound-force 16) Figure 2: Unit Definitions Thus, conversion to a desired unit is accomplished by multiplying the source quantity by a f d For example, to convert a measurement in terms of feet to centimeters, the factor would be 0:01 Given the factor algorithm shown in Fig. 1 that computes the factor for any simple or composite unit, it is easy to convert any combination of units to any equivalent combination. Since a number is also defined as a unit, a numeric quantity of a given unit can also be converted. The convert function takes as arguments the source unit and desired unit; it returns the conversion factor f , or NIL if the conversion is undefined or incorrect. ?(convert 'foot 'centimeter)?(convert 'meters 'feet)?(convert '(/ pi (* micro fortnight)) '(/ inch sec))?(convert '(* acre foot) 'tablespoons)?(convert '(/ (* mega pound-force) acre) 'kilopascals)?(convert 'kilograms 'meters) The last example, an attempt to make an incorrect conversion from kilograms to meters, gives a result of NIL. Dimensional analysis, as described in the next section, is used to verify that a requested conversion is correct; if not, a result of NIL is returned rather than a numeric conversion factor. There are certain conversions that, while not strictly correct, deserve a special note. The pound, for example, which is a unit of mass, is often used as the name of the force unit that is properly called pound-force. [The same confusion exists with other mass units such as the ounce and the kilogram.] Conversion of pounds to pounds-force involves an apparent conversion from mass to force. Similarly, in particle physics the mass of a particle is often described using energy units such as gigaelectronvolts (GeV). As described below, the dimensional analysis system can either perform strict dimensional checking and prohibit force-mass and energy-mass conversion, or it can be made to detect and allow these specific conversions, with strict checking otherwise. In the latter case, the following conversions are allowed: ?(convert 'pound-force 'kilogram)?(convert '(* 1.67e-27 kg) 'gev)The conversion from mass units to the corresponding force units requires multiplication by g, the conventional value for the acceleration of free fall at the earth's surface (sometimes called "gravity"), while the conversion from mass units to energy units requires multiplication by the square of the speed of light. Each of these is a physical constant, expressed in SI units. 5 Dimensional Analysis If the above algorithms are to produce meaningful results, it must be verified that the requested conversion is legitimate; it is clearly impossible, for example, to convert kilograms to meters. Correctness of unit conversion is verified by the long-established technique of dimensional analysis [6]: the source and goal units must have the same dimensions. Formally, we define a dimension as an 8-vector of integral powers of eight base quantities. The base quantities are shown in Fig. 3 together with the base unit that is used for each quantity in the SI system [18] [14]. We have added money, which is not part of the SI system, as a dimension. Figure 3: Base Quantities and Units The dimension of a product of units is the vector sum of the dimensions of its components, while the dimension of a quotient of units is the vector difference of the dimensions of its components. It can be verified that conversion from one unit to another is legitimate by showing that the dimension vectors of the two units are equal, or equivalently, that their difference is a zero vector. The powers of base quantities that are encountered in practice are usually small: they are seldom outside the range \Sigma4. While a dimension can be represented as a vector of eight integer values, with dimension checking done by operations on vectors, this is somewhat expensive computationally. Since the integers in the vector are small, it might be more efficient to pack them into bit fields within an integer word. In this section, we describe a variation of this packing technique. A dimension vector is encoded within a single 32-bit integer, which we call a dimension integer, using the algorithms presented below. Using this encoding, dimensions can be added, subtracted, or compared using ordinary scalar integer arithmetic. It may be helpful to consider the analogy of doing vector arithmetic by encoding vectors as decimal integers. For example, the vector operation [1 2 3] be simulated using decimal integers: 123 This technique will work as long as it can be guaranteed that there will not be a "carry" from one column of the decimal integer to another. We use a similar method to encode a dimension vector as a 32-bit integer. A careful justification of the conditions under which use of the integer encoding is correct is presented following the algorithms. Finally, we argue that these conditions will be satisfied in practice, so that use of the integer encoding for dimension checking is justified. We define two 8-vectors and an integer constant (shown in decimal notation) as follows: 800000 8000000 80000000] We assume for purposes of this presentation that the 8-vectors are indexed beginning with 0; the index into an 8-vector for each kind of quantity corresponds to the Index column shown in Fig. 3. The vector dimsizes gives the size of the field assigned to each quantity; e.g., dimsizes[0] is 20, corresponding to a field size of 20 and an allowable value range of \Sigma9 for the power of length. The vector dimvals gives multipliers that can be used to move a vector value to its proper field position; it is defined as follows: dimvals dimvals The integer dimbias is a value that, when added to a dimension integer, will make it positive and will bias each vector component within its field by half the size of the field. dimbias is defined as: dimbias =X dimvals i \Delta dimsizes iGiven these definitions, algorithms are easily defined to convert between an 8-vector form of dimension and the equivalent dimension integer. A dimension integer is easily derived from a dimension vector v as the vector dot product of v and dimvals: dimint(v) =X For example, the dimension integer for a force can be calculated as follows: A dimension integer can be converted back to an 8-vector by adding dimbias to it and then extracting the values from each field. This algorithm is not needed for unit conversion, procedure dimvect (n, v) integer m, sz, mm; for to 7 do begin sz := dimsizes[i]; Figure 4: Conversion of Dimension Integer to Dimension Vector but is provided for completeness. The algorithm, shown in Fig. 4, has as arguments a dimension integer n and an 8-vector v; it stores the dimension values derived from n into v. This procedure uses truncated division to extract the biased value from each field of the integer encoding. The bias value, sz / 2, is then removed to yield the signed field value. Dividing by the field size is then used to bring the next field into the low-order position. Our algorithm uses dimension integers, rather than dimension vectors, to check the correctness of requested unit conversions. Addition, subtraction, and comparison of dimension vectors are simulated by scalar addition, subtraction, and comparison of corresponding dimension integers. We can state the following theorems regarding dimension integers: Theorem 1 If u and v are dimension vectors, then: if These results follow immediately from the definition of dimint. Theorem 2 If u and v are dimension vectors, and dimint(u) = dimint(v), and dimsizes dimsizes i then Proof: Suppose that dimint(u) = dimint(v) but u 6= v. Suppose that u 0 6= v 0 . By the definition of dimint and dimvals, Therefore, and by the triangle inequality, j u but this is contrary to our assumptions that j u 0 j! dimsizes 0and j v 0 j! dimsizes 0. Therefore, it must be the case that u inductive repetition of this argument on the remaining elements of u and v , it must be the case that These theorems show that checking the dimensions of unit conversions by means of dimension integers is correct so long as the individual dimension quantities are less than half the field sizes given in the dimsizes vector. We justify the use of the integer encoding of dimension vectors as follows. The powers of dimension quantities that are found in units that are used in practice are generally small - usually within the range \Sigma4. If a field size of 20 is assigned to length, time, and temperature, and a field size of 10 is assigned to the others, the dimension vector will fit within a 32-bit integer. The representation allows a power of \Sigma( dimsizes i\Gamma 1) for each quantity. As long as each element of a dimension vector is within this range, two dimension vectors are equal if and only if their corresponding dimension integers are equal; furthermore, integer addition and subtraction of dimension integers produce results equal to the dimension integers of the vector sum and difference of the corresponding dimension vectors. Our representation allows a power of \Sigma9 for length, time, and temperature, and a power of \Sigma4 for mass, current, substance, luminosity, and money. This should be quite adequate. We note that dimension vectors are used only in tests of equality: unequal dimensions of source and goal units indicate an incorrect conversion. An "overflow" from a field of the vector in the integer representation will not cause an error to be indicated when correct unit conversions are performed, because the two dimension values will still be equal despite the overflow. Two unequal dimension vectors will appear unequal, despite an overflow, unless the incorrect dimension integer corresponds to a very different kind of unit that has a dimension value that happens to be exactly equal; this is most unlikely to happen accidentally. For example, if the user attempts to convert a 20th power of length into a time, the system will fail to detect an error. This is such an unlikely occurrence that we consider the use of the more efficient integer encoding to be justified. Note, however, that 8-vectors could be used for dimension checking instead if desired. Cunis [8] describes an alternative representation of dimensions. He represents dimensions as a rational number in Lisp, i.e., as a ratio of integers that represent the positive and negative powers of dimensions. Each base quantity, such as length, is assigned a distinct small prime; the product of these, raised to the appropriate powers, forms the integer used in the ratio. This method requires somewhat more storage and computation than the method we present, and arithmetic overflow could be a problem if extended-precision arithmetic is not used; since Lisp provides extended-precision integers, this is not a problem in Lisp. 5.1 Unit Conversion Checking The dimension integer corresponding to a unit can be found as follows. The dimension of a constant is 0; this is also the case for units such as radian or nano 2 . The dimension of a base quantity is given by the corresponding value in the vector dimvals; for example, the dimension of time is dimvals[1] or 20. The dimension integer of a product of units is the sum of their dimension integers (using ordinary 32-bit integer arithmetic), and the dimension of a quotient of units is the difference of their dimensions. Dimensions of common abstract units such as force are found by computing the dimension of their expansion in terms of base abstract units; for force this expansion is: We also define an abstract unit dimensionless with dimension integer 0. When a unit symbol is defined to the system, its dimension is determined from the abstract unit specified for it; thus, in Fig. 2, meter receives the dimension of length. When a unit is defined by an expansion in terms of other units, the dimension of the expansion is verified by comparison with the dimension of its abstract unit. When convert is called to convert one unit to another, it also computes the dimension of the source unit minus the dimension of the goal unit. If the difference is 0, the dimensions are the same, and the conversion is legitimate. A nonzero value indicates a difference in dimensions of the source and goal units. If strict conversion is desired, any difference in dimension is treated as an error. In some cases, however, it may be desired to allow automatic conversion between mass and force or between mass and energy. Each of these conversions will produce a unique difference signature, which can be recognized; the conversions and corresponding dimension differences (source - goal) are shown in Fig. 5. If the difference matches the integer signature, the conversion factor should be multiplied by the additional factor shown in the table. For example, in converting kilograms (mass) to newtons (force), ?(convert 'kilogram 'newton)the dimension of kilogram is 8000 and the dimension of newton is 7961, so the difference is and the proper multiplier is 9:80665 . Although these multipliers are expressed in SI units, the conversion works for all unit systems. 2 Constants can be considered to have a dimension of unity, whose logarithmic representation is zero; such units are sometimes referred to as "dimensionless" [18]. Conversion Vector Integer Factor mass to weight [ weight to mass [ mass to energy [ energy to mass [ Figure 5: Dimension Conversions 6 Units in Programming Languages Although most modern programming languages require specification of data types and feature compile-time type checking, units generally are not included as part of types. This is unfortunate, since use of incorrect units must be considered to be a type error. Some commonly used procedures have implicit requirements on the units of their arguments; for example, the system sin function may require that its argument be expressed in floating-point radians. Karr and Loveman [15] advocated the inclusion of units in programming languages; although the ATLAS language [5] incorporates units, to our knowledge no widely-used programming language does so. We have implemented the use of units in the GLISP language. GLISP ("Generic Lisp") [19, 20] is a high-level language with abstract data types that is compiled into Lisp (or into C by an additional translation step); the GLISP compiler is implemented in Common Lisp [28]. GLISP has a data description language that can describe Lisp data structures or data structures in other languages. GLISP is described only briefly here; for more detail, see [21] and [19]. In the sections below, we describe both the language features needed to include units in a programming language and the compiler operations necessary to perform unit checking and conversion. Karr and Loveman [15] suggested that units be implemented as reserved words that could be used as multipliers in arithmetic expressions. Instead, we have implemented units as part of data types. The implementation of units within a programming language involves several different aspects: 1. inclusion of units as part of the type specification language 2. type checking of uses of data that have units 3. derivation of the units of the result of an arithmetic operation 4. coercion of data into appropriate units when necessary 5. a syntax for expressing numeric constants together with their units Each of these aspects is described below. 6.1 Units as Part of Types The types usually used to describe numeric data, such as integer, real, etc., describe only the method of encoding numeric values. The units denoted by the numeric values are an independent issue. Therefore, both the numeric type and unit must be specified as part of a data type. We have adopted a simple syntax to specify the two together: (units numeric-type unit ) For example, a floating-point number denoting a quantity of meters would have the type: (units real meters) type specification of this form may be used wherever a numeric type specification such as real would otherwise be used. Since the unit specification language allows constants to be included as part of a unit, it is possible to specify unusual units that might be used by hardware devices. For example, suppose that an optical shaft encoder provides the angular position of a shaft as an 8-bit integer, so that a circle is broken into 256 equal parts. This unit can be expressed as: (units integer (/ (* 2 pi radians) 256)) 6.2 Results of Operations and Coercion If unit checking and conversion are to be performed, it is necessary to determine the unit of the result of an arithmetic operation. In general, it is necessary to create and perhaps simplify new symbolic unit descriptions. There are several classes of operations, which are handled differently. The units produced by multiplication and division are easily derived by creating new units that symbolically multiply or divide, respectively, the source units. For example, if a quantity whose unit is (/ meter second) is multiplied by a quantity whose unit is second, the resulting unit is: (* (/ meter second) second) This unit could be simplified to meter, but in most cases it is not necessary for a compiler to perform such simplification: usually only the numeric conversion factor and dimension of the unit are used, and these are not affected by redundancy in the unit specification. Exponentiation to integer powers can be treated as multiplication or division. The function sqrt is a special case: the dimension vector of the argument unit must contain only multiples of 2, and it is necessary to produce an output unit that is "half" the input unit; this may require unit simplification, as discussed below. There are differences of opinion regarding coercion of types by a compiler. Some languages allow coercion within an arithmetic expression; for example, if an integer and a real are added, the integer will be converted to real prior to the addition. Other languages allow coercion only across an assignment operator. The most strict languages have no coercion and treat type differences as errors. The same issues and arguments can be raised regarding automatic coercion of units, and the same implementation options are available. Note, however, that if no coercion is allowed, the language must furnish some construct to allow the programmer to invoke type conversion explicitly. We describe below how automatic coercion can be implemented if it is desired. In the case of addition, subtraction, comparison, and assignment operations, the units of the two arguments must be the same if the operation is to be meaningful. If the units are unequal, an attempt is made to convert the unit of the right-hand argument to the unit of the left-hand argument. If a conversion factor f is not returned by the convert algorithm, the operation is illegitimate (e.g., an attempt to add kilograms to meters), and an error should be signaled by the compiler. (gldefun t1 (x: (units real meters) y: (units real kilograms)) glisp error detected by GLCOERCEUNITS in function Cannot apply op + to METERS and KILOGRAMS in expression: (X If the conversion factor f is 1:0, no compiler action is needed; this can occur if the units are equivalent but unsimplified. If the conversion factor is other than 1:0, a multiplication of the right-hand operand by the conversion factor must be inserted by the compiler. The following example illustrates how the GLISP compiler inserts such a conversion for an addition operation: (gldefun t2 (x: (units real meters) y: (units real feet)) result type: (UNITS REAL METERS) (LAMBDA (X Y) In this example, the variable y, which has units feet, is added to the value of the variable x, which has units meters. In this case, the compiler has inserted a multiplication by the appropriate factor to convert feet to meters prior to the addition. The result type is the type of the left-hand argument; this convention causes the type of a variable that is on the left-hand side of an assignment statement to take precedence. In some cases, it may be known that an argument of a procedure is required to have certain units; in such cases, procedure arguments can be type-checked and coerced if needed. For example, a library sin function may require an argument in radians; if the unit of the existing data is as described above for the shaft encoder example, conversion will be required: (gldefun t3 (x: (units integer (/ (* 2 pi radians) result type: REAL We have not described any language mechanism to allow the programmer to explicitly convert units to a desired form. Such a conversion can be accomplished by assigning a value to a variable that has the desired unit. The units used for intermediate results within an arithmetic expression may be somewhat unusual, but will always be converted to a programmer-specified unit upon assignment to a variable. Conversion of units may generate extra multiplication operations; however, if the compiler performs constant folding [3], these operations and their conversion factors can often be combined with other constants. Human programmers usually write programs in such a way that intermediate results have reasonable units and reasonable numeric values. When automatic coercion of units is performed, it is possible that intermediate values may have unusual units and very large or very small numeric values. It is possible that compiler-generated unit conversions might cause a loss of accuracy compared to code written by humans that does the unit conversions explicitly. For this reason, it is advisable that automatic coercion of units be used only with floating-point representations with high accuracy, such as the 64-bit IEEE Standard representation. While a human programmer who is aware of unit conversions can always force the desired units to be used, a compiler that performs conversions automatically might allow a careless programmer to overlook a potential accuracy problem. We have found that inclusion of units in programs tends to be "all or nothing". That is, if units are specified for some variables, then units need to be specified for other variables that appear in expressions with those variables to avoid type errors. 6.3 Constants with Units There may be a need to include physical constants, i.e., numbers with attached units, as part of a program. We have adopted a syntax that allows a numeric constant and unit to be packaged together: '(q number unit ) The quoted q form indicates a quantity with units. The type of the result is the type of the numeric constant combined with the specified unit. For example, the speed of light could be '(q 2.99792458e8 (/ meter second)) 6.4 Unit Simplification There are some cases in which unit simplification is needed. For example, it is desirable to simplify a unit that describes the result of a function. An algorithm for unit simplification should be able to handle any combination of units, including mixtures of units from different systems. The form of a unit that is considered to be "simplified" may depend on the needs of the user: an electrical engineer might consider (* kilowatt hour) to be simplified, while a physicist might prefer joule. We present below an algorithm that works well in simplifying units for several commonly used systems of units; in addition, it allows some customization by specifying new unit systems. A unit system is a set of base units that are by convention taken as dimensionally independent, and a set of derived units, formed from the base units by multiplication and division, that are by convention used with the unit system. Other units that are used for historical reasons may be associated with a unit system by defining them in terms of a numeric conversion factor and a combination of base units. We have implemented three unit systems: si (the Syst'eme International or SI system), cgs (centimeter-gram-second), and english (slug-foot-second). For each commonly used kind of unit (e.g., length, force, pressure, etc.) we define the standard unit for that kind of unit in each system (e.g., meter, newton, and pascal, respectively, for the si system). Our algorithm for symbolic simplification of a unit is as follows: 1. The desired system for the simplified result may be specified as a parameter. If it is unspecified, the dominant system of the input unit is determined by counting the number of occurrences of units associated with known systems; if a dominant system cannot be determined, si is used. 2. The input unit is "flattened" so that it consists of a quotient of two products. At the same time, input units are recursively expanded to their equivalents in terms of base units (length, mass, time, etc. Units that are equivalent to numbers (have dimensionality 0), such as mega or degree, are converted to numbers. 3. Any base units in the numerator and denominator product lists that are not in the goal system are converted to the corresponding units in the goal system. The conversion factors are accumulated. 4. The numerator and denominator product lists are sorted alphabetically. 5. Corresponding duplicate units are removed from the lists in a linear pass down the two lists; this cancels units that appear in both numerator and denominator. 6. The standard units that are defined for the goal system are examined. If the multisets represented by the numerator and denominator of the standard unit's expansion are contained in the numerator and denominator, then the standard unit can be a factor of the simplified unit. (The standard unit is also tested as an inverse factor.) The largest standard unit factor (with size greater than one base unit) is chosen, and it replaces its expansion in the unit that is being simplified. This process is continued until no further replacements can be made; it must terminate, since each replacement makes the unit expansion smaller. As an example, we show how the algorithm simplifies the unit expression: ?(simplifyunit '(/ joule watt)) The units joule and watt are defined in terms of base units: (* SECOND SECOND)) (* SECOND SECOND SECOND)) The quotient of these two units is flattened as a quotient of two products: (/ (* METER METER KILOGRAM SECOND SECOND SECOND) (* SECOND SECOND METER METER KILOGRAM)) The two product lists are sorted: (KILOGRAM METER METER SECOND SECOND SECOND) (KILOGRAM METER METER SECOND SECOND) Duplicated units in the two sorted lists are removed: In this case, the result is just a single unit: SECOND. This algorithm has the advantage of being universal: by completely breaking its input down to base units, canceling any duplicates, and then making a new unit from the result, it can accept any combination of units as input. It is also deterministic: it produces the same result for any way of stating the same unit. The algorithm is also reasonably fast. Since the algorithm works with a definition of a unit system in terms of a set of preferred units, it is possible for a user to define a modified unit system in which the user specifies the units that are preferred as the result of simplification. Some examples of unit simplification are shown below. ?(simplifyunit '(/ meter foot))?(simplifyunit '(/ joule watt)) ?(simplifyunit '(/ joule horsepower)) ?(simplifyunit '(/ (* kilogram meter) (* second second))) NEWTON ?(simplifyunit 'atm) (* 101325.0 PASCAL) ?(simplifyunit 'atm 'english) (* 14.695948775721259 POUNDS-PER-SQUARE-INCH) ?(simplifyunit '(/ (* amp second) volt)) FARAD ?(simplifyunit '(/ (* newton meter) (* ampere second))) ?(simplifyunit '(/ (* volt volt) (* lbf (/ (* atto parsec) (* 26250.801011041247 OHM) It was mentioned above that determining the type returned by the sqrt function requires making a unit that is "half" the input unit; for example, if the input unit is (* meter meter), the output unit would be meter. The process for determining the unit returned by sqrt is the same as the process of unit simplification described above, except for the last step. After the initial steps of simplification, the input unit will be represented by flat, sorted numerator and denominator lists containing base units of the same unit system, and possibly a numeric factor. Both lists must consist of adjacent pairs of identical units; otherwise, the input unit is in error. The output unit is determined by collecting every other member of the input lists (checking to make sure the alternate member is identical) and making a new unit from these lists and the square root of the numeric factor. 6.5 Units and Generic Procedures We have done research on the reuse of generic procedures [22] [23]; a generic procedure is one that can be used for a variety of data types. When the arguments of a generic procedure include units, automatic checking and conversion of units are essential for correct reuse. In the GLISP language [19] [21], it is not necessary to declare the type of every variable. When a variable is assigned a value, type inference is used to determine the type of the value, and the variable's type becomes the type of the value assigned to it. (Assignment of values of different types to the same variable will cause an error to be reported by the compiler.) This feature is useful in writing generic procedures: it is only necessary to specify the main types that are used (often just the types of input parameters); other types can be derived from those types. Because the types of local variables are specified indirectly, a single generic procedure can be specialized for a variety of input types. This is especially useful in the case of types that include units. Figure Calculation of Position of Aircraft from Radar Data We have developed a system, called VIP [24] (for View Interactive Programming) that generates programs from graphical connections of physical and mathematical models. A program is generated from equations associated with the physical models. Typically, only the types and units of inputs and outputs are specified; the units and types of intermediate values are derived by type and unit inference. This system is illustrated in the diagram shown in Fig. 6. The problem used as an example is a small but realistic numerical problem: the calculation of the position of an aircraft from data provided by an air search radar. We assume that the radar provides as input the time difference between transmission and return of the radar pulse, as well as the angle of the radar antenna at the time the return pulse is detected. When the radar illuminates the aircraft, we assume that the aircraft transponder transmits the identity of the aircraft and its altitude. The position and altitude of the radar station are assumed to be known. These items comprise the input data provided to the program. We assume that the units of measurement of the input data are externally specified (e.g., by hardware devices), so that the program is required to use the given units. In creating the program, the user of VIP is able to select from a variety of predefined physical and mathematical models, constant values, and operators. Initially, the VIP display consists of a set of boxes representing the input data, and an output box. In our example, the user first decides to model the travel of the radar beam as an instance of uniform-motion. The user selects the Physics command, then kinematics from the Physics menu, then uniform-motion from the kinematics menu. The input value TIME-DIFF is connected to the time button t of the motion. Next, the user selects Constant and obtains the constant for the speed of light, denoted C, and connects it to the velocity v of the motion. The distance d of the motion then gives the total (out-and-back) distance from the radar to the aircraft; by dividing this distance by 2, the one-way distance is obtained. This distance is connected to the hypotenuse of a Geometry object, right-triangle. The difference between the altitude of the aircraft and the altitude of the radar is connected to the y of this triangle. The x of this triangle is then the distance to a point on the ground directly underneath the aircraft. This distance and the angle of the radar give a range and bearing to the aircraft from the by connecting these to another right triangle, x and y offsets of the aircraft from the radar are obtained. These are collected to form a relative position vector, RELPOS, which is added to the radar's UTM (universal transverse mercator) coordinates to form the output. While the process described above is rather lengthy when described in words, the time taken by an experienced user to create this program using VIP was less than two minutes. Note that this problem involves several instances of conversion of units of measurement, a physical constant, and algebraic manipulation of several equations; all of these were hidden and performed automatically. Fig. 7 shows the GLISP program produced by VIP. Fig. 8 shows the program after it has been compiled and mechanically translated into C. In this example, unit conversion is a major part of the application program. However, the user only needed to specify the input units; all unit conversion and checking was performed automatically by the compiler, so that this source of programming difficulty and potential error was eliminated. (LAMBDA (TIME-DIFF: (UNITS INTEGER (* 100 NANOSECOND)) RADAR-UTM: UTM-CVECTOR) (D2 := (* '(Q 2.997925E8 (/ M S)) TIME-DIFF)) (X4 := (* X3 (COS RADAR-ANGLE))) (RELPOS := Figure 7: GLISP Program Generated by VIP for Radar Problem 7 Conclusions and Future Work We have described algorithms for conversion of units, for compiler checking of units used in arithmetic operations and for coercing units when necessary, and for symbolic simplification of combinations of units. The unit conversion algorithms are as simple as possible: they require only one multiply or divide per unit for conversion, and one add or subtract per unit for dimension checking. These algorithms have been implemented in a compiler that allows units as part of data type specifications and that performs automatic unit checking and conversion. Unit conversion is a problem that will not go away, even if the United States converts to the SI system. Workers in particular fields will continue to use units such as parsec or micron rather than meter, both because of tradition and because such units are convenient in size for the measurements typically used in practice. The compiler algorithms that we have described are relatively easy to implement, so that units could be incorporated into a variety of programming languages. These algorithms make it feasible to implement essentially all known units of measurement, so that users may use any units they find convenient. We agree with Karr and Loveman [15] that scientific programming languages should support the use of units; we hope that presentation of these algorithms will encourage such a trend. The ARPA Knowledge-Sharing Project [17] focuses on combining data from distributed databases and knowledge bases. The algorithms described in this paper can be used for conversion when these databases use different units. We have included money as a dimension, since it is often important to convert units such as (/ dollar kilowatt-hour) that include monetary units. Of course, the conversion CUTM *tqc (time-diff, aircraft-altitude, radar-altitude, radar-angle, long time-diff, aircraft-altitude, radar-altitude, radar-angle; long out1; float d1, out2, x1, y1, x2; CUTM *relpos, *glvar1621; relpos-?east glvar1621-?east relpos-?east return output; Figure 8: Radar Program Compiled and Converted to C factors for different currencies are not constant; however, by updating the conversion factors periodically, useful approximate conversions can be obtained. Our algorithms do not handle units that include additive constants; the common examples of such units are the Celsius and Fahrenheit temperature scales. Other features of the GLISP language can be used to handle these cases. Note that it is only possible to convert from a pure temperature unit to another temperature unit; it would be incorrect to multiply a non-absolute temperature by another unit. The kelvin and the degree Rankine are linearly related and can be converted by our algorithms. Ruey-Juin Chang implemented an Analyst's Workbench [7] to aid in making analytical models. She included substance as an additional part of a quantity, along with numeric quantity and unit; for example, "10 gallons of gasoline" has gasoline as the substance. Engineering and scientific calculations often involve conversions that depend on the substance as well as the quantity and units. For example, "10 gallons of gasoline" can be converted into volume (10 gallons), mass, weight, energy, money, or energy equivalent in kilograms of anthracite coal. The algorithms presented in this paper might usefully be extended to include these kinds of conversions as well. 8 Software Available The unit conversion software described in this paper is available free by anonymous ftp from ftp.cs.utexas.edu/pub/novak/units/ . It is written in Common Lisp. An on-line demonstration of the software, which requires a workstation running X windows, is available on the World Wide Web via http://www.cs.utexas.edu/users/novak . Acknowledgment I thank the anonymous reviewers for their suggestions for improving this paper. --R Handbook of Mathematical Functions Physical Measurements and the International (SI) System of Units Compilers: Principles American National Standard for Metric Practice IEEE Standard C/ATLAS Dimensional Analysis "Cliche-Based Modeling for Expert Problem-Solving Systems" "A Package for Handling Units of Measure in Lisp" "An Ontology for Engineering Mathematics" "Units of Measure as a Data Attribute" "A Value Transmission Method for Abstract Data Types" "An Ada Package for Dimensional Analysis" Conversion Tables of Units in Science and Engineering Quantities and Units "Incorporation of Units into Programming Languages" "IDL: Sharing Intermediate Representations" "Enabling Technology for Knowledge Sharing" "The International System of Units (SI)" "GLISP: A LISP-Based Programming System With Data Abstraction" "Negotiated Interfaces for Software Reuse" "Software Reuse through View Type Clusters" "Software Reuse by Specialization of Generic Procedures through Views" "Generating Programs from Connections of Physical Models" Fundamental Measures and Constants for Science and Technology "Writing Applications for Uniform Operation on a Mainframe or PC: A Metric Conversion Program" Conversion Tables for SI Metrication A Metrication Handbook for Engineers --TR --CTR Sharon L. Greene , Tracy Lou , Paul Matchen, Dynamic dimensional feedback: an interface aid to business rule creation, CHI '05 extended abstracts on Human factors in computing systems, April 02-07, 2005, Portland, OR, USA Tarun Rathnam , Christiaan J. J. Paredis, Developing federation object models using ontologies, Proceedings of the 36th conference on Winter simulation, December 05-08, 2004, Washington, D.C. Raya Khanin, Dimensional analysis in computer algebra, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.201-208, July 2001, London, Ontario, Canada Tudor Antoniu , Paul A. Steckler , Shriram Krishnamurthi , Erich Neuwirth , Matthias Felleisen, Validating the Unit Correctness of Spreadsheet Programs, Proceedings of the 26th International Conference on Software Engineering, p.439-448, May 23-28, 2004 Jie Liu , Elaine Cheong , Feng Zhao, Semantics-based optimization across uncoordinated tasks in networked embedded systems, Proceedings of the 5th ACM international conference on Embedded software, September 18-22, 2005, Jersey City, NJ, USA Jiang , Zhendong Su, Osprey: a practical type system for validating dimensional unit correctness of C programs, Proceeding of the 28th international conference on Software engineering, May 20-28, 2006, Shanghai, China Gordon S. Novak Jr., Creation of Views for Reuse of Software with Different Data Representations, IEEE Transactions on Software Engineering, v.21 n.12, p.993-1005, December 1995 Gordon S. Novak, Jr., Software Reuse by Specialization of Generic Procedures through Views, IEEE Transactions on Software Engineering, v.23 n.7, p.401-417, July 1997

dimensional analysis;unit of measurement;data type;unit conversion

631185

TLA in Pictures.

Predicate-action diagrams, which are similar to standard state-transition diagrams, are precisely defined as formulas of TLA (the Temporal Logic of Actions). We explain how these diagrams can be used to describe aspects of a specificationand those descriptions then proved correcteven when the complete specification cannot be written as a diagram. We also use the diagrams to illustrate proofs.

Introduction Pictures aid understanding. A simple flowchart is easier to understand than the equivalent programming-language text. However, complex pictures are confusing. A large, spaghetti-like flowchart is harder to understand than a properly structured program text. Pictures are inadequate for specifying complex systems, but they can help us understand particular aspects of a system. For a picture to provide more than an informal comment, there must be a formal connection between the complete specification and the picture. The assertion that the picture is a correct description of (some aspect of) the system must be a precise mathematical statement. We use TLA (the Temporal Logic of Actions) to specify systems. In TLA, a specification is a logical formula describing all possible correct behaviors of the system. As an aid to understanding TLA formulas, we introduce here a type of picture called a predicate-action diagram. These diagrams are similar to the various kinds of state-transition diagrams that have been used for years to describe sys- tems, starting with Mealy and Moore machines [1], [2]. We relate these pictures to TLA specifications by interpreting a predicate-action diagram as a TLA formula. A diagram denoting formula D is a correct description of a system with specification S iff (if and only if) S implies D. We therefore provide a precise statement of what it means for a diagram to describe a specification. We use predicate-action diagrams in three ways that we believe are new for a precisely defined formal notation: ffl To describe aspects of a specification even when it is not feasible to write the complete specification as a diagram. ffl To draw different diagrams that provide complementary views of the same system. ffl To illustrate formal correctness proofs. Section II is a brief review of TLA; a more leisurely introduction to TLA appears in [3]. Section III describes predicate-action diagrams, using an n-input Muller C-element as an example. It shows how diagrams are used to describe aspects of a complete specification, and to provide complementary views of a system. Section IV gives another example of how predicate-action diagrams are used to describe a system, and shows how they are used to illustrate a proof. II. TLA We now describe the syntax and semantics of TLA. The description is illustrated with the formulas defined in Figure 1. (The symbol \Delta means equals by definition.) We assume an infinite set of variables (such as x and y) and a class of semantic values. Our variables are the flexible variables of temporal logic, which are analogous to variables in a programming language. TLA also includes the rigid variables of predicate logic, which are analogous to constant parameters of a program, but we ignore them here. The class of values includes numbers, strings, sets, and functions. A state is an assignment of values to variables. A behavior is an infinite sequence of states. Semantically, a TLA formula is true or false of a behavior. Syntactically, TLA formulas are built up from state functions using Boolean operators (:, -, ) [implication], and j [equivalence]) and the operators 0 and 2, as described below. TLA also has a hiding operator 999 999, which we do not use here. A state function is a nonBoolean expression built from variables, constants, and constant operators. Semantically, it assigns a value to each state-for example assigns to state s one plus the value that s assigns to the variable x. A state predicate (often called just a predicate) is a Boolean expression built from variables, constants, and constant operators such as +. Semantically, it is true or false for a state-for example the predicate Init \Phi is true of state s iff s assigns the value zero to both x and y. An action is a Boolean expression containing primed and unprimed variables. Semantically, an action is true or false of a pair of states, with primed variables referring to the second state-for example, action M 1 is true for hs; ti iff the value that state t assigns to x equals one plus the value that state s assigns to x, and the values assigned to y by states s and t are equal. A pair of states satisfying an action A is called an A step. Thus, an M 1 step is one that increments x by one and leaves y unchanged. If f is a state function or state predicate, we write f 0 for the expression obtained by priming all the variables of f . For example \Phi equals 0). For an action A and a state function v, Init \Phi Fig. 1. The TLA formula \Phi describing a simple program that repeatedly increments x or y. we define [A] v to equal A- (v so a [A] v step is either an A step or a step that leaves the value of v unchanged. Thus, a [M 1 step is one that increments x by one and leaves y unchanged, or else leaves the ordered pair hx; yi unchanged. Since a tuple is unchanged iff each component is unchanged, a [M 1 step is one that increments x by one and leaves y unchanged, or else leaves both x and y unchanged. We define hAi v to equal A - (v 0 6= v), so an step is an M 1 step that changes x or y. Since an unchanged, an hM 1 i hx;yi step is a step that increments x by 1, changes the value of x, and leaves y unchanged. We say that an action A is enabled in state s iff there exists a state t such that hs; ti is an A step. For example, M 1 is enabled iff it is possible to take a step that increments x by one, changes x, and leaves y unchanged. Since natural number x, action hM 1 i hx;yi is enabled in any state in which x is a natural number. If is not enabled in a state in which x equals 1. A TLA formula is true or false of a behavior. A predicate is true of a behavior iff it is true of the first state. An action is true of a behavior iff it is true of the first pair of states. As usual in temporal logic, if F is a formula then 2F is the formula meaning that F is always true. Thus, 2Init OE is true of a behavior iff x and y equal zero for every state in the behavior. The formula 2[M] hx;yi is true of a behavior iff each step (pair of successive states) of the behavior is a step. Using 2 and "enabled" predicates, we can define fairness operators WF and SF. The asserts of a behavior that there are infinitely many hAi v steps, or there are infinitely many states in which hAi v is not enabled. In other words, WF v (A) asserts that if hAi v becomes enabled forever, then infinitely many hAi v steps occur. The strong fairness formula SF v (A) asserts that either there are infinitely many hAi v steps, or there are only finitely many states in which hAi v is enabled. In other words, asserts that if hAi v is enabled infinitely of- ten, then infinitely many hAi v steps occur. The standard form of a TLA specification is Init - Init is a predicate, N is an action, v is a state function, and L is a conjunction of fairness con- ditions. This formula asserts of a behavior that (i) Init is true for the initial state, (ii) every step of the behavior is an N step or leaves v unchanged, and (iii) L holds. Formula \Phi of Figure 1 is in this form, asserting that (i) initially x and y both equal zero, (ii) every step either increments x by one and leaves y unchanged, increments y by one and leaves x unchanged, or leaves both x and y unchanged, and (iii) the fairness condition WF hx; yi holds. Formula WF hx; yi (M 1 ) asserts that there are infinitely many steps or hM 1 i hx;yi is infinitely often not enabled. Since (i) and (ii) imply that x is always a natural number, hM 1 i hx;yi is always enabled. Hence, WF hx; yi (M 1 ) implies that there are infinitely many steps, so x is incremented infinitely often. Simi- larly, WF hx; yi (M 2 ) implies that y is incremented infinitely r e l e e out Fig. 2. A Muller C-element. often. Putting this all together, we see that \Phi is true of a behavior iff (i) x and y are initially zero, (ii) every step increments either x or y by one and leaves the other unchanged or else leaves both x and y unchanged, and (iii) both x and y are incremented infinitely many times. The formula Init - 2[N ] v is a safety property [4]. It describes what steps are allowed, but it does not require anything to happen. (The formula is satisfied by a behavior satisfying the initial condition in which no variables ever change.) Fairness conditions are used to specify that something must happen. III. Predicate-Action Diagrams A. An Example We take as an example a Muller C-element [5]. This is a circuit with n binary inputs one binary output out , as shown in Figure 2. As the figure indicates, we are considering the closed system consisting of the C-element together with its environment. Initially, all the inputs and the output are equal. The output becomes 0 when all the inputs are 0, and it becomes 1 when all the inputs are 1. After an input changes, it must remain stable until the output changes. The behavior of a 2-input C-element and its environment is described by the predicate-action diagram of Figure 3(a), where C is defined by The short arrows, with no originating node, identify the nodes labeled C(0; 0; 0) and C(1; 1; 1) as initial nodes. They indicate that the C-element starts in a state satisfying 1). The arrows connecting nodes indicate possible state transitions. For example, from a state satisfying C(1; 1; 1), it is possible for the system to go to a state satisfying either C(0; 1; 1) or C(1; 0; 1). More precisely, these arrows indicate all steps in which the triple hin[1]; in[2]; outi changes-that is, transitions in which at least one of in[1], in[2], and out changes. Steps that change other variables-for example, variables representing circuit elements inside the environment-but leave hin[1]; in[2]; outi unchanged are also possible. The predicate-action diagram of Figure 3(a) looks like a standard state-transition diagram. However, we interpret it formally not as a conventional state machine, but as the (a) A predicate-action diagram. \Delta- AU A AU \Delta- \Delta- AU A AU \Delta- @ @R @ @R (b) The corresponding TLA formula. Fig. 3. Predicate-Action diagram of hin[1]; in[2]; outi for a 2-input C-element, and the corresponding TLA formula. TLA formula of Figure 3(b). 1 This formula has the form Init is a state predicate and there is one conjunct F o for each node o. The predicate Init is Each F describes the possible state changes starting from a state described by node o. For example, the formula F o for the node labeled C(1; 1; 0) is A predicate-action diagram represents a safety property; it does not include any fairness conditions. Figure 3(a) is a reasonable way to describe a 2-input C-element. However, the corresponding diagram for a 3- input C-element would be quite complicated; and there is no way to draw such a diagram for an n-input circuit. The general specification is written directly as a TLA formula in Figure 4. The array of inputs is represented formally by a variable in whose value is a function with domain brackets denote function applica- tion. (Formally, n is a rigid variable-one whose value is constant throughout a behavior.) We introduce two pieces of notation for representing functions: denotes the function f with domain S such that f [i] equals e(i) for every i in S. ffl [f except denotes the function g that is the same as f except that g[i] equals e. The formulas defined in Figure 4 have the following interpretation Init C A state predicate asserting that out is either 0 or 1, and that in is the function with domain ng such that in[i] equals out for all i in its domain. Input(i) An action that is enabled iff in [i] equals out . It complements in[i], leaves in [j] unchanged for j 6= i, and leaves out unchanged. (The symbol i is a param- eter.) Output An action that is enabled iff all the in[i] are different from out . It complements out and leaves in unchanged. list of formulas bulleted by - or - denotes their conjunction or disjunction; - and - are also used as ordinary infix operators. Init C ng 7! out ] - in ng : in [i] 6= out - in Next ng : Input(i) -WF hin ;outi (Output) Fig. 4. A TLA specification of an n-input C-element. Next An action that is the disjunction of Output and all the Input (i) actions, for ng. Thus, a Next step is either an Output step or an Input(i) step for some input line i. \Pi C A temporal formula that is the specification of the C-element (together with its environment). It asserts that (i) Init C holds initially, (ii) every step is either a Next step or else leaves hin ; outi unchanged, and (iii) Output cannot be enabled forever without an Output step occurring. The fairness condition (iii) requires the output to change if all the inputs have; inputs are not required to change. (Since predicate-action diagrams describe only safety properties, the fairness condition is irrelevant to our explanation of the dia- grams.) The specification \Pi C is short and precise. However, it is not as reader-friendly as a predicate-action diagram. We therefore use diagrams to help explain the specification, beginning with the predicate-action diagram of Figure 5. It is a diagram of the state function hin [i]; outi, meaning that it describes transitions that change hin[i]; outi. It is a diagram for the formula \Pi C , meaning that it represents a formula that is implied by \Pi C . The diagram shows the synchronization between the C-element's ith input and its output. We can draw many different predicate-action diagrams oe ae oe ae oe ae oe ae \Omega \Omega \Omega OE J \Omega \Omega \Omega AE Fig. 5. A predicate-action diagram of hin[i]; outi for the specification \Pi C of an n-input C-element, where 1 in Y Fig. 6. Another predicate-actiondiagram of hin[i]; outi for \Pi C , where for the same specification. Figure 6 shows another diagram of hin [i]; outi for \Pi C . It is simpler than the one in Figure 5, but it contains less information. It does not indicate that the values of in[i] and out are always 0 or 1, and it does not show which variable is changed by each transition. The latter information is added in the diagram of Figure 7(a), where each transition is labeled with an action. The label Input (i) on the left-to-right arrow indicates that a transition from a state satisfying in out to a state satisfying in[i] 6= out is an Input(i) step. This diagram represents the TLA formula of Figure 7(b). Even more information is conveyed by a predicate-action diagram of hin; outi, which also shows transitions that leave in[i] and out unchanged but change in[j] for some j 6= i. Such a diagram is drawn in Figure 8(a). Figure 8(b) gives the corresponding TLA formula. There are innumerable predicate-action diagrams that can be drawn for a specification. Figure 9 shows yet another diagram for the C-element specification \Pi C . Since we are not relying on these diagrams as our specification, but simply to help explain the specification, we can show as much or as little information in them as we wish. We can (a) A predicate-action diagram of hin[i]; outi. in Y Output (b) The corresponding TLA formula. Fig. 7. A more informative predicate-action diagram of hin[i]; outi for \Pi C , and the corresponding TLA formula. (a) A predicate-action diagram of hin; outi. in - in [i] 6= out Y Output U U (b) The corresponding TLA formula. - in Fig. 8. A predicate-action diagram of hin; outi for \Pi C , and the corresponding TLA formula, where 1 in - in [i] 6= out Y in out U U Fig. 9. Yet another predicate-action diagram of hin; outi for \Pi C . draw multiple diagrams to illustrate different aspects of a system. Actual specifications are written as TLA formulas, which are much more expressive than pictures. B. A Formal Treatment B.1 Definition We first define precisely the TLA formula represented by a diagram. Formally, a predicate-action diagram consists of a directed graph, with a subset of the nodes identified as initial nodes, where each node is labeled by a state predicate and each edge is labeled by an action. We assume a given diagram of a state function v and introduce the following notation. N The set of nodes. I The set of initial nodes. E(n) The set of edges originating at node n. d(e) The destination node of edge e. Pn The predicate labeling node n. e The action labeling edge e. The formula \Delta represented by the diagram is defined as follows. Init \Delta An When no explicit label is attached to an edge e, we take e to be true. When no set of initial nodes is explicitly indicated, we take I to be N . With the usual convention for quantification over an empty set, An is defined to equal false if there are no edges originating at node n. B.2 Another Interpretation Another possible interpretation of the predicate-action diagram is the formula b \Delta, defined by This is perhaps a more obvious interpretation-especially if the diagram is viewed as a description of a next-state relation. We now show that \Delta always implies b \Delta, and that the converse implication holds if the predicates labeling the nodes are disjoint. \Delta. simple invariance proof, using rule INV1 of [3, Figure 5, page 888], shows that \Delta implies 2(9 n We then have: conjunction and 8, and is equivalent to [ by predicate logic, since B ) C implies 2[B]v holds for all m, n in N with m 6= n, then implies \Delta. propositional logic, the hypothesis implies The result then follows from simple temporal reasoning, essentially by the reverse of the string of equivalences and implication used to prove (A). We usually label the nodes of a predicate-action diagram with disjoint predicates, in which case (A) and (B) imply that the interpretations \Delta and b are equivalent. Diagrams with nondisjoint node labels may occasionally be useful; \Delta is the more convenient interpretation of such diagrams. C. Proving a Predicate-Action Diagram Saying that a diagram is a predicate-action diagram for a specification \Pi asserts that \Pi implies the formula \Delta represented by the diagram. Formula \Pi will usually have the form Init \Pi - 2[M] u - L, where L is a fairness condition. To prove \Pi ) \Delta, we prove: 1. Init \Pi 2. Init \Pi - 2[M] u for each node n. The first condition is an assertion about predicates; it is generally easy to prove. To prove the second condition, one usually finds an invariant Inv such that Init \Pi - 2[M] u implies 2Inv , so \Pi implies 2[M- Inv ] u . The second condition is then proved by showing that [M - Inv ] u implies for each node n. Usually, u and v are tuples and every component of v is a component of u, so v. In this case, one need show only that M-Inv implies [P n for each n. By definition of An , this means proving for each node n. This formula asserts that an M step that starts with Pn and Inv true and changes v is an Em step that ends in a state satisfying P d(m) , for some edge m originating at node n. IV. Illustrating Proofs In TLA, there is no distinction between a specification and a property; they are both formulas. Verification means proving that one formula implies another. A practical, relatively complete set of rules for proving such implications is described in [3]. We show here how predicate-action diagrams can be used to illustrate these proofs. We take as our example the same one treated in [3], that the specification \Psi defined in Section IV-A below implies the specification \Phi defined in Section II above. A. Another Specification We define a TLA formula \Psi describing a program with two processes, each of which repeatedly loops through the sequence of operations P (sem); increment ; V (sem), where one process increments x by one and the other increments y by one. Here, P (sem) and V (sem) denote the usual operations on a semaphore sem. To describe this program formally, we introduce a variable pc that indicates the control state. Each process has three control points, which we call "a", "b", and "g". (Quotes indicate string values.) We motivate the definition of \Psi with the three predicate- action diagrams for \Psi in Figure 10. In these diagrams, the predicate PC (p; q) asserts that control is at p in process 1 and at q in process 2. Figure 10(a) shows how the control state changes when the P (sem), V (sem), and increment actions are performed. Variables other than pc not mentioned in an edge label are left unchanged by the indicated steps-for example, steps described by the edge labeled leave y and sem unchanged-but this is not asserted by the diagram. The next-state action N is written as the disjunction N 1 - N 2 of the next-state actions of each process; and each N i is written as the disjunction Figure 10(b) illustrates this decomposition. Finally, the predicate-action diagram of Figure 10(c) describes how the semaphore variable sem changes. To write the specification \Psi, we let pc be a function with domain f1; 2g, with pc[i] indicating where control resides in process i. The formula PC (p; q) can then be defined by (a) \Phi \Phi \Phi \Phi \Phi* \Upsilon R (b) \Phi \Phi \Phi \Phi \Phi* \Upsilon R (c) Y U Fig. 10. Three predicate-action diagrams of hx; The semaphore actions P and V are defined by sem sem sem Missing from Figure 10 are a specification of the initial values of x and y, which we take to be zero, and a fairness con- dition. One could augment predicate-action diagrams with some notation for indicating fairness conditions. However, the conditions that are easy to represent with a diagram are not expressive enough to describe the variety of fairness requirements that arise in practice. The WF and SF formu- las, which are expressive enough, are not easy to represent graphically. So, we have not attempted to represent fairness in our diagrams. We take as the fairness condition for our specification \Psi strong fairness on the next-state action N i of each process. The complete definition of \Psi appears in Figure 11. B. An Illustrated Proof The proof of \Psi ) \Phi is broken into the proof of three conditions: 1. Init \Psi ) Init \Phi 2. Init \Psi - 2[N 3. \Psi ) WF hx;yi We illustrate the proofs of conditions 2 and 3 with the predicate-action diagram of hx; for \Psi in Figure 12, where Q is defined by and Nat is the set of natural numbers. First, we must show that the diagram in Figure 12 is a predicate-action diagram for \Psi. This is easy using the definition in Section III-B.1; no invariant is needed. For example, the condition to be proved for the node labeled is that an N step that starts with true is an M 1 step (one that increments x and leaves y unchanged) that makes Q 0 ("g"; "a") true. This follows easily from the definitions of Q and N , since an N step starting with PC ("b"; "a") true must be a fi 1 step. To prove condition 2, it suffices to prove that every step allowed by the diagram of Figure 12 is a [M] hx;yi step. The steps not shown explicitly by the diagram are ones that leave w unchanged. Such steps leave hx; yi unchanged, so they are [M] hx;yi steps. The actions labeling all the edges of the diagram imply [M] hx;yi , so all the steps shown ex- 7Init \Psi sem sem Fig. 11. The specification \Psi. \Phi \Phi \Phi \Phi \Phi* \Upsilon R Fig. 12. Another predicate-action diagram of hx; plicitly by the diagram are also [M] hx;yi steps. This proves condition 2. We now sketch the proof of condition 3. To prove WF hx;yi suffices to show that infinitely many steps occur. We first observe that each of the predicates labeling a node in the diagram implies that either enabled. The fairness condition of \Psi then implies that a behavior cannot remain forever at any node, but must keep moving through the diagram. Hence, the behavior must infinitely often pass through the node. The predicate Q 1 ("a"; "a") implies that both hN 1 i w and hN 2 i w are enabled. Hence, the fairness condition implies that infinitely many hN 1 i w steps and infinitely many hN 2 i w steps must occur. Action hN 1 i w is enabled only in the three nodes of the top loop. Taking infinitely many hN 1 i w steps is therefore possible only by going around the top loop infinitely many times, which implies that infinitely many M 1 steps occur, each starting in a state with Q 0 ("b"; "a") true. Since Nat, an M 1 step starting with so it is an hM 1 i hx;yi step. Hence, infinitely many hM 1 i hx;yi steps occur. Similarly, taking infinitely many hN 2 i w steps implies that infinitely many steps occur. This completes the proof of condition 3. Using the predicate-action diagram does not simplify the proof. If we were to make the argument given above rigor- ous, we would go through precisely the same steps as in the proof described in [3]. However, the diagram does allow us to visualize the proof, which can help us to understand it. V. Conclusion We have described three uses of diagrams that we believe are new for diagrams with a precise formal semantics: ffl To describe particular aspects of a complex specification with a simple diagram. An n-input C-element cannot be specified with a simple picture. However, we explained the specification with diagrams describing the synchronization between the output and each individual input. ffl To provide complementary views of the same system. Diagrams (b) and (c) of Figure 10 look quite different, but they are diagrams for the same specification. ffl To illustrate proofs. The disjunction of the predicates labeling the nodes in Figure 12 equals the invariant I of the proof in Section 7.2 of [3]. The diagram provides a graphical representation of the invariance proof. TLA differs from traditional specification methods in two important ways. First, all TLA specifications are interpreted over the same set of states. Instead of assigning values just to the variables that appear in the specifica- tion, a state assigns values to all of the infinite number of variables that can appear in any specification. Second, TLA specifications are invariant under stuttering. A formula can neither require nor rule out finite sequences of steps that do not change any variables mentioned in the formula. (The state-function subscripts in TLA formulas are there to guarantee invariance under stuttering.) These two differences lead to two major differences between traditional state-transition diagrams and predicate- action diagrams. In traditional diagrams, each node represents a single state. Because states in TLA assign values to an infinite number of variables, it is impossible to describe a single state with a formula. Any formula can specify the values of only a finite number of variables. To draw diagrams of TLA formulas, we let each node represent a predicate, which describes a set of states. In traditional di- agrams, every possible state change is indicated by an edge. Because TLA formulas are invariant under stuttering, we draw diagrams of particular state functions-usually tuples of variables. TLA differs from most specification methods because it is a logic. It uses simple logical operations like implication and conjunction instead of more complicated automata-based notions of simulation and composition [6]. Everything we have done with predicate-action diagrams can be done with state-transition diagrams in any purely state-based formalism. However, conventional formalisms must use some notion of homomorphism between diagrams to describe what is expressed in TLA as logical implication. Most formalisms employing state-transition diagrams are not purely state-based, but use both states and events. Nodes represent states, and edges describe input and output events. The meaning of a diagram is the sequence of events it allows; the states are effectively hidden. In TLA, there are only states, not events. Systems are described in terms of changes to interface variables rather than in terms of interface events. Variables describing the internal state are hidden with the existential quantifier 999 999 described in [3]. Changes to any variable, whether internal or interface, can be indicated by node labels or edge labels. Hence, a purely state-based approach like TLA allows more flexibility in how diagrams are drawn than a method based on states and events. --R "A method for synthesizing sequential circuits," "Gedanken-experiments on sequential machines," "The temporal logic of actions," "Defining liveness," Introduction to VLSI Systems "Conjoining specifications," --TR --CTR M. Lusini , E. Vicario, Engineering the usability of visual formalisms: a case study in real time logics, Proceedings of the working conference on Advanced visual interfaces, May 24-27, 1998, L'Aquila, Italy Harold Thimbleby, User interface design with matrix algebra, ACM Transactions on Computer-Human Interaction (TOCHI), v.11 n.2, p.181-236, June 2004 Jules Desharnais , Marc Frappier , Ridha Khdri , Ali Mili, Integration of sequential scenarios, ACM SIGSOFT Software Engineering Notes, v.22 n.6, p.310-326, Nov. 1997 Jules Desharnais , Marc Frappier , Ridha Khdri , Ali Mili, Integration of Sequential Scenarios, IEEE Transactions on Software Engineering, v.24 n.9, p.695-708, September 1998 L. E. Moser , Y. S. Ramakrishna , G. Kutty , P. M. Melliar-Smith , L. K. Dillon, A graphical environment for the design of concurrent real-time systems, ACM Transactions on Software Engineering and Methodology (TOSEM), v.6 n.1, p.31-79, Jan. 1997

specification;concurrency;state-transition diagrams;temporal logic

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Creation of Views for Reuse of Software with Different Data Representations.

Software reuse is inhibited by the many different ways in which equivalent data can be represented. We describe methods by which views can be constructed semi-automatically to describe how application data types correspond to the abstract types that are used in numerical generic algorithms. Given such views, specialized versions of the generic algorithms that operate directly on the application data can be produced by compilation. This enables reuse of the generic algorithms for an application with minimal effort. Graphical user interfaces allow views to be specified easily and rapidly. Algorithms are presented for deriving, by symbolic algebra, equations that relate the variables used in the application data to the variables needed for the generic algorithms. Arbitrary application data structures are allowed. Units of measurement are converted as needed. These techniques allow reuse of a single version of a generic algorithm for a variety of possible data representations and programming languages. These techniques can also be applied in data conversion and in object-oriented, functional, and transformational programming.

Introduction An algorithm should be like a mathematical theorem in the sense that once the algorithm has been developed, it should be reusable and should never have to be recoded manually. Like other engineering artifacts, however, algorithms that are used in an application must be adapted to fit the other parts of the application. Almost everyone is willing to reuse the standard algorithm for sqrt, but algorithms such as testing whether a point is inside a polygon are less likely to be reused. The argument and result types of sqrt match application types; however, there are many possible polygon representations, so an application is unlikely to use the same data types as a library program. Strong typing is an important optimization: by checking types statically, a compiler can avoid generation of runtime type-checking code, thus providing type safety while saving time and storage. Unfortunately, rigidity of types inhibits reuse. Traditional ways of making application data match a procedure that is to be reused are costly and discourage reuse [24]. An effective method of reuse must minimize two costs: 1. Human cost: the time required by the programmer to find the program to be reused, to understand its documentation, and to adapt the reused program and/or application so that they fit. 2. Computational cost: the added cost of running a reused program, compared to a hand-written version. This paper describes methods for reuse using views that describe how application data types correspond to the abstract types used in generic algorithms. Algorithms based on algebra create the views from correspondences that are specified via an easily used graphical interface. Given the views, a compiler produces efficient specialized versions of generic algorithms that operate directly on the application data. A single copy of a generic algorithm can be specialized for a variety of application data representations and for a variety of programming languages. The efficiency, encapsulation, and type safety of strong type checking are retained, while the barriers to reuse caused by type rigidity are eliminated. The techniques described here are an enabling technology that makes software reuse easy and practical. The graphical interface is illustrated in Fig. 1. In this example, we assume that the user has existing data that describes a Christmas tree; the user would like to reuse a small program, which calculates the area of the side of a cone, on this data. A view of the type xmas-tree as a cone is made using the program mkv ("make view"): (mkv 'cone 'xmas-tree) Figure 1: Viewing a Christmas Tree as a Cone The view is specified by correspondences between the application data type and the abstract representation of a cone. The user first selects a "button" on the cone diagram, then selects a corresponding item within the menu of fields of the xmas-tree type. The system draws lines between the selected items to show the correspondences. Using these correspondences together with equations associated with a cone, the system constructs a view of a xmas-tree as a cone. Any generic procedure associated with cone can then be specialized and used for the xmas-tree data: (gldefun t1 (tree:xmas-tree) This function, written in a Lisp-like functional form, requests the side-area of the cone view of the tree. The function is compiled into Lisp code that computes the side-area directly from the application data structure: (LAMBDA (TREE) (* 3.1415926535897931 (* Mechanical translation can produce code for applications in other languages, such as C: float t1c (tree) return 3.1415926535897931 * tree-?base-radius * sqrt(square(tree-?base-radius) The user obtains this specialized code without having to understand the algorithm and without having to understand the implementation of the cone abstract data type. The user only needs to select the items in the diagram that correspond to the application data. In this paper, we concentrate on numerical data of the kinds used in scientific and engineering programs. We have previously described [33] [34] [35] techniques for reuse of generic algorithms that deal with discrete data structures such as linked lists and trees; those methods, together with the methods described in this paper, allow reuse of algorithms that involve both discrete data structures and numerical data. The generic algorithms that can be specialized range from simple formulas, such as the formula for the area of a circle, to larger procedures such as testing whether a point is inside a polygon. Section 2 gives a formal definition of views. Section 3 presents the algorithms that construct views from correspondences using algebraic manipulation of equations. Section 4 describes the graphical user interface for specifying views. Section 5 describes several ways in which views enable software reuse: by specialization of generic procedures, by translation of data to a desired form, and in object-oriented, functional, or transformational programming. Section 6 surveys related work, and a Section 7 presents conclusions. Finally, an on-line demonstration of the program is described. Representation and Views Traditional data types combine two issues that should be separated to facilitate reuse: 1. the way in which application data are represented 2. procedures associated with a conceptual kind of object These are termed implementation inheritance and interface inheritance in object-oriented programming [9]. When the two are combined, the code of a procedure implicitly states assumptions about details of data representation. This inhibits software reuse, since any assumptions made in writing a procedure become requirements that must be met if the procedure is to be reused. For effective reuse, assumptions must be minimized. Object-oriented programming and functional programming partly separate the two issues, but still make some representation assumptions and also incur performance penalties, as we discuss later. Views make a clean separation between representation and procedures; compilation techniques yield good performance and can produce code in multiple target languages. There are many ways in which representations of equivalent data can differ: 1. names of individual variables, 2. data representations and units of measurement of variables, 3. data structures used to aggregate variables, 4. the set of variables chosen to represent an object, and 5. the conceptual method, or ontology [30], of a representation. For example, a vector could be represented using Cartesian coordinates or polar coordinates. In order to reuse a generic algorithm for application data, it must be possible either to translate the data into the form expected by the algorithm, or to modify the algorithm to work with the existing data; both can be done using views. A view describes how an application data type corresponds to an abstract type. In effect, the view encapsulates the application data type and makes it appear to be of the abstract Fig. 2 illustrates how an application type pipe can be viewed as a circle that is defined in terms of radius. The radius of the circle corresponds to the inside-diameter of the pipe divided by 2. The view makes visible only the name radius, hiding the names of pipe; procedures defined for circle can be inherited through the view. Note that it is not the case that a pipe is a circle; rather, a circle is a useful view of a pipe. A second view of a pipe as a circle, using outside-diameter, is also useful. length inside-diameter outside-diameter material radius pipe-as-circle pipe Figure 2: View as Encapsulation of Application Data We formalize the notion of views as follows. An abstract type is considered to be an abstract record containing a set of basis variables. A view encapsulates the application type, hiding its names, and presents an external interface that consists of the basis variables of the abstract type. The view allows the basis variables to be both "read" and "written". The view emulates the abstract record by maintaining two properties: 1. Storage property: After a value z is "written" into basis variable v i , a "read" of v i will yield the value z. 2. Independence property: If a "read" of basis variable v i yields the value z, and a value is then "written" into some basis variable v j a "read" of v i will still yield z. These two properties express the behavior normally expected of a record: stored values can be retrieved, and storing into one field does not change the values of other fields. If these properties are maintained, then the view faithfully emulates a record consisting of the basis variables, using the application type as its internal storage. The view inherits all the generic procedures associated with the abstract type. The view thus makes the application type appear to be a full-fledged implementation of the abstract type. Smith [45] uses the term theory morphism for a similar notion; Gries [17] uses the term coupling invariant and cites the term coordinate transformation used by Dijkstra [7]. The example of Fig. 2 illustrates a simple view in which a pipe is viewed as a circle in terms of the basis variable radius. A "read" of radius is accomplished by dividing the inside-diameter by 2; a "write" of radius is implemented by multiplying the value to be written by 2 and storing it into the inside-diameter. Source code that accesses the radius through the view and equivalent compiled code in C are shown below. (gldefun t5 (p:pipe) (radius (circle p)) ) float t5 (p) return p-?inside-diameter / 2; - (gldefun t6 (p:pipe r:real) ((radius (circle p)) := r) return Because the view makes the application data appear to be exactly like the abstract type, any generic procedure that is defined for the abstract type will produce the same results using the application data through the view. With optimized in-line compilation of the code for access to data through views, the specialized versions of generic algorithms operate directly on the application data and are efficient. In practice, it is often necessary to relax the storage and independence properties slightly: 1. The storage and independence properties are assumed to hold despite representation inaccuracy, e.g. floating-point round-off error. For example, if an application type that uses polar coordinates is viewed as a Cartesian vector, the value of x that is "read" may be slightly different from the value of x that was "written". 2. A view may be partial, i.e., may define only those basis variables that are used. For example, to compute the area of a circle, only the radius is needed. Any use of an undefined basis variable is detected as an error by the compiler. 3 Creation of Views from Correspondences A view contains a set of procedures to read and write each basis variable of the abstract type. These procedures could be written by hand. However, it is nontrivial to ensure that the procedures are complete, consistent, efficient, and satisfy the storage and independence properties. The procedures to view an application type as a line-segment (Fig. 5 below) are 66 lines of code; it would be difficult to make such a view manually. This section describes algorithms to derive the procedures for a view from correspondences, using symbolic algebra. A later section describes the graphical user interface that makes it easy to specify the correspondences. 3.1 Basis Variables and Equations (setf (get 'circle 'basis-vars) '(radius)) (setf (get 'circle 'equations) '((= diameter (* 2 radius)) (= circumference (* pi diameter)) (= area (* pi (expt radius 2))))) Figure 3: Equations for Circle Each abstract type defines a set of basis variables and a set of equations. The basis variables and equations for a simple circle abstract type are shown in Fig. 3; those for a line-segment are shown in Fig. 4. The equations are written in a fully parenthesized Lisp notation with the operator appearing first in each subexpression. The basis variables are specified by the designer of the abstract data type and constitute a contract between the implementer and user of the generic procedures: the implementer can assume that every use of the generic procedures will behave as if the set of basis variables is directly implemented, and the user can assume that if the view of the application data emulates a direct implementation of the basis variables, the generic procedures will work for the application data. A view is created from the correspondences provided by the graphical interface and from the equations. Each correspondence is a pair, (abstract-var application-var), that associates a variable of the abstract type with a corresponding variable of the application are processed sequentially. A view must have a procedure to read and write each basis variable, while maintaining the storage and independence properties. The procedures are derived by symbolic algebra. Although powerful packages such as Mathematica [54] exist, we use a simple equation solver Our interface actually allows an algebraic expression in terms of variables or computed quantities from the application type. (setf (get 'line-segment 'basis-vars) '(p1x p1y p2x p2y)) (setf (get 'line-segment 'equations) (= p1y (y p1)) (= p2y (y p2)) (= deltay (- p2y p1y)) (= slope (/ deltay (float deltax))) (= slope (tan theta)) (= slope (/ 1.0 (tan phi))) (= length (sqrt (+ (expt deltax 2) (expt deltay 2)))) (= theta (atan deltay deltax)) (= phi (atan deltax deltay)) (= deltay (* length (sin theta))) (= deltax (* length (cos theta))) (= deltay (* length (cos phi))) Figure 4: Equations for Line-Segment and a relatively large (and possibly redundant) set of simple equations, as in Fig. 4. The equations that describe abstract data types typically are simple, so that a simple equation solver has been sufficient. Equations are solved by algebraic manipulations. The equation solver is given a formula and a desired variable. If the left-hand side of the formula is the desired variable, the equation is solved. Otherwise, an attempt is made to invert the right-hand side, using algebraic laws, to find the desired variable; for example, the equation (= x (+ y z)) is equivalent to (= (- x y) z) or (= (- x z) y). These manipulations are performed recursively until the desired variable is isolated. Of course, this procedure assumes that the desired variable occurs exactly once in the equation. The equations that specify a variable as a tuple of other variables are used to describe grouping relationships. Application data might specify a point in terms of separate x and y components, or as a data structure that represents a point as a whole (either by containing x and y components directly or by having a view that defines x and y). The user of the graphical interface should be able to specify either the whole representation of the point, or one or more of its components; however, it would be incorrect to specify both the whole and a component. The treatment of tuple equations, described below, guarantees that only correct combinations can be specified. Corresponding to each tuple definition are equations that define how to extract the components from the tuple. 3.2 Incremental Equation Solving An equation set is initialized by making a copy of the equations of the target abstract type. As each correspondence between an abstract variable and an application variable is processed, the equation set is examined to see whether any equations can be solved. This incremental solution of the equations accomplishes several goals: 1. It produces equations for computing variables of the abstract type in terms of stored variables of the application type. These equations are later compiled into code. 2. It produces efficient ways of computing the variables, as described below. 3. The algorithm returns a list of all variables that are defined or computable based on the correspondences entered thus far. The buttons for these variables are removed from the user interface, preventing the user from entering a contradictory specification. The following algorithm, which we call var-defined, is performed for each correspondence, (abstract-variable application-variable). 1. The abstract variable is added to a list of defined variables and to a list of solved variables, i.e., variables that are computable from the correspondences specified so far. 2. Each equation is examined to determine what unsolved variables it contains: (a) If there is exactly one unsolved variable, and the equation can be solved to produce a new equation defining that variable, then i. The right-hand side of the new equation is symbolically optimized using a pattern-matching optimizer. ii. The new equation is saved for later use in generating code. iii. The variable is added to the list of solved variables. iv. The equation is deleted. (b) If there are no unsolved variables, the equation is deleted. This case can occur if the equation set contains multiple equations for computing the same variable. Assuming that the equation set is consistent, the equation will represent a mathematical identity among variables that have already been defined. (c) If the equation defines a tuple variable, and some component of the tuple has been solved, the variable is added to a list of deleted tuple variables, and the equation is deleted. (d) If the equation contains a deleted tuple variable, the equation is deleted. A deleted tuple variable can never become defined. 3. If any variables were solved in step 2.a., step 2 is repeated. 4. Finally, a list of variables is returned; this list includes the newly defined variable, any variables that were solved using equations, and any deleted tuple variables. Figure 5: Viewing LS1 as a Line Segment Fig. 5 shows correspondences between an application type LS1 and a line-segment. When the correspondences are specified using the graphical user interface, var-defined is called as each individual correspondence is specified. The correspondences are saved so that a view can be remade without using the graphical interface. Each correspondence is a pair of an abstract variable and a field of the application type; for the example of Fig. 5 the correspondences are: (LENGTH SIZE) Fig. 6 shows the calls to var-defined and the actions performed as each correspondence is processed; each action is labeled with the number of the step in the algorithm. After all correspondences have been processed, two results have been produced: 1. a list of abstract variables that are defined as references to the application type. 2. a list of equations that define other abstract variables. 1. Enter var-defined, 2c. deleting tuple 2d. deleting eqn (= P1X (X P1)) 2d. deleting eqn (= P1Y (Y P1)) 4. exit, vars (P1 P1Y) 1. Enter var-defined, 4. exit, vars (LENGTH) 1. Enter var-defined, 2a. solved eqn (= SLOPE (TAN THETA)) 2a. solved eqn (= PHI (- (/ PI 2.0) THETA)) 2a. solved eqn (= DELTAY (* LENGTH (SIN THETA))) 2a. solved eqn (= DELTAX (* LENGTH (COS THETA))) 3. repeating step 2. 2a. solved eqn (= DELTAY (- P2Y P1Y)) giving (= P2Y (+ DELTAY P1Y)) 2b. deleting eqn (= SLOPE (/ DELTAY 2b. deleting eqn (= SLOPE (/ 1.0 (TAN PHI))) 2b. deleting eqn (= LENGTH 2b. deleting eqn (= (ATAN DELTAY DELTAX)) 2b. deleting eqn (= PHI (ATAN DELTAX DELTAY)) 2b. deleting eqn (= DELTAY (* LENGTH (COS PHI))) 2b. deleting eqn (= DELTAX (* LENGTH (SIN PHI))) 3. repeating step 2. 2c. deleting tuple 2d. deleting eqn (= P2X (X P2)) 2d. deleting eqn (= P2Y (Y P2)) 4. exit, vars (P2 P2Y DELTAX DELTAY PHI 1. Enter var-defined, 2a. solved eqn (= DELTAX (- P2X P1X)) giving (= P1X (- P2X DELTAX)) 3. repeating step 2. 4. exit, vars (P1X P2X) Figure Incremental Equation Solving Abstract Type \Gamma\Psi @ @ @ @ @R \Theta \Theta \Theta \Theta Figure 7: Variable Dependency Graph for LS1 The equations form a directed acyclic graph that ultimately defines each variable on the left-hand side of an equation in terms of references to the application type; Fig. 7 shows the graph for the LS1 example. If each variable on the right-hand side of an equation is replaced by its defining equation, if any, and the process is repeated until no further replacements are possible, the result will be an expression tree whose leaves are references to the application type. 2 This is easily proved by induction. Initially, the only solved variables are those that are defined by correspondence with the application type. Each variable that becomes solved via an equation is defined in terms of previously solved variables. Therefore, the graph of variable references is acyclic, and replacement of variables by their equational definitions will result in an expression tree whose leaf nodes are references to the application type. In general, the abstract type will define procedures that compute all of the variables shown in the diagram, as functions of the basis variables. Therefore, it is only strictly necessary for the view type to define procedures to compute the basis variables; all other variables could be derived from those. However, this approach might be inefficient. For example, it would be inefficient to calculate the LENGTH from the basis variables for the application type, since the LS1 type stores the LENGTH directly as the field SIZE. It is desirable to compute each variable as directly as possible from the application data, i.e., using as few data references and operations as possible. The var-defined algorithm operates incrementally and produces an equation to calculate each abstract variable as soon as it is possible to do so; the equations therefore are close to references to the application type. It would be possible to guarantee optimal computation of each variable by implementing a search algorithm: This replacement process is performed by the GLISP compiler when code that uses the view is compiled. 1. Assign to each abstract variable that is defined as a field of the application type a cost of 1 and mark it solved. 2. Examine each equation in the equation set to determine which variables can be solved in terms of existing solved variables. Assign, as the cost of such a solution, the sum of the costs of its components and the cost of each operator. If the variable is unsolved or has a higher cost, adopt the new equation as its definition and the new cost as its cost. 3. Repeat step 2 until no further redefinitions occur. We have not implemented this algorithm because the var-defined algorithm approximates it and has produced excellent results in practice; this algorithm would be useful if some operators had much higher cost than others. 3.3 Storing Basis Variables Some generic procedures both read and write data; thus, it is necessary to define methods that "store" into basis variables through the view. We assume that values can be stored only into basis variables; this is a reasonable restriction, since it corresponds to a record consisting of the basis variables in an ordinary programming language. Each storing method is a small procedure whose arguments are an instance of the application type and a value that is to be "stored" into the basis variable. The procedure must update the application data in such a way that the storage and independence properties are maintained. Without this constraint, the method used to "store" a variable would be ambiguous, and generic algorithms might behave differently with different data implementations. The variables of the abstract type that correspond directly to fields of the application type are called transfer variables; a list of these is saved. In Fig. 7, the transfer variables are P1Y, LENGTH, THETA, and P2X. Storing new values for all transfer variables following a change in value of a basis variable would accomplish a storing of the basis variable. Such a procedure could be derived in a trivial way: 1. compute the values of all basis variables, other than the one to be stored, from the application data; 2. compute values of the transfer variables from the basis variables; 3. store these values into the application data structure. However, such a procedure might be inefficient. It is desirable to update the smallest possible subset of stored variables. The algorithm below accomplishes this. 1. A set of basis equations is created. This is done by initializing an equation set with all equations of the abstract type, then calling var-defined for each basis variable. The result is a set of equations for computing each non-basis variable in terms of basis variables and, implicitly, a dependency graph that shows the dependency of all variables on basis variables. 2. The set xfers is computed; this is the subset of the transfer variables that depend on the basis variable that is to be stored. Dependency is determined by recursively computing the union of the leaf nodes of the expression tree for the transfer variable, as implicitly defined by the basis equations. 3. The set dep is computed; this is the subset of the basis variables that some member of xfers depends on. 4. Code is generated to compute each basis variable in dep, other than the basis variable to be stored, from the application data. 5. Code is generated to compute each transfer variable in xfers and to store it into the corresponding field of the application structure. Temporary variables are generated for intermediate variables that are used in computing the transfer variables if they are used more than once; otherwise, the intermediate variables are expanded using the basis equations. The result of this algorithm is a procedure that revises the application data structure only as much as necessary to emulate a "store" into a basis variable while leaving the values of other basis variables unchanged. One such procedure is created for each basis variable. In the special case where a basis variable corresponds exactly to a transfer variable and does not affect the value of any other transfer variable, no procedure needs to be created. (GLAMBDA (VAR-LS1 P1Y) (DELTAY := (- P2Y P1Y)) (DELTAX := (- P2X P1X)) Figure 8: Method to Store P1Y into a LS1 Data Structure An example of a procedure to store the basis variable P1Y into an LS1 is shown in Fig. 8. Although P1Y corresponds directly to the LOW field, it is also necessary to update the values of the SIZE and ANGLE fields in order to leave the value of the basis variable P2Y unchanged. The RIGHT field does not need to be updated. 3.4 Creating Application Data from Basis Variables Some generic procedures create new data structures; for example, two vectors can be added to produce a new vector. Therefore, a view must have a procedure that can create an instance of the application type from a set of basis variable values. This is similar to storing a basis variable, except that all basis variables are stored simultaneously. (GLAMBDA (SELF P1X P1Y P2X P2Y) (LET (DELTAY DELTAX) (DELTAY := (- P2Y P1Y)) (DELTAX := (- P2X P1X)) ANGLE (ATAN DELTAY DELTAX) Figure 9: Method to Create a LS1 Data Structure from Basis Variables Fig. 9 shows the procedure that creates a LS1 data structure from a set of line-segment basis variables. Two local variables, DELTAX and DELTAY, have been created since these variables are used more than once. When compiled by GLISP, the A function produces an instance of the LS1 data structure with the specified component values. When tuple substructures are involved, additional A functions are inserted to create them as well. The GLISP compiler invokes this method when compiling an A function; as a result, the use of views can be recursive. For example, a line-segment could be specified by two points P1 and P2, each of which is in polar coordinates r and theta with a view as a Cartesian vector. In this case, creating a new line-segment instance would also create new polar components. 3.5 Data Translation through Views Suppose that there are two data types t 1 and t 2 , each of which has a view as abstract type a. Then it is easy to translate data d 1 of type t 1 1. For each basis variable v i of the abstract type a, compute its value from data d 1 using the view from type t 1 to a. 'i 'i 'i view view Figure 10: Application Data that Share a View 2. Create a data structure of type t 2 from the set of basis variables using the view from to a. However, this algorithm might not be efficient. For example, types t 1 and t 2 might store the same non-basis variable, which could be transferred directly without computing basis variable values. For this reason, we have developed another algorithm for creating data translation procedures. The algorithm begins by finding the unique abstract type a that is the intersection of the views from the source and goal types (the name of the view to be used for each type can be specified if the intersection is not unique). Next, the set xfers is computed; this is the set of transfer variables of the goal type, i.e., the variables of the abstract type that correspond to stored fields in the goal type. Code is then generated to create an instance of the goal type using the values of the transfer variables from the shared view, a, of the source type. Since the view computes these transfer variables as close to the source data as possible, the resulting code is often more efficient, and never worse, than a version that computes basis variables first. Optimal code could be guaranteed by a search process, as described earlier. 3.6 Unit Conversion Application data can use various units of measurement. Most programming languages omit units of measurement entirely: there is no way to state the units that are used, much less to check consistency of units. GLISP allows units to be specified, and it automatically performs unit conversion [37] and checks validity. Therefore, unit conversion is performed automatically for all uses of views described in this paper. Checking A user could specify a partial set of correspondences between the abstract type and application type: some generic procedures defined by the abstract type may involve only a subset of the basis variables. For example, it is possible to compute the slope of a line-segment if only the abstract variable theta is defined. However, any actual errors are detected by our system. After the user enters correspondences, mkv issues a warning if any basis variables remain unsolved. A method to store a basis variable is produced only if that variable was solved and all basis variables in the set dep (described in section 3.3) were solved. A method to create an application data structure from basis variables is produced only if all basis variables were solved. An attempt to specialize a generic procedure that uses missing parts of a view type will result in errors detected by the GLISP compiler. It is possible for the user to specify a correspondence that can be computed, but cannot be "stored". For example, a variable of the abstract type could be defined as the sum of two stored fields of the application type; it is not possible to determine unambiguously how to "store into" this variable. An attempt to store into such a variable will result in an error detected by the GLISP compiler. Views do not replace or evade strong typing. Indeed, all of the code that is produced is correctly typed and can be mechanically translated to a strongly typed language. Views enhance type checking by checking units of measurement, and they provide encapsulation: when a view is used, only the operations defined by the view are available. The view mechanism provides the benefits of encapsulation while enhancing reusability and producing more efficient code than other encapsulation mechanisms [9]. 4 Graphical User Interface A graphical user interface makes it easy to specify correspondences between an application type and an abstract type using a mouse pointing device. The program, called mkv (for "make view"), is called with the goal abstract type and source type as arguments. mkv opens a window and draws a menu of the values available from the source type and a diagram or menu of variables of the abstract type (Fig. 1). The user selects items from the goal diagram by clicking the mouse on labeled "buttons" on the diagram. The interface program highlights a button when the mouse pointer is moved near it; clicking the mouse selects the item. The user then selects a corresponding item from the menu that represents the application data; a line is drawn between the two items to show the correspondence. The user can also specify an algebraic expression, involving one or more application data fields, by selecting OP from the command menu and specifying an expression tree of operators and operands. A diagram can present buttons for many ways to represent a given kind of data. 3 With such a diagram, it is likely that there will be buttons that correspond directly to the existing form of the data. Fig. 11 shows the diagram for a line-segment. Even though line-segment is a simple concept, a line segment could be specified in many ways: by two end points, or by one end point, a length, and an angle, etc. The diagram is intended to present virtually all the reasonable possibilities as buttons. This interface has several advantages: 1. Diagrams are easily and rapidly perceived by humans [26], and they are widely used in engineering and scientific communication [11]. 3 If there were alternative representations that were sufficiently different to require different diagrams, a menu to select among alternative diagrams could be presented to the user first. Figure Initial Diagram for a Line Segment 2. The interface is self-documenting: the user does not need to consult a manual, or know details of the abstract data type, to specify a view. 3. The interface is very fast, requiring only a few mouse clicks to create a view. 4. The user does not have to perform error-prone algebraic manipulations. It is easy to add diagrams for new abstract types. A drawing program allows creation of diagrams, including buttons. The only other thing that is needed is a specification of the basis variables and equations for the abstract type, as shown in Figs. 3 and 4. Achieving Reuse with Views Views are important for practical reuse because they achieve a clean separation between the representation of application data and the abstract data types used in generic procedures. The user obtains the benefits of reuse without having to understand and conform to a standard defined by someone else. In this section, we describe several ways in which views can be applied to achieve reuse: 1. by specialization of generic procedures through views 2. by translation of data 3. by creation of wrappers or transforms for use with object-oriented, functional, or transformational programming. 5.1 Specialization of Generic Procedures The GLISP compiler [31, 32] can produce a specialized version of a generic procedure by compiling it relative to a view. The result is a self-contained procedure that performs the action of the generic procedure directly on the application data. The specialized procedure can be used as part of the application program. GLISP is a high-level language with abstract data types that is compiled into Lisp; it is implemented in Common Lisp [47]. GLISP types include data structures in Lisp and in other languages. GLISP is described only briefly here; for more detail, see [33] and [31]. Data representation is a barrier to reuse in most languages because the syntax of program code depends on the data structures that are used and depends on whether data is stored or is computed. This prevents reuse of code for alternative implementations of data. GLISP uses a single syntax, similar to a Lisp function call, to access features of an object [33]: The interpretation of this form depends on the compile-time type of the object: 1. If feature is the name of a data field of object, code to access that field is compiled. 2. If feature is a message selector that is defined for the type of object, then (a) a runtime message send can be compiled, or (b) the procedure that implements the message can be specialized or compiled in-line, recursively. 3. If feature is the name of a view defined for the type of object, then the type of object is locally changed to the view type. 4. If feature is defined as a function, the code is left unchanged as a function call. 5. Otherwise, a warning message is generated that feature is undefined. In this way, a generic program can access a feature of an application type without making assumptions about how that feature is implemented. This is similar to object-oriented programming; however, since actual data types are known at compile time, message sending can be eliminated and replaced by in-line compilation or by calls to specialized procedures. This is equivalent to making transformations to the code, and it significantly improves efficiency of the compiled code. A view is implemented as a GLISP type whose stored form is the application data; the abstract type associated with the view is a super-class of the view type. The view type defines messages to compute the basis variables of the abstract type from the application type. As a simple example, consider a (handwritten, for simplicity) view of a pipe as a circle (Fig. 2): (pipe-as-circle (p pipe) prop ( (radius (inside-diameter p) / supers (circle)) The stored form of pipe-as-circle is named p and is of type pipe. The basis variable radius is defined as the inside-diameter of the pipe divided by 2. circle is a superclass, so all methods of circle are inherited by pipe-as-circle: (gldefun t7 (p:pipe) (area (circle p))) float t7 (p) return 0.78539816 The function t7 has an argument p whose type is pipe. The code (circle p) changes the type of p to the view type pipe-as-circle. The definition of area is inherited from circle and compiled in-line; this definition is in terms of the radius, which is expanded in-line as the inside-diameter of the pipe divided by 2. The use of the view has zero cost at runtime because the optimizer has combined the translation from the view with the constant -, producing a new constant -=4. Once a view has been made, all of the generic procedures of the abstract type are available by automatic specialization. Thus, a single viewing process allows reuse of many procedures. As a larger example, we consider a function that finds the perpendicular distance of a point to the left of a directed line segment (if the point is to the right, the distance will be negative). Although this is not a large function, it is not easy to find an effective algorithm in a reference book or to derive it by hand: graduate students assigned to produce such a procedure by hand for the LS1 data type report that it takes from 20 minutes to over an hour. Books often omit important features: [3] assumes that a human will determine the sign of the result. Even if a formula is found, it may not be expressed in terms of the available data. Some versions of the formula involve division by numbers that can be nearly zero. Reuse of a carefully developed generic procedure is faster, less costly, and less error-prone than writing one by hand. Code that results from the viewing and compilation process is presented below. We do not expect a user of our system to read or understand such code. While some authors [42] have proposed that the user read and edit code that is produced by an automatic programming system, we do not: it is not easy to read someone else's code, and this is especially true for machine-generated code that has been optimized. We expect that users will treat the output of our system as a "black box", as is often done with library subroutines. Fig. 12 shows the generic function line-segment-leftof-distance. Fig. 13 shows a specialized version of it for a LS1 record in C. This code was produced by GLISP compilation (gldefun line-segment-leftof-distance (ls:line-segment p:vector) (deltay ls) * ( Figure 12: Generic Function: Distance of Point to the Left of a Line Segment float lsdist (l, p) return cos(l-?angle) * (p-?x (l-?right l-?size * cos(l-?angle))); Figure 13: leftof-distance specialized for LS1 in C followed by mechanical translation into C; the resulting C code has no dependence on Lisp. Since the LS1 type is quite different from the abstract type line-segment, this example illustrates that a single generic procedure can be reused for a variety of quite different implementations of data. The specialized version is efficient (two multiplies and division by the length were removed by algebraic optimization) and is expressed in terms of the application data, without added overhead. While repeated subexpressions sometimes appear in specialized code, these can be removed by the well-understood compiler technique of common subexpression elimination [43]. A useful viewpoint is that there is a mapping between an abstract data type and the corresponding application type, and an isomorphism between the two based on a generic algorithm and a specialized version of that algorithm. Just as compilation of a program in an ordinary programming language produces an equivalent program in terms of lower-level operations and data implementations, specialization of a generic procedure produces an equivalent procedure that is more tightly bound to a specific implementation of the abstract data. Similar viewpoints are found in mathematical definitions of isomorphism (e.g., [39], p. 129), in denotational semantics [15], and in work on program transformation [45] [17]. [13] describes views in terms of such isomorphisms. However, we note that many applications use approximations that do not satisfy the strict mathematical definition of isomorphism. 5.2 Reuse by Data Translation One way to reuse a program with data in a different format is to translate the data into the right form; translation is also required to combine separate data sets that have different formats. As use of computer networks increases, users will often need to combine data from different sources or use data with a program that assumes a different format. The ARPA Knowledge-Sharing Project [30] addresses the problem of sharing knowledge bases that were developed using different ontologies. Writing data translation programs by hand requires human understanding of both data formats. [40] presents a language for describing parameter lists and a system that produces interface modules that translate from a source calling sequence to a target calling sequence. Our paper [33] described automatic construction of translation procedures from correspondences; the techniques in this paper extend those. Standardization of data representations and formats is one way to achieve interoperability. However, it is difficult to find standards that fit everyone's needs, and conformity to standards is costly for some users. Views provide the benefits of standardization without the costs. As described in Section 3.5, if there are views from two application types to a common abstract type, a data conversion procedure to convert from one application type to the other can be generated automatically. Thus, interchange of data requires only that the owner of each data set create a view that describes how the local data format corresponds to an abstract type. Only n views are needed to translate among any of n data formats, and knowledge of others' data formats is unnecessary. Use of remote procedure calls to servers across a network could be facilitated by advertising an abstract data type expected by the remote procedure; if users create views of their data as the abstract data, the translation of data can be performed automatically. Materialization of a new data set may be computationally expensive, but the cost is minor for small data sets. It is also reasonable to translate a large data set incrementally, or as a whole if a large amount of computation will be performed on it; [25] found that translation of data between phases of a large compiler was a minor cost. 5.3 Object-Oriented, Functional, and Transformational Program- ming The algorithms in this paper can be used with other styles of program development. One benefit of object-oriented programming (OOP) is reuse of methods that have been pre-defined. At the same time, OOP systems often impose restrictions that application objects must meet, e.g., that an application object must have the same stored data as its superclass or must provide certain methods with the same names that are used in the existing methods. Thus, reuse with OOP requires conformance to existing standards. Views allow new kinds of objects to be used with existing methods. The term adapter or wrapper [9] denotes an object class that makes application data appear to be an instance of a target class. The wrapper object contains a pointer to the application data and performs message translation to implement the messages required of a member of the target class. In the pipe example shown in Fig. 2, a wrapper class pipe-as-circle would implement the radius message expected by the class circle by sending an inside-diameter message to the pipe and returning the result of this message divided by 2. The algorithms and user interfaces described in this paper could be used to create wrapper classes. The messages expected by the target class correspond to basis variables. The equations produced in making views can easily be converted into methods in the appropriate syntax for the OOP system. We have previously described [35] the use of wrapper objects to allow display and direct-manipulation editing of user data by generic editor programs. There are two disadvantages of using wrapper objects [9]. It is necessary to allocate a wrapper object at runtime, which costs time and storage. Second, since translation of data is performed interpretively, there is overhead of additional message sending, and the same translation may be performed many times during execution. However, in cases where these costs are tolerable, wrapper objects are an easy way to achieve reuse. Views can be used in a related way with functional programming. Functions can be created to calculate the value of each basis variable from data of the application type. In the pipe example, a function radius(p:pipe) would be created that returns the value of the inside-diameter divided by 2. Storing of basis variables could be implemented by storing into the application data, for functional languages that allow this, or by creating new data with the updated values in the case of strict functional languages. The Polya language [17] [8] allows a user to specify a set of transformations that are made to the intermediate code of a generic algorithm; these transformations are equivalent to the transformations performed by the GLISP compiler [31]. The algorithms presented here could be used to generate transforms for a language such as Polya. 6 Related Work 6.1 Software Reuse Krueger [24] is an excellent survey of software reuse; it also gives criteria for effective software reuse. Biggerstaff and Perlis [4] contains papers on theory and applications of software reuse. Mili [29] provides an extensive survey of approaches to software reuse, emphasizing the technical challenges of reuse for software production. Artificial intelligence approaches to software engineering are described in [1], [28], and [41]. Some papers from these sources are reviewed individually in this section. 6.2 Software Components Weide [52] proposes a software components industry analogous to the electronic components industry, based on formally specified and unchangeable components with rigid interfaces. Because the components would be verified, unchangeable, and have rigid interfaces, errors in using or modifying them would be prevented. Views, as described in this paper, allow components to be adapted to fit the application. 6.3 Languages with Generic Procedures Programming languages such as Ada, Modula-2 [21] [27], and C++ [48] allow parameterized modules; by constructing a module containing generic procedures for a parameterized abstract data type, the user obtains a specialized version of the module and its procedures. However, these languages allow much less parameterization than is possible using views; for example, it is not possible to define a procedure that works for either Cartesian or polar vectors, and it is not even possible to state units of measurement in these languages. 6.4 Functional and Set Languages ML [53] [38] is like a strongly typed Lisp; it includes polymorphic functions (e.g., functions over lists of an arbitrary type) and functors (functions that map structures, composed of types and functions, to structures). ML also includes references (pointers) that allow imperative programming. ML functors can instantiate generic modules such as container types. However, ML does not allow generics as general as those described here. Our system allows storing into a data structure through a view; for example, a radius value can be "stored" into a pipe through a view. Our system also allows composition of views. Miranda [50] is a strongly typed, purely functional language that supports higher-order functions. While this allows generic functions to be written, it is often difficult to write efficient programs in a purely functional language [38]: a change to data values requires creation of a new structure in a functional language. 6.5 Transformation Systems Transformation systems generate programs starting from an abstract algorithm specification; they repeatedly apply transformations that replace parts of the abstract algorithm with code that is closer to an implementation, until executable code is finally reached. Our views specify transformations from features of abstract types to their implementations. Kant et al. [23] describe the Sinapse system for generating scientific programs involving simulation of differential equations over large spatial grids for applications such as seismic analysis. Sinapse accepts a relatively small program specification and generates from it a much larger program in Fortran or other languages by repeatedly applying transformations within Mathematica [54]. This system appears to work well within its domain of applicability. KIDS [46] can transform general algorithms into executable versions that are highly efficient for combinatorial problems. The user selects transformations to be used and supplies a formal theory for the domain of application. This system is interesting and powerful, but its user must be mathematically sophisticated. Gries and Prins [16] proposed a system that would use syntactic transformations to specify the implementation of abstract algorithms. [31] describes related techniques that were implemented earlier. Volpano [51] and Gries [17] describe transformation of programs by syntactic coordinate transformations for variables or for patterns involving uses of vari- ables. Our views require fewer specifications because they operate at semantic (type-based and algebraic) levels, rather than at the syntactic level; most patterns of use are handled automatically by the algebraic optimization of the compiler. Berlin and Weise [2] used partial evaluation to improve the efficiency of scientific pro- grams. Using information that some features of a problem are constant, their compiler performs as many constant calculations as possible at compile time, resulting in a program that is specialized and runs faster. Our system incorporates partial evaluation by means of in-line compilation and symbolic optimization. 6.6 Views Goguen describes a library interconnection language called LIL [12] and has implemented the language OBJ3 [13] [14] that incorporates parameterized programming and views. A view in OBJ3 is a mapping between a theory T and a module M that consistently maps sorts (types) of T to sorts of M and operations of T to operations of M. OBJ3 is based on formal logical theory using order-sorted algebra; it operates as a theorem prover in which computation is performed by term rewriting. The authors state ([14], p. 56): OBJ3 is not a compiler, but is rather closer to an interpreter. The associa- tive/commutative rewrite engine is not efficient enough for very large problems. Tracz [49] describes LILEANNA, which implements LIL for construction of Ada packages; views in LILEANNA map types, operations, and exceptions between theories. Our system creates views from correspondences between application types and mathematical objects; the possible correspondences are more general than the one-to-one correspondences specified in OBJ3 views. Our system produces efficient specialized procedures in ordinary programming languages and is intended to be used as a program generation system. Garlan [10] and Kaiser [22] use views to allow multiple tools in a program development environment to access a common set of data about the program being developed. Their MELD system can combine features, which are collections of object classes and methods, to allow additive construction of a system from selected component features. Hailpern and Ossher [18] describe views in OOP that are subsets of the methods defined for a class. They use views to restrict use of certain methods; for example, a debugger could use methods that were unavailable to ordinary programs. This system has been used in the development environment [19]. 6.7 Data Translation IDL (Interface Description Language) [25] allows exchange of large structured data, possibly including structure sharing, between separately written components of a large software system such as a compiler. IDL performs representation translation, so that different representations of data can be used by the different components. Use of IDL requires that the user write precise specifications of the source and target data structures. Herlihy and Liskov [20] describe a method for transmission of structured data over a net- work, with a possibly different data representation at the destination. Their method employs user-written procedures to encode and decode the data into transmissible representations. They also describe a method for transmission of shared structures. [5] describes a system that automatically generates stub programs to interconnect processing modules that are in different languages or processors; this work is complementary to the techniques presented in this paper. The Common Object Request Broker Architecture (CORBA) [6] includes an Interface Definition Language and can automatically generate stubs to allow interoperability of objects across distributed systems and across languages and machine architectures. Purtilo and Atlee [40] describe a system that translates calling sequences by producing small interface modules that reorder and translate parameters as necessary for the called procedure. In all of these cases, the emphasis is on relatively direct translation of data, focusing on issues of record structure, number representation, etc. The techniques described in this paper could be used to extend these approaches to cases where the ontology, or method of description of objects, differs. Specialization of generic algorithms is more efficient than interpretive conversion of data. 6.8 Object-oriented Programming OOP is popular as a mechanism for software reuse; Gamma et al. [9] describe design patterns that are useful for OOP. We have described how views could be used to construct wrapper objects that make application objects appear to be members of a desired class. Use of views with the GLISP compiler extends good ideas in OOP: 1. OOP makes the connection between a message and the corresponding procedure at runtime; this is often a significant cost [9]. C++ [48] has relatively efficient message dispatching, at some cost in flexibility. Because GLISP [31] can specialize a method in-line and optimize the resulting code in context, the overhead of interpretation is eliminated, and often there is no extra cost. 2. Interpretation of messages in OOP postpones error checking to runtime. With views and GLISP, type inference and in-line expansion cause this checking to be done statically 3. Views provide a clean separation between application data and the abstract type, while some OOP systems require conformance between an instance and a superclass, e.g., the instance may have to contain the same data variables. With views, there can be multiple views as the same type, e.g. a pipe can be viewed as a circle in two distinct ways. Views allow a partial use of an abstract type, e.g. the area of a circle can be found without specifying its center. 4. In OOP, the user must learn the available classes and messages. Some OOP operating systems involve over 1,500 classes, so this is nontrivial. With views, the user does not have to understand the abstract type, but only has to indicate correspondences between the application type and the abstract type. The user interface is self-documenting. 5. Our system allows translation to a separate application language, such as C, without requiring that the application be written in a particular language. 7 Discussion and Conclusions We believe that generation of application code by compilation through views is an efficient and practical technique. Expansion of code through views is recursive at compile time, so that composition of views is possible. Generic algorithms are often written in terms of other generics. While the examples presented in this paper have been small ones for clarity, the approach does scale to larger algorithms. We have produced specialized versions of algorithms that comprise about 200 lines of code in C, e.g., finding the convex hull of a set of points (which uses the leftof-distance generic function defined for a line-segment), finding the perimeter, area, and center of mass of a polygon, etc. Compilation and translation to C of a 200-line program takes approximately 10 seconds on a workstation; this is much faster than human coding. The output code is efficient, often better than code produced by human programmers. Because of the efficiency of the generated code, we consider specialized compilation of generic procedures to be the best way to achieve reuse using views. Our approach has several significant advantages: 1. The user interface is self-documenting and allows views to be created quickly and easily. 2. Specialized versions of generic algorithms for an application can be created in seconds. 3. The specialized algorithms can be produced in a standard programming language that is independent of our system. 4. The code that is produced is optimized and efficient. 5. Static error checking is performed at compile time. Related techniques have been used to create programs by connecting diagrammatic representations of physical and mathematical laws [36]. The sizes of the various components of the system, in non-comment lines of Lisp source code, are shown in the following table: Component: Lines: Make Views: mkv 265 Symbolic Algebra 1,003 Graphical Interface 1,020 GLISP Compiler 9,097 Translation to C 831 Total 12,216 The Symbolic Algebra component includes algorithms used by mkv and described in this paper. The amount of code required for making views is not too large. Translation of specialized code into languages other than C should not be difficult: the C translation component is not large, and much of it could be reused for other languages; the translation uses patterns that are easily changed. Acknowledgments Computer equipment used in this research was furnished by Hewlett Packard and IBM. I thank the anonymous reviewers for their helpful suggestions for improving this paper. --R IEEE Trans. "Compiling Scientific Code Using Partial Eval- uation," CRC Standard Mathematical Tables and Formulae Software Reusability (2 vols. "A Packaging System for Heterogeneous Execution Environments," "The Common Object Request Broker: Architecture and Specification," A Discipline of Programming "On Program Transformations," Design Patterns: Elements of Reusable Object-Oriented Software "Views for Tools in Integrated Environments," "Reusing and Interconnecting Software Components," "Principles of Parameterized Programming," "Introducing OBJ," The Denotational Description of Programming Languages "A New Notion of Encapsulation," "The Transform - a New Language Construct," "Extending Objects to Support Multiple Interfaces and Access Control," Integrating Tool Fragments," "A Value Transmission Method for Abstract Data Types," Data Abstraction and Program Development using Modula-2 "Synthesizing Programming Environments from Reusable Features," "Scientific Programming by Automated Synthesis," "Software Reuse," "IDL: Sharing Intermediate Representations" "Why a Diagram is (Sometimes) Worth 10,000 Words," The Modula-2 Software Component Library Automating Software Design "Reusing Software: Issues and Research Directions," "Enabling Technology for Knowledge Sharing," "GLISP: A LISP-Based Programming System With Data Abstraction," "Negotiated Interfaces for Software Reuse," "Software Reuse through View Type Clusters," "Software Reuse by Compilation through View Type Clusters," "Generating Programs from Connections of Physical Models," "Conversion of Units of Measurement," ML for the Working Programmer Introduction to Discrete Structures "Module Reuse by Interface Adaptation," Readings in Artificial Intelligence and Software Engineering The Programmer's Apprentice A Mathematical Theory of Global Program Optimization Models of Thought "KIDS: A Semiautomatic Program Development System," Knowledge-based Software Development System," the Language "LILEANNA: A parameterized programming language," "An Overview of Miranda," "The Templates Approach to Software Reuse," "Reusable Software Components," Mathematica: a System for Doing Mathematics by Computer Advances in Computers --TR --CTR Jeffrey Parsons , Chad Saunders, Cognitive Heuristics in Software Engineering: Applying and Extending Anchoring and Adjustment to Artifact Reuse, IEEE Transactions on Software Engineering, v.30 n.12, p.873-888, December 2004 Ted J. Biggerstaff, A perspective of generative reuse, Annals of Software Engineering, 5, p.169-226, 1998 Don Batory , Bart J. Geraci, Composition Validation and Subjectivity in GenVoca Generators, IEEE Transactions on Software Engineering, v.23 n.2, p.67-82, February 1997 Gordon S. Novak, Jr., Software Reuse by Specialization of Generic Procedures through Views, IEEE Transactions on Software Engineering, v.23 n.7, p.401-417, July 1997

visual programming;abstract data type;symbolic algebra;software reuse;data conversion;program transformation;generic algorithm

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Compositional Programming Abstractions for Mobile Computing.

AbstractRecent advances in wireless networking technology and the increasing demand for ubiquitous, mobile connectivity demonstrate the importance of providing reliable systems for managing reconfiguration and disconnection of components. Design of such systems requires tools and techniques appropriate to the task. Many formal models of computation, including UNITY, are not adequate for expressing reconfiguration and disconnection and are, therefore, inappropriate vehicles for investigating the impact of mobility on the construction of modular and composable systems. Algebraic formalisms such as the -calculus have been proposed for modeling mobility. This paper addresses the question of whether UNITY, a state-based formalism with a foundation in temporal logic, can be extended to address concurrent, mobile systems. In the process, we examine some new abstractions for communication among mobile components that express reconfiguration and disconnection and which can be composed in a modular fashion.

Introduction The UNITY [1] approach to concurrency has been influential in the study of distributed systems in large part because of its emphasis on design aspects of the programming process, rather than simply serving as a tool for veri- fication. The technique has been used to derive concurrent algorithms for a wide range of problems, and to specify and verify correctness even in large software systems [2]. How- ever, because of the essentially static structure of computations that can be expressed, standard UNITY is not a suitable tool for addressing the problems faced by the designers of mobile computing systems, such as cellular telephone networks. This paper addresses the problem of modeling dynamically reconfiguring distributed systems with an extension of the UNITY methodology, which we refer to as Mobile UNITY. While formal models capable of expressing reconfiguration have been explored from the algebraic perspective [3] and from a denotational perspective [4], [5], very few state-based models can naturally express reconfiguration of com- ponents. Also, while algebraic models such as the - calculus may be adequate for expressing reconfiguration, it is not so clear how to handle the issue of disconnection. Recent work has recognized the importance of introducing location and failures as concepts in mobile process algebras [6], [7], [8], but because the authors are primarily concerned with modeling mobile software agents and the effect of host failures on such systems, they do not directly P. McCann is with Lucent Technologies, Naperville, Illinois. E-mail: [email protected] . G.-C. Roman is with the Department of Computer Science, Washington University, St. Louis, Missouri. E-mail: [email protected]. address disconnection of components that continue to function correctly but independently. In addition to directly modeling reconfiguration and dis- connection, Mobile UNITY attempts to address design issues raised by mobile computing. These issues stem from both the characteristics of the wireless connection and the nature of applications and services that will be demanded by users of the new technologies. Broadly speaking, mobile computing leads to systems that are decoupled and context dependent, and brings new challenges to implementing the illusion of location-transparency. By examining the trends in applications and services currently being implemented by system designers, we hope to gain insight into the fundamentals of the new domain and outline opportunities for extensions to models of computation such as UNITY. Decoupling. The low bandwidth, frequent disconnec- tion, and high latency of a wireless connection lead to a de-coupled style of system architecture. Disconnections may be unavoidable as when a host moves to a new location, or they may be intentional as when a laptop is powered off to conserve battery life. Systems designed to work in this environment must be decoupled and opportunis- tic. By "decoupled," we mean that applications must be able to run while disconnected from or weakly connected to servers. "Opportunistic" means that interaction can be accomplished only when connectivity is available. These aspects are already apparent in working systems such as filesystems and databases that relax consistency so that disconnected hosts can continue to operate [9], [10]. Decoupling corresponds to the issue of modularity in system design, although in the case of mobility modularity is taken to a new extreme. Because of user demands, components must continue to function even while disconnected from the services used. Also, components must be ready to interface with whatever services are provided at the current location; the notion that a component is statically composed with a fixed set of services must be abandoned. The separation of interfaces from component implementations has long been advocated in the programming language community, but these notions need to be revisited from the more dynamic perspective of mobile computing. Context Dependencies. In addition to being weakly connected, mobile computers change location frequently, which leads to demand for context dependent services. A simple example is the location dependent World Wide Web browser of Voelker et al [11]. This system allows the user to specify location-dependent queries for information about the current surroundings and the services available. A more general point of view is evidenced in [12], which notes that application behavior might depend on the to- IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XX, NO. X, MONTH 1998 tality of the current context, including the current location and the nearness of other components, like the identity of the nearest printer or the group of individuals present in a room. The dynamic nature of interaction among components brings with it unprecedented challenges analogous to those of open software systems. Components must function correctly in any of the myriad configurations that might oc- cur. They must also continue to function as components are reconfigured. In Mobile UNITY, although interaction is specified on a basis and is usually conditioned on the proximity of two components, the model in general can express inter-action that is conditioned on arbitrary global predicates, such as the willingness of two components to participate in an interaction, the presence of other components, or the presence of interference or noise on a wireless link. Also, any given collection of pairwise interactions compose naturally to produce compound interactions that may span many components. Location Transparency. While some systems will be mobile-aware and require explicit reasoning about location and context, other applications naturally make use of location-transparent messaging. For example, Mobile IP [13] attempts to provide this in the context of the In- ternet. It is illustrative of the mobility management issues that must be addressed by designers. Our previous work [14] modeled Mobile IP in Mobile UNITY. Other location registration schemes have also been dealt with for- mally, for example with the -calculus [15] and with standard UNITY [16]. Although the latter work is similar to ours in that it is an application of UNITY to mobility, it deals mainly with the part of the algorithm running in the fixed network and only indirectly with communication between the mobile nodes and the fixed network. From the perspective of formal modeling, location registration provides a rich source of problems to use as ex- amples. Such mobility management algorithms show that even if the goal is transparent mobility, the designers of such a protocol must face the issues brought on by mobil- ity. Explicit reasoning about location and location changes are required to argue that a given protocol properly implements location transparency. Also, location registration protocols may form the very basis for location- and context-dependent services, which might make use of location information for purposes other than routing. The remainder of the paper is organized as follows. Section II presents a brief introduction to standard UNITY and the modifications we have made to express context-dependent interactions. The last part of the section gives a proof logic that accommodates the changes. Section III makes use of the new notation to express a new abstraction for communication called transient sharing. Section IV continues by introducing and formally expressing a new communication mechanism called transient synchroniza- tion. Concluding remarks are presented in Section V. II. Mobile UNITY Notation Our previous work [17] presented a notation and logic for interactions among mobile components. The pair-wise limitation simplified some aspects of the discussion, for instance, side-effects of an assignment were limited to only those components directly interacting with the component containing that assignment. However, the proof logic presented was very complex and operational, sometimes relying on sequencing of operations to define precise semantics. In this paper we present a simpler expression of transient interactions that focuses attention on the implications that component mobility has for the basic atomicity assumptions made by a system model, and provide a proof logic that is much more concise than our earlier work. In the process, we generalize interactions to include multiple participants. The new model is developed in the context of very low-level wireless communication, in order to focus on the essential details of transient interaction among mobile com- ponents. The key concept introduced in this section is the reactive statement, which allows for the modular specification of far-reaching and context-dependent side effects that a statement of one component may have. Using this primitive and a few others as a basis, in subsequent sections we present high-level language constructs that may be specified and reasoned about. A. Standard UNITY Recall that UNITY programs are simply sets of assignment statements which execute atomically and are selected for execution in a weakly fair manner-in an infinite computation each statement is scheduled for execution infinitely often. Two example programs, one called sender and the other receiver , are shown in Figure 1. Program sender starts off by introducing the variables it uses in the declare section. Abstract variable types such as sets and sequences can be used freely. The initially section defines the allowed initial conditions for the program. If a variable is not referenced in this section, its initial value is constrained only by its type. The heart of any UNITY program is the assign section consisting of a set of assignment statements. The execution of program sender is a weakly-fair interleaving of its two assignment statements. The assignment statements here are each single assignment statements, but in general they may be multiple-valued, assigning different right-hand expressions to each of several left-hand vari- ables. Such a statement could be written ~x := ~e, where ~x is a comma-separated list of variables and ~e is a comma-separated list of expressions. All right-hand side expressions are evaluated in the current state before any assignment to variables is made. Execution of a UNITY program is a nondeterministic but fair infinite interleaving of the assignment statements. Each produces an atomic transformation of the program state and in an infinite execution, each is selected infinitely often. The program sender , for example, takes the variable bit through an infinite sequence of 1's and 0's. The next value assigned to bit is chosen MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 3 program sender declare initially assign program receiver declare sequence of boolean initially assign history Fig. 1. Two standard UNITY programs. nondeterministically, but neither value may be forever excluded The program receiver has two variables, one of which is a sequence of boolean values, but contains only one assignment statement. The statement uses the notation history \Delta bit to denote the sequence resulting from appending bit to the end of history . This expression is evaluated on the right-hand side and assigned to history on the left hand side, essentially growing the history sequence by one bit on each execution. Note that we have not yet introduced the notion of composition, so the two programs should be considered completely separate entities for now. A.1 Proof Logic Rather than dealing directly with execution sequences, the formal semantics of UNITY are given in terms of program properties that can be proven from the text. The fair interleaving model leads to a natural definition of safety and liveness properties, based on quantification over the set of assignment statements. We choose to use the simplified form of these definitions presented in [18] and [19] for the operators co and transient. Each operator may be applied to simple non-temporal state predicates constructed from variable names, constants, mathematical operators, and standard boolean connectives. For example, if p and q are state predicates, the safety property p co q means that if the program is in a state satisfying p, the very next state after any assignment is executed must satisfy q. Proof of this property involves a universal quantification over all statements s, showing that each will establish q if executed in a state satisfying p. The notation fpgsfqg is the standard Hoare-triple notation [20]. In addition to the quantification, we are also obligated to show p ) q, which in effect takes the special statement skip (which does nothing) to be part of the quantification. Without this qualification, some cases of co would not be true safety properties, because they could be violated by the execution of an action which does nothing. As an example of co, consider the property Clearly, this property holds of program sender , because every statement, when started in a state where bit is 1 or 0, leaves bit in a state satisfying 1 or 0. This property is an example of a special case of co because the left- and right-hand sides are identical. Such a property may be abbreviated with the operator stable, as in stable 1. Because the initial conditions satisfy the predicate, i.e., an invariant, written invariant it is not clear from context to which program a property applies, it can be specified explicitly, as in invariant Progress properties can be expressed with the notation transient p, which states that the predicate p is eventually falsified. Under UNITY's weak fairness assumption, this can be defined using quantification as transient which denotes the existence of a statement which, when executed in a state satisfying p, produces a state that does not satisfy p. For example, the property transient 0 can be proven of the program sender , because of the statement that sets bit to 1. The transient operator can be used to construct other liveness properties. The reader may be more familiar with the ensures operator from UNITY, which is really the conjunction of a safety and a liveness property. ensures q - transient (p - :q) The ensures operator expresses the property that if the program is in a state satisfying p, it remains in that state unless q is established, and, in addition, it does not remain forever in a state satisfying p but not q. While ensures can express simple progress properties that are established by a single computational step, proofs of more complicated progress properties often require the use of induction to show that the program moves through a whole sequence of steps in order to achieve some goal. This notion is captured with the leads-to operator, written 7!. Informally, the property p 7! q means that if the program is in a state satisfying p, it will eventually be in a state satisfying q, although it may not happen in only one step and the property p may be falsified in the meantime. For 4 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XX, NO. X, MONTH 1998 example, consider the property receiver (1) which states that if receiver is ever in a state where the length of the history sequence is 3, it will eventually be in a state where this length is 5. In the meantime, however, the length of the sequence may (and does!) take on the value 4, which satisfies neither side of the relation. Proofs of leads-to properties are carried out inductively, with ensures as a base case. Formally, the rules of inference can be summarized as: (basis) p ensures q where S is any set of predicates. The rules are written in hypothesis-conclusion form; each has an assumption above the line, and a deduction below the line. The basis rule, for instance, allows one to conclude p 7! q from p ensures q. The transitivity rule could be used in a proof of Equation 1, taking p to be the formula q to be the formula and r to be the formula 5. The disjunction rule is useful for breaking up a complicated proof into cases. The proof rules introduced above come from standard UNITY, but they are also a part of Mobile UNITY. How- ever, the notion of what is a basic state transition is different in the two models, because Mobile UNITY can express the location- and context-dependent state transitions that typify mobile computing. Although this means the basic Hoare triple must be redefined, the rest of the UNITY inference toolkit, including other rules for carrying out high-level reasoning which are not shown here, are preserved. A.2 Composition. Before giving the new composition mechanisms of Mobile UNITY, we should first describe the standard UNITY mechanisms for program composition. The most basic composition mechanism is known as program union, and we can use the UNITY union operator, [], to construct a new system, denoted by sender [] receiver . Operationally, the new system consists of the union of all the program variables, i.e., variables with the same name refer to the same physical memory; the union of all the assignment statements, which are executed in a fair atomic interleav- ing; and the intersection of the initial conditions. Communication between sender and receiver thus takes place via the shared variable bit . The sender writes an infinite sequence of 1's and 0's to this variable, fairly inter- leaved, and the receiver occasionally reads from this variable to build its history sequence. Note that the receiver may not see every value written by the sender, because execution is a fair interleaving, not turn-taking. Also, the resulting history generated by the receiver may have duplicate entries because the assignment statements of sender may be excluded from execution for a finite amount of time. Another way to compose systems is through the use of superposition, which combines the components by synchronizing statements rather than sharing variables. Superposition on an underlying program F proceeds by adding new statements and variables to F such that the new statements do not assign to any of the original underlying variables of F , and each of the new statements is synchronized with some statement of F . This allows for (1) the maintenance of history variables that do not change the behavior of the underlying program but are needed for certain kinds of proofs, and (2) the construction of layered systems, where the underlying layers are not aware of the higher layer variables For example, the receiver, instead of being composed via program union, could have used superposition to synchronize its assignment to history with the assignments in the sender that update bit , thus ensuring that it would receive an exact history of the values written to bit . However, superposition is limited because communication can take place in only one direction. Also, like program union, it is an essentially static form of composition that provides a fixed relationship between the components. It also would require that the single statement in the program receiver be broken up into two statements, one for recording 1's and the other for recording 0's. The challenge of mobile computing is to model the system in a more modular fash- ion, where the receiver does not know about the internal workings of the sender, and which allows the receiver to be temporarily decoupled from the sender during periods of disconnection. Towards this end we must investigate novel constructs for expressing coordination among the compo- nents, so that for instance, the receiver can get an exact history sequence while the components are connected, but may lose information while disconnected. A major contribution of [1] was the examination of program derivation strategies using union and superposition as basic construction mechanisms. From a purely theoretical standpoint, it is natural to ask whether we can re-think these two forms of program composition by reconsidering the fundamentals of program interaction and what abstractions should be used for reasoning about composed programs. B. System Structuring If the two programs sender and receiver represent mobile components, or software running on mobile hardware, then it is not appropriate to represent the resulting system as a static composition sender [] receiver . Mobile computing systems exhibit reconfiguration and disconnection of the components, and we would like to capture these essentially new features in our model. Composition with standard UNITY union would share the variable bit throughout system execution and would prohibit dynamic reconfiguration and disconnection of the components. In this section we introduce a syntactic structure that MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 5 makes clear the distinction between parameterized program types and processes which are the components of the sys- tem. A more radical departure from standard UNITY is the isolation of the namespaces of the individual processes. We assume that variables associated with distinct processes are distinct even if they bear the same name. For example, the variable bit in the sender from the earlier example is no longer automatically shared with the bit in the receiver- they should be thought of as distinct variables. To fully specify a process variable, its name should be prepended with the name of the component in which it appears, for example sender.bit or receiver.bit . The separate namespaces for programs serve to hide variables and treat them as internal by default, instead of universally visible to all other components. This will facilitate more modular system specifications, and will have an impact on the way program interactions are specified for those situations where programs must communicate. Figure 2 shows the system sender-receiver which embodies these concepts. System sender-receiver program sender at - program receiver at - Components receiver at - 0 [] sender at - 0 Interactions Fig. 2. Example system notation. The system starts out by declaring its name, in this case sender-receiver . Then, a set of programs are given, each of which is structured like a standard UNITY program, the details of which are elided here. A new feature of these programs is the addition of a program variable - which stands for the current location of the program. It could have been placed in the declare section with the other program vari- ables, but it is promoted here to the same line as the program name to emphasize its importance when reasoning about mobile computations. The precise semantics of the location variable will be discussed in Section II-C. Assume for now that the internals of each program are as given in Figure 1. In Figure 2, these programs are really type declarations that are instantiated in the Components section. In general, the program types may parameters that are bound by the instan- tiations; for example, the receiver could have been declared as receiver (i), and might have been instantiated as receiver(1) at - 1 . A whole range of receiver s could have been instantiated in this way. The transient interactions among the program instances should be given in the Interactions section. Constructs used for specifying interactions are unique to Mobile UNITY and will be introduced in Section II-D. For now we leave the details of this section blank, but it will be developed further to capture the location- and context-dependent aspects of communication between the components C. Location Mobile computing systems must operate under conditions of transient connectivity. Connectivity will depend on the current location of components and therefore location is a part of our model. Just as standard UNITY does not constrain the types of program variables, we do not place restrictions on the type of the location variable -. It may be discrete or continuous, single or multi-dimensional. This might correspond to latitude and longitude for a physically mobile platform, or it may be a network or memory address for a mobile agent. A process may have explicit control over its own location which we model by assignment of a new value to the variable modeling its location. For instance, a mobile receiver might contain the statement - := NewLoc(-), where the function NewLoc returns a new location, given the current location. In general, such an assignment could compute a new location based on arbitrary portions of the program state, not just the current location. In a physically moving system, this statement would need to be compiled into a physical effect like actions on mo- tors, for instance. In a mobile code (agents) scenario, this statement would have the effect of migrating an executing program to a new host. Even if the process does not exert control over its own location we can still model movement by an internal assignment statement that is occasionally selected for execution; any restrictions on the movement of a component should be reflected in this statement. Also, - may still appear on the right-hand side of some assignment statements if there is any location-dependent behavior internal to the program. D. Interactions When disconnected, components should behave as ex- pected. This means that the components must not be made too aware of the other programs with which they interface. The sender, for example, must not depend on the presence of a receiver when it transmits a value. It is unrealistic for the sender to block when no receiver is present. However, there are constraints that the two programs must satisfy when they are connected. We wish to express these constraints when the programs are composed, while not cluttering up the individual components in such a way that they must be aware of and dependent on the existence of other programs. This argues for the development of a coordination language sufficiently powerful to express these interactions and to preserve the modularity of a single program running in isolation. As we will see in the sections that follow, this composition mechanism will have certain aspects in common with UNITY union and other traits characteristic of superposition. The new constructs presented here, although they are 6 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XX, NO. X, MONTH 1998 primarily motivated by the need to manage context-dependent coordination of the components in the Interactions section, are really orthogonal to the system structur- ing; one can put these in stand-alone UNITY programs as well. In fact, the formal proof logic abstracts away from the structuring conventions and assumes a flat set of program variables (properly qualified by the name of the program in which they appear) and assignment statements. However, for now, we present each construct and give an example of how each may be used in the Interactions section of the system sender-receiver . D.1 Extra Statements. Suppose that the sender and receiver can only communicate when they are at the same location, and we wish to express the fact that sender.bit is copied to receiver.bit when this is true. We might begin the Interactions section with receiver.bit := sender.bit when This kind of interaction can be treated like an extra program statement that is executed in an interleaved fashion with the existing program statements. The predicate following when is treated like a guard on the statement (when can be read as if ). The statement as written copies the value from sender.bit to receiver.bit when the two programs are at the same location. Here "at the same loca- tion" is taken to mean that the programs can communicate, but in general the when predicate may take into account arbitrary factors such as the distance between the components or the presence of other components. Note that this interaction alone is not guaranteed to propagate every value written by the sender to the receiver; it is simply another interleaved statement that is fairly selected for execution from the pool of all statements. Therefore, the sender may write several values to bit before the extra statement executes once even when the programs are co-located. The receiver may of course move away (by assigning a new value to receiver .- before any value is copied. Also, the construction of receiver.history is not necessarily an accurate account of the history of bits written to receiver.bit , because the execution of the history-recording action is completely unconstrained with respect to the extra statement and will be interleaved in a fair but arbitrary order. D.2 Reactions. A reactive statement provides a mechanism for making certain that each and every value written to sender.bit also appears at receiver.bit . Such a statement would appear in the Interactions section as receiver.bit := sender.bit reacts-to Operationally, the reactive statement is scheduled to execute whenever the predicate following reacts-to is true. In this sense it is a statement with higher priority than the other statements in the system. In general, there may be other reactive statements implementing other interactions. Informally, all of the reactive statements have equal priority and are executed in an interleaved fashion, much like a standard UNITY program. The set of reactive statements, sometimes denoted with the symbol R, continues to execute until no statement would have an effect if executed. Formally this is known as the fixed point of R. Note that this particular statement is idempotent, so if there is no interference from other reactive statements, it reaches fixed point after one execution. In Section II-E we show how this construct can be captured in an axiomatic semantics. Because this propagation occurs after every step of either component, it effectively presents a read-only shared- variable abstraction to the receiver program, when the two components are co-located. Later we will show how to generalize this notion so that variables shared in a read/write fashion by multiple components can be modeled. In gen- eral, reactive statements allow for the modeling of side effects that a given non-reactive statement may have when executed in a given context, such as a particular arrangement of components in space. D.3 Inhibitions. Note that even with reactive propagation of updates to sender.bit , the receiver still will not construct an accurate history of the values that appear on receiver.bit . Because of the nondeterministic interleaving of statements, several values may be written to receiver.bit between executions of the statement that updates receiver.history . In a real wireless communication system, closely synchronized clocks and timing considerations would ensure that values are read at the proper moment so as not to omit or duplicate any bits in the sequence. An inhibitor provides a mechanism for constraining UNITY's nondeterministic scheduler when execution of some statement would be undesirable in a certain global context. Adding a label to a statement lets us express inhibition in a modular way, without modifying the original statement. For example, consider a new sender program, given in Figure 3. The two statements each now carry a program sender at - declare initially assign Fig. 3. A new version of program sender that counts bits. label; we can refer to the first as sender.s0 and the second as sender.s1 . Each also updates the integer counter to re- MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 7 flect the number of bits written so far. This counter serves as an abstraction for a real-time clock, virtual or actual, that may be running on program sender . If we assume that the statement of program receiver is labeled read , for example, read :: history := history \Delta bit then we might add the following set of clauses to the Interactions section: inhibit sender.s0 when sender.counter ? length(receiver.history) when sender.counter ? length(receiver.history) [] inhibit receiver.read when length(receiver.history) - sender.counter . The net effect of inhibit s when p is a strengthening of the guard on statement s by conjoining it with :p and thus inhibiting execution of the statement when p is true. The inhibitions given above constrain execution so that the receiver reads exactly one bit for every bit written by the sender. Note that the constraint applies equally well when the components are disconnected; this is not unrealistic because we can assume that realtime clocks can remain roughly synchronized even after disconnection, even though the reactive propagation of values will cease. Reactive statements must not be inhibited. D.4 Transactions. With reactive propagation and inhibitions as given above, execution of the system will append values to receiver:history even when the two components are disconnected. Thus there will be subsequences of receiver:history containing redundant copies of the last value written by the sender. In an actual wireless transmission system, the receiver does have some indication of receipt of a transmission, and would not build a history that depended only on timing constraints. In our model, this might be represented as an extra third state that can be taken on by the wireless transmission medium. Assume that the bit in each component was therefore declared: and initialized to ?, which means simply that no transmission is currently taking place. Transmission of a bit by the sender then involves placing a value on the communications medium, and then returning it to a quiescent state. The receiver may then reactively record the value written. A transaction provides a form of sequential execution, and can be used by the statements in sender that write new values to sender.bit : A transaction consists of a sequence of assignment state- ments, enclosed in angle brackets and separated by semi- colons, which must be scheduled in the specified order with no other nonreactive statements interleaved in between. The assignment statements of standard UNITY may be viewed as singleton transactions. Note that reactive statements are allowed to execute to fixed point at each semi-colon and at the end of the transaction; this lets us write a new receiver program, shown in Figure 4. Here the first program receiver at - declare sequence of boolean initially assign flag := history reacts-to bit 6=? -:flag [] flag := 0 reacts-to bit =? Fig. 4. A new version of program receiver that reacts to transactions. reactive statement records values written to the shared bit , and the variable flag is added to make the reactive recording idempotent. Another reactive statement is added to reset flag when the communications medium returns to a quiescent state. Transactions may be inhibited, but may not be reactive. E. Proof Logic Now we give a logic for proving properties of programs that use the above constructs. The execution model has assumed that each non-reactive statement is fairly selected for execution, is executed if not inhibited, and then the set of reactive statements, denoted R, is allowed to execute until it reaches fixed point, after which the next non-reactive statement is scheduled. In addition, R is allowed to execute to fixed point between the sub-statements of a transaction. These reactively augmented statements thus make up the basic atomic state transitions of the model and we denote them by s , for each non-reactive statement s. We denote the set of non-reactive statements (including transactions) by N . Thus, the definitions for basic co and transient properties become: and transient Even though s is really a possibly inhibited statement augmented by reactions, we can still use the Hoare triple notation fpgs fqg to denote that if s is executed in a state satisfying p, it will terminate in a state satisfying q. The Hoare triple notation is appropriate for any terminating computation. We first deal with statement inhibition. The following rule holds for non-reactive statements s, whether they are 8 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XX, NO. X, MONTH 1998 transactions or singleton statements: fpgs fqg (2) We define i(s) to be the disjunction of all when predicates of inhibit clauses that name statement s. Thus, the first part of the hypothesis states that if s is inhibited in a state satisfying p, then q must be true of that state also. The notation s R denotes the statement s extended by execution of the reactive statement set R. For singleton, non-transactional statements, frgs R fqg can be deduced from frgs R fqg (3) where H may be computed as the strongest postcondition of r with respect to s, or guessed at as appropriate. We take frgsfHg from the hypothesis to be a standard Hoare triple for the non-augmented statement s. The notation FP (R) denotes the fixed-point predicate of the set of re-active statements, which can be determined from its text. The "in R" must be added because the proof of termination is to be carried out from the text of the reactive statements, ignoring other statements in the system. This can be accomplished with a variety of standard UNITY techniques. For statements that consist of multiple steps in a trans- action, we have the rule where w may be guessed at or derived from r and q as appropriate. This represents sequential composition of a reactively-augmented prefix of the transaction with its last sub-action. Then Equation 3 can be applied as a base case. This rule may seem complicated, but it represents standard axiomatic reasoning for ordinary sequential pro- grams, where each sub-statement is a predicate transformer that is functionally composed with others. The notation and proof logic presented above provide tools for reasoning about concurrent, mobile systems. Apart from the redefinition of the basic notion of atomic transitions, we keep the rest of the UNITY inference toolkit which allows us to derive more complex properties in terms of these primitives. In the following sections, we will show how the programming notation can be used to construct systems of mobile components that exhibit much more dynamic behavior than could be easily expressed with standard UNITY. III. Transient Sharing In the previous sections, we presented a notation and logic for reasoning about systems of mobile components. In this section and in Section IV, we attempt to build higher level abstractions out of those low-level primitives that will contribute to the design of systems that are decoupled and context-sensitive. To be successful, such abstractions should be familiar to designers, should take into account the realities of mobile computing, should be implementable, and should have a strong underlying formal foundation. An obvious starting point is the communication mechanisms from standard UNITY, namely shared variables and statement synchro- nization. This section examines a variant of sharing suited to mobile computing systems and gives an underlying semantics in terms of the notation we have already developed. In the mobile setting, variables from two independently moving programs are not always connected, and this is reflected in the model by the isolation of each of the names- paces, as was the case with sender.bit and receiver.bit from our earlier example. However, with the addition of a re-active propagation statement to the Interactions section, these two variables took on some of the qualities of a shared variable. While the two components were at the same lo- cation, any value written by the sender was immediately visible to the receiver . The semantics of reactive statements guarantee that such propagation happens in the same atomic step as the statement sender.s0 or sender.s1 . Sharing may also be an appropriate abstraction for communication at a coarser granularity; for example, one might think of two mobile hosts as communicating via a (virtual) shared packet, instead of a single shared bit. This is realistic because of the lower level protocols, such as exponential back-off, that are providing serialized access to the communications medium. At an even coarser (more abstract) level, there might be data structures that are replicated on each host, access to which is serialized by a distributed algorithm implementing mutual exclusion. Of course, no such algorithm can continue to guarantee both mutual exclusion and progress in the presence of disconnection, but our (so far informal) notion of a transiently shared variable does not require consistency when disconnected. In what follows, we package these notions into a coordination construct that can be formally specified and reasoned about. As a running example, we consider a queue of documents to be output on a printer. Assume that a lap-top computer, connected by some wireless communication medium, is wandering in and out of range of the printer, so it maintains a local cache of this queue. When the laptop is in range of the printer, updates to the queue are atomically propagated, expressed as a transient sharing of the queue. This may be denoted by the expression The operations on the queue could include the laptop appending or deleting items from the queue, and the printer deleting items from the head of the queue as it finishes each job. The - relationship can be defined formally in terms of reactive statements that propagate changes. Because the sharing is bidirectional, there is slightly more complexity than the earlier example where a single reactive statement could propagate values in one direction. In the present case, we need a mechanism for detecting changes and selectively propagating only new values. Therefore we add MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 9 additional variables to each program that model the previous state of the queue. In program laptop, this variable is called q printer:q , and in program printer , this variable is called . The reactive statements that detect and propagate changes are printer.q , printer.q laptop:q , laptop.q printer:q := laptop.q , laptop.q , laptop.q reacts-to laptop.q 6= laptop.q printer:q laptop.q , laptop.q printer:q , printer.q laptop:q := printer.q , printer.q , printer.q reacts-to printer.q 6= printer.q laptop:q which execute when any history variable is different from the current value of the variable which it is tracking, when the components are connected. Each statement updates both history variables as well as the remote copy of the queue. This can be thought of as "echo cancellation," in that the remote copy is kept the same as its history vari- able, and the reverse reaction is kept disabled. In addition, we add statements that simply update the history vari- ables, without propagating values, when the components are disconnected: laptop.q printer:q := laptop.q reacts-to laptop .- 6= printer .- printer.q laptop:q := printer.q reacts-to printer .- 6= laptop .- These statements reflect the fact that because disconnection may take place at any moment, one component cannot know that its change actually did propagate to the remote component and so the local behavior (update of the history variable) must be exactly the same in both cases. Although the reactions given above may meet our informal expectations for a shared variable while connection is continuous, there are some subtle issues that arise when disconnection and reconnection are allowed. For instance, when disconnection takes place, the laptop and printer each have separate identical copies of the queue. If changes are made independently, for instance, if the laptop adds a few items and the printer deletes a few items, an inconsistent state arises which may present a problem upon re- connection. The semantics given above are well defined for this case: whichever component makes the first assignment to the reconnected queue will have its copy propagated to the other component. This may be undesirable; documents which have already been printed may be re-inserted into the queue, or documents which have been added by the laptop while disconnected may be lost. Instead of wiping out these changes we would like to integrate them according to some programmer-specified policy. For inspiration we can look to filesystems and databases like [9] and [10] that operate in a disconnected mode. Here the program variables would be replicated files or records of a database, and update propagation is possible only when connectivity is available. These systems also provide a way for the programmer to specify reintegration policies, which indicate what values the variables should take on when connectivity is re-established after a period of disconnection. We call this an engage value. The programmer may also wish to specify what values each variable should have upon disconnection. We call these disengage values. For exam- ple, the print queue example may be extended with the following laptop.q - printer.q when laptop engage laptop.q \Delta printer.q disengage ffl, printer.q The engage value specifies that upon reconnection, the shared queue should take on the value constructed by appending printer.q to laptop.q . The disengage construct contains two values; the first is assigned to laptop.q and the second is assigned to printer.q upon disconnection. The values given empty laptop.q and leave the printer.q untouched. This is justified because the queue would realistically reside on the printer during periods of disconnection and the laptop would have no access to it. However, any documents appended to the queue by the laptop are appended to the print queue upon reconnection. Formally, the above construct would translate into reactions that take place when a change in the connection status is detected. Again we add an auxiliary history variable, this time to record the status of the connection, denoted by status laptop:q;printer:q . For engagement, we add laptop.q , printer.q , status laptop:q;printer:q := laptop.q which integrates both values when the connection status changes from down to up. For disengagement, we add laptop.q , printer.q , status laptop:q;printer:q := ffl, printer.q , false reacts-to status laptop:q;printer:q printer .- 6= laptop .- which assigns different values to each variable when the connection status changes from up to down. In the absence of interference, each of these statements executes once and is then disabled. Systems like [9] and [10] have a definite notion of reintegration policies like engage values when a client reconnects to a fileserver or when two replicas come into contact. Specification of disengage values may be of less practical significance unless disconnection can be predicted in advance. Although this is not feasible for rapidly reconfiguring systems like mobile telephone networks, it may in fact be a good abstraction for the file hoarding policies of [9], which can be carried out as a user prepares to take his laptop home at the end of a workday, for instance. Predictable disconnection is not possible in every situa- tion, for example when we try to model directly a mobile telephone system. Users travel between base stations at will and without warning. Also, an operating system for a wireless laptop may be attempting to hide mobility from Construct Description Definition Read-only transient sharing (A.x is read by B.y) B.y , B.y A:x , A.xB:y := A.x , A.x , A.x reacts-to A.x 6= A.xB:y - p A.xB:y := A.x reacts-to :p Read-write transient sharing engage(A.x , B.y) when p value e Engagement 1 A.x , B.y , status A:x;B:y := e, e, true disengage(A.x , B.y) when p value Disengagement 2 A.x , B.y , status A:x;B:y := d 1 , d 2 , false reacts-to status A:x;B:y - :p I Transient sharing notational constructs. its users and should be written in such a way that it can handle sudden, unpredictable disconnection. Such a system would also be more robust against network failures not directly related to location. Because of well known results on the impossibility of distributed consensus in the presence of failures [21], providing engage and disengage semantics in these settings is possible only in a probabilistic sense. This may be adequate; consider, for instance, the phenomenon of metastable states [22]. Almost every computing device in use today is subject to some probability of failure due to metastability, but the probability is so low that it is almost never considered in reasoning about these systems. A similarly robust implementation of engage and disengage may be possible. However, the basic semantics of - do not imply distributed consensus and are in fact implementable. The transient sharing construct given above is a relationship between two variables, but it is compositional in a very natural way. For instance, suppose we would like to distribute the print jobs among two different printers. This could be accomplished by simply adding another sharing relationship of the form printer.q - printer2.q when true which specifies that the queue should be shared with printer2 always. Each printer would have atomic access to this shared queue and could remove items from the head as they are printed. Because all reacts-to statements are executed until fixed point, any change to one of the three variables is propagated to the other two, when the laptop is co-located with the printer. This transitivity is a major factor contributing to the construction of modular systems, as it allows the statement of one component to have far-reaching implicit effects that are not specified explicitly in the program code for that component. A summary of the notation developed in this section appears in Table I, which also breaks up the - construct into two uni-directional sharing relationships. In general these can be combined in arbitrary ways, but remember that any proofs of correctness require proof that R terminates, so not every construction will be correct. For example, if there are any cycles in the sharing relationship, and if two different variables on a given cycle are set to distinct values in the same assignment statement, then it is not possible to prove termination. This is analagous to UNITY's restriction that each statement assigns a unique value to each left-hand variable. Also, termination of R may be difficult to prove if some engagement or disengagement values cause other when predicates to change value. The transient sharing abstraction presented here has shown promise as a way to manage the complexity of con- current, mobile systems. Based on a familiar programming paradigm, that of shared memory, it provides a mechanism for expressing highly decoupled and context-dependent sys- tems. The abstraction is apparently a good one for low-level wireless communication, and mutual exclusion protocols that implement the abstraction at a coarser level of granularity may be simple generalizations of existing repli- cation, transaction, or consistency algorithms. This section presented a formal definition for the concept that facilitates reasoning about systems that make use of it. IV. Transient Synchronization The previous section presented new abstractions for shared state among mobile components, where such sharing is necessarily transient and location dependent, and where the components involved execute asynchronously. How- ever, synchronous execution of statements is also a central part of many models of distributed systems. In this section we investigate some new high-level constructs for synchronizing statements in a system of mobile components, trying to generalize the synchronization mechanisms of existing non-mobile models. For example, CSP [23] provides 1 If engagement is used without a corresponding disengagement, an extra reaction must be added to reset status A:x;B:y to false when p becomes false. Similarly, if disengagement is used without a corresponding en- gagement, an extra reaction must be added to set status A:x;B:y to true when p becomes true. MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 11 a general model in which computation is carried out by a static set of sequential processes, and communication (in- cluding pure synchronization) is accomplished via blocking, asymmetric, synchronous, two-party interactions called In- put/Output Commands. The I/O Automata model [24] communication via synchronization of a named output action with possibly many input actions of the same name. Statement synchronization is also a part of the UNITY model, where it provides a methodology for proving properties of systems that can only be expressed with history variables and for the construction of layered sys- tems. Synchronous composition can also be used in the refinement process [25], although our emphasis here is on composition of mobile programs rather than refinement. In UNITY, synchronous execution is expressed via su- perposition, in which a new system is constructed from an underlying program and a collection of new statements. Because the goal is to preserve all properties of the underlying program in the new system, the new statements must not assign values to any of the variables of the underlying program. In this way, all execution behaviors that were allowed by the underlying program executing in isolation are also allowed by the new superposed system, and any properties of the underlying program that mention only underlying variables and were proven only from the text of the underlying program are preserved. The augmented statements may be used to keep histories of the underlying variables or to present an abstraction of the underlying system as a service to some higher layer environment. UNITY superposition is an excellent example of how synchronization can be used as part of a design methodology for distributed systems. It also shows an important distinction between our notion of synchronization, which is the construction of new, atomic statements from two or more simpler atomic statements by executing them in par- allel, and the notion of synchronous computing which is a system model characterized by bounded communication and computation delays [26]. While the latter is a very important component of our current understanding of distributed systems, and in many circumstances is perhaps a prerequisite to the implementation of the former, it is not our focus here. Rather, we examine mechanisms that allow us to compose programs and to combine a group of statements into a new one through parallel execution. This idea of statement co-execution was inspired by UNITY superposition However, superposition is limited in two important ways. First, a superposed system is statically defined and synchronization relationships are fixed throughout the execution of the system. Continuing the theme of modeling mobility with a kind of transient program composition, we would like the ability to specify dynamically changing and location dependent forms of synchronization where the participants may enter into and leave synchronization relationships as the computation evolves. Static forms of statement synchronization are more limited as discussed in [27]. Second, superposition is an asymmetric relationship that subsumes one program to another and disallows any communication from the superposed to the underlying program. While this is the source of the strong formal results about program properties, such a restriction may not be appropriate in the mobile computing domain where two programs may desire to make use of each other's services and carry out bi-directional communication while making use of some abstraction for synchronization. Inspired by UNITY superposition, which combines statements into new atomic actions, we will now explore some synchronization mechanisms for the mobile computing do- main. These take the form of coordination constructs involving statements from each of two separate programs. Informally, the idea is to allow the programmer to specify that the two statements should be combined into one atomic action when a given condition is true. For exam- ple, consider two programs A and B , where A contains the integer x and B contains the integer y . Assume there is a statement named increment in each program, where A.increment is increment and B.increment is increment :: y Let us assume the programs are mobile, so each contains a variable -, and that they can communicate only when co-located. Also, assume that the counters represent some value that must be incremented simultaneously when the two hosts are together. We might use the following notation in the Interactions section to specify this coordination A.increment ed B.increment when Note that this does not prohibit the statements from executing independently when the programs are not co- located. If the correctness criteria state that the counters must remain synchronized at all times, we could add the following two inhibit clauses to the Interactions section: inhibit A.increment when (A.- inhibit B.increment when (A.- As distinct from standard UNITY superposition, the ed construct is a mechanism for synchronizing pairs of statements rather than specifying a transformation of an underlying program. Also, the interaction is transient and location dependent, instead of static and fixed throughout system execution. To reason formally about transient statement synchro- nization, we must express it using lower-level primitives. The basic idea is that each statement should react to selection of the other for execution, so that both are executed in the same atomic step. We can accomplish this by separating the selection of a statement from its actual execution, and assume for example that a statement A.s is of the form: A.s.driver :: hA.s phase := GO; A.s phase := IDLEi A.s.actionkA.s f := false reacts-to A.s phase A.s f := true reacts-to A.s phase = IDLE (5) where A.s phase is an auxiliary variable that can hold a value from the set fGO, IDLEg, and A.s.action is the actual assignment that must take place. Note that A.s.action reacts to a value of GO in A.s phase and that A.s f is simultaneously set to false so that the action executes only once. When A.s phase returns to IDLE, the flag A.s f is reset to true so that the cycle can occur again. The non-reactive statement A.s.driver will be selected fairly along with all other non-reactive statements, and because A.s.action reacts during this transaction, the net effect will be the same as if A.s.action were listed as a simple non-reactive state- ment. However, expressing the statement with the three lines above gives us access to and control over key parts of the statement selection and execution process. Most im- portantly, we can provide for statement synchronization by simply sharing the phase variable between two statements. Assuming both statements are of the form given in Equation 5, we can define ed as: A.s ed B.t when r \Delta Then, whenever one of the statements is selected for execution by executing A.s.driver or B.t.driver , the corresponding phase variable will propagate to the other statement and reactive execution of B.t.action or A.s.action will pro- ceed. Also, transitive and multi-way sharing will give us transitive and multi-way synchronization. Note that if we wish to disable the participants from executing, as we did with inhibit above, we must be sure to inhibit all participants at the level of the named driver transactions. If we inhibit only some of them, they may still fire reactively if ed is used to synchronize them with statements that are not inhibited. We call the ed operator coselection because it represents simultaneous selection of both statements for execution. When used in the Interactions section of a sys- tem, it embodies the assumption that statement execution is controlled by a phase variable, as in Equation 6. The semantics of Equations 5 and 6 do not really guarantee simultaneous execution of the statements in the same sense as UNITY "k", but rather that the statements will be executed in some interleaved order during R. In many cases this will be equivalent to simultaneous execution because neither statement will evaluate variables that were assigned to by the other. However, there may be cases when we desire both statements to evaluate their right- hand-sides in the old state without using values that are set by the other statement. For these cases, we can add another computation phase to Equation 5 which models the evaluation of right-hand sides as a separate step from assignment to left-hand variables: A.s.driver :: hA.s phase := LOAD; A.s phase := STORE; A.s phase := IDLEi A.s.loadkA.s lf := false reacts-to A.s phase = LOAD - A.s lf A.s.storekA.s sf := false reacts-to A.s phase = STORE - A.s sf reacts-to A.s phase = IDLE (7) Here the phase variables may hold values from the set fLOAD, STORE, IDLEg and the original A.s.action is split into two statements, one for evaluating and one for assigning. A.s.load is assumed to evaluate the right-hand side of A.s.action and store the results in some internal variables that are not given explicitly here. A.s.store is assumed to assign these values to the left-hand variables of A.s.action . In this way, statements can still be synchronized by sharing phase variables as in Equation 6, but now all statements will evaluate right-hand sides during the LOAD phase, will assign to left-hand variables during the STORE phase, and will reset the two flags during the IDLE phase. This prevents interference between any two synchronized statements, even if the two are connected indirectly through a long chain of synchronization relationships, and even if variables assigned to by the statements are shared indirectly. For example, we return to the increment example and consider the following set of interactions: A.increment ed B.increment when Here the statements A.increment and B.increment are syn- chronized, but the variable A.x is indirectly shared with the variable B.y , via the intermediate variable C.z , when the predicate r is true. If the increment statements are of the form given in Equation 5, then changes to one variable may inadvertently be used in computing the incremented value of the other variable, which seems to violate the intuitive semantics of simultaneous execution. In con- trast, increment statements of the form given in Equation 7 have a separate LOAD phase for computing the right-hand sides of assignment statements and shared variables are not assigned to during this phase. Assignment to shared variables, and the associated reactive propagation of those values, is reserved until the STORE phase. This isolates the assignment statements from one another and prevents unwanted communication. There may be situations, however, where we do wish the two statements to communicate during synchronized exe- cution. This strategy is central to models like CSP and I/O Automata, where communication occurs along with synchronized execution of statements. In CSP, a channel is used to communicate a value from a single sender to a single receiver. I/O automata can pass arbitrary parameters from an output statement to all same-named input MCCANN AND ROMAN: COMPOSITIONAL PROGRAMMING ABSTRACTIONS FOR MOBILE COMPUTING 13 statements. As an example, we will now give a construction that expresses I/O Automata-style synchronization. Recall that each automaton has a set of input actions, a set of internal actions, and a set of output actions. The execution of any action may modify the state of the machine to which it belongs; in addition, the execution of any output action takes place simultaneously with the execution of all input actions of the same name in all other machines. We can assume that output actions are of a form similar to Equation 5, but with an important addition: A.s.driver :: hA.s params := exp; A.s phase := GO; A.s phase := IDLEi A.s.actionkA.s f := false reacts-to A.s phase A.s f := true reacts-to A.s phase = IDLE (8) We have added the assignment A.s params := exp as the first statement of the transaction. This assignment models binding of the output parameters to a list of auxiliary variables A.s params . Here exp is assumed to be a vector of expressions that may reference other program variables. For example, assume for a moment that the function of the A.increment statement from the earlier example is to increment the variable by a value e which is a function of the current state of A. Assume also that B.increment must increment B.y by the same amount when the two are co-located. This value could be modeled as a parameter A.increment p of the synchronization, and A.increment would then be of the form given in Equation 8: A.increment.driver :: hincrement p := e; increment phase := GO; increment phase := IDLEi x increment p kA.s f := false reacts-to A.s phase A.s f := true reacts-to A.s phase = IDLE And B.increment could also be of this form, with perhaps a different expression e for the increment value. We could then express the sharing of the parameter with the interaction A.increment and the synchronization of the statements with A.increment ed B.increment when Thus, either statement may execute its driver transaction, which in the first phase assigns a value to the parameter which is propagated to the other component, in the second phase triggers execution of both statements, and in the third resets the flags associated with each statement. We can easily make use of the guards on the statements to specify many different and interesting forms of synchronization with the use of appropriately tailored inhibit clauses. For example, in addition to the definition of coselection defined by Equations 5 and 6, we can specify a notion of coexecution which has the added meaning that when co-located, the statements may only execute when both guards are enabled. This might be defined as coexecute(A.s, B.t, r) A.s phase - B.t phase when r inhibit A.s.driver when r - :(A.s.guard - B.t.guard ) inhibit B.t.driver when r - :(A.s.guard - B.t.guard ) which still allows the statements to execute in isolation when not co-located. In contrast, we might require that the statements may not execute in isolation when disconnected. We call this exclusive coexecution and it could be specified as xcoexecute(A.s, B.t, r) A.s phase - B.t phase when r inhibit A.s.driver when :(r - A.s.guard - B.t.guard ) inhibit B.t.driver when :(r - A.s.guard - B.t.guard ) A similar notion of exclusive coselection could be defined if we ignore the guards on the statements xcoselect(A.s, B.t, r) A.s phase - B.t phase when r inhibit A.s.driver when :r inhibit B.t.driver when :r Each of these constructions could be generalized to pass parameters from a sender to a receiver. In general, if both driver statements are of the form specified in Equation 8, either statement may bind parameters and propagate them to the other as long as the sharing specified is bi-directional and the driver statement itself is not inhibited. Because the semantics of a transaction mean that it will finish before another is allowed to begin, there is no ambiguity about which statement is currently executing and no conflict in assigning parameters to the synchronized execution. Also, any of the above could be used with statements of the form in Equation 7 instead of Equation 5 to provide truly simultaneous access instead of interleaved access to any other shared variables that might be referenced by or assigned to by the various actions. In contrast to the symmetric forms of synchronization considered so far, the coordination between actions in models such as I/O Automata and CSP is asymmetric. In each model, actions are divided into input and output classes; parameters are passed from output actions to input actions. The two models differ in the number of participants in a synchronization. I/O Automata are capable of expressing one-to-many synchronization styles, while CSP emphasizes pairwise rendezvous of output actions with input actions. All of these synchronization styles can be expressed in Mobile UNITY with appropriate use of transactions, variable sharing, and inhibitions. For example, one-to-many synchronization with parameter passing can be simulated by 14 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XX, NO. X, MONTH 1998 a quantified set of one-way variable sharing relationships, where the output statement executes as a transaction and the input statements are simply reactive. For rendezvous style synchronization, the phase variable must be propagated to at most one input action, which can be ensured by a flag which is set by the first (nondeterministically chosen) reactive statement and which prevents other propagations from taking place. Of course, other aspects of CSP, such as the dynamic creation and deletion of terms, could not be so easily captured in a UNITY-style model because of fundamental differences in the underlying approaches. Similarly, the discussion of I/O Automata has assumed that it is acceptable to model a whole set of IOA actions with one parameterized UNITY action, which may not be appropriate in every case. Even so, modeling the basic synchronization mechanisms of the two models in Mobile UNITY can be a useful exercise. Our point in examining the many different forms of synchronization is to show the versatility and broad applicability of the model. Because the field of mobile computing is so new, we cannot predict which high-level abstractions will become dominant and gain acceptance in the research community. However, we believe that the examples above show that Mobile UNITY can at least formalize direct generalizations to the mobile setting of existing mechanisms for synchronous statement execution in models of non-mobile concurrency, and there is good reason to believe it is capable of expressing new constructs that may be proposed in the future. V. Discussion The Mobile UNITY notation and logic presented in this paper is the result of a careful reevaluation of the implications of mobility on UNITY. We took as a starting point the notion that mobile components should be modeled as programs (by the explicit addition of an auxiliary variable representing location), and that interactions between components should be modeled as a form of dynamic program composition (with the addition of coordination constructs). The UNITY-style composition, including union and super- position, led to a new set of basic programming constructs amenable to a dynamic and mobile setting. Previous work extended the UNITY proof logic to handle pairwise forms of such interaction. This paper presented a more modular and compositional construction of transient sharing and synchronization that allowed for multi-party interactions among components. We applied these constructs to a very low-level communication task in an attempt to show that the basic notation is useful for realistic specifications involving discon- nection. The seemingly very strong reactive semantics matched well the need to express dynamically changing side-effects of atomic actions. Finally, we explored the expressive power of the new notation by examining new transient forms of shared variables and synchronization, mostly natural extensions of the comparable non-mobile abstractions of interprocess communication-indeed others may propose radically different communication abstractions for mobile computing. The notation was able to express formally all extensions that were considered and promises to be a useful research tool for investigating whatever new abstractions may appear. Plans for future work include the application of Mobile UNITY to distributed databases with consistency semantics, capable of continuing operation in the presence of disconnection. These problems only recently have received attention in the engineering and re-search community, and formal reasoning has an important role to play in communicating and understanding proposed solutions as well as the assumptions made by each. VI. Acknowledgements This paper is based upon work supported in part by the National Science Foundation of the United States under Grant Numbers CCR-9217751 and CCR-9624815. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation --R Parallel Program Design: A Foundation "Formal derivation of concurrent pro- grams: An example from industry," "A calculus of mobile processes. I," "Foundations of actor semantics," Actors: A Model of Concurrent Computation in Distributed Systems "An asynchronous model of locality, fail- ure, and process mobility," "Distributed processes and location failures," "Experience with disconnected operation in a mobile computing environment," "Managing update conflicts in Bayou, a weakly connected replicated storage system," "Mobisaic: An information system for a mobile wireless computing environment," "Context-aware computing applications," "IP mobility support," "Mobile UNITY coordination constructs applied to packet forwarding for mobile hosts," "An algebraic verification of a mobile network," "Specification and proof of an algorithm for location management for mobile communication devices," "Mobile UNITY: Reasoning and specification in mobile comput- ing," "A logic for concurrent programming: Safety," "A logic for concurrent programming: Progress," "An axiomatic basis for computer programming," "Impossibility of distributed consensus with one faulty process," "Anomalous behavior of synchronizer and arbiter circuits," "Communicating sequential processes," "An introduction to in- put/output automata," "Fundamentals of object-oriented specification and modeling of collective behaviors," "On the minimal synchronism needed for distributed consensus," "Dynamic synchrony among atomic actions," --TR --CTR Gruia-Catalin Roman , Amy L. Murphy, Rapid development of dependable applications over Ad hoc networks, ACM SIGSOFT Software Engineering Notes, v.25 n.1, p.77-78, Jan 2000 Gian Pietro Picco , Amy L. Murphy , Gruia-Catalin Roman, Developing mobile computing applications with LIME, Proceedings of the 22nd international conference on Software engineering, p.766-769, June 04-11, 2000, Limerick, Ireland Gruia-Catalin Roman , Christine Julien , Jamie Payton, Modeling adaptive behaviors in Context UNITY, Theoretical Computer Science, v.376 n.3, p.185-204, May, 2007 Camelia Zlatea , Tzilla Elrad, A design methodology for mobile distributed applications based on UNITY formalism and communication-closed layering, Proceedings of the eighteenth annual ACM symposium on Principles of distributed computing, p.284, May 04-06, 1999, Atlanta, Georgia, United States Cecilia Mascolo , Gian Pietro Picco , Gruia-Catalin Roman, A fine-grained model for code mobility, ACM SIGSOFT Software Engineering Notes, v.24 n.6, p.39-56, Nov. 1999 A. L. Murphy , G.-C. Roman , G. Varghese, An Exercise in Formal Reasoning about Mobile Communications, Proceedings of the 9th international workshop on Software specification and design, p.25, April 16-18, 1998 Gian Pietro Picco , Amy L. Murphy , Gruia-Catalin Roman, LIME: Linda meets mobility, Proceedings of the 21st international conference on Software engineering, p.368-377, May 16-22, 1999, Los Angeles, California, United States Gruia-Catalin Roman , Peter J. McCann, A Notation and Logic for Mobile Computing, Formal Methods in System Design, v.20 n.1, p.47-68, January 2002 Guido Menkhaus, An Architecture for Supporting Multi-Device, Client-Adaptive Services, Annals of Software Engineering, v.13 n.1-4, p.309-327, June 2002 Tore Fjellheim , Stephen Milliner , Marlon Dumas , Julien Vayssire, A process-based methodology for designing event-based mobile composite applications, Data & Knowledge Engineering, v.61 n.1, p.6-22, April, 2007 Cecilia Mascolo , Gian Pietro Picco , Gruia-Catalin Roman, CODEWEAVE: Exploring Fine-Grained Mobility of Code, Automated Software Engineering, v.11 n.3, p.207-243, June 2004 S. K. S. Gupta , P. K. Srimani, Adaptive Core Selection and Migration Method for Multicast Routing in Mobile Ad Hoc Networks, IEEE Transactions on Parallel and Distributed Systems, v.14 n.1, p.27-38, January Gruia-Catalin Roman , Jamie Payton, A principled exploration of coordination models, Theoretical Computer Science, v.336 n.2-3, p.367-401, 26 May 2005 Peter J. McCann , Gruia-Catalin Roman, Modeling mobile IP in mobile UNITY, ACM Transactions on Software Engineering and Methodology (TOSEM), v.8 n.2, p.115-146, April 1999 Ichiro Satoh, Building and Selecting Mobile Agents for Network Management, Journal of Network and Systems Management, v.14 n.1, p.147-169, March 2006 Gianluigi Ferrari , C. Montangero , L. Semini , S. Semprini, Mark, a Reasoning Kit for Mobility, Automated Software Engineering, v.9 n.2, p.137-150, April 2002 Michel Wermelinger , Jos Luiz Fiadeiro, A graph transformation approach to software architecture reconfiguration, Science of Computer Programming, v.44 n.2, p.133-155, August 2002 Gian Pietro Picco , Gruia-Catalin Roman , Peter J. McCann, Reasoning about code mobility with mobile UNITY, ACM Transactions on Software Engineering and Methodology (TOSEM), v.10 n.3, p.338-395, July 2001 Michel Wermelinger , Jos Luiz Fiadeiro, Algebraic software architecture reconfiguration, ACM SIGSOFT Software Engineering Notes, v.24 n.6, p.393-409, Nov. 1999 Wolfgang Emmerich , Cecilia Mascolo , Anthony Finkelstein, Implementing incremental code migration with XML, Proceedings of the 22nd international conference on Software engineering, p.397-406, June 04-11, 2000, Limerick, Ireland Amy L. Murphy , Gian Pietro Picco , Gruia-Catalin Roman, LIME: A coordination model and middleware supporting mobility of hosts and agents, ACM Transactions on Software Engineering and Methodology (TOSEM), v.15 n.3, p.279-328, July 2006 Dianxiang Xu , Jianwen Yin , Yi Deng , Junhua Ding, A Formal Architectural Model for Logical Agent Mobility, IEEE Transactions on Software Engineering, v.29 n.1, p.31-45, January P. Ciancarini , F. Franz , C. Mascolo, Using a coordination language to specify and analyze systems containing mobile components, ACM Transactions on Software Engineering and Methodology (TOSEM), v.9 n.2, p.167-198, April 2000 Gruia-Catalin Roman , Gian Pietro Picco , Amy L. Murphy, Software engineering for mobility: a roadmap, Proceedings of the Conference on The Future of Software Engineering, p.241-258, June 04-11, 2000, Limerick, Ireland

shared variables;formal methods;mobile computing;Mobile UNITY;transient interactions;synchronization;weak consistency

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Incremental Design of a Power Transformer Station Controller Using a Controller Synthesis Methodology.

AbstractIn this paper, we describe the incremental specification of a power transformer station controller using a controller synthesis methodology. We specify the main requirements as simple properties, named control objectives, that the controlled plant has to satisfy. Then, using algebraic techniques, the controller is automatically derived from this set of control objectives. In our case, the plant is specified at a high level, using the data-flow synchronous Signal language, and then by its logical abstraction, named polynomial dynamical system. The control objectives are specified as invariance, reachability, ... properties, as well as partial order relations to be checked by the plant. The control objectives equations are synthesized using algebraic transformations.

Introduction Motivations The Signal language [8] is developed for precise specification of real-time reactive systems [2]. In such systems, requirements are usually checked a posteriori using property verification and/or simulation techniques. Control theory of Discrete Event Systems (DES) allows to use constructive methods, that ensure, a priori, required properties of the system behavior. The validation phase is then reduced to properties that are not guaranteed by the programming process. There exist different theories for control of Discrete Event Systems since the 80's [14, 1, 5, 13]. Here, we choose to specify the plant in Signal and the control synthesis as well as verification are performed on a logical abstraction of this program, called a polynomial dynamical system (PDS) over Z= 3Z . The control This work was partially supported by lectricit de France (EDF) under contract number M64/7C8321/E5/11 and by the Esprit SYRF project 22703. of the plant is performed by restricting the controllable input values with respect to the control objectives (logical or optimal). These restrictions are obtained by incorporating new algebraic equations into the initial system. The theory of PDS uses classical tools in algebraic geometry, such as ideals, varieties and mor- phisms. This theory sets the basis for the verification and the formal calculus tool, Sigali built around the Signal environment. Sigali manipulates the system of equations instead of the sets of solutions, avoiding the enumeration of the state space. This abstract level avoids a particular choice of set implementations, such as BDDs, even if all operations are actually based on this representation for sets. Fig. 1. Description of the tool The methodology is the following (see Figure 1). The user first specifies in Signal both the physical model and the control/verification objectives to be ensured/checked. The Signal compiler translates the Signal program into a PDS, and the control/verification objectives in terms of polynomial relation- s/operations. The controller is then synthesized using Sigali. The result is a controller coded by a polynomial and then by a Binary Decision Diagram. To illustrate our approach, we consider in this paper the application to the specification of the automatic control system of a power transformer station. It concerns the response to electric faults on the lines traversing it. It involves complex interactions between communicating automata, interruption and pre-emption behaviors, timers and timeouts, reactivity to external events, among others. The functionality of the controller is to handle the power interruption, the redirection of supply sources, and the re-establishment of the power following an interruption. The objective is twofold: the safety of material and uninterrupted best service. The safety of material can be achieved by (automatic) triggering circuit-breakers when an electric fault occurs on lines, whereas the best quality service can be achieved by minimizing the number of costumers concerned by a power cut, and re-establishment of the current as quickly as possible for the customers hit by the fault (i.e, minimizing the failure in the distribution of power in terms of duration and size of the interrupted sub-network). 2 Overview of the power transformer station In this section, we make a brief description of the power transformer station network as well as the various requirements the controller has to handle. 2.1 The power transformer station description lectricit de France has hundreds of high voltage networks linked to production and medium voltage networks connected to distribution. Each station consists of one or more power transformer stations to which circuit-breakers are connected. The purpose of an electric power transformer station is to lower the voltage so that it can be distributed in urban centers to end-users. The kind of transformer (see Figure we consider, receives high voltage lines, and feeds several medium voltage lines to distribute power to end-users. Fig. 2. The power transformer station topology. For each high voltage line, a transformer lowers the voltage. During operation of this system, several faults can occur (three types of electric faults are considered: phase PH, homopolar H, or wattmetric W), due to causes internal or external to the station. To protect the device and the environment, several circuit breakers are placed in a network of cells in different parts of the station (on the arrival lines, link lines, and departure lines). These circuit breakers are informed about the possible presence of faults by sensors. Power and Fault Propagation: We discuss here some physical properties of the power network located inside the power transformer station controller. It is obvious that the power can be seen by the different cells if and only if all the upstream circuit-breakers are closed. Consequently, if the link circuit-breaker is opened, the power is cut and no fault can be seen by the different cells of the power transformer station. The visibility of the fault by the sensors of the cells is less obvious. In fact, we have to consider two major properties: On one hand, if a physical fault, considered as an input of our system, is seen by the sensors of a cell, then all the downstream sensors are not able to see some physical faults. In fact, the appearance of a fault at a certain level (the departure level in Figure 3(a) for example) increases the voltage on the downstream lines and masks all the other possible faults. (a) The fault masking (b) The fault propagation Fig. 3. The Fault properties - On the other hand, if the sensors of a cell at a given level (for example the sensors of one of the departure cells as illustrated in Figure 3(b)) are informed about the presence of a fault, then all the upstream sensors (here the sensors of the arrival cell) detect the same fault. Consequently, it is the arrival cell that handle the fault. 2.2 The controller The controller can be divided into two parts. The first part concerns the local controllers (i.e., the cells). We chose to specify each local controller in Signal, because they merge logical and numerical aspects. We give here only a brief description of the behavior of the different cells (more details can be found in [12, 7]). The other part concerns more general requirements to be checked by the global controller of the power transformer station. That specification will be described in the following. The Cells: Each circuit breaker controller (or cell) defines a behavior beginning with the confirmation and identification of the type of the fault. In fact, a variety of faults are transient, i.e., they occur only for a very short time. Since their duration is so short that they do not cause any danger, the operation of the circuit-breaker is inhibited. The purpose of this confirmation phase is let the transient faults disappear spontaneously. If the fault is confirmed, the handling consists in opening the circuit-breaker during a given delay for a certain number of periods and then closing it again. The circuit-breaker is opened in consecutive cycles with an increased duration. At the end of each cycle, if the fault is still present, the circuit-breaker is reopened. Finally, in case the fault is still present at the end of the last cycle, the circuit-breaker is opened definitively, and control is given to the remote operator. The specification of a large part of these local controllers has been performed using the Signal synchronous language [12] and verified using our formal calculus system, named Sigali [7]. Some global requirements for the controller: Even if is quite easy to specify the local controllers in Signal, some other requirements are too informal, or their behaviors are too complex to be expressed directly as programs. 1. One of the most significant problems concerns the appearance of two faults (the kind of faults is not important here) at two different departure cells, at the same time. Double faults are very dangerous, because they imply high defective currents. At the place of the fault, this results in a dangerous path voltage that can electrocute people or cause heavy material damages. The detection of these double faults must be performed as fast as possible as well as the handling of one of the faults. 2. Another important aspect is to know which of the circuit breakers must be opened. If the fault appears on the departure line, it is possible to open the circuit breaker at departure level, at link level, or at arrival level. Obviously, it is in the interest of users that the circuit be broken at the departure level, and not at a higher level, so that the fewest users are deprived of power. 3. We also have to take into account the importance of the departure circuit- breaker. Assume that some departure line, involved in a double faults prob- lem, supplies a hospital. Then, if the double faults occur, the controller should not open this circuit-breaker, since electricity must always delivered to a hospital. The transformer station network as well as the cells are specified in Signal. In order to take into account the requirements (1), (2) and (3), with the purpose of obtaining an optimal controller, we rely on automatic controller synthesis that is performed on the logical abstraction of the global system (network cells). 3 The Signal equational data flow real-time language Signal [8] is built around a minimal kernel of operators. It manipulates signals X, which denote unbounded series of typed values indexed by time t in a time domain T . An associated clock determines the set of instants at which values are present. A particular type of signals called event is characterized only by its presence, and has always the value true (hence, its negation by not is always false). The clock of a signal X is obtained by applying the operator event X. The constructs of the language can be used in an equational style to specify the relations between signals i.e. , between their values and between their clocks. Systems of equations on signals are built using a composition construct, thus defining processes. Data flow applications are activities executed over a set of instants in time. At each instant, input data is acquired from the execution environment; output values are produced according to the system of equations considered as a network of operations. 3.1 The Signal language. The kernel of the Signal language is based on four operations, defining primitive processes or equations, and a composition operation to build more elaborate processes in the form of systems of equations. Functions are instantaneous transformations on the data. The definition of a signal Y t by the function f : 8t; Y Xng. Y, are required to have the same clock. Selection of a signal X according to a boolean condition C is: Y := X when C. If C is present and true, then Y has the presence and value of X. The clock of Y is the intersection of that of X and that of C at the value true. Deterministic merge noted: Z := X default Y has the value of X when it is present, or otherwise that of Y if it is present and X is not. Its clock is the union of that of X and that of Y. Delay gives access to past values of a signal. E.g., the equation ZX with initial value V 0 defines a dynamic process. It is encoded by: ZX := X$1 with initialization ZX init V0. X and ZX have equal clocks. Composition of processes is noted "-" (for processes P 1 and P 2 , with paren- It consists in the composition of the systems of e- quations; it is associative and commutative. It can be interpreted as parallelism between processes. The following table illustrates each of the primitives with a trace: Derived features: Derived processes have been defined on the base of the primitive operators, providing programming comfort. E.g., the instruction specifies that signals X and Y are synchronous (i.e., have equal clocks); when B gives the clock of true-valued occurrences of B. For a more detailed description of the language, its semantic, and applica- tions, the reader is referred to [8]. The complete programming environment also features a block-diagram oriented graphical user interface and a proof system for dynamic properties of Signal programs, called Sigali (see Section 4). 3.2 Specification in Signal of the power transformer station The transformer station network we are considering contains four departure, t- wo arrival and one link circuit-breakers as well as the cells that control each circuit-breaker [7]. The process Physical Model in Figure 4 describes the power and fault propagation according to the state of the different circuit- breakers. It is composed of nine subprocesses. The process Power Propagation describes the propagation of power according to the state of the circuit-breakers (Open/Closed). The process Fault Visibility describes the fault propagation and visibility according to the other faults that are potentially present. The remaining seven processes encode the different circuit-breakers. Fig. 4. The main process in Signal The inputs of this main process are booleans that encode the physical fault- s: Fault Link M, Fault Arr i M (i=1,2), Fault Dep j M (j =1,.,4). They encode faults that are really present on the different lines. The event inputs close . and req open . indicate opening and closing requests of the various circuit-breakers. The outputs of the main process are the booleans Fault Link, Fault Arr i, Fault Dep j, representing the signals that are sent to the different cells. They indicate whether a cell is faulty or not. These outputs represents the knowledge that the sensors of the different cells have. We will now see how the subprocesses are specified in Signal. The circuit-breaker: A circuit-breaker is specified in Signal as follows: The process Circuit-Breaker takes two sensors inputs: Req Open and Req Close. They represent opening and closing requests. The output Close represents the status of the circuit-breaker. (- Close := (ReqClose default (false when ReqOpen) default ZClose ZClose := Close $1 init true Close "= Tick (ReqClose when ReqOpen) "= when (not ReqOpen) -) Fig. 5. The Circuit-breaker in Signal The boolean Close becomes true when the process receives the event close, and false when it receives the event Req open, otherwise it is equal to its last value (i.e. Close is true when the circuit-breaker is closed and false other- wise). The constraint Req Close when Req Open "= when not Req Close says that the two events Req Close and Req Open are exclusive. Power Propagation: It is a filter process using the state of the circuit-breakers. Power propagation also induces a visibility of possible faults. If a circuit-breaker is open then no fault can be detected by the sensors of downstream cells. Fig. 6. specification in Signal of the power propagation This is specified in the process Power Propagation shown in Figure 6. The inputs are booleans that code the physical faults and the status of the circuit-breakers. For example, a fault could be detected by the sensor of the departure cell 1 (i.e. Fault Dep 1 E is true) if there exists a physical if the upstream circuit-breakers are closed (ie, Close Link=true and Close Arr 1=true and Close Dep 1=true). Fault visibility and propagation: The Fault Visibility process in Figure 7, specifies fault visibility and propagation. As we explained in Section 2.1, a fault could be seen by the sensors of a cell only if no upstream fault is present. Fig. 7. Specification in Signal of the fault propagation and visibility For example, a fault cannot be detected by the sensor of the departure cell 1 (i.e. Fault Dep 1 is false), even if a physical fault exists at this level exists at the link level K=true) or at the arrival level 1 It is thus, true just when the departure cell 1 detects a physical fault E) and no upstream fault exists. A contrario, if a fault is picked up by a cell, then it is also picked up by the upstream cells. This is for example the meaning of Fault Link := (when Fault link K. 4 Verification of Signal programs The Signal environment contains a verification and controller synthesis tool- box, Sigali. This tool allows us to prove the correctness of the dynamical behavior of the system. The equational nature of the Signal language leads to the use of polynomial dynamical equation systems (PDS) over Z= 3Z , i.e. integers modulo 3: f-1,0,1g, as a formal model of program behavior. The theory of PDS uses classical concepts of algebraic geometry, such as ideals, varieties and co- morphisms [6]. The techniques consist in manipulating the system of equations instead of the sets of solutions, which avoids enumerating state spaces. To model its behavior, a Signal process is translated into a system of polynomial equations over Z= 3Z [7]. The three possible states of a boolean signal X (i.e. , present and true, present and false, or absent) are coded in a signal variable x by (present and true ! 1, present and false ! \Gamma1, and absent ! 0). For the non-boolean signals, we only code the fact that the signal is present or absent: (present Each of the primitive processes of Signal are then encoded as polynomial equations. Let us just consider the example of the selection operator. C := A when B means "if 0". It can be rewritten as a polynomial Indeed, the solutions of this equation are the set of possible behaviors of the primitive process when. For example, if the signal B is true (i.e. , b=1), then to 1 Note that this fault has already be filtered. It can only be present if all the upstream circuit-breakers are closed The delay $, which is dynamical, is different because it requires memorizing the past value of the signal into a state variable x. In order to encode we have to introduce the three following equations: where x 0 is the value of the memory at the next instant. Equation (1) describes what will be the next value x 0 of the state variable. If a is present, x 0 is equal to a (because is equal to the last value of a, memorized by x. Equation (2) gives to b the last value of a (i.e. the value of x) and constrains the clocks b and a to be equal. Equation (3) corresponds to the initial value of x, which is the initial value of b. Table shows how all the primitive operators are translated into polynomial equations. Remark that for the non boolean expressions, we just translate the synchronization between the signals. Boolean expressions B := not A C := A or B non-boolean expressions Table 1. Translation of the primitive operators. Any Signal specification can be translated into a set of equations called polynomial dynamical system (PDS), that can be reorganized as follows: (1) are vectors of variables in Z= 3Z and components of the vectors X and X 0 represent the states of the system and are called state variables. They come from the translation of the delay operator. Y is a vector of variables in Z= 3Z , called event variables. The first equation is the state transition equation; the second equation is called the constraint equation and specifies which events may occur in a given state; the last equation gives the initial states. The behavior of such a PDS is the following: at each instant t, given a state x t and an admissible y t , such that Q(x t ; y t the system evolves into state x Verification of a Signal program: We now explain how verification of a Signal program (in fact, the corresponding PDS) can be carried out. Using algebraic operations, it is possible to check properties such as invariance, reachability and attractivity [7]. Note that most of them will be used in the sequel as control objectives for controller synthesis purposes. We just give here the basic definitions of each of this properties. Definition 1. 1. A set of states E is invariant for a dynamical system if for every x in E and every y admissible in x, P (x; y) is still in E. 2. A subset F of states is reachable if and only if for every state x 2 F there exists a trajectory starting from the initial states that reaches x. 3. A subset F of states is attractive from a set of states E if and only if every state trajectory initialized in E reaches F . ffl For a more complete review of the theoretical foundation of this approach, the reader may refer to [6, 7]. Specification of a property: Using an extension of the Signal language, named Signal+, it is possible to express the properties to be checked, as well as the control objectives to be synthesized (see section 5.2), in the Signal program. The syntax is The keyword Sigali means that the subexpression has to be evaluated by Si- gali. The function Verif Objective (it could be invariance, reachability, attractivity, etc) means that Sigali has to check the corresponding property according to the boolean PROP, which defines a set of states in the corresponding PDS. The complete Signal program is obtained composing the process specifying the plant and the one specifying the verification objectives in parallel. Thus, the compiler produces a file which contains the polynomial dynamical system resulting from the abstraction of the complete Signal program and the algebraic verification objectives. This file is then interpreted by Sigali. Suppose that, for example, we want, in a Signal program named "system", to check the attractivity of the set of states where the boolean PROP is true. The corresponding Signal+ program is then: (- system() (the physical model specified in Signal) definition of the boolean PROP in Signal The corresponding Sigali file, obtained after compilation of the Signal pro- gram, is: read("system.z3z"); =? loading of the PDS Compute the states where PROP is true =? Check for the attractivity of SetStates from the initial states The file "system.z3z" contains in a coded form the polynomial dynamical system that represents the system. Set States is a polynomial that is equal to 0 when the boolean PROP is true. The methods consist in verifying that the set of states where the polynomial Set States takes the value 0 is attractive from the initial states (the answer is then true or false): Attractivity(S, Set States).f This file is then interpreted by Sigali that checks the verification objective. 4.1 Verification of the power transformer network In this section, we apply the tools to check various properties of our Signal implementation of the transformer station. After the translation of the Signal program, we obtain a PDS with 60 state variables and 35 event variables. Note that the compiler also checks the causal and temporal concurrency of our program and produces an executable code. We will now describe some of the different properties, which have been proved. (1) "There is no possibility to have a fault at the departure, arrival and link level when the link circuit-breaker is opened." In order to check this property, we add to the original specification the following code (- or FaultArr1 or FaultArr1 or FaultDep1 or FaultDep2 or FaultDep3 or FaultDep4) when OpenLink) default false The Error signal is a boolean which takes the value true when the property is violated. In order to prove the property, we have to check that there does not exist any trajectory of the system which leads to the states where the Error signal is true (Reachable(True(Error))). The produced file is interpreted by Sigali that checks whether this set of states is reachable or not. In this case, the result is false, which means that the boolean Error never takes the value true. The property is satisfied 2 . In the same way, we proved similar properties when one of the arrival or departure circuit-breakers is open. (2) "If there exists a physical fault at the link level and if this fault is picked up by its sensor then the arrival sensors can not detect a fault". We show here the property for the arrival cell 1. It can be expressed as an invariance of a set of states. (- 2 Alternatively, this property could be also expressed as the invariance of the boolean False(Error), namely Sigali(Invariance(False(Error))). We have proved similar properties for a departure fault as well as when a physical fault appears at the arrival level and at the departure level at the same time. (3) We also proved using the same methods the following property: "If a fault occurs at a departure level, then it is automatically seen by the upstream sensors when no other fault exists at a higher level." All the important properties of the transformer station network have been proved in this way. Note that the cell behaviors have also been proved (see [7] for more details). 5 The automatic controller synthesis methodology 5.1 Controllable polynomial dynamical system Before speaking about control of polynomial dynamical systems, we first need to introduce a distinction between the events. From now on, we distinguish between the uncontrollable events which are sent by the system to the controller, and the controllable events which are sent by the controller to the system. A polynomial dynamical system S is now written as: (2) where the vector X represents the state variables; Y and U are respectively the set of uncontrollable and controllable event variables. Such a system is called a controllable polynomial dynamic system. Let n, m, and p be the respective dimensions of X , Y , and U . The trajectories of a controllable system are sequences and x include an uncontrollable component y t and a controllable one u t 3 . We have no direct influence on the y t part which depends only on the state x t , but we observe it. On the other hand, we have full control over u t and we can choose any value of u t which is admissible, i.e. , such that Q(x t To distinguish the two components, a vector called an event and a vector u 2 (Z= 3Z ) p a control . From now on, an event y is admissible in a state x if there exists a control u such that such a control is said compatible with y in x. The controllers: A PDS can be controlled by first selecting a particular initial state x 0 and then by choosing suitable values for We will here consider control policies where the value of the control u t is instantaneously computed from the value of x t and y t . Such a controller is called a static controller . It is a system of two equations: 3 This particular aspect constitutes one of the main differences with [14]. In our case, the events are partially controllable, whereas in the other case, the events are either controllable or uncontrollable. the equation C determines initial states satisfying the control objectives and the other one describes how to choose the instantaneous controls; when the controlled system is in state x, and when an event y occurs, any value u such that Q(x; can be chosen. The behavior of the system composed with the controller is then modeled by the system S c : However, not every controller (C; CO ) is acceptable. First, the controlled system SC has to be initialized ; thus, the equations Q must have common solutions. Furthermore, due to the uncontrollability of the events Y , any event that the system S can produce must be admissible by the controlled system SC . Such a controller is said to be acceptable. 5.2 Traditional Control Objectives We now illustrate the use of the framework for solving a traditional control synthesis problem we shall reuse in the sequel. Suppose we want to ensure the invariance of a set of states E. Let us introduce the operator pre, defined by: for any set of states F , pre Consider now the sequence pre (E) (4) The sequence (4) is decreasing. Since all sets E i are finite, there exists a j such that . The set E j is then the greatest control-invariant subset of j be the polynomial that has E j as solution, then C is an admissible feed-back controller and the system SC : verifies the invariance of the set of states E. Using similar methods, we are also able to to compute controllers (C; C 0 ) that ensure - the reachability of a set of states from the initial states of the system, - the attractivity of a set of states E from a set of states F . - the recurrence of a set of states E. We can also consider control objectives that are conjunctions of basic properties of state trajectories. However, basic properties cannot, in general, be combined in a modular way. For example, an invariance property puts restrictions on the 4 the solutions of the polynomial P (g) are the triples (x; that satisfy the relation is solution of the polynomial g". set of state trajectories which may be not compatible with an attractivity prop- erty. The synthesis of a controller insuring both properties must be effected by considering both properties simultaneously and not by combining a controller insuring safety with a controller insuring attractivity independently. For more details on the way controllers are synthesized, the reader may refer to [4]. Specification of the control objectives: As for verification (Section 4), the control objectives can be directly specified in Signal+ program, using the key-word Sigali. For example, if we add in the Signal program the line Sigali(S Attractivity(S,PROP)), the compiler produces a file that is interpreted by Sigali which computes the controller with respect to the control objective. In this particular case, the controller will ensure the attractivity of the set of states Set States, where Set States is a polynomial that is equal to zero when the boolean PROP is true. The result of the controller synthesis is a polynomial that is represented by a Binary Decision Diagram (BDD). This BDD is then saved in a file that could be used to perform a simulation [9]. Application to the transformer station: We have seen in the previous sec- tion, that one of the most critical requirements concerns the double fault prob- lem. We assume here that the circuit-breakers are ideal, i.e. they immediately react to actuators (i.e. , when a circuit-breaker receives an opening/closing re- quest, then at the next instant the circuit-breaker is opened/closed). With this assumption, the double fault problem can be rephrased as follows: "if two faults are picked up at the same time by two different departure cells, then at the next instant, one of the two faults (or both) must disappear." In order to synthesize the controller, we assume that the only controllable events are the opening and closing requests of the different circuit-breakers. The other events concern the appearance of the faults and cannot be considered controllable. The specification of the control objective is then: (- 2Fault := when (FaultDep1 and FaultDep2) default when (FaultDep1 and FaultDep3) default when (FaultDep1 and FaultDep4) default when (FaultDep2 and FaultDep3) default when (FaultDep2 and FaultDep4) default when (FaultDep3 and FaultDep4) default false The boolean 2 Fault is true, when two faults are present at the same time and is false otherwise. The boolean Error is true when two faults are present at two consecutive instants. We then ask Sigali to compute a controller that forces the boolean Error to be always false (i.e., whatever the behavior, there is no possibility for the controlled system to reach a state where Error is true). The Signal compiler translates the Signal program into a PDS, and the control objectives in terms of polynomial relations and polynomial operations. Applying the algorithm, described by the fixed-point computation (4), we are able to synthesize a controller (C ensures the invariance of the set of states where the boolean Error is true, for the controlled system The result is a controller coded by a polynomial and a BDD. Using the controller synthesis methodology, we solved the double fault prob- lem. However, some requirements have not been taken into account (importance of the lines, of the circuit-breakers,. This kind of requirements cannot be solved using traditional control objectives such as invariance, reachability or at- tractivity. In the next section, we will handle this kind of requirements, using control objectives expressed as order relations. 5.3 Numerical Order Relation Control Problem We now present the synthesis of control objectives that considers the way to reach a given logical goal. This kind of control objectives will be useful in the sequel to express some properties of the power transformer station controller, as the one dealing with the importance of the different circuit-breakers. For this purpose we introduce cost functions on states. Intuitively speaking, the cost function is used to express priority between the different states that a system can reach in one transition. Let S be a PDS as the one described by (2). Let us suppose that the system evolves into a state x, and that y is an admissible event at x. As the system is generally not deterministic, it may have several controls u such that Q(x; be two controls compatible with y in x. The system can evolve into either x Our goal is to synthesize a controller that will choose between u 1 and u 2 , in such a way that the system evolves into either x 1 or x 2 according to a given choice criterion. In the sequel, we express this criterion as a cost function relation. Controller synthesis method: Let be the state variables of the system. Then, a cost function is a map from (Z= 3Z ) n to N, which associates to each x of (Z= 3Z ) n some integer k. Definition 2. Given a PDS S and a cost function c over the states of this system, a state x 1 is said to be c-better than a state x 2 (denoted x 1 c x 2 ), if and only if, c(x 2 In order to express the corresponding order relation as a polynomial relation, let us consider The following sets of states are then computed ig: The sets partition of the global set of states. Note that some A i could be reduced to the empty set. The proof of the following property is straightforward: Proposition 1. x kmax be the polynomials that have the sets A solutions 5 . The order relation c defined by the proposition 1 can be expressed as polynomial relation: Corollary 1. x c x 0 , Rc (x; x Y Y As we deal with a non strict order relation, from c , we construct a strict order relation, named c defined as: x c x 0 , fx c x 0 "q(x 0 c x)g. Its translation in terms of polynomial equation is then given by: We now are interested in the direct control policy we want to be adopted by the system; i.e. , how to choose the right control when the system S has evolved into a state x and an uncontrollable event y has occurred. Definition 3. A control u 1 is said to be better compared to a control u 2 , if and only if x Using the polynomial approach, it gives Rc (P (x; In other words, the controller has to choose, for a pair (x; y), a compatible control with y in x, that allows the system to evolve into one of the states that are maximal for the relation Rc . To do so, let us introduce a new order relation A c defined from the order relation c . In other words, a triple (x; y; u) is "better" than a triple (x; the state P (x; reached by choosing the control u is better than the state reached by choosing the control u 0 . We will now compute the maximal triples of this new order relation among all of the triples. To this effect, we use I = f(x; 0g the set of admissible triples (x; u). The maximal set of triples I max is then provided by the following relation: I The characterization of the set of states I max in terms of polynomials is the 5 To compute efficiently such polynomials, it is important to use the Arithmetic Decision Diagrams (ADD) developed, for example, by [3]. Proposition 2. The polynomial C that has I max as solutions is given by: where the solutions of 9elimU 0 are given by the set Using this controller, the choice of a control u, compatible with y in x, is reduced such that the possible successor state is maximal for the (partial) order relation c . Note that if a triple (x; y; u) is not comparable with the maximal element of the order relation A c , the control u is allowed by the controller (i.e. , u is compatible with the event y in the state x). Without control, the system can start from one of the initial states of I 0g. To determine the new initial states of the system, we will take the ones that are the maximal states (for the order relation Rc ) among all the solutions of the equation Q This computation is performed by removing from I 0 all the states for which there exist at least one smaller state for the strict order relation c . Using the same method as the one previously described for the computation of the polynomial C, we obtain a polynomial C 0 . The solutions of this polynomial are the states that are maximal for the order relation A c . Theorem 1. With the preceding notations, (C; C 0 ) is an acceptable controller for the system S. Moreover, the controlled system adopts the control policy of Definition 3. ffi Some others characterization of order relations in terms of polynomials can be found in [11]. Finally, note that the notion of numerical order relation has been generalized over a bounded states trajectory of the system, retrieving the classical notion of Optimal Control [10]. Application to the power transformer station controller: We have seen in Section 5.2 how to compute a controller that solves the double fault problem. However, even if this particular problem is solved, other requirements had not been taken into account. The first one is induced by the obtained controller itself. Indeed, several solutions are available at each instant. For example, when two faults appear at a given instant, the controller can choose to open all the circuit-breakers, or at least the link circuit-breaker. This kind of solutions is not admissible and must not be considered. The second requirements concerns the importance of the lines. The first controller (C does not handle this kind of problems and can force the system to open the bad circuit-breakers. As consequences, two new requirements must be added in order to obtain a real controller: 1. The number of opened circuit-breaker must be minimal 2. The importance of the lines (and of the circuit-breakers) has to be different. These two requirements introduce a quantitative aspect to the control objectives. We will now describe the solutions we proposed to cope with these problems. First, let us assume that the state of a circuit-breaker is coded with a state variable according to the following convention: the state variable i is equal to 1 if and only if the corresponding circuit-breaker i is closed. CB is then a vector of state variables which collects all the state variables encoding the states of the circuit-breakers. To minimize the number of open circuit-breaker and to take into account the importance of the line, we use a cost function . We simply encode the fact that the more important is the circuit-breaker, the larger is the cost allocated to the state variable which encodes the circuit-breaker. The following picture summarizes the way we allocate the cost. The cost allocated to each state variable corresponds to the cost when the corresponding circuit-breaker is opened. When it is closed, the cost is equal to 0. The cost of a global state is simply obtained by adding all the circuit-breaker costs. With this cost function, it is always more expensive to open a circuit- breaker at a certain level than to open all the downstream circuit-breakers. Moreover, the cost allocated to the state variable that encodes the second departure circuit-breaker (encoded by the state variable X dep2 )) is bigger than the others because the corresponding line supplies a hospital (for example). Finally note that the cost function is minimal when the number of open circuit-breaker is minimal. Let us consider the system SC1 . We then introduce an order relation over the states of the system: a state x 1 is said to be better compared to a state only if for their corresponding sub-vectors CB 1 and CB 2 , we have CB 1 w c CB 2 . This order relation is then translated in an algebraic relation Rwc , following Equation (5) and by applying the construction described in proposition 2 and 1, we obtain a controller (C which the controlled system respects the control strategy. 6 Conclusion In this paper, we described the incremental specification of a power transformer station controller using the control theory concepts of the class of polynomial dynamical systems over Z= 3Z . As this model results from the translation of a Signal program [8], we have a powerful environment to describe the model for a synchronous data-flow system. Even if classical control can be used, we have shown that using the algebraic framework, optimal control synthesis problem is possible. The order relation controller synthesis technique can be used to synthesize control objectives which relate more to the way to get to a logical goal, than to the goal to be reached. Acknowledgment : The authors gratefully acknowledge relevant comments from the anonymous reviewers of this paper. --R Supervisory control of a rapid thermal multiprocessor. Verification of Arithmetic Functions with Binary Dia- grams Control of polynomial dynamic systems: an exam- ple A survey of Petri net methods for controlled discrete event systems. Polynomial dynamical systems over finite fields. Formal verification of signal programs: Application to a power transformer station controller. Programming real-time applications with signal A design environment for discrete-event controllers based on the Signal language On the optimal control of polynomial dynamical systems over z/pz. Partial order control of discrete event systems modeled as polynomial dynamical systems. Synchronous design of a transformer station controller with Signal. Controller synthesis for the production cell case study. The control of discrete event systems. --TR --CTR Kai-Yuan Cai , Yong-Chao Li , Wei-Yi Ning , W. Eric Wong , Hai Hu, Optimal and adaptive testing with cost constraints, Proceedings of the 2006 international workshop on Automation of software test, May 23-23, 2006, Shanghai, China Xiang-Yun Wang , Wenhui Zhang , Yong-Chao Li , Kai-Yuan Cai, A polynomial dynamic system approach to software design for attractivity requirement, Information Sciences: an International Journal, v.177 n.13, p.2712-2725, July, 2007

optimal control;discrete event systems;signal;sigali;polynomial dynamical system;power plant;supervisory control problem

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Quantitative Analysis of Faults and Failures in a Complex Software System.

AbstractThe dearth of published empirical data on major industrial systems has been one of the reasons that software engineering has failed to establish a proper scientific basis. In this paper, we hope to provide a small contribution to the body of empirical knowledge. We describe a number of results from a quantitative study of faults and failures in two releases of a major commercial system. We tested a range of basic software engineering hypotheses relating to: The Pareto principle of distribution of faults and failures; the use of early fault data to predict later fault and failure data; metrics for fault prediction; and benchmarking fault data. For example, we found strong evidence that a small number of modules contain most of the faults discovered in prerelease testing and that a very small number of modules contain most of the faults discovered in operation. However, in neither case is this explained by the size or complexity of the modules. We found no evidence to support previous claims relating module size to fault density nor did we find evidence that popular complexity metrics are good predictors of either fault-prone or failure-prone modules. We confirmed that the number of faults discovered in prerelease testing is an order of magnitude greater than the number discovered in 12 months of operational use. We also discovered fairly stable numbers of faults discovered at corresponding testing phases. Our most surprising and important result was strong evidence of a counter-intuitive relationship between pre- and postrelease faults: Those modules which are the most fault-prone prerelease are among the least fault-prone postrelease, while conversely, the modules which are most fault-prone postrelease are among the least fault-prone prerelease. This observation has serious ramifications for the commonly used fault density measure. Not only is it misleading to use it as a surrogate quality measure, but, its previous extensive use in metrics studies is shown to be flawed. Our results provide data-points in building up an empirical picture of the software development process. However, even the strong results we have observed are not generally valid as software engineering laws because they fail to take account of basic explanatory data, notably testing effort and operational usage. After all, a module which has not been tested or used will reveal no faults, irrespective of its size, complexity, or any other factor.

Introduction Despite some heroic efforts from a small number of research centres and individuals (see, for example [Carman et al 1995], [Kaaniche and Kanoun 1996], [Khoshgoftaar et al 1996], [Ohlsson N and Alberg 1996], [Shen et al 1985]) there continues to be a dearth of published empirical data relating to the quality and reliability of realistic commercial software systems. Two of the best and most important studies [Adams 1984] and [Basili and Perricone 1984] are now over 12 years old. Adams' study revealed that a great proportion of latent software faults lead to very rare failures in practice, while the vast majority of observed failures are caused by a tiny proportion of the latent faults. Adams observed a remarkably similar distribution of such fault 'sizes' across nine different major commercial systems. One conclusion of the Adams' study is that removing large numbers of faults may have a negligible effect on reliability; only when the small proportion of 'large' faults are removed will reliability improve significantly. Basili and Pericone looked at a number of factors influencing the fault and failure proneness of modules. One of their most notable results was that larger modules tended to have a lower fault density than smaller ones. Fault density is the number of faults discovered (during some pre-defined phase of testing or operation) divided by a measure of module size (normally KLOC). While the fault density measure has numerous weaknesses as a quality measure (see [Fenton and Pfleeger 1996] for an in-depth discussion of these) this result is nevertheless very surprising. It appears to contradict the very basic hypotheses that underpin the notions of structured and modular programming. Curiously, the same result has been rediscovered in other systems by [Moeller and Paulish 1995]. Recently Hatton provided an extensive review of similar empirical studies and came to the conclusion: 'Compelling empirical evidence from disparate sources implies that in any software system, larger components are proportionally more reliable than smaller components' [Hatton 1997]. Thus the various empirical studies have thrown up results which are counter-intuitive to very basic and popular software engineering beliefs. Such studies should have been a warning to the software engineering research community about the importance of establishing a wide empirical basis. Yet these warnings were clearly not heeded. In [Fenton et al 1994] we commented on the almost total absence of empirical research on evaluating the effectiveness of different software development and testing methods. There also continues to be an almost total absence of published benchmarking data. In this paper we hope to provide a small contribution to the body of empirical knowledge by describing a number of results from a quantitative study of faults and failures in two releases of a major commercial system. In Section 2 we describe the background to the study and the basic data that was collected. In Section 3 we provide pieces of evidence that one day (if a reasonable number of similar studies are published) may help us test some of the most basic of software engineering hypotheses. In particular we present a range of results and examine the extent to which they provide evidence for or against following hypotheses: . Hypotheses relating to the Pareto principle of distribution of faults and failures 1a) a small number of modules contain most of the faults discovered during pre-release 1b) if a small number of modules contain most of the faults discovered during pre-release testing then this is simply because those modules constitute most of the code size 2a) a small number of modules contain the faults that cause most failures 2b) if a small number of modules contain most of the operational faults then this is simply because those modules constitute most of the code size. . Hypotheses relating to the use of early fault data to predict later fault and failure data (at the module level): higher incidence of faults in function testing implies a higher incidence of faults in system testing higher incidence of faults in pre-release testing implies higher incidence of failures in operation. We tested each of these hypotheses from an absolute and normalised fault perspective. . Hypotheses about metrics for fault prediction (such as LOC) are good predictors of fault and failure prone modules. Complexity metrics are better predictors than simple size metrics of fault and failure-prone modules . Hypotheses relating to benchmarking figures for quality in terms of defect densities Fault densities at corresponding phases of testing and operation remain roughly constant between subsequent major releases of a software system systems produced in similar environments have broadly similar fault densities at similar testing and operational phases. For the particular system studied we provide very strong evidence for and against some of the above hypotheses and also explain how some previous studies that have looked at these hypotheses are flawed. Hypotheses 1a and 2a are strongly supported, while 1b and 2b are strongly rejected. Hypothesis 3 is weakly supported, while curiously hypothesis 4 is strongly rejected. Hypothesis 5 is partly supported, but hypotheses 6 is weakly rejected for the popular complexity metrics. However, certain complexity metrics which can be extracted from early design specifications are shown to be reasonable fault predictors. Hypothesis 7 is partly supported, while 8 can only be tested properly once other organisations publish analogous results. We discuss the results in more depth in Section 4. 2 The basic data The data presented in this paper is based on two major consecutive releases of a large legacy project developing telecommunication switching systems. We refer to the earlier of the releases as release n, and the later release as release n+1. For this study 140 and 246 modules respectively from release n and n+1 were selected randomly for analysis from the set of modules that were either new or had been modified. The modules ranged in size from approximately 1000 to 6000 LOC (as shown in Table 1). Both releases were approximately the same total system size. Table 1. Distribution of modules by size.LOC Release n Release n+1 <1000 23 26 1001-2000 58 85 2001-3000 37 73 3001-4000 15 38 Total 140 246 2.1 Dependent variable The dependent variable in this study was number of faults. Faults are traced to unique modules. The fault data were collected from four different phases: . function test . system test (ST) . first 26 weeks at a number of site tests (SI) . first year (approx) operation (OP) Therefore, for each module we have four corresponding instances of the dependent variable. The testing process and environment used in this project is well established within the company. It has been developed, maintained, taught and applied for a number of years. A team separated from the design and implementation organisation develop the test cases based on early function specifications. Throughout the paper we will refer to the combination of FT and ST faults collectively as testing faults. We will refer to the combination of SI and OP faults collectively as operational faults. We shall also refer at times to failures. Formally, a failure is an observed deviation of the operational system behaviour from specified or expected behaviour. All failures are traced back to a unique (operational) fault in a module. Observation of distinct failures that are traced to the same fault are not counted separately. This means, for example, that if 20 OP faults are recorded against module x, then these 20 unique faults caused the set of all failures observed (and which are traced back to faults in module x) during the first year of operation. The Company classified each fault found at any phase according to the following: a) the fault had already been corrected; b) the fault will be corrected; c) the fault requires no action (i.e. not treated as a fault); d) the fault was due to installation problems. In this paper we have only considered faults classified as b. Internal investigations have shown that the documentation of faults and their classification according to the above categories is reliable. A summary of the number of faults discovered in each testing phase for each system release is shown in Table 2. pre-release faults post-release faults Release Function test System test Site test Operation (sample size 140 modules) n+1 (sample size 246 modules) Table 2. Distribution of faults per testing phase 2.2 Independent variables Various metrics were collected for each module. These included: . Lines of code (LOC) as the main size measure . McCabe's cyclomatic complexity. . Various metrics based on communication (modelled with signals) between modules and within a module During the specification phase, the number of new and modified signals are similar to messages) for each module were specified. Most notably, the metric SigFF is the count of the number of new and modified signals. This metric was also used as a measure of interphase complexity. [Ohlsson and Alberg, 1996] provides full details of these metrics and their computation. The complexity metrics were collected automatically from the actual design documents using a tool, ERIMET [Ohlsson, 1993]. This automation was possible as each module was designed using FCTOOL, a tool for the formal description language FDL which is related to SDL's process diagrams [Turner, 1993]. The metrics are extracted direct from the FDL- graphs. The fact that the metrics were computed from artefacts available at the design stage, is an important point. It has often been asserted that computing metrics from design documents is far more valuable than metrics from source code [Heitkoetter et al 1990]. However, there have been very few published attempts to do so. [Kitchenham et al, 1990] reported on using design metrics, based on Henry and Kafura's information and flow metrics [1981 and 1984], for outlier analysis. [Khoshgoftaar et al, 1996] used a subset of metrics that "could be collected from design documentation", but the metrics were extracted from the code. Numerous studies, such as [Ebert and Liedtke, 1995]; and [Munson and Khoshgoftaar, 1992] have reported using metrics extracted from source code, but few have reported promising prediction results based on design metrics. 3 The hypotheses tested and results Since the data were collected and analysed retrospectively there was no possibility of setting up any controlled experiments. However, the sheer extent and quality of the data was such that we could use it to test a number of popular software engineering hypotheses relating to the distribution and prediction of faults and failures. In this section we group the hypotheses into four categories. In Section 3.1 we look at hypotheses relating to the Pareto principle of distribution of faults and failures. It is widely believed, for example, that a small number of modules in any system are likely to contain the majority of the total system faults. This is often referred to as the '20-80 rule' in the sense that 80% of the faults are contained in 20% of the modules. We show that there is strong evidence to support the two most commonly cited Pareto principles. The assumption of the Pareto principle for faults has led many practitioners to seek methods for predicting the fault-prone modules at the earliest possible development and testing phases. These methods seem to fall into two categories: 1. use of early fault data to predict later fault and failure data; 2. use of product metrics to predict fault and failure data Given our evidence to support the Pareto principle we therefore test a number of hypotheses which relate to these methods of early prediction of fault-prone modules. In Section 3.2, we test hypotheses concerned with while in Section 3.3 we test hypotheses concerned with 2). Finally, in Section 3.4 we test some hypotheses relating to benchmarking fault data, and at the same time provide data that, can themselves, be valuable in future benchmarking studies. 3.1 Hypotheses relating to the Pareto principle of distribution of faults and failures The main part of the total cost of quality deficiency is often found to be caused by very few faults or fault types [Bergman and Klefsjo 1991]. The Pareto principle [Juran 1964], also called the 20-80 rule, summarises this notion. The Pareto principle is used to concentrate efforts on the vital few, instead of the trivial many. There are a number of examples of the Pareto principle in software engineering. Some of these have gained widespread acceptance, such as the notion that in any given software system most faults lie in a small proportion of the software modules. Adams [1984] demonstrated that a small number of faults were responsible for a large number of failures. [Munson et al 1992] motivated their discriminative analysis by referring to the 20-80 rule, even though their data demonstrated a rule. [Zuse 1991] used Pareto techniques to identify the most common types of faults found during function testing. Finally, [Schulmeyer and MacManus 1987] described how the principle supports defect identification, inspection and applied statistical techniques. We investigated four related Pareto hypotheses: Hypothesis 1a: a small number of modules contain most of the faults discovered during pre-release testing (phases FT and ST); Hypothesis 1b: if a small number of modules contain most of the faults discovered during pre-release testing then this is simply because those modules constitute most of the code size. Hypothesis 2a: a small number of modules contain most of the operational faults (meaning failures as we have defined them above observed in phases SI and OP); Hypothesis 2b: if a small number of modules contain most of the operational faults then this is simply because those modules constitute most of the code size. We now examine each of these in turn. 3.1.1 Hypothesis 1a: a small number of modules contain most of the faults discovered during testing (phases FT and ST) Figure 1 illustrates that 20% of the modules were responsible for nearly 60% of the faults found in testing for release n. An almost identical result was obtained for release n+1 but is not shown here. This is also almost identical to the result in earlier work where the faults from both testing and operation were considered [Ohlsson et al 1996]. This, together with other results such as [Munson et al 1992], provides very strong support for hypothesis 1a), and even suggests a specific Pareto distribution in the area of 20-60. This 20-60 finding is not as strong as the one observed by [Compton and Withrow, 1990] (they found that 12% of the modules, referred to as packages, accounted for 75% of all the faults during system integration and test), but is nevertheless important.2060100 %of Modules %of Faults Figure 1: Pareto diagram showing percentage of modules versus percentage of faults for Release n 3.1.2 Hypothesis 1b: if a small number of modules contain most of the faults discovered during pre-release testing then this is simply because those modules constitute most of the code size. Since we found strong support for hypothesis 1a, it makes sense to test hypothesis 1b. It is popularly believed that hypothesis 1a is easily explained away by the fact that the small proportion of modules causing all the faults actually constitute most of the system size. For example, [Compton and Withdraw, 1990] found that the 12% of modules accounting for 75% of the faults accounted for 63% of the LOC. In our study we found no evidence to support hypotheses 1b. For release n, the 20% of the modules which account for 60% of the faults (discussed in hypothesis 1a) actually make up just 30% of the system size. The result for release n+1 was almost identical. 3.1.3 Hypothesis 2a: a small number of modules contain most of the operational faults (meaning failures as we have defined them above, namely phases CU and OP) We discovered not just support for a Pareto distribution, but a much more exaggerated one than for hypothesis 1a. Figure 2 illustrates this Pareto effect in release n. Here 10% of the modules were responsible for 100% of the failures found. The result for release n+1 is not so remarkable but is nevertheless still quite striking: 10% of the modules were responsible for 80% of the failures.2060100 % of Failures % of Modules Figure 2: Pareto diagram showing percentage of modules versus percentage of failures for Release n 3.1.4 Hypothesis 2b: if a small number of modules contain most of the operational faults then this is simply because those modules constitute most of the code size. As with hypothesis 1a, it is popularly believed that hypothesis 2a is easily explained away by the fact that the small proportion of modules causing all the failures actually constitute most of the system size. In fact, not only did we find no evidence for hypothesis 2, but we discovered very strong evidence in favour of a converse hypothesis: most operational faults are caused by faults in a small proportion of the code For release n, 100% of the operational faults are contained in modules that make up just 12% of the entire system size. For release n+1 60% of the operational faults were contained in modules that make up just 6% of the entire system size, while 78% of the operational faults were contained in modules that make up 10% of the entire system size. 3.2 Hypotheses relating to the use of early fault data to predict later fault and failure data Given the likelihood of hypotheses 1a and 2a there is a strong case for trying to predict the most fault-prone modules as early as possible during development. In this and the next subsection we test hypotheses relating to methods of doing precisely that. First we look at the use of fault data collected early as a means of predicting subsequent faults and failures. Specifically we test the hypotheses: Hypothesis 3: Higher incidence of faults in function testing (FT) implies higher incidence of faults in system testing (ST) Hypothesis 4: Higher incidence of faults in all pre-release testing (FT and ST) implies higher incidence of faults in post-release operation (SI and OP). We tested each of these hypotheses from an absolute and normalised fault perspective. We now examine the results. 3.2.1 Hypothesis 3: Higher incidence of faults in function testing (FT) implies higher incidence of faults in system testing (ST) The results associated with this hypothesis are not very strong. In release n (see Figure 3), 50% of the faults in system test occurred in modules which were responsible for 37% of the faults in function test. 0% 20% 40% 80% 100% 15% 30% 45% 60% 75% 90% FT % of Modules % of Accumalated Faults in ST Figure 3: Accumulated percentage of the absolute number faults in system test when modules are ordered with respect to the number of faults in system test and function test for release n. From a prediction perspective the figures indicate that the most fault-prone modules during function test will, to some extent, also be fault-prone in system test. However, 10% of the most fault-prone modules in system test are responsible for 38% of the faults in system test, but 10% of the most fault-prone modules in function test is only responsible for 17% of the faults in system test. This is persistent up to 75% of the modules. This means that nearly 20% of the faults in system test need to be explained in another way. The same pattern was found when using normalised data (faults/LOC) instead of absolute, even though the percentages were general lower and the prediction a bit poorer. The results were only slightly different for release n+1, where we found: . 50% of the faults in system test occurred in modules which were responsible for 25% of the faults in function test . 10% of the most fault-prone modules in system test are responsible for 46% of the faults in system test, but 10% of the most fault-prone modules in function test is only responsible for 24% of the faults in system test. These results and also when using normalised data instead of absolute are very similar to the result in release n. 3.2.2 Hypothesis 4: Higher incidence of faults in all pre-release testing (FT and ST) implies higher incidence of faults in post-release operation (SI and OP). The rationale behind hypothesis 4 is that the relatively small proportion of modules in a system that account for most of the faults are likely to be fault-prone both pre- and post release. Such modules are somehow intrinsically complex, or generally poorly built. 'If you want to find where the faults lie, look where you found them in the past' is a very common and popular maxim. For example, [Compton and Withrow, 1990] have found as much as six times greater post delivery defect density when analysing modules with faults discovered prior to delivery. In many respects the results in our study relating to this hypothesis are the most remarkable of all. Not only is there no evidence to support the hypothesis, but again there is strong evidence to support a converse hypothesis. In both release n and release n+1 almost all of the faults discovered in pre-release testing appear in modules which subsequently reveal almost no operation faults. Specifically, we found: . In release n (see Figure 4), 93% of faults in pre-release testing occur in modules which have NO subsequent operational faults (of which there were 75 in total). Thus 100% of the 75 failures in operation occur in modules which account for just 7% of the faults discovered in pre-release testing. Post-release faults261014 Pre-release faults Figure 4: Scatter plot of pre-release faults against post-release faults for version n (each dot represents a module) . In release n+1 we observed a much greater number of operational faults, but a similar phenomenon to that of release n (see Figure 5). Some 77% of pre-release faults occur in modules which have NO post-release faults. Thus 100% of the 366 failures in operation occur in modules which account for just 23% of the faults discovered in function and system test. These remarkable results are also exciting because they are closely related to the Adams' phenomenon. The results have major ramifications for one of the most commonly used software measures, fault density. Specifically it appears that modules with high fault density pre-release are likely to have low fault-density post-release, and vice versa. We discuss the implications at length in Section 4. 3.3 Hypotheses about metrics for fault prediction In the previous subsection we were concerned with using early fault counts to predict subsequent fault prone modules. In the absence of early fault data, it has been widely proposed that software metrics (which can be automatically computed from module designs or code) can be used to predict fault prone modules. In fact this is widely considered to be the major benefit of such metrics [Fenton and Pfleeger 1997]. We therefore attempted to test the basic hypotheses which underpin these assumptions. Specifically we tested: Hypothesis 5: Size metrics (such as LOC) are good predictors of fault and failure prone modules. Hypothesis Complexity metrics are better predictors than simple size metrics, especially at predicting fault-prone modules 3.3.1 Hypothesis 5: Size metrics (such as LOC) are good predictors of fault and failure prone modules. Strictly speaking, we have to test several different, but closely, related hypotheses: Hypothesis 5a: Smaller modules are less likely to be failure prone than larger ones Hypothesis 5b Size metrics (such as LOC) are good predictors of number of pre-release faults in a module Hypothesis 5c: Size metrics (such as LOC) are good predictors of number of post-release faults in a module Post-release faults5152535 Pre-release faults Figure 5: Scatter plot of pre-release faults against post-release faults for version (each dot represents a module) Hypothesis 5d: Size metrics (such as LOC) are good predictors of a module's (pre-release) fault-density Hypothesis 5e: Size metrics (such as LOC) are good predictors of a module's (post-release) fault-density Hypothesis 5a underpins, in many respects, the principles behind most modern programming methods, such as modular, structured, and objected oriented. The general idea has been that smaller modules should be easier to develop, test, and maintain, thereby leading to fewer operational faults in them. On the other hand, it is also accepted that if modules are made too small then all the complexity is pushed into the interface/communication mechanisms. Size guidelines for decomposing a system into modules are therefore desirable for most organisations. It turns out that the small number of relevant empirical studies have produced counter-intuitive results about the relationship between size and (operational) fault density. Basili and Pericone [1984] reported that fault density appeared to decrease with module size. Their explanation to this was the large number of interface faults spread equally across all modules. The relatively high proportion of small modules were also offered as an explanation. Other authors, such as [Moeller and Paulish 1995] who observed a similar trend, suggested that larger modules tended to be under better configuration management than smaller ones which tended to be produced 'on the fly'. In fact our study did not reveal any similar trend, and we believe the strong results of the previous studies may be due to inappropriate analyses. We begin our results with a replication of the key part of the [Basili and Pericone 1984] study. Table 3 (which compare with Basili and Perricone's Table III) shows the number of modules that had a certain number of faults. The table also displays the figures for the different types of modules and the percentages. The data set analysed in this paper has, in comparison with [Basili and Pericone 1984] a lower proportion of modules with few faults and the proportion of new modules is lower. In subsequent analysis all new modules have been excluded. The modules are also generally larger than those in [Basili and Pericone 1984], but we do not believe this introduces any bias. The scatter plots Figure 6, for lines of code versus the number of pre- and post-release faults does not reveal any strong evidence of trends for release n+1. Neither could any strong trends be observed when line of code versus the total number of faults were graphed, Figure 7. The results for release n were reasonably similar.20601001400 2000 4000 6000 8000 10000 Post-release Faults Faults Lines of code Lines of code Figure Scatterplots of LOC against pre- and post-release faults forrelease n+1 (each dot represents a module).Release n Release n+1 Fault Mod New Percent modified modules Mod New Splitted Percent modified modules 11 to 15 21 to 26 to 31 to 36 to 40 Table 3. Number of Modules Affected by a fault for Release n (140 modules, 1815 Faults) and Release n+1 (246 modules, 3795 faults ). Faults Lines of code Figure 7: Scatterplots of LOC against all faults for release n+1 (each dotrepresents a module).When Basili and Pericone could not see any trend they calculated the number of faults per 1000 executable lines of code. Table 4 (which compares with table VII in [Basili and Pericone 1984]) shows these results for our study. Release n Release n+1 Module size Frequency Faults/1000 Lines Frequency Faults/1000 Lines 1000 15 4.77 17 6 2000 3000 22 5.74 37 5 Table 4. Faults/1000 Lines of code release n and n+1. Superficially, the results in table 4 for release n+1 appear to support the Basili and Pericone finding. In release n+1 it is clear that the smallest modules have the highest fault density. However, the fault density is very similar for the other groups. For release n the result is the opposite of what was reported by Basili and Perricone. The approach to grouping data as done in [Basili and Perricone 1984] is highly misleading. What Basili and Pericone failed to show was a simple plot of fault density against module size, as we have done in Figure 9 for release n+1. Even though the grouped data for this release appeared to support the Basili and Pericone findings, this graph shows only a very high variation for the small modules and no evidence that module size has a significant impact on fault-density. Clearly other explanatory factors, such as design, inspection and testing effort per module, will be more important. Lines of code Figure 8: Scatter plot of module fault density against size for release n+1 The scatter plots assumes that the data belong to an interval or ratio scale. From a prediction perspective it is not always necessary. In fact, a number of studies are built on the Pareto principle, which often only require that we have ordinal data. In the tests of hypothesis above we have used a technique that is based on ordinal data, called Alberg diagrams [Ohlsson and Alberg 1996], to evaluate the independent variables' ability to rank the dependent variable. The LOC ranking ability is assessed in Figure 9. The diagram reveals that, even though previous analysis did not indicate any predictability, LOC is quite good at ranking the most fault-prone modules, and for the most fault prone-modules (the 20 percent) much better than any previous ones. 0% 20% 40% 80% 100% 10% 20% 40% 60% 80% 100% all faults % of Accumalated Faults % of Modules Figure 9. Accumulated percentage of the absolute number of all faults when modules are ordered with respect to LOC for release n+1. 3.3.2 Hypothesis Complexity metrics are better predictors than simple size metrics of fault and failure-prone modules 'Complexity metrics' is the rather misleading term used to describe a class of measures that can be extracted directly from source code (or some structural model of it, like a flowgraph representation). Occasionally (and more beneficially) complexity metrics can be extracted before code is produced, such as when the detailed designs are represented in a graphical language like SDL (as was the case for the system in this study). The archetypal complexity metric is McCabe's cyclomatic number [McCabe, 1976], but there have in fact been many dozens that have been published [Zuse 1991]. The details, and also the limitations of complexity metrics, have been extensively documented (see [Fenton and Pfleeger 1996]) and we do not wish to re-visit those issues here. What we are concerned with here is the underlying assumption that complexity metrics are useful because they are (easy to extract) indicators of where the faults lie in a system. For example, Munson and Khosghoftaar asserted: 'There is a clear intuitive basis for believing that complex programs have more faults in them than simple programs', [Munson and Khosghoftaar, 1992] An implicit assumption is that complexity metrics are better than simple size measures in this respect (for if not there is little motivation to use them). We have already seen, in section 3.3.1, that size is a reasonable predictor of number of faults (although not of fault density). We now investigate the case of complexity metrics such as the cyclomatic number. We demonstrated in testing the last hypothesis the problem with comparing average figures for different size intervals. Instead of replicating the relevant analysis in [Basili and Pericone 1984] by calculating the average cyclomatic number for each module size class, and than plotting the results we just generated scatter plots and Alberg diagrams. When the cyclomatic complexity and the pre- and post-release faults were graphed for release n+1 (Figure 10) we observed a number of interesting trends. The most complex modules appear to be more fault-prone in pre-release, but appear to have nearly no faults in post-release. The most fault-prone modules in post-release appear to be the less complex modules. This could be explained by how test effort is distributed over the modules: modules that appear to be complex are treated with extra care than simpler ones. Analysing in retrospect the earlier graphs for size versus faults reveal a similar pattern. The scatter plot for the cyclomatic complexity and the total number of faults (Figure 11) shows again some small indication of correlation. The Alberg diagrams were similar as when size was used.20601001400 1000 2000 3000 Faults Cyclomatic complexity Figure 11: Scatterplot of cyclomatic complexity against all faults for release n+1 (each dot represents a module). To explore the relations further the scatter plots were also graphed with normalised data Figure 12). The result showed even more clearly that the most-fault prone modules in pre-release have nearly no post-release faults.20601001400 1000 2000 3000 Post-release Faults Faults Cyclomatic complexity Cyclomatic complexity Figure 10: Scatterplots of cyclomatic complexity against number of pre-and post-release faults for release n+1 (each dot represents a module). In order to determine whether or not large modules were less dense or complex than smaller modules [Basili and Perricone, 1984] plotted the cyclomatic complexity versus module size. Following the same pattern in earlier analysis they failed to see any trends, and therefore they analysed the relation by grouping modules according to size. As illustrated above this can be very misleading. Instead we graphed scatter plots of the relation and calculated the correlation (Figure 13). The relation may not be linear. However, there is a good linear correlation between cyclomatic complexity and LOC 2 . Earlier studies [Ohlsson and Alberg, 1996] have suggested that other design metrics could be used in combination or on their own to explain fault-proneness. Therefore, we did the same analysis using the SigFF measure instead of cyclomatic complexity. 0,000 Pre-release 0,000 0,004 0,006 Post-release Cyclomatic complexity Cyclomatic complexity Figure 12: Scatterplots of cyclomatic complexity against fault density (pre-and post-release) for release n+1 (each dot represents a module).10003000500070009000 Cyclomatic complexity Figure 13: Complexity versus Module Size The scatterplots using absolute numbers (Figure 14), or normalised data did not indicate any new trends. In earlier work the product of cyclomatic complexity and SigFF was shown to be a good predictor of fault-proneness. To evaluate CC*SigFF predictability the Alberg diagram was graphed (Figure 15). The combined metrics appear to be better than both SigFF and Cyclomatic Complexity on there own, and also better than the size metric. 0% 20% 40% 80% 100% 20% 40% 60% 10% 80% 100% All faults Dec % of Accumulated Faults % of Modules Figure 15. Accumulated percentage of the absolute number of all faults when modules are ordered with respect to LOC for release n+1. The above results do not paint a very glowing report of the usefulness of complexity metrics, but it can be argued that 'being a good predictor of fault density' is not an appropriate validation criteria for complexity metrics. This is discussed in section 4. Nevertheless there are some positive aspects. The combined metric CC*SigFF is again shown to be a reasonable predictor of fault-prone modules. Also, measures like SigFF are, unlike LOC, available at a very early stage in the software development. The fact that it correlates so closely with the final LOC, and is a good predictor of total number of faults, is a major benefit. 3.4 Hypotheses relating to benchmarking One of the major benefits of collecting and publicising the kind of data discussed in this paper is to enable both intra- and inter-company comparisons. Despite the incredibly vast20601001400 100 200 300 400 Post-release Faults Faults Interphase complexity Interphase complexity Figure 14: Scatterplots of SigFF against number of pre-and post-release faults for release n+1 (each dot represents a module). volumes of software in operation throughout the world there is no consensus about what constitutes, for example, a good, bad, or average fault density under certain fixed conditions of measurement. It does not seem unreasonable to assume that such information might be known, for example, for commercial C programs where faults are defined as operational faults (in the sense of this paper) during the first 12 months of use by a typical user. Although individual companies may know this kind of data for their own systems, almost nothing has ever been published. The 'grey' literature (as referenced, for example, in [Pfleeger and Hatton 1997] seems to suggest some crude (but unsubstantiated guidelines) such as the following for fault density in first 12 months of typical operational use: . less that 1 fault per KLOC is very good (and typically only achieved by companies using state-of-the-art development and testing methods) . between 4 to 8 faults per KLOC is typical . greater than 12 faults per KLOC is bad When pre-release faults only are considered there is some notion that 10-30 faults per KLOC is typical for function, system and integration testing combined. For reasons discussed already high values of pre-release fault density is not indicative of poor quality (and may in fact suggest the opposite). Therefore it would be churlish to talk in terms of 'good' and `bad' densities because, as we have already stressed, these figures may be explained by key factors such as the effort spent on testing. In this study we can consider the following hypothesis Hypothesis 7: Fault densities at corresponding phases of testing and operation remain roughly constant between subsequent major releases of a software system since we have data on successive releases. The results we present, being based only on one system, represents just a single data-point, but nevertheless we believe it may also be valuable for other researchers. In a similar vein we consider: Hypothesis 8: software systems produced in similar environments have broadly similar fault densities at similar testing and operational phases. Really we are hoping to build up an idea of the range of fault densities that can reasonably be expected. We compare our results with some other published data. 3.4.1 Fault densities at corresponding phases of testing and operation remain roughly constant between subsequent major releases of a software system FT ST SI OP Rel n 3.49 2.60 0.07 0.20 Rel n+1 4.15 1.82 0.43 0.20 Table 5: Fault densities at the four phases of testing and operation As table 5 shows, there is some support for the hypothesis that the fault-density remains roughly the same between subsequent releases. The only exceptional phase is SI. As well as providing some support for the hypothesis the result suggests that the development process is stable and repeatable with respect to the fault-density. This has interesting implications for the software process improvement movement, as epitomised by the Capability Maturity Model CMM. A general assumption of CMM is that a stable and repeatable process is a necessary pre-requisite for continuous process improvement. For an immature organisation (below level it is assumed to take many years to reach such a level. In CMM's terminology companies do not have the kind of stable and repeatable process indicated in the above figures until they are at level 3. Yet, like almost every software producing organisation in the world, the organisation in this case study project is not at level 3. The results reflects a stability and repeatability that according to CMM should not be the case. At such we question the CMM's underlying assumption about what constitutes an organisation that should have a stable and repeatable process. 3.4.2 Software systems produced in similar environments have broadly similar fault densities at similar testing and operational phases. To test this hypothesis we compared the results of this case study with other published data. For simplicity we restricted our analysis to the two distinct phases: 1) pre-release fault density; and 2) post-release fault density. First, we can compare the two results of the two separate releases in the cases study (Table 6). Pre-release Post-release All Rel n 6.09 0.27 6.36 Rel n+1 5.97 0.63 6.60 Table densities pre-and post-release for the case study system The overall fault densities are similar to those reported for a range of systems in [Hatton 1995], while [Agresti and Evanco, 1992] reported similar ball-park figures in a study of Ada programs, 3.0 to 5.5 faults/KLOC. The post-release fault densities seem to be roughly in line of those reported studies of best practice. More interesting is the difference between the pre- and post-release fault densities. In both versions the pre-release fault density is an order of magnitude higher than the post-release density. Of the few published studies that reveal the difference between pre- and post-release fault density, [Pfleeger and Hatton, 1997] also report 10 times as many faults in pre-release (although the overall fault density is lower. [Kitchenham et al 1986] reports a higher ratio of pre-release to post-release. Their study was an investigation into the impact of inspections; combining the inspected and non-inspected code together reveals a pre-release fault density of approx per KLOC and a post-release fault density of approximately 0.3 per KLOC. However, it is likely that the operational time here was not as long. Thus, from the small amount of evidence we conclude that there appears to be 10-30 times as many faults pre-release as post release. 4 Discussion and conclusions Apart from the usual quality control angle, a very important perceived benefit of collecting fault data at different testing phases is to be able to move toward statistical process control for software development. For example, this is the basis for the software factory approach proposed by Japanese companies such as Hitachi [Yasuda and Koga 1995] in which they build fault profiles that enable them to claim accurate fault and failure prediction. Another important motivation for collecting the various fault data is to enable us to evaluate the effectiveness of different testing strategies. In this paper we have used an extensive example of fault and failure data to test a range of popular software engineering hypotheses. The results we have presented come from just two releases of a major system developed by a single organisation. It may therefore be tempting for observers to dismiss their relevance for the broader software engineering community. Such an attitude would be dangerous given the rigour and extensiveness of the data-collection, and also the strength of some of the observations. The evidence we found in support of the two Pareto principles 1a) and 2a) is the least surprising. It does seem to be inevitable that a small number of the modules in a system will contain a large proportion of the pre-release faults and that a small proportion of the modules will contain a large proportion of the post-release faults. However, the popularly believed explanations for these two phenomena appear to be quite wrong: . It is not the case that size explains in any significant way the number of faults. Many people seem to believe (hypotheses 1b and 2b) that the reason why a small proportion of modules account for most faults is simply because those fault-prone modules are disproportionately large and therefore account for most of the system size. We have shown this assumption to be false for this system. . Nor is it the case that 'complexity' (or at least complexity as measured by 'complexity metrics') explains the fault-prone behaviour (hypothesis 6). In fact complexity is not significantly better at predicting fault and failure prone modules than simple size measures. . It is also not the case that the set of modules which are especially fault-prone pre-release are going to be roughly the same set of modules that are especially fault-prone post-release (hypothesis 4). Yet this view seems to be widely accepted, partly on the assumption that certain modules are 'intrinsically' difficult and will be so throughout their testing and operational life. Our strong rejection of hypothesis 4 is a very important observation. Many believe that the first place to look for modules likely to be fault-prone in operation is in those modules which were fault prone during testing. In fact our results relating to hypothesis 4 suggest exactly the opposite testing strategy as the most effective. If you want to find the modules likely to be fault-prone in operation then you should ignore all the modules which were fault-prone in testing! In reality, the danger here is in assuming that the given data provides evidence of a causal relationship. The data we observed can be explained by the fact that the modules in which few faults are discovered during testing may simply not have been tested properly. Those modules which reveal large numbers of faults during testing may genuinely be very well tested in the sense that all the faults really are 'tested out of them'. The key missing explanatory data in this case is, of course, testing effort. The results of hypothesis 4 also bring into question the entire rationale for the way software complexity metrics are used and validated. The ultimate aim of complexity metrics is to predict modules which are fault-prone post-release. Yet we have found that there is no relationship between the modules which are fault-prone pre-release and the modules which are fault-prone post-release. Most previous 'validation' studies of complexity metrics have deemed a metric 'valid' if it correlates with the (pre-release) fault density. Our results suggest that 'valid' metrics may therefore be inherently poor at predicting what they are supposed to predict. The results of hypothesis 4 also highlight the dangers of using fault density as a de-facto measure of user perceived software quality. If fault density is measured in terms of pre-release faults (as is very common), then at the module level this measure tells us worse than nothing about the quality of the module; a high value is more likely to be an indicator of extensive testing than of poor quality. Our analysis of the value of 'complexity' metrics is mixed. We confirmed some previous studies' results that popular complexity metrics are closely correlated to size metrics like LOC. While LOC (and hence also the complexity metrics) are reasonable predictors of absolute number of faults, they are very poor predictors of fault density (which is what we are really after). However, some complexity metrics like SigFF are, unlike LOC, available at a very early stage in the software development process. The fact that it correlates so closely with the final LOC, is therefore very useful. Moreover, we argued [Fenton and Pfleeger 1996], that being a good predictor of fault-proneness may not be the most appropriate test of 'validity' of a complexity metric. It is more reasonable to expect complexity metrics to be good predictors of module attributes such as comprehensibility or maintainability. We investigated the extent to which benchmarking type data could provide insights into software quality. In testing hypotheses 7 and 8, we showed that the fault densities are roughly constant between subsequent major releases and our data indicates that there are 10- times as many pre-release faults as post-release faults. Even if readers are uninterested in the software engineering hypotheses (1-6) they will surely value the publication of these figures for future comparisons and benchmarking. We believe that there are no 'software engineering laws' as such, because it is always possible to construct a system in an environment which contradicts the law. For example, the studies summarised in [Hatton 1997] suggest that larger modules have a lower fault density than smaller ones. Apart from the fact that we found no clear evidence of this ourselves (hypothesis 5) and also found weaknesses in the studies, it would be very dangerous to state this as a law of software engineering. You only need to change the amount of testing you do to 'buck' this law. If you do not test or use a module you will not observe faults or failures associated with it. Again this is because the association between size and fault density is not a causal one. It is for this kind of reason that we recommend more complete models that enable us to augment the empirical observations with other explanatory factors, most notably, testing effort and operational usage. In this sense our results justify the recent work on building causal models of software quality using Bayesian Belief Networks, rather than traditional statistical methods which are patently inappropriate for defects prediction.[Neil and Fenton 1996]. In the case study system described in this paper, the data-collection activity is considered to be a part of routine configuration management and quality assurance. We have used this data to shed light on a number of issues that are central to the software engineering discipline. If more companies shared this kind of data, the software engineering discipline could quickly establish the empirical and scientific basis that it so sorely lacks. Acknowledgements We are indebted to Martin Neil for his valuable input to this work and to Pierre-Jacques Courtois, Karama Kanoun, Jean-Claude Laprie, and Stuart Mitchell for their valuable review comments. The work was supported, in part, by the EPSRC-funded project IMPRESS, the ESPRIT-funded projects DEVA and SERENE, the Swedish National Board for Industrial and Technical Development, and Ericsson Utvecklings AB. --R Project Software Defects From Analyzing Ada Designs Estimating the fault content of software suing the fix-on-fix model Prediction and Control of ADA Software Defects An integrated approach to criticality prediction. A Rigorous and Practical Approach (2nd Edition) tapping the wheels of software Design metrics and aids to their automatic collection Software structure metrics based on information flow. 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Gunes Koru , Jeff (Jianhui) Tian, Comparing High-Change Modules and Modules with the Highest Measurement Values in Two Large-Scale Open-Source Products, IEEE Transactions on Software Engineering, v.31 n.8, p.625-642, August 2005 Zhenmin Li , Lin Tan , Xuanhui Wang , Shan Lu , Yuanyuan Zhou , Chengxiang Zhai, Have things changed now?: an empirical study of bug characteristics in modern open source software, Proceedings of the 1st workshop on Architectural and system support for improving software dependability, p.25-33, October 21-21, 2006, San Jose, California Raimund Moser , Barbara Russo , Giancarlo Succi, Empirical analysis on the correlation between GCC compiler warnings and revision numbers of source files in five industrial software projects, Empirical Software Engineering, v.12 n.3, p.295-310, June 2007 Matthew S. Harrison , Gwendolyn H. Walton, Identifying high maintenance legacy software, Journal of Software Maintenance: Research and Practice, v.14 n.6, p.429-446, November 2002 Michael Ellims , James Bridges , Darrel C. Ince, The Economics of Unit Testing, Empirical Software Engineering, v.11 n.1, p.5-31, March 2006 Khaled El Emam , Sada Benlarbi , Nishith Goel , Walcelio Melo , Hakim Lounis , Shesh N. Rai, The Optimal Class Size for Object-Oriented Software, IEEE Transactions on Software Engineering, v.28 n.5, p.494-509, May 2002 Lenin Singaravelu , Calton Pu , Hermann Hrtig , Christian Helmuth, Reducing TCB complexity for security-sensitive applications: three case studies, ACM SIGOPS Operating Systems Review, v.40 n.4, October 2006 Avner Engel , Mark Last, Modeling software testing costs and risks using fuzzy logic paradigm, Journal of Systems and Software, v.80 n.6, p.817-835, June, 2007 Norman E. Fenton , Martin Neil, Software metrics: roadmap, Proceedings of the Conference on The Future of Software Engineering, p.357-370, June 04-11, 2000, Limerick, Ireland Mohammad Alshayeb , Wei Li, An Empirical Validation of Object-Oriented Metrics in Two Different Iterative Software Processes, IEEE Transactions on Software Engineering, v.29 n.11, p.1043-1049, November Yair Wiseman, Advanced non-distributed operating systems course, ACM SIGCSE Bulletin, v.37 n.2, June 2005 Norman E. Fenton , Martin Neil, A Critique of Software Defect Prediction Models, IEEE Transactions on Software Engineering, v.25 n.5, p.675-689, September 1999 Kalhed El Emam , Sada Benlarbi , Nishith Goel , Shesh N. Rai, The Confounding Effect of Class Size on the Validity of Object-Oriented Metrics, IEEE Transactions on Software Engineering, v.27 n.7, p.630-650, July 2001 Niels Veerman , Ernst-Jan Verhoeven, Cobol minefield detection, SoftwarePractice & Experience, v.36 n.14, p.1605-1642, November 2006 Khaled El-Emam, Object-oriented metrics: A review of theory and practice, Advances in software engineering, Springer-Verlag New York, Inc., New York, NY, 2002

software faults and failures;empirical studies;software metrics

631253

Architecture-Based Performance Analysis Applied to a Telecommunication System.

AbstractSoftware architecture plays an important role in determining software quality characteristics, such as maintainability, reliability, reusability, and performance. Performance effects of architectural decisions can be evaluated at an early stage by constructing and analyzing quantitative performance models, which capture the interactions between the main components of the system as well as the performance attributes of the components themselves. This paper proposes a systematic approach to building Layered Queueing Network (LQN) performance models from a UML description of the high-level architecture of a system and more exactly from the architectural patterns used for the system. The performance model structure retains a clear relationship with the system architecture, which simplifies the task of converting performance analysis results into conclusions and recommendations related to the software architecture. In the second part of the paper, the proposed approach is applied to a telecommunication product for which an LQN model is built and analyzed. The analysis shows how the performance bottleneck is moving from component to component (hardware or software) under different loads and configurations and exposes some weaknesses in the original software architecture, which prevent the system from using the available processing power at full capacity due to excessive serialization.

Introduction Performance characteristics (such as response time and throughput) are an integral part of the quality attributes of a software system. There is a growing body of research that studies the role of software architecture in determining different quality characteristics in general [12], [1], and performance characteristics in special [15], [16]. Architectural decisions are made very early in the software development process, therefore it would be helpful to be able to assess their effect on software performance as soon as possible. This paper contributes toward bridging the gap between software architecture and early performance analysis. It proposes a systematic approach to building performance models from the high-level software architecture of a system, which describes the main system components and their interactions. The architectural descriptions on which the construction of a performance model is based must capture certain issues relevant to performance, such as concurrency and parallelism, contention for software resources (as, for example, for software servers or critical sections), synchronization and serialization, etc. Frequently used architectural solutions are identified in literature as architectural patterns (such as pipeline and filters, client/server, client/broker/server, layers, master-slave, blackboard, etc.) [3], [12]. A pattern introduces a higher-level of abstraction design artifact by describing a specific type of collaboration between a set of prototypical components playing well-defined roles, and helps our understanding of complex systems. The paper proposes a systematic approach to building a performance model by transforming each architectural pattern employed in a system into a performance sub-model. The advantage of using patterns is that they are already identified and catalogued, so we can build a library of transformation rules for converting patterns to performance models. If, however, not all components and interactions of a high-level architecture are covered by previously identified architectural patterns, we can still describe the remaining interactions as UML mechanisms [2] and proceed by defining ad-hoc transformations into performance models. The formalism used for building performance models is the Layered Queueing Network (LQN) model [11, 17, 18], an extension of the well-known Queueing Network model. LQN was developed especially for modelling concurrent and/or distributed software systems. Some LQN components represent software processes, others hardware devices. One of the most interesting performance characteristics of such systems is that a software process may play a dual role, acting both as a client to some processes /devices, and as a server to others (see section 3 for a more detailed description). Since a software server may have many clients, important queueing delays may arise for it. The server may become a software bottleneck, thus limiting the potential performance of the system. This can occur even if the devices used by the process are not fully utilized. The analysis of an LQN model produces results such as response time, throughput, queueing delays and utilization of different software and hardware components, and indicates which components are the system bottleneck(s). By understanding the cause for performance limitations, the developers will be able to concentrate on the system's ``trouble spots'' in order to eliminate or mitigate the bottlenecks. The analysis of LQN models for various alternatives will help in choosing the "right" changes, so that the system will eventually meet its performance requirements. Software Performance Engineering (SPE) is a technique introduced in [14] that proposes to use quantitative methods and performance models in order to assess the performance effects of different design and implementation alternatives, from the earliest stages of software development throughout the whole lifecycle. LQN modelling is very appropriate for such a use, due to fact that the model structure can be derived systematically from the high-level architecture of the system, as proposed in this paper. Since the high-level architecture is decided early in the development process and does not change frequently afterwards, the structure of the LQN model is also quite stable. However, the LQN model parameters (such execution times of the high-level architectural components on behalf of different types of system requests) depend on low-level design and implementation decisions. In the early development stages, the parameter values are estimations based on previous experience with similar systems, on measurement of reusable components, on known platform overheads (such as system call execution times) and on time budgets allocated to different components. As the development progresses and more components are implemented and measured, the model parameters become more accurate, and so do the results. In [14] it is shown that early performance modelling has definite advantages, despite its inaccurate results, especially when the model and its parameters are continuously refined throughout the software lifecycle. LQN was applied to a number of concrete industrial systems (such as database applications [5], web servers [7], telecommunication systems, etc.) and was proven to be useful for providing insights into performance limitations at software and hardware levels, for suggesting performance improvements in different development stages, for system sizing and for capacity planning. In this paper, LQN is applied to a real telecommunication system. Although the structure of the LQN model was derived from the high-level architecture of the system, which was chosen in the early development stage, we used model parameters obtained from prototype measurements, which are more accurate than the estimated values available in pre-implementation phases. The reason is that we became involved with the project when the system was undergoing performance tuning, and so we used the best data available to analyze the high-level 5architecture of the system (which was unchanged from the early design stages). We found some weaknesses in the original architecture due to excessive serialization, and used the LQN model to assess different architectural alternatives, in order to improve the performance by removing or mitigating software bottlenecks. The paper proceeds as follows: section 2 discusses architectural patterns and the UML notation [2] used to represent them. Section 3 gives a brief description of the LQN model. Section 4 proposes transformations of the architectural patterns into LQN sub-models. Section 5 presents the telecommunication system case study and its LQN model. Section 6 analyzes the LQN model under various loads and configurations, shows how the bottleneck moves around the system and proposes improvements to the system. Section7 gives the conclusions of the work. 3. architectural Patterns According to [1], a software architecture represents a collection of computational components that perform certain functions, together with a collection of connectors that describe the interactions between components. A component type is described by a specification defining its functions, and by a set of ports representing logical points of interaction between the component and its environment. A connector type is defined as a set of roles explaining the expected behaviour of the interacting parties, and a glue specification showing how the interactions are coordinated. A similar, even though less formal, view of a software architecture is described in the form of architectural patterns [3, 13] which identify frequently used architectural solutions, such as pipeline and filters, client/server, client/broker/server, master-slave, blackboard, etc. Each architectural pattern describes two inter-related aspects: its structure (what are the components) and behaviour (how they interact). In the case of high-level architectural patterns, the components are usually concurrent entities that execute in different threads of control, compete for resources, and interact in a prescribed manner which may require some kind of synchronization. These are aspects that contribute to the performance characteristics of the system, and therefore must be captured in the performance model. The paper proposes to use high-level architectural patterns as a basis for translating software architecture into performance models. A subset of such patterns, which are later used in the case study, are described in the paper in the form of UML collaborations (not to be confused with UML collaboration diagrams, a type of interaction diagrams). According to the authors of UML, a collaboration is a notation for describing a mechanism or pattern, which represents "a society filter1 UpStreamFilter filter2 DownStreamFilter UpStreamFilter DownStreamFilter filter1 filter2 wait for item proc_item() wait for next item proc_item() Figure 1. Structural and behavioural views of the collaboration for PIPELINE WITH MESSAGE filter1 buffer filter2 proc_item() return read() {sequential} buffer filter1 1.n filter2 1.n WITH BUFFER UpStreamFilter DownStreamFilter Buffer UpStreamFilter DownStreamFilter Buffer return read() wait for item wait for next item Figure 2. Structural and behavioural view of the collaboration PIPELINE WITH BUFFER Figure of classes, interfaces, and other elements that work together to provide some cooperative behaviour that is bigger than the sum of all of its parts" [2]. A collaboration has two aspects: structural and behavioural. Fig. 1 and 2 illustrate these aspects for two alternatives of the pipeline and filters pattern. Each figures contains on the left a UML class/object diagram describing the pattern structure, and on the right a UML sequence diagram illustrating the pattern behaviour. A brief explanation of the UML notation used in the paper is given below (see [2] for more details). The notation for a class or object is a rectangle indicating the class/object name (the name is underlined for objects); the rectangle may contain optionally a section for the class/object operations, and another one for its attributes. The multiplicity of the class/object is represented in the upper right corner. A rectangle with thick lines represents an active class/object, which has its own thread of control, whereas a rectangle with thin lines represents a passive one. An active object may be implemented either as a process or as a thread (identified by the stereotype < > or < >, respectively). The constraint {sequential} attached to the operations of a passive object, as in Fig. 2, indicates that the callers must coordinate outside the passive object (for example, by the means of a semaphore) so that only one calls the passive object's operations at any given time. The UML symbol for collaboration is an ellipse with a dashed line that may have an "embedded" rectangle showing template classes. The collaboration symbol is connected with the classes/objects with dashed lines, whose labels indicate the roles played by each component. A line connecting two objects, named link, represents a relationship between the two objects which interact by exchanging messages. Depending on the kind of interacting objects (passive or active), UML messages may represent either operation calls, or actual messages sent between different flows of control. Links between objects may be optionally annotated with arrows showing the name and type of messages exchanged. For example, in Fig.1 an arrow with a half arrowhead between the active objects filter1 and filter2 represents an asynchronous message, whereas in Fig.2 the arrows with filled solid arrowheads labeled "write()" and "read()" represent synchronous messages implemented as calls to the operations indicated by the label. When relevant, the "object flow" carried by a message is represented by a little arrow with a circle (as in Fig.2), while the message itself is an arrow without circle. A synchronous message implies a reply, therefore it can carry objects in both directions. For example, in Fig.2, the object flow carried by the message read() goes in the reverse direction than the message itself. A sequence diagram, such as on the right side of Fig.1 and 2, shows the messages exchanged between a set of objects in chronological order. The objects are arranged along the horizontal axis, and the time grows along the vertical axis, from top to bottom. Each object has a lifeline running parallel with the time axis. On the lifeline one can indicate the period of time during which an object is performing an action as a tall thin rectangle called "focus of control", or the state of the object as a rectangle with rounded corners called "state mark". The messages exchanged between objects (which can be asynchronous or synchronous) are represented as horizontal directed lines. An object can also send a message to itself, which means that one of its operations invokes another operation of the same object. Architectures using the pipeline and filters pattern divide the overall processing task into a number of sequential steps which are implemented as filters, while the data between filters flows through unidirectional pipes. We are interested here in active filters [3] that are running concurrently. Each filter is implemented as a process or thread that loops through the following steps: "pulls" the data (if any) from the preceding pipe, processes it, then "pushes" the results down the pipeline. The way in which the push and pull operations are implemented may have performance consequences. In Fig.1 the filters communicate through asynchronous messages. A filter "pulls" an item by accepting the message sent by the previous filter, processes the item by invoking its own operation proc_item(), passes the data on to the next filter by sending an asynchronous message, after which goes into a waiting state for the next item. In Fig.2, the filters communicate through a shared buffer (one pushes by writing to the buffer, and the other pulls by reading it). Whereas the filters are active objects with a multiplicity of one or higher, the buffer itself is a passive object that offers two operations, read() and write(), which must be used one at a time (as indicated by the constraint {sequential} ). When defining the transformations from architectural patterns into LQN sub-models, we use both the structural and the behavioural aspect of the respective collaborations. The structural part is used directly, in the sense that each software component has counterpart(s) in the structure of the LQN model (the mapping is not bijective). However, the behavioural part is used indirectly, in the sense that it is matched by the behaviour of the LQN model, but is not represented graphically. 3. LQN Model LQN was developed as an extension of the well-known Queueing Network (QN) model, at first independently in [17, 18] and [11], then as a joint effort [4]. The LQN toolset presented in [4] includes both simulation and analytical solvers that merge the best previous approaches. The main difference of LQN with respect to QN is that a server, to which customer requests are arriving and queueing for service, may become a client to other servers from which it requires nested services while serving its own clients. An LQN model is represented as an acyclic graph whose nodes (named also tasks) are software entities and hardware devices, and whose arcs denote service requests (see Fig.3). The software entities are drawn as parallelograms, and the hardware devices as circles. The nodes with outgoing and no incoming arcs play the role of pure clients. The intermediate nodes with incoming and outgoing arcs play both the role of client and of server, and usually represent software components. The leaf nodes are pure servers, and usually servers (such as processors, I/O devices, communication network, etc.) A software or hardware server node can be either a single-server or a multi-server (composed of more than one identical servers that work in parallel and share the same request queue). A LQN task may offer more than one kind of service, each modelled by a so-called entry, drawn as a parallelogram "slice". An entry has its own execution time and demands for other services (given as model parameters). Although not explicitly illustrated in the LQN notation, each server has an implicit message queue, where the incoming requests are waiting their turn to be served. Servers with more then one entry still have a single input queue, where requests for different entries wait together. The default scheduling policy of the queue is FIFO, but other policies are also supported. Fig. 3 shows a simple example of an LQN model for a three-tiered client/server system: at the top there are two client classes, each with a known number of stochastic identical clients. Each client sends requests for a specific service offered by a task named Application, which represents the business layer of the system. Each Application entry requires services from two different entries of the Database task, which offers in total three kinds of services. Every software task is running on a processor node, drawn as a circle; in the example, all clients of the same class share a processor, whereas Application and Database share another processor. Database uses also two disk devices, as shown in Fig.3. It is worth mentioning Application Database Proc2 Proc1 Proc3 Figure 3. Simple LQN model that the word layered in the name of LQN does not imply a strict layering of the tasks (for example, a task may call other tasks in the same layer, or skip over layers). There are three types of LQN messages, synchronous, asynchronous and forwarding, whose effect is illustrated in Fig.4. A synchronous message represents a request for service sent by a client to a server, where the client remains blocked until it receives a reply from the provider of service (see Fig.4.a). If the server is busy when a request arrives, the request is queued and waits phase1 (service) Client Server synchronous message busy reply included services phase2 phase3 (autonomous phases) idle Client Server a) LQN synchronous message Client Server busy included services phase1 phase2 phase3 idle asynchronous message Client Server asynchronous message forwarding Client synchronous message reply to original client busy phase1 phase2 idle Client c) LQN forwarding message busy idle phase1 phase2 idle Figure 4. Different types of LQN messages its turn. After accepting a message, the server starts to serve it by executing a sequence of phases (one or more). At the end of phase 1, the server replies to the client, which is unblocked and continues its work. The server continues with the following phases, if any, working in parallel with the client, until the completion of the last phase. (In Fig.4.a, a case with three phases is shown). After finishing the last phase, the server begins to serve a new request from the queue, or becomes idle if the queue is empty. During any phase, the server may act as a client to other servers, asking for and receiving so-called "included services". In the case of an asynchronous message, the client does not block after sending the message, and the server does not reply back, only executes its phases, as shown in Fig.4b. The third type of LQN message, named forwarding message (represented as a dotted request arc) is associated with a synchronous request that is served by a chain of servers, as illustrated in Fig. 4.c. The client sends a synchronous request to which begins to process the request, then forwards it to Server2 at the end of phase1. proceeds normally with the remaining phases in parallel with Server2, then goes on to another cycle. The client, however, remains blocked until the forwarded request is served by which replies to the client at the end of its phase 1. A forwarding chain can contain any number of servers, in which case the client waits until it receives a reply from the last server in the chain. The parameters of a LQN model are as follows: customer (client) classes and their associated populations or arrival rates; - for each software task entry: average execution time per phase; - for each software task entry seen as a client to a device (i.e., for each request arc from a task entry to a device): average service time at the device, and average number of visits per phase of the requesting entry; - for each software task entry seen as a client to another task entry (i.e., for each request arc from a task entry to another task entry): average number of visits per phase of the requesting entry; - for each request arc: average message delay; - for each software and hardware server: scheduling discipline. Typical results of an LQN model are response times, throughput, utilization of servers on behalf of different types of requests, and queueing delays. The LQN results may be used to identify the software and/or hardware bottlenecks that limit the system performance under different workloads and configurations. Understanding the cause for performance limitations helps the development team to come up with appropriate remedies. 4. Transformation from architecture to performance models A software system contains many components involved in various architectural connection instances (each described by a pattern/collaboration), and a component may play different roles in connections of various types. The transformation of the architecture into a performance model is done in a systematic way, pattern by pattern. As expected, the performance of the system depends on the performance attributes of its components and on their interactions (as described by patterns/collaborations). Performance attributes are not central to the software architecture itself, and must be supplied by the user as additional information. They describe the demands for hardware resources by the software components: allocation of processes to processors, execution time demands for each software component on behalf of different types of system requests, demands for other resources such as I/O devices, communication networks, etc. We will specify more clearly what kind of performance attributes must be provided for each pattern/collaboration. The tranformations from the architecture to the performance modelling domain are discussed next. LQN model for Pipeline and Filters. Fig. 5 and 6 show the transformation into LQN submodels of the two Pipeline and Filters collaborations described in Fig.1 and 2, respectively. The translation takes into account, on one side, the structural and behavioural information provided by the UML collaboration, and on the other side the allocation of software components filter1 filter2 filter1 UpStreamFilter filter2 DownStreamFilter UpStreamFilter DownStreamFilter Figure 5. Transformation of the PIPELINE WITH MESSAGE into an LQN submodel filter1 filter2 semaphore and buffer read proc a) All filters are running on the same processor node filter1 filter2 semaphore proc1 proc2 read b) The filters are running on different processor nodes read() {sequential} buffer filter1 1.n filter2 1.n WITH BUFFER UpStreamFilter DownStreamFilter Buffer UpStreamFilter DownStreamFilter Buffer Figure 6. Transformation of the PIPELINE WITH BUFFER into an LQN submodel to processors, which will lead to different LQN submodels for the same pattern (see Fig.6). The tansformation rules are as follows: a) Each active filter from Fig.5 and 6 becomes an LQN software server with a single entry, whose service time includes the processing time of the filter. The filter tasks will receive an asynchronous message (described in Fig. 4.b) and will execute its phases in response to it. A typical distribution of the work into phases is to receive the message in phase 1, to process it in phase 2, and to send it to the next filter in phase 3. b) The allocation of LQN tasks to processors mimics the real system. The way the filters are allocated on the same or on different processor nodes does not make any difference for the pipeline with message (reason for which the processors are not represented in Fig.5), but it affects the model for a pipeline with buffer, as explained below. c) In the case of a pipeline with message aspects related to the pipeline connector between the two filters are completely modelled by the LQN asynchronous message. The CPU times for send and receive system calls are added to the execution times of the phases in which the respective operations take place. If we want to model a network delay for the message, it can be represented by a delay attached to the request arc. d) In the case of a pipeline with buffer (Fig.6), an asynchronous LQN arc is necessary, but is not sufficient to model all the aspects concerning the pipeline connector. Additional LQN elements are required to take into account the serialization delay introduced by the constraint that buffer operations must be mutually exclusive. A third task that plays the role of semaphore will enforce this constraint, due to the fact that any task serializes the execution of its entries. The task has as many entries as the number of critical sections executed by the filters accessing the buffer (two in this case, "write" and "read"). Since the execution of each buffer operation takes place in the thread of control of the filter initiating the operation, the allocation of filters to processors matters. If both filters are running on the same processor node (which may have more than one processor) as in Fig.6.a, then the read and write operations will be executed on the same processor node. Thus, they can be modelled as entries of the semaphore task that is, obviously, co-allocated with the filters. If, however, the filters are running on different processor nodes, as in Fig.6.b, the mutual-exclusive operations read and write will be executed on different processor nodes, so they cannot be modelled as entries of the same task. (In LQN, all entries of a task are executed on the same processor node). The solution is shown in Fig.6.b: we keep the semaphore task for enforcing the mutual exclusion, but its entries are only used to delegate the work to two new tasks, each one responsible for a buffer operation. Each new task is allocated on the same processor as the filter initiating the respective operation. The LQN models for both pipeline and filters collaborations from Fig. 5 and 6 can be generated with a forwarding message (as in Fig.4.c) instead of an asynchronous one (as in Fig.4.b), if the source of requests for the first filter in a multi-filter architecture is closed instead of open. A closed source is composed of a set of client tasks sending synchronous requests to the first filter, and waiting for a reply from the last filter. Since a LQN task may send a forwarding message exactly at the end of its phase1, all the work done by a filter task must take place in the first phase. Client-Server Pattern is very frequently used in today's distributed systems. In Fig.7 is illustrated a case where the client communicates directly with the server through synchronous requests (described in Fig.4.a). A server may offer a wide range of services (represented in the architectural view as the server's class methods) each one with its own performance attributes. From a performance modelling point of view it is important not only to identify theses services, but also to find out who is invoking them and how frequently. The UML class diagram contains a single association between a client and a server, no matter how many server methods the client may invoke. Therefore, we indicate here in addition to the line that represents the client-server association, the messages sent by the client to the server (used mostly in collaboration diagrams) to indicate all the services a client will invoke at one time or another. There are other ways in which client/server connections may be realized, which are not described in the paper as they do not apply to our case study. A well-known example is the use of midware technology, such as CORBA, to interconnects clients and servers running on heterogeneous platforms across local or wide-area networks. CORBA connections introduce very interesting performance implications and modelling issues [9]. LQN was originally created to model client/server systems, so the transformation from the client-server pattern to LQN is quite straightforward. An LQN server may offer a range of services (object methods in the architectural view), each with its own CPU time and number of visits to other servers (these are performance attributes that must be provided). Each service is modelled as an entry of the server task, as shown in Figure 7, and will contribute differently to the clientclientserver service1 service2 server client2 1.n client1 1.n Client Client Server CLIENT SERVER Client Server Figure 7. Transformation of the client/server pattern into a LQN submodel response time, utilization, and throughput of the server. A client may invoke more than one of these services at different times. The performance attributes for the clients include their average time demands, and the average number of calls for each entry of the server. As in the pipeline connection case, the CPU time required to execute the system call for send/receive/reply are added to the service times of the corresponding entries. The allocation of tasks to processors is not shown in Fig. 7, because the transformation does not depend on it. Each LQN task is allocated exactly as its architectural component counterpart. Critical section. This is a collaboration at a lower-level of abstraction than the previous architectural patterns, but very frequently used. It describes the case where two or more active objects share the same passive object. The constraint {sequential} attached to the methods of the shared object indicates that the callers must coordinate outside the shared object (for example, by the means of a semaphore) to insure correct behaviour. Such synchronization introduces performance delays, and must be represented in the LQN model. For simplicity reasons, Fig.8 illustrates a case where each user invokes only a method of the shared object, but this can be extended easily to allow each user to call a subset of methods. The transformation of the critical section collaboration produces either the model given in Fig. 8.a or in Fig. 8.b, depending on the allocation of user processes to processor nodes (similar to the pipeline with buffer case). The premise is that the shared object operations are mutually exclusive, that an LQN task cannot change its processor node, and that all the entries of a task are executed on the task's processor node. In the case where all users are running on the same processor node, the shared object operations can be modelled as entries of a task that plays the role of semaphore (see Fig.8.a), which is running on the same processor node as the users. The generalization for allowing a user to call a subset of operations (entries) is straightforward: the user is connected by a request arcs to every entry in the subset. If the users are running on different processor nodes as in Fig. 8.b, then the shared object operations (i.e., critical sections) are executed by different threads of controls corresponding to different users that are running on different processors. Therefore each operation is modelled as an entry of a new task responsible for that operation that is running on its user's node. (If a user is to call more shared operations, its new associated task will have an entry for every such operation. This means that an operation called by more than one user will be represented by more than one entry.) However, these new tasks must be prevented from running simultaneously, shared userN user1 Accessor Accessor Shared CRITICAL SECTION Accessor Shared user1 userN proc semaphore and critical sections f1 f2 . fN user1 userN semaphore proc1 procN a) All users are running on the same processor node b) The users are running on different processor nodes Figure 8. Transformation of the critical section collaboration to a LQN submodel so a semaphore task, with one entry for each user, is used to enforce the mutual exclusion. An entry of the semaphore task delegates the work to the entries modelling the required operations. The performance attributes to be provided for each user must specify the average execution times for each user outside and inside the critical section separately. Co-allocation collaboration. Fig. 9 shows a the so-called co-allocation collaboration, where two active objects are contained in a third active object, and are constrained to execute only one at a time. The container object may be implemented as a process. This is an example of architectural connection from our case-study system, which is not necessarily an architectural pattern, but is quite frequently used. The most obvious solution to model the two contained objects as entries of the same task presents a disadvantage: it cannot represent the case where each of the two contained objects has its own request queue. (An LQN task has a unique message queue, where requests for all entries are waiting together). One reason for which we may need separate queues is to avoid cyclic graphs, which could not be accepted by the LQN solver used for this paper. The solution presented in Fig.9 represents each contained active object as a separate "dummy" task that delegates all the work to an entry of the container task, which serializes all its entries. The dummy tasks are allocated on a dummy processor (not to interfere with the scheduling of the "real" processor node). dummyActive1 dummyActive2 container active1 active2 dummy proc proc active_container active1 active2 Contained COALLOCATION Container Contained Contained Container Figure 9. LQN submodel of COALLOCATION collaboration 5. LQN Model of a Telecommunication System We conducted performance modelling and analysis of an existing telecommunication system which is responsible for developing, provisioning and maintaining various intelligent network services, as well as for accepting and processing real-time requests for these services. According to the Software Performance Engineering methodology [14], we first identified the critical scenarios with the most stringent performance constraints (which correspond in this case to real-time processing of service requests). Next we identified the software components involved in, and the architectural patterns exercised by the execution of the chosen scenarios (see Fig.10). CLIENT SERVER Client Server WITH BUFFER UpStrmFilter DownStrmFilter Buffer UpStrmFilter DownStrmFilter COALLOCATION Container Contained RequestHandler 1.n IO IOin IOout Stack StackIn StackOut doubleBuffer inBuffer outBuffer COALLOCATION Container Contained WITH BUFFER UpStrmFilter DownStrmFilter Buffer UpStrmFilter DownStrmFilter CRITICAL SECTION Accessor Shared CRITICAL SECTION Accessor Shared DataBase Figure 10. UML model of a telecommunication system The real time scenario we have modelled starts from the moment a request arrives to the system and ends after the service was completely processed and a reply was sent back. As shown in Fig. 10, a request is passed through several filters of a pipeline: from Stack process to IO process to RequestHandler and all the way back. The main processing is done by the RequestHandler (as it can be seen from Fig.12 and Table 1), which accesses a real-time database to fetch an execution "script" for the desired service, then executes the steps of the script accordingly. The script may vary in size and types of operations involved, and hence the workload varies largely from one type of service to another (by one or two orders of magnitude). Based on experience and intuition, the designers decided from the beginning to allow for multiple replications of the RequestHandler process in order to speed up the system. Two shared objects, ShMem1 and ShMem2, are used by the multiple RequestHandler replications. The system was intended to be run either on a single-processor or on a multi-processor with shared memory. Processor scheduling is such that any process can run on any free processor (i.e., the processors were not dedicated to specific tasks). Therefore, the processor node was modelled as a multi-server. By Dummy Proc read IOin Request Handler IOout DataBase Proc IOexec StackIn StackOut Buffer ShMem1 l update Figure 11. LQN base case model of the telecommunication system applying systematically the transformation rules described in the previous section to the architectural patterns/collaborations used in the system, as shown in Fig. 10, the LQN model shown in Fig.11 was obtained. The next step was to determine the LQN model parameters (average service time for each entry, and average number of visits for each request arc) and to validate the model. We have made use of measurements using Quantify [19] and the Unix utility top. The measurements with Quantify were obtained at very low arrival rates of around a couple of requests/second. Quantify is a profiling tool which uses data from the compiler and run-time information to determine the user and kernel execution times for test cases chosen by the user. Since we wanted to measure average execution times for different software components on behalf of a system request (see Table 1 in the Appendix), we have measured the execution of 2000 requests repeated in a loop, then computed the average per request. Although we have not computed confidence intervals on the measurements, repeated experiments were in close agreement. The top utility provided us with utilization figures for very high loads of hundreds of requests/second, close to the actual Execution time demands per system request StackIn StackOut IOin IOout RequestHandler DataBase Execution time (msec) Non-critical section Critical sect:Buffer Critical sect:ShMem1 Critical sect:ShMem2 Total Figure 12. Distribution of the total demand for CPU time per request over different software components operating point. These measurements were done on a prototype in the lab for two different hardware configurations, with one and four processors. Again, repeated measurements were in close agreement. We have used the execution times measured with Quantify (given in Table 1 in the Appendix) to determine the model parameters, and the utilization results from top to validate our model. The utilization values obtained by solving the model were within 5% of the measured values. Unfortunately, a more rigorous validation was hindered by the lack of response time measurements. 6. Performance Analysis of the Telecommunication System Although the LQN toolset [4] offers both analytical and simulation solvers, the model results used in this section were obtained by simulation. The reason is that one of the system features, namely the scheduling policy by polling used for the RequestHandler multi-server, could not be handled by the analytical solver. All the simulation results were obtained with a confidence interval of plus/minus1% at the 95% level. Maximum Throughput Vs. replication factor of the RequestHandler (RH)2006001000n=1 processor n=4 processors n=6 processors Configurations Throughput Figure 13. Maximum achievable throughput for different hardware and software configurations and a single class of service requests The first question of interest to developers was to find the "best" hardware and software configuration that can achieve the desired throughput for a given mix of services. By "configuration" we understand more specifically the number of processors in the multiprocessor system and the number of RequestHandler software replications. We tried to answer this question by exploring a range of configurations for a given service mix, determining for each the highest achievable throughput (as in Fig. 13). Then the configurations with a maximum throughput lower then the required values are discarded. The cheapest configuration that can insure satisfactory throughput and response time at an operating point below saturation will be chosen. Solving the LQN model is a more efficient way to explore different configurations under a wide range of workloads then to measure the real system for all these cases. Although we have modelled the system for two classes of services, we selected to report here only results for a single class because they illustrate more clearly how the bottleneck moves around from hardware to software for different configurations. The model was analyzed for three hardware configurations: with one, four and six processors, respectively. We chose one and four processors since the actual system had been run for such configurations, and six processors to see how the software architecture scales up. When running the system on a single processor (see Fig. 14), the replication of the RequestHandler does not increase the maximum achievable throughput. This is due to the fact that the processor is the system bottleneck. As known from [8], the replication of software processes brings performance rewards only if there is unused processing capacity (which is not the case here). A software server is said to be "utilized" when is doing effective work and when is waiting to be served by lower level servers (including the queueing for the included services). Fig. 17 and show different contributions to the utilization of two tasks, IOout and RH, when the system is saturated (i.e., works at the highest achievable throughput given in Fig. 13) for different numbers of RH replications. It is easy to see that for more than five RH copies, one task (i.e., IOout) has a very high utilization even though it does little useful work, whereas at the same time another task (i.e., a RH copy) has a lower utilization and does more useful work. Interestingly enough, in the case of 4- processor configuration we notice that with more processing capacity in the system, the processors do not reach the maximum utilization level, as shown in Fig.15. Instead, two software tasks IOout and IOin (which are actually responsible for little useful work on behalf of a system reach critical levels of utilization due to serialization constraints in the software 4-Processor Configuration: Base Case0.20.61 Number of RequestHandler replications Utilization IOout Processor IOExec Database RequestHandler Figure 15. Task Utilizations for 4-proc base case 6-Processor Configuration: Base Case0.20.61 Number of RequestHandler Replications Utilization IOout IOexec Processor DataBase RequestHandler Figure 16. Task utilizations for 6-proc base case 1-Processor Configuration: Base Case0.20.61 Number of RequestHandlers Utilization RequestHandler Processor IOExec Database Figure 14. Task utilizations for 1-proc base case architecture. There are two reasons for serialization: i) IOin and IOout are executed by a single thread of control (which in LQN means waiting for task IOexec), and ii) contend for the same buffer. Thus, with increasing processing power, the system bottleneck is moving from hardware to software. This trend is more visible in the case of 6- processors configuration, where the processor utilization reaches only 86.6%, as shown in Fig.16. The bottleneck has definitely shifted from hardware to software, where the limitations in performance are due to constraints in the software architecture. We tried to eliminate the serialization constraints in two steps: first by making each filter a process on its own (i.e., by removing StackExec and IOexec tasks in the LQN model), then by splitting the pipeline buffer sitting between IO process and RequestHandler in two buffers. The LQN model obtained after the 2-step modifications is shown in Fig. 19. The results of the "half- way" modified system (after the first step only) are given in Fig. 20, and show no major performance improvement (the processor is still used below capacity). By examining again the utilization components of IOin and IOout (which are still the bottleneck) we found that they 4-Proc Base Case: Contributions to IOout utilization0.20.613 4 5 6 7 Number of RequestHandler replications Utilization IOout waiting for IOexec IOexec waiting for semaphore processor Useful work: read from buffer Figure 17. Contributions to IOout utilization when the system is saturated, in function of the RH replication level Contributions to RequestHandler Utilization0.20.613 4 5 6 7 Number of RequestHandler Reprlications Utilization Useful work by each RequestHandler Useful work by others on behalf of RequestHandler RequestHandler is busy while waiting for nested services Figure 18. Contributions to the utilization of a RH copy when the system is saturated, in function of the RH replication level wait most of the time to gain access to the Buffer (IOout is waiting about 90% of the time and IOin 80%). After applying both modification steps, though, the software bottleneck due to excessive serialization in the pipeline was removed, and the processor utilization went up again, as shown in Fig. 21 for the 4-processor and in Fig. 22 for the 6-processor configuration. As expected, the maximum achievable throughput increased as well. The throughput increase was rather small in the case of 4 processors (only 2.7%), and larger in the case of 6 processors (of 10.3%), where there was more unused processing power. We also realized that in the case of 6 processors a new software bottleneck has emerged, namely the Database process (which is now 100% utilized). The new bottleneck, caused by a low-level server, has propagated upwards, saturating all the software processes that are using it (all RequestHandler replications). The final conclusion of the performance analysis is that different configurations will have different bottlenecks, and by solving the bottleneck in one configuration, we shift the problem somewhere else. What performance modelling has to offer is the ability to explore a range of design alternatives, configurations and workload mixes, and to detect the causes of performance limitations just by analyzing the model, before proceeding to change the real system. IOin Request Handler IOout DataBase Proc StackIn StackOut l BufferIn ShMem1 read write BufferOut update Figure 19. LQN model of the modified system 7. Conclusions This paper contributes toward bridging the gap between software architecture and performance analysis. It proposes a systematic approach to building performance models from the high-level software architecture of a system, by transforming each architectural pattern employed in the system into a performance sub-model. There is on-going work to formalize the kind of transformations presented in the paper from the architecture to the performance domain by using formal graph transformations based on the The paper illustrates the proposed approach to building LQN models by applying it to an existing telecommunication system. The performance analysis exposes weaknesses in the original 4-Processors Configuration: half-way modified system0.20.612 4 Number of RequestHandler replications Utilization IOout / IOin Processor DataBase RequestHandler StackOut StackIn Figure 20. Task utilizations for the 4-proc half-way modified system 4-processor configuration: fully modified system0.20.612 4 Number of RH replications Utilization Processor DataBase RequestHandler IOout IOin StackIn StackOut Figure 21. Task utilizations for the 4-proc fully modified system 6-Processor Configuration: fully modified system0.20.61 Number of RequestHandler replications Utilization RequestHand ler Processor DataBase IOout StackOut IO in StackIn Figure 22. Task utilization for the 6-proc fully modified architecture due to excessive serialization, which show up when more processing power is added to the system. Surprisingly, software components that do relatively little work on behalf of a system request can become the bottleneck in certain cases, whereas components that do most of the work do not. After removing the serialization constraints, a new software bottleneck emerges, which leads to the conclusion that the software architecture as it is does not scale up well. This case study illustrates the usefulness of applying performance modelling and analysis to software architectures. --R "A Formal Basis for architectural Connection" The Unified Modeling Language User Guide A System of Patterns "A toolset for Performance Engineering and Software Design of Client-Server Systems" "Performance Analysis of Distributed Server Systems" "Performance of Multi-Level Client-Server Systems with Parallel Service Operations" "Measuremnt Tool and Modelling Techniques for Evaluating Web Server Performance" "Software bottlenecking in client-server systems and rendezvous networks" "Deriving Software Performance Models from architectural Patterns by Graph Transformations" "A Multi-Layer Client-Server Queueing Network Model with Synchronous and Asynchronous Messages" "The Method of Layers" Perspectives on an Emerging Discipline "Some Patterns for Software Architecture" Performance Engineering of Software Systems "Architecture-Based Performance Analysis" "Performance Evaluation of "Throughput Calculation for Basic Stochastic Rendezvous Networks" "The Stochastic Rendezvous Network Model for Performance of Synchronous Client-Server-like Distributed Software" --TR --CTR performance model for a BPI middleware, Proceedings of the 4th ACM conference on Electronic commerce, June 09-12, 2003, San Diego, CA, USA Software Performance Engineering of a Web service-based Clinical Decision Support infrastructure, ACM SIGSOFT Software Engineering Notes, v.29 n.1, January 2004 Gordon P. 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layered queueing networks;Unified Modeling Language UML;software architecture;software performance analysis;architectural patterns

633045

Scalable application layer multicast.

We describe a new scalable application-layer multicast protocol, specifically designed for low-bandwidth, data streaming applications with large receiver sets. Our scheme is based upon a hierarchical clustering of the application-layer multicast peers and can support a number of different data delivery trees with desirable properties.We present extensive simulations of both our protocol and the Narada application-layer multicast protocol over Internet-like topologies. Our results show that for groups of size 32 or more, our protocol has lower link stress (by about 25%), improved or similar end-to-end latencies and similar failure recovery properties. More importantly, it is able to achieve these results by using orders of magnitude lower control traffic.Finally, we present results from our wide-area testbed in which we experimented with 32-100 member groups distributed over 8 different sites. In our experiments, average group members established and maintained low-latency paths and incurred a maximum packet loss rate of less than 1% as members randomly joined and left the multicast group. The average control overhead during our experiments was less than 1 Kbps for groups of size 100.

INTRODUCTION Multicasting is an efficient mechanism for packet delivery in one-many data transfer applications. It eliminates redundant packet replication in the network. It also decouples the size of the receiver set from the amount of state kept at any single node and therefore, is an useful primitive to scale multi-party applications. However, deployment of network-layer multicast [11] has not beenwidely adopted by most commercial ISPs, and thus large parts of the Internet are still incapable of native multicast more than a decade after the protocols were developed. Application-Layer Multicast protocols [10, 12, 7, 14, 15, 24, 18] do not change the network infrastructure, instead they implement multicast forwarding functionality exclusively at end-hosts. Such application-layer multicast protocols and are increasingly being used to implement efficient commercial content- distribution networks. In this paper, we present a new application-layer multicast protocol which has been developed in the context of the NICE project at the University of Maryland 1 . NICE is a recursive acronym which stands for NICE is the Internet Cooperative Environment. In this paper, we refer to the NICE application-layer multicast protocol as simply the NICE protocol. This protocol is designed to support applications with large receiver sets. Such applications include news and sports ticker services suchas Infogate (http://www.infogate.com) andESPN Bottomline (http://www.espn.com); real-time stock quotes and updates, e.g. the Yahoo! Market tracker, and popular Internet Radio sites. All of these applications are characterized by very large (potentially tens of thousands) receiver sets and relatively low bandwidth soft real-time data streams that can withstand some loss. We refer to this class of large receiver set, low bandwidth real-time data applications as data stream applications. Data stream applications present an unique challenge for application-layer multicast protocols: the large receiver sets usually increase the control overhead while the relatively low-bandwidth data makes amortizing this control overhead difficult. NICE can be used to implement very large data stream applications since it has a provably small (con- stant) control overhead and produces low latency distribution trees. It is possible to implement high-bandwidth applications using NICE as well; however, in this paper, we concentrate exclusively on low bandwidth data streams with large receiver sets. 1.1 Application-Layer Multicast The basic idea of application-layer multicast is shown in Figure 1. Unlike native multicast where data packets are replicated at routers inside the network, in application-layer multicast data packets are replicated at end hosts. Logically, the end-hosts form an overlay network, and the goal of application-layer multicast is to construct and maintain an efficient overlay for data transmission. Since appli- A Network Layer Multicast Application Layer Multicast Figure 1: Network-layer and application layer multicast. Squarenodesare routers, and circular nodesare end-hosts. The dotted lines represent peers on the overlay. cation-layer multicast protocols must send the identical packetsover the same link, they are less efficient than native multicast. Two intuitive measures of "goodness"for application layer multicast over- lays, namely stress and stretch, were defined in [10]). The stress metric is defined per-link and counts the number of identical packets sent by a protocol over each underlying link in the network. The stretch metric is defined per-member and is the ratio of path-length from the source to the member along the overlay to the length of the direct unicast path. Consider an application-layer multicast protocol in which the data source unicasts the data to each receiver. Clearly, this "multi-unicast" protocol minimizes stretch, but doesso at a cost of O(N) stress at links near the source (N is the number of group members). It also requires O(N)control overhead at some single point. However, this protocol is robust in the sense that any number of group member failures do not affect the other members in the group. In general, application-layer multicast protocols can be evaluated along three dimensions: ffl Quality of the data delivery path: The quality of the tree is measured using metrics such as stress, stretch, and node degrees ffl Robustnessof the overlay: Since end-hosts are potentially less stable than routers, it is important for application-layer multicast protocols to mitigate the effect of receiver failures. The robustnessof application-layer multicast protocols is measured by quantifying the extent of the disruption in data delivery when different members fail, and the time it takes for the protocol to restore delivery to the other members. We present the first comparison of this aspect of application-layer multicast protocols. ffl Control overhead: For efficient use of network resources, the control overhead at the members should be low. This is an important cost metric to study the scalability of the scheme to large member groups. 1.2 NICE Trees Our goals for NICE were to develop an efficient, scalable, and distributed tree-building protocol which did not require any underlying topology information. Specifically, the NICE protocol reduces the worst-case state and control overheadat any member to O(log N), maintains a constant degree bound for the group members and approach the O(log N) stretch bound possible with a topology-aware centralized algorithm. Additionally, we also show that an average member maintains state for a constant number of other members, and incurs constant control overheadfor topology creation andmain- tenance. In the NICE application-layer multicast scheme, we create a hier- archically-connectedcontrol topology. The data delivery path is implicitly defined in the way the hierarchy is structured and no additional route computations are required. Along with the analysis of the various bounds, we also present a simulation-based performance evaluation of NICE. In our simula- tions, we compare NICE to the Narada application-layer multicast protocol [10]. Narada was first proposed as an efficient application-layer multicast protocol for small group sizes. Extensions to it have subsequently been proposed [9] to tailor its applicability to high-bandwidth media-streaming applications for these groups, and have been studied using both simulations and implementation. Lastly, we present results from a wide-area implementation in which we quantify the NICE run-time overheads and convergence properties for various group sizes. 1.3 Roadmap The rest of the paper is structured as follows: In Section 2, we describe our general approach, explain how different delivery trees are built over NICE and present theoretical bounds about the NICE protocol. In Section 3, we present the operational details of the pro- tocol. We present our performance evaluation methodology in Section 4, and present detailed analysis of the NICE protocol through simulations in Section 5 and a wide-area implementation in Section 6. We elaborate on related work in Section 7, and conclude in Section 8. 2. SOLUTION OVERVIEW The NICE protocol arranges the set of end hosts into a hierarchy; the basic operation of the protocol is to create and maintain the hi- erarchy. The hierarchy implicitly defines the multicast overlay data paths, as described later in this section. The member hierarchy is crucial for scalability, since most members are in the bottom of the hierarchy and only maintain state about a constant number of other members. The members at the very top of the hierarchy maintain (soft) state about O(log N) other members. Logically, each member keeps detailed state about other members that are near in the hierarchy, and only has limited knowledge about other members in the group. The hierarchical structure is also important for localizing the effect of member failures. The NICE hierarchy describedin this paper is similar to the member hierarchy used in [3] for scalable multicast group re-keying. How- ever, the hierarchy in [3], is layered over a multicast-capable net-work and is constructed using network multicast services (e.g. scoped expanding ring searches). We build the necessary hierarchy on a unicast infrastructure to provide a multicast-capable network. In this paper, we use end-to-end latency as the distance metric between hosts. While constructing the NICE hierarchy, members that are "close" with respect to the distance metric are mapped to the same part of the hierarchy: this allows us to produce trees with low stretch. In the rest of this section, we describe how the NICE hierarchy is defined, what invariants it must maintain, and describe how it is used to establish scalable control and data paths. 2.1 Hierarchical Arrangement of Members The NICE hierarchy is created by assigning members to different levels (or layers) as illustrated in Figure 2. Layers are numbered sequentially with the lowest layer of the hierarchy being layer zero (denoted by L0 ). Hosts in each layer are partitioned into a set of clusters. Each cluster is of size between k and 3k \Gamma 1, where k is a constant, and consists of a set of hosts that are close to each other. We explain our choice of the cluster size bounds later in this paper (Section 3.2.1). Further, each cluster has a cluster leader. The protocol distributedly chooses the (graph-theoretic) center of the clus- Figure 3: Control and data delivery paths for a two-layer hierarchy. All A i hosts are members of only L0 clusters. All B i hosts are members of both layers L0 and L1 . The only C host is the leader of the L1 cluster comprising of itself and all the B hosts. A Cluster-leaders of Cluster-leaders of layer 0 form layer 1 Topological clusters joined to layer 0 All hosts are G Figure 2: Hierarchical arrangement of hosts in NICE. The layers are logical entities overlaid on the same underlying physical network. ter to be its leader, i.e. the cluster leader has the minimum maximum distance to all other hosts in the cluster. This choice of the cluster leader is important in guaranteeing that a new joining member is quickly able to find its appropriate position in the hierarchy using a very small number of queries to other members. Hosts are mapped to layers using the following scheme: All hosts are part of the lowest layer, L0 . The clustering protocol at L0 partitions these hosts into a set of clusters. The cluster leaders of all the clusters in layer L i join layer L i+1 . This is shown with an example in Figure 2, using 3. The layer L0 clusters are [ABCD], [EFGH] and [JKLM] 2 . In this example, we assume that C , F and M are the centers of their respective clusters of their L0 clusters, and are chosen to be the leaders. They form layer L1 and are clustered to create the single cluster, [CFM], in layer L1 . F is the center of this cluster, and hence its leader. Therefore F belongs to layer L2 as well. TheNICE clusters and layers are created using a distributed algorithm described in the next section. The following properties hold for the distribution of hosts in the different layers: ffl A host belongs to only a single cluster at any layer. ffl If a host is present in some cluster in layer L i , it must occur in one cluster in each of the layers, In fact, it is the cluster-leader in each of these lower layers. ffl If a host is not present in layer, L i , it cannot be present in any layer ffl Each cluster has its size bounded between k and 3k \Gamma 1. The leader is the graph-theoretic center of the cluster. ffl There are at most log k N layers, and the highest layer has only a single member. We denote a cluster comprising of hosts We also define the term super-cluster for any host, X . Assume that host, X , belongs to layers no other layer, and let [.XYZ.] be the cluster it belongs it in its highest layer (i.e. layer its leader in that cluster. Then, the super-cluster of X is defined as the cluster, in the next higher layer (i.e. L i ), to which its leader Y belongs. It follows that there is only one supercluster defined for every host (except the host that belongs to the top-most layer, which does not have a super-cluster), and the supercluster is in the layer immediately above the highest layer that H belongs to. For example, in Figure 2, cluster [CFM] in Layer1 is the super-cluster for hosts B, A, and D. In NICE each host maintains state about all the clusters it belongs to (one in each layer to which it belongs) and about its super-cluster. 2.2 Control and Data Paths The host hierarchy can be used to define different overlay structures for control messages and data delivery paths. The neighbors on the control topology exchange periodic soft state refreshes and do not generate high volumes of traffic. Clearly, it is useful to have a structure with higher connectivity for the control messages, since this will cause the protocol to converge quicker. In Figure 3, we illustrate the choices of control and data paths using clusters of size 4. The edges in the figure indicate the peerings between group members on the overlay topology. Each set of four hosts arranged in a 4-clique in Panel 0 are the clusters in layer L0 . Hosts and C0 are the cluster leaders of these four L0 clusters andform the single cluster in layer L1 . Host C0 is the leader of this cluster in layer L1 . In the rest of the paper, we use to denote the cluster in layer L j to which member X belongs. It is defined if and only if X belongs to layer L j . The control topology for the NICE protocol is illustrated in Figure 3, Panel 0. Consider a member, X , that belongs only to layers Its peers on the control topology are the other members of the clusters to which X belongs in each of these layers, i.e. members of clusters (X). Using the example(Figure 3, Panel 0), member A0 belongs to only layer L0 , and therefore, its control path peers are the other members in its L0 cluster, i.e. and B0 . In contrast, member B0 belongs to layers L0 and L1 and therefore, its control path peers are all the other members of its L0 cluster (i.e. A0 ; A1 and A2 ) and L1 cluster (i.e. B1 ; B2 and C0 ). In this control topology, each member of a cluster, therefore, exchanges soft state refreshes with all the remaining members of the cluster. This allows all cluster members to quickly identify changes in the cluster membership, and in turn, enables faster restoration of a set of desirable invariants (described in Section 2.4), which might be violated by these changes. The delivery path for multicast data distribution needs to be loop- free, otherwise, duplicate packet detection and suppression mecha- clusters for j in end for Figure 4: Data forwarding operation at a host, h, that itself received the data from host p. nisms need to be implemented. Therefore, in the NICE protocol we choose the data delivery path to be a tree. More specifically, given a data source, the data delivery path is a source-specific tree, and is implicitly defined from the control topology. Each member executes an instance of the Procedure MulticastDataForward given in Figure 4, to decide the set of members to which it needs to forward the data. Panels 1, 2 and 3 of Figure 3 illustrate the consequent source-specific trees when the sources are at members A0 ; A7 and C0 respectively. We call this the basic data path. To summarize, in each cluster of each layer, the control topology is a clique, and the data topology is a star. It is possible to choose other structures, e.g. in each cluster, a ring for control path, and a balanced binary tree for data path. 2.3 Analysis Each cluster in the hierarchy has between k and 3k \Gamma 1 members. Then for the control topology, a host that belongs only to layer L0 peers with O(k) other hosts for exchange of control messages. In general, a host that belongs to layer L i and no other higher layer, peers with O(k) other hosts in eachof the layers fore, the control overhead for this member is O(k i). Hence, the cluster-leader of the highest layer cluster (Host C0 in Figure 3), peers with a total of O(k log N) neighbors. This is the worst case control overhead at a member. It follows using amortized cost analysis that the control overhead at an average member is a constant. The number of members that occur in layer L i and no other higher layer is boundedby O(N=k i ). Therefore, the amortized control overhead at an average member is log N O( log N with asymptotically increasing N . Thus, the control overhead is O(k) for the average member, and O(k log N) in the worst case. The same holds analogously for stress at members on the basic data path 3 . Also, the number of application-level hops on the basic data path between any pair of members is O(log N). While an O(k log N) peers on the data path is an acceptableupper- bound, we have definedenhancementsthat further reduce the upper-bound of the number of peers of a member to a constant. The stress at eachmember on this enhanceddata path (created using local transformations of the basic data path) is thus reducedto a constant, while the number of application-level hops between any pair of members still remain bounded by O(log N). We outline this enhancement to the basic data path in [4]. 2.4 Invariants All the properties described in the analysis hold as long as the hierarchy is maintained. Thus, the objective of NICE protocol is to 3 Note that the stress metric at members is equivalent to the degree of the members on the data delivery tree. scalably maintain the host hierarchy as new members join and existing members depart. Specifically the protocol described in the next section maintains the following set of invariants: ffl At every layer, hosts are partitioned into clusters of size between k and 3k \Gamma 1. ffl All hosts belong to an L0 cluster, and each host belongs to only a single cluster at any layer ffl The cluster leaders are the centers of their respective clusters and form the immediate higher layer. 3. PROTOCOL DESCRIPTION In this section we describe the NICE protocol using a high-level description. Detailed description of the protocol (including packet formats and pseudocode) can be found in [4]. We assume the existence of a special host that all members know of a-priori. Using nomenclature developed in [10], we call this host the RendezvousPoint (RP). Each host that intends to join the application-layer multicast group contacts the RP to initiate the join pro- cess. For ease of exposition, we assume that the RP is always the leader of the single cluster in the highest layer of the hierarchy. It interacts with other cluster members in this layer on the control path, and is bypassedon the data path. (Clearly, it is possible for the RP to not be part of the hierarchy, and for the leader of the highest layer cluster to maintain a connection to the RP, but we do not belabor that complexity further). For an application such as streaming media delivery, the RP could be a distinguished host in the domain of the data source. The NICE protocol itself has three main components: initial cluster assignment as a new host joins, periodic cluster maintenanceand refinement, and recovery from leader failures. We discuss these in turn. 3.1 New Host Joins When a new host joins the multicast group, it must be mapped to some cluster in layer L0 . We illustrate the join procedure in Figure 5. Assume that host A12 wants to join the multicast group. First, it contacts the RP with its join query (Panel 0). The RP responds with the hosts that are present in the highest layer of the hierarchy. The joining host then contacts all members in the highest layer (Panel 1) to identify the member closest to itself. In the example, the highest layer L2 has just onemember, C0 , which by default is the closest member to A12 amongst layer L2 members. Host C0 informs A12 of the three other members (B0 ; B1 and B2 ) in its L1 cluster. A12 then contacts each of these members with the join query to identify the closest member among them (Panel 2), and iteratively uses this procedure to find its L0 cluster. It is important to note that any host, H , which belongs to any layer L i is the center of its L i\Gamma1 cluster, and recursively, is an approximation of the center among all members in all L0 clusters that are below this part of the layered hierarchy. Hence, querying each layer in succession from the top of the hierarchy to layer L0 results in a progressive refinement by the joining host to find the most appropriate layer L0 cluster to join that is close to the joining member. The outline of this operation are presented in pseudocode as Procedure BasicJoinLayer in Figure 6. We assume that all hosts are aware of only a single well-known host, the RP, from which they initiate the join process. Therefore, overheads due to join query-response messages is highest at the RP and descreasesdown the layers of the hierarchy. Under a very rapid sequenceof joins, the RP will need to handlea large number of such join query-responsemessages. Alternate andmore scalable join schemes Join L0 L2:{ C0 } Join L0 { B0,B1,B2 } Attach Figure 5: Host A12 joins the multicast group. while (j ? i) Find y s.t. dist(h; y) - dist(h; x); x; y 2 Decrement j, endwhile Join cluster Cl j Figure Basic join operation for member h, to join layer L i . new member. If i ? 0, then h is already part of layer L seeks the membership information of Cl j\Gamma1 (y) from member y. Query(RP; \Gamma) seeks the membership information of the topmost layer of the hierarchy, from the RP . are possible if we assume that the joining host is aware of some other "nearby"host that is already joined to the overlay. In fact, both Pastry [19] and Tapestry [23] alleviate a potential bottleneck at the RP for a rapid sequence of joins, based on such an assumption. 3.1.1 Join Latency The joining process involves a message overhead of O(k log N) query-response pairs. The join-latency depends on the delays incurred in this exchanges, which is typically about O(log N) round-trip times. In our protocol, we aggressively locate possible "good" peers for a joining member, and the overhead for locating the appropriate attachments for any joining member is relatively large. To reducethe delay betweena member joining the multicast group, and its receipt of the first data packet on the overlay, we allow joining members to temporarily peer, on the data path, with the leader of the cluster of the current layer it is querying. For example, in Figure 5, when A12 is querying the hosts B0 ; B1 and B2 for the closest point of attachment, it temporarily peers with C0 (leader of the layer L1 cluster) on the data path. This allows the joining host to start receiving multicast data on the group within a single round-trip latency of its join. 3.1.2 Joining Higher Layers An important invariant in the hierarchical arrangement of hosts is that the leader of a cluster be the center of the cluster. Therefore, as members join and leave clusters, the cluster-leader may occasionally change. Considera changein leadership of a cluster, C , in layer . The current leader of C removes itself from all layers L j+1 and higher to which it is attached. A new leader is chosen for each of these affected clusters. For example, a new leader, h, of C in layer L j is chosen which is now required to join its nearest L j+1 cluster. This is its current super-cluster (which by definition is the cluster in layer L j+1 to which the outgoing leader of C was joined to), i.e. the new leader replaces the outgoing leader in the super-cluster. How- ever, if the super-cluster information is stale and currently invalid, then the new leader, h, invokes the join procedure to join the nearest L j+1 cluster. It calls BasicJoinLayer(h;j and the routine terminates when the appropriate layer L j+1 cluster is found. Also note that the BasicJoinLayer requires interaction of the member h with the RP. The RP, therefore, aids in repairing the hierarchy from occasional overlay partitions, i.e. if the entire super-cluster information becomes stale in between the periodic HeartBeat messages that are exchanged between cluster members. If the RP fails, for correct operation of our protocol, we require that it be capable of recovery within a reasonable amount of time. 3.2 Cluster Maintenance and Refinement Each member H of a cluster C , sends a HeartBeat message every h seconds to each of its cluster peers (neighbors on the control topology). The messagecontains the distance estimate of H to each other member of C . It is possible for H to have inaccurate or no estimate of the distance to some other members, e.g. immediately after it joins the cluster. The cluster-leader includes the complete updated cluster membership in its HeartBeat messagesto all other members. This allows existing members to set up appropriate peer relationships with new cluster members on the control path. For each cluster in level L i , the cluster-leader also periodically sends the its immediate higher layer cluster membership(which is the super-cluster for all the other members of the cluster) to that L i cluster. All of the cluster member state is sent via unreliable messages and is kept by each cluster member as soft-state, refreshed by the periodic HeartBeat messages. A member H is declared no longer part of a cluster independently by all other members in the cluster if they do not receive a message from H for a configurable number of HeartBeat message intervals. 3.2.1 Cluster Split and Merge A cluster-leader periodically checks the size of its cluster, and appropriately splits or merges the cluster when it detects a size bound violation. A cluster that just exceeds the cluster size upper bound, into two equal-sized clusters. For correct operation of the protocol, we could have chosen the cluster size upper bound to be any value - 2k \Gamma 1. However, if waschosenas the upperbound, then the cluster would require to split when it exceeds this upper bound (i.e. reaches the size 2k). Subsequently, an equal-sized split would create two clusters of size k each. However, a single departure from any of these new clusters would violate the size lower bound and require a cluster merge operation to be performed. Choosing a larger upper bound (e.g. 3k-1) avoids this problem. When the cluster exceeds this upper bound, it is split into two clusters of size at least 3k=2, and therefore, requires at least k=2 member departures before a merge operation needs to be invoked. The cluster leader initiates this cluster split operation. Given a set of hosts and the pairwise distances between them, the cluster split operation partitions them into subsetsthat meet the size bounds,such that the maximum radius (in a graph-theoretic sense) of the new set of clusters is minimized. This is similar to the K-center problem (known to be NP-Hard) but with an additional size constraint. We use an approximation strategy - the leader splits the current cluster into two equal-sized clusters, such that the maximum of the radii among the two clusters is minimized. It also chooses the centers of the two partitions to be the leaders of the new clusters and transfers leadership to the new leaders through LeaderTransfer messages. If these new clusters still violate the size upper bound, they are split by the new leaders using identical operations. If the size of a cluster, Cl i (J) (in layer below k, the leader J , initiates a cluster merge operation. Note, J itself belongs to a layer L i+1 cluster, Cl i+1 (J). J chooses its closest cluster-peer, K , in Cl i+1(J) . K is also the leader of a layer L i cluster, initiates the merge operation of C i with by sending a ClusterMergeRequest message to K . J updates the members of Cl i (J) with this merge information. K similarly updates the members of Cl i (K). Following the merge, J removes itself from layer L i+1 (i.e. from cluster Cl i+1 (J). 3.2.2 Refining Cluster Attachments When a member is joining a layer, it may not always be able to locate the closest cluster in that layer (e.g. due to lost join query or join response, etc.) and instead attaches to some other cluster in that layer. Therefore, eachmember, H , in any layer (say L i ) periodically probes all members in its super-cluster (they are the leaders of layer L i clusters), to identify the closest member (say J ) to itself in the super-cluster. If J is not the leader of the L i cluster to which H belongs then such an inaccurate attachment is detected. In this case, H leaves its current layer L i cluster and joins the layer L i cluster of which J is the leader. 3.3 Host Departure and Leader Selection When a host H leaves the multicast group, it sends a Remove message to all clusters to which it is joined. This is a graceful-leave. However, if H fails without being able to send out this message all cluster peers of H detects this departure through non-receipt of the periodic HeartBeat message from H . If H was a leader of a clus- ter, this triggers a new leader selection in the cluster. Each remaining member, J , of the cluster independently select a new leader of the cluster, depending on who J estimates to be the center among these members. Multiple leaders are re-conciled into a single leader of the cluster through exchange of regular HeartBeat messages using an appropriate flag (LeaderTransfer) each time two candidate leaders detect this multiplicity. We present further details of these operations in [4]. It is possible for members to have an inconsistent view of the cluster membership, and for transient cycles to develop on the data path. These cycles are eliminated once the protocol restores the hierarchy invariants and reconciles the cluster view for all members. 4. EXPERIMENTAL METHODOLOGY We have analyzed the performance of the NICE protocol using detailed simulations and a wide-area implementation. In the simulation environment, we compare the performance of NICE to three other schemes: multi-unicast, native IP-multicast using the Core Based Tree protocol [2], and the Narada application-layer multicast protocol (as given in [10]). In the Internet experiments, we benchmark the performance metrics against direct unicast paths to the member hosts. Clearly, native IP multicast trees will have the least (unit) stress, since each link forwards only a single copy of each data packet. Unicast paths have the lowest latency and so we consider them to be of unit stretch 4 . They provide us a reference against which to compare the application-layer multicast protocols. 4.1 Data Model In all these experiments, we model the scenario of a data stream source multicasting to the group. We chose a single end-host, uniformly at random, to be the data source generating a constant bit rate data. Each packet in the data sequence, effectively, samples the data path on the overlay topology at that time instant, and the entire data packet sequence captures the evolution of the data path over time. 4.2 Performance Metrics We compare the performance of the different schemes along the following dimensions: ffl Quality of data path: This is measured by three different metrics - tree degree distribution, stress on links and routers and stretch of data paths to the group members. ffl Recovery from host failure: As hosts join and leave the multicast group, the underlying data delivery path adapts accordingly to reflect these changes. In our experiments, we modeled member departures from the group as ungraceful depar- tures, i.e. members fail instantly and are unable to send appropriate leave messages to their existing peers on the topol- ogy. Therefore, in transience, particularly after host failures, path to some hosts may be unavailable. It is also possible for multiple paths to exist to a single host and for cycles to develop temporarily. To study these effects, we measured the fraction of hosts that correctly receive the data packets sent from the source as the group membership changed. We also recorded the number of duplicates at each host. In all of our simulations, for both the application-layer multicast protocols, the number of duplicates was insignificant and zero in most cases. ffl Control traffic overhead: We report the mean, variance and the distribution of the control bandwidth overheads at both routers and end hosts. 5. SIMULATION EXPERIMENTS We have implemented a packet-level simulator for the four different protocols. Our network topologies were generated using the Transit-Stub graphmodel, using the GT-ITM topology generator [5]. All topologies in these simulations had 10; 000 routers with an average node degree between 3 and 4. End-hosts were attached to a set of routers, chosen uniformly at random, from among the stub- domain nodes. The number of such hosts in the multicast group were varied between 8 and 2048 for different experiments. In our simulations, we only modeled loss-less links; thus, there is no data loss due to congestion, and no notion of background traffic or jit- ter. However, data is lost whenever the application-layer multicast 4 There are some recent studies [20, 1] to show that this may not always be the case; however, we use the native unicast latency as the reference to compare the performance of the other schemes. protocol fails to provide a path from the source to a receiver, and duplicates are received whenever there is more than one path. Thus, our simulations study the dynamics of the multicast protocol and its effects on data distribution; in our implementation, the performance is also affected by other factors such as additional link latencies due to congestion and drops due to cross-traffic congestion. For comparison,we haveimplemented the entire Narada protocol from the description given in [10]. The Narada protocol is a "mesh- first" application-layer multicast approach, designed primarily for small multicast groups. In this approach the members distributedly construct a mesh which is an overlay topology where multiple paths exists between pairs of members. Each member participates in a routing protocol on this overlay mesh topology to generate source-specific trees that reach all other members. In Narada, the initial set of peer assignments to create the overlay mesh is done randomly. While this initial data delivery path may be of "poor" quality, over time Narada adds "good" links and discards "bad" links from the overlay. Narada has O(N 2 ) aggregate control overhead because of its mesh-first nature: it requires each host to periodically exchange updates and refreshes with all other hosts. The protocol, as defined in [10], has a number of user-defined parameters that we needed to set. These include the link add/drop thresholds, link add/drop probe frequency, the periodic refresh rates, the mesh degree, etc. We present detailed description of our implementation of the Narada protocol, including the impact of different choices of parameters, in [4]. 5.1 Simulation Results We havesimulated a wide-range of topologies, group sizes, member join-leave patterns, and protocol parameters. For NICE, we set the cluster size parameter, k, to 3 in all of the experiments presented here. Broadly, our findings can be summarized as follows: ffl NICE trees have data paths that have stretch comparable to Narada. ffl The stress on links and routers are lower in NICE, especially as the multicast group size increases. ffl The failure recovery of both the schemes are comparable. ffl NICE protocol demonstratesthat it is possibleto provide these performance with orders of magnitudelower control overhead for groups of size ? 32. We begin with results from a representative experiment that captures all the of different aspects comparing the various protocols. 5.1.1 Simulation Representative Scenario Thisexperiment hastwo different phases: a join phaseand a leave phase. In the join phase a set of 128 members 5 join the multicast group uniformly at random between the simulated time 0 and 200 seconds. These hosts are allowed to stabilize into an appropriate overlay topology until simulation time 1000 seconds. The leave phase starts at time 1000 seconds: hosts leave the multicast group over a short duration of 10 seconds. This is repeated four more times, at 100 second intervals. The remaining 48 members continue to be part of the multicast group until the end of simulation. All member departures are modeled as host failures since they have the most damaging effect on data paths. We experimented with different numbers of member departures, from a single member to 16 members leaving over the ten secondwindow. Sixteen departures from agroup 5 We show results for the 128 member case because that is the group size used in the experiments reported in [10]; NICE performs increasingly better with larger group sizes. of size 128 within a short time window is a drastic scenario, but it helps illustrate the failure recovery modes of the different protocols better. Member departures in smaller sizes cause correspondingly lower disruption on the data paths. We experimented with different periodic refresh rates for Narada. For a higher refresh rate the recovery from host failures is quicker, but at a cost of higher control traffic overhead. For Narada, we used different values for route update frequencies and periods for probing other mesh members to add or drop links on the overlay. In our re- sults, we report results from using route update frequencies of once every 5 seconds (labeled Narada-5), and once every seconds (la- beled Narada-30). The second update period corresponds to the what was used in [10]; we ran with the 5 second update period since the heartbeat period in NICE was set to 5 seconds. Note that we could run with a much smaller heartbeat period in NICE without significantly increasing control overheadsince the control messages are limited within clusters and do not traverse the entire group. We also varied the mesh probe period in Narada and observed data path instability effect discussedabove. In these results, we set the Narada mesh probe period to 20 seconds. Data Path Quality In Figures 7 and 8, we show the average link stress and the average path lengths for the different protocols as the data tree evolves during the member join phase. Note that the figure shows the actual path lengths to the end-hosts; the stretch is the ratio of average path length of the members of a protocol to the average path length of the members in the multi-unicast protocol. As explained earlier, the join procedure in NICE aggressivelyfinds good points of attachment for the members in the overlay topology, and the NICE tree converges quicker to a stable value (within 350 seconds of simulated time). In contrast, the Narada protocols gradually improve the mesh quality, and consequently so does the data path over a longer duration. Its average data path length converges to a stable value of about 23 hops between 500 and 600 seconds of the simulated time. The corresponding stretch is about 2.18. In Narada path lengths improve over time due to addition of "good" links on the mesh. At the same time, the stress on the tree gradually increases since the Narada decides to add or drop overlay links based purely on the stretch metric. The cluster-based data dissemination in NICE reduces average link stress, and in general, for large groups NICE converges to trees with about 25% lower average stress. In this experiment, the NICE tree had lower stretch than the Narada tree; however, in other experiments the Narada tree had a slightly lower stretch value. In gen- eral, comparing the results from multiple experimentsover different group sizes, (See Section 5.1.2), we concluded that the data path lengths to receivers were similar for both protocols. In Figures 9 and 10, we plot a cumulative distribution of the stress and path length metrics for the entire member set (128 members) at a time after the data paths have converged to a stable operating point. The distribution of stress on links for the multi-unicast scheme has a significantly large tail (e.g. links close to the source has a stress of 127). This should be contrasted with better stress distribution for both NICE and Narada. Narada uses fewer number of links on the topology than NICE, since it is comparably more aggressive in adding overlay links with shorter lengths to the mesh topology. However, due to this emphasis on shorter path lengths, the stress distribution of the links hasa heavier-tail than NICE. More than 25% of the links have a stress of four and higher in Narada, compared to 5% in NICE. The distribution of the path lengths for the two protocols are comparable. 1.92.12.3 100 200 300 400 500 600 700 800 900 Average link stress Time (in secs) 128 end-hosts joinJoin Narada-5 Figure 7: Average link stress (simulation)1525100 200 300 400 500 600 700 800 900 Average receiver path length Time (in secs) 128 end-hosts joinJoin Narada-5 IP Multicast Unicast Figure 8: Average path length Number of links Link stress Cumulative distribution of link stress after overlay stabilizes (Unicast truncated Extends to stress = 127) Narada-5 Unicast Figure 9: Stress distribution Number of hosts Overlay path length (hops) Cumulative distribution of data path lengths after overlay stabilizes Unicast IP Multicast Narada-5 Figure 10: Path length distribution (simulation) Failure Recovery and Control Overheads To investigate the effect of host failures, we present results from the second part of our scenario: starting at simulated time 1000 sec- onds, a set of 16 members leave the group over a 10 second period. We repeat this procedure four more times and no members leave after simulated time 1400 seconds when the group is reduced to 48 members. When members leave, both protocols "heal" the data distribution tree and continue to send data on the partially connected topology. In Figure 11, we show the fraction of members that correctly receive the data packets over this duration. Both Narada-5 and NICE have similar performance, and on average, both protocols restore the data path to all (remaining) receivers within onds. We also ran the same experiment with the period for Narada. The lower refresh period caused significant disruptions on the tree with periods of over 100 seconds when more than 60% of the tree did not receive any data. Lastly, we note that the data distribution tree used for NICE is the least connected topology possible; we expect failure recovery results to be much better if structures with alternate paths are built atop NICE. In Figure 12, we show the byte-overheads for control traffic at the access links of the end-hosts. Each dot in the plot represents the sum of the control traffic (in Kbps) sent or received by eachmember in the group, averaged over 10 second intervals. Thus for each 10 second time slot, there are two dots in the plot for each (remaining) host in the multicast group corresponding to the control overheads for Narada and NICE. The curves in the plot are the average control overhead for each protocol. As can be expected, for groups of size 128, NICE has an order of magnitude lower average overhead, e.g. at simulation time 1000 seconds, the average control overhead for NICE is 0.97 Kbps versus 62.05 Kbps for Narada. At the same time instant, Narada-30 (not shown in the figure) had an average control overhead of 13.43 Kbps. Note that the NICE control traffic includes all protocol messages, including messages for cluster formation, cluster splits, merges, layer promotions, and leader elections 5.1.2 Aggregate Results We present a set of aggregate results as the group size is varied. The purpose of this experiment is to understand the scalability of the different application-layer multicast protocols. The entire set of members join in the first 200 seconds, and then we run the simulation for 1800secondsto allow the topologies to stabilize. In Table 1, we compare the stress on network routers and links, the overlay path lengths to group members and the average control traffic overheads at the network routers. For each metric, we present the both mean and the standard deviation. Note, that the Narada protocol involves an aggregate control overhead of O(N 2 ), where N is the size of the group. Therefore, in our simulation setup, we were unable to simu- 0Fraction of hosts that correctly received data Time (in secs) 128 end-hosts join followed by periodic leaves in sets of 16 Leave Narada-5 Figure 11: Fraction of members that receiveddata packetsover the duration of member failures. (simulation)103050700 200 400 600 800 1000 1200 1400 1600 1800 2000 Control traffic bandwidth (Kbps) Time (in secs) Control traffic bandwidth at the access linksJoin Leave Narada-5 (Avg) Figure 12: Control bandwidth required at end-host accesslinks Group Router Stress Link Stress Path Length Bandwidth Overheads (Kbps) Size Narada-5 NICE Narada-5 NICE Narada-5 NICE Narada-30 NICE Table 1: Data path quality and control overheads for varying multicast group sizes (simulation) late Narada with groups of size 1024 or larger since the completion time for these simulations were on the order of a day for a single run of one experiment on a 550 MHz Pentium III machine with 4 GB of RAM. NaradaandNICE tend to converge to trees with similar path lengths. The stress metric for both network links and routers, however, is consistently lower for NICE when the group size is large (64 and greater). It is interesting to observe the standard deviation of stress as it changes with increasing group size for the two protocols. The standard deviation for stress increased for Narada for increasing group sizes. In contrast, the standard deviation of stress for NICE remains relatively constant; the topology-basedclustering in NICE distributes the data path more evenly among the different links on the underlying links regardless of group size. The control overhead numbers in the table are different than the ones in Figure 12; the column in the table is the average control traffic per network router as opposed to control traffic at an end- host. Since the control traffic gets aggregated inside the network, the overhead at routers is significantly higher than the overhead at an end-host. For these router overheads, we report the values of the version in which the route updatefrequency set to onds. Recall that the Narada-30 version has poor failure recovery performance, but is much more efficient (specifically 5 times less overhead with groups of size 128) than the Narada-5 version. The HeartBeat messages in NICE were still sent at 5 second intervals. For the NICE protocol, the worst case control overheads at members increase logarithmically with increase in group size. The control overheads at routers (shown in Table 1), show a similar trend. Thus, although we experimented with upto 2048 members in our simulation study, we believe that our protocol scales to even larger groups. 6. WIDE-AREA IMPLEMENTATION We have implemented the complete NICE protocol and experimented with our implementation over a one-month period, with to 100 member groups distributed across 8 different sites. Our experimental topology is shown in Figure 13. The number of members at each site was varied between 2 and different experi- ments. For example, for the member experiment reported in this section, we had 2 members each in sites B, G and H, 4 each at A, in C and 8 in D. Unfortunately, experiments with much larger groups were not feasible on our testbed. However, our implementation results for protocol overheads closely match our simulation experiments, and we believe our simulations provide a reasonable indication of how the NICE implementation would behave with larger group sizes. 6.1 Implementation Specifics We haveconductedexperiments with data sourcesat different sites. In this paper, we present a representative set of the experimentswhere the data stream source is located at site C in Figure 13. In the fig- ure, we also indicate the typical direct unicast latency (in millisec- onds) from the site C, to all the other sites. These are estimated one-way latencies obtained using a sequence of application layer (UDP) probes. Data streams were sent from the source host at site C, to all other hosts, using the NICE overlay topology. For our implementa- GHFEDCBA39.44.60.533.3 Source A: cs.ucsb.edu B: asu.edu C: cs.umd.edu F: umbc.edu G: poly.edu Figure 13: Internet experiment sites and direct unicast latencies from C tion, we experimented with different HeartBeat rates; in the results presented in this section, we set the HeartBeat message period to 10 seconds. In our implementation, we had to estimate the end-to-end latency between hosts for various protocol operations, including member joins, leadership changes, etc. We estimated the latency between two end-hosts using a low-overhead estimator that sent a sequence of application-layer (UDP) probes. We controlled the number of probes adaptively using observed variance in the latency estimates. Further, instead of using the raw latency estimates as the distance metric, we used a simple binning scheme to map the raw latencies to a small set of equivalence classes. Specifically, two latency estimates were considered equivalent if they mappedto the same equivalence class, and this resulted in faster convergence of the overlay topology. The specific latency ranges for each class were 0-1 ms, 1-5 ms, 5-10 ms, 10-20 ms, 20-40 ms, 40-100 ms, 100-200 ms and greater than 200 ms. To compute the stretch for end-hosts in the Internet experiments, we used the ratio of the latency from between the source and a host along the overlay to the direct unicast latency to that host. In the wide-area implementation, when a host A receives a data packet forwarded by memberB along the overlay tree, A immediately sends back a overlay-hop acknowledgment back to B. B logs the round-trip latency between its initial transmission of the data packet to A and the receipt of the acknowledgment from A. After the entire experiment is done, we calculated the overlay round-trip latencies for each data packet by adding up the individual overlay-hop latencies available from the logs at each host. We estimated the one-way overlay latency as half of this round trip latency. We obtained the unicast latencies using our low-overhead estimator immediately after the overlay experiment terminated. This guaranteed that the measurements of the overlay latencies and the unicast latencies did not interfere with each other. 6.2 Implementation Scenarios The Internet experiment scenarios have two phases: a join phase and a rapid membership change phase. In the join phase, a set of member hosts randomly join the group from the different sites. The hosts are then allowed to stabilize into an appropriate overlay de-0.750.850.951 Fraction of members Stress Cumulative distribution of stress members members members Figure 14: Stress distribution (testbed) livery tree. After this period, the rapid membership change phase starts, where host members randomly join and leave the group. The average member lifetime in the group, in this phase was set to seconds. Like in the simulation studies, all member departures are ungraceful and allow us to study the worst case protocol behavior. Finally, we let the remaining set of members to organize into a stable data delivery tree. We present results for three different groups of size of 32, 64, and 96 members. Data Path Quality In Figure 14, we show the cumulative distribution of the stress metric at the group members after the overlay stabilizes at the end of the join phase. For all group sizes, typical members have unit stress (74% to 83% of the members in these experiments). The stress for the remaining members vary between 3 and 9. These members are precisely the cluster leaders in the different layers (recall that the cluster size lower and upperbounds for these experiments is 3 and 9, respectively). The stress for these members can be reduced further by using the high-bandwidth data path enhancements,described in [4]. For larger groups, the number of members with higher stress (i.e. between 3 and 9 in these experiments) is more, since the number of clusters (and hence, the number of cluster leaders) is more. How- ever, as expected, this increase is only logarithmic in the group size. In Figure 15, we plot the cumulative distribution of the stretch metric. Instead of plotting the stretch value for each single host, we group them by the sites at which there are located. For all the member hosts at a given site, we plot the mean and the 95% confidence intervals. Apart from the sites C, D, and E, all the sites have near unit stretch. However, note that the source of the data streams in these experiments were located in site C and hosts in the sites C, D, and E had very low latency paths from the source host. The actual end-to-end latencies along the overlay paths to all the sites are shown in Figure 16. For the sites C, D and E these latencies were 3.5 ms, 3.5 ms and 3.0 ms respectively. Therefore, the primary contribution to these latencies are packet processing and overlay forwarding on the end-hosts themselves. In Table 2, we present the mean and the maximum stretch for the different members, that had direct unicast latency of at least 2 ms from the source (i.e. sites A, B, G and H), for all the different sizes. The mean stretch for all these sites are low. However, in some cases we do see relatively large worst case stretches (e.g. Stretch Sites Distribution of stretch (64 members) Figure 15: Stretch distribution (testbed)515253545 Overlay end-to-end latency (in Sites Distribution of latency (64 members) Figure Latency distribution (testbed) in the 96-member experiment there was one member that for which the stretch of the overlay path was 4.63). Failure Recovery In this section, we describethe effects of groupmembershipchanges on the data delivery tree. To do this, we observe how successful the overlay is in delivering data during changesto the overlay topology. We measured the number of correctly received packets by different members during the rapid membershipchangephase of the experiment, which begins after the initial member set has stabilized into the appropriate overlay topology. This phase lasts for 15 minutes. Members join and leave the grou at random such that the average lifetime of a member in the group is seconds. In Figure 17 we plot over time the fraction of members that successfully received the different data packets. A total of membership changes happened over the duration. In Figure 18 we plot the cumulative distribution of packet losses seen by the different membersover the entire 15 minute duration. Themaximum number of packet losses seen by a member was 50 out of 900 (i.e. about 5.6%), and30% of the members did not encounterany packetlosses. Even under this rapid changes to the group membership, the largest continuous duration of packet losses for any single host was 34 sec- onds, while typical members experienced a maximum continuous0.50.70.90 100 200 300 400 500 600 700 800 900 Fraction of hosts that correctly receive data Time (in secs) Distribution of losses for packets in random membership change phase members Average member lifetime = Figure 17: Fraction of members that received data packets as group membership continuously changed (testbed)0.10.30.50.70.90 Fraction of members Fraction of packets lost Cumulative distribution of losses at members in random membership change phase members Average member lifetime = Figure Cumulative distribution of fraction of packets lost for different members out of the entire sequence of 900 packets during the rapid membership change phase (testbed) data loss for only two seconds - this was true for all but 4 of the members. These failure recovery statistics are good enough for use in most data stream applications deployed over the Internet. Note that in this experiment, only three individual packets (out of 900) suffered heavy losses: data packets at times 76 s, 620 s, and 819 s were not received by 51, 36 and 31 members respectively. Control Overheads Finally, we present the control traffic overheads(in Kbps) in Table 2 for the different group sizes. The overheads include control packets that were sent as well as received. We show the average and maximum control overhead at any member. We observed that the control traffic at most members lies between 0.2 Kbps to 2.0 Kbps for the different group sizes. In fact, about80% of the members require less than 0.9 Kbps of control traffic for topology management. More in- terestingly, the average control overheads and the distributions do not change significantly as the group size is varied. The worst case control overhead is also fairly low (less than 3 Kbps). Group Stress Stretch Control overheads (Kbps) Size Mean Max. Mean Max. Mean Max. Table 2: Average and maximum values of of the different metrics for different group sizes(testbed) 7. RELATED WORK A number of other projects have explored implementing multi-cast at the application layer. They can be classified into two broad categories: mesh-first (Narada [10], Gossamer[7]) and tree-first protocols (Yoid [12], ALMI [15], Host-Multicast [22]). Yoid and Host- Multicast defines a distributed tree building protocol between the end-hosts, while ALMI uses a centralized algorithm to create a minimum spanning tree rooted at a designated single source of multi-cast data distribution. The Overcast protocol [14] organizes a set of proxies (called Overcast nodes) into a distribution tree rooted at a central source for single source multicast. A distributed tree-building protocol is used to create this source specific tree, in a manner similar to Yoid. RMX [8] provides support for reliable multicast data delivery to end-hosts using a set of similar proxies, called Reliable Multicast proXies. Application end-hosts are configured to affiliate themselves with the nearest RMX. The architecture assumes the existence of an overlay construction protocol, using which these proxies organize themselves into an appropriate data delivery path. TCP is used to provide reliable communicationbetween each pair of peer proxies on the overlay. Some other recent projects (Chord [21], Content AddressableNet- works (CAN) [17], Tapestry [23] and Pastry [19]) havealso addressed the scalability issue in creating application layer overlays, and are therefore, closely related to our work. CAN definesa virtual d-dimensional Cartesian coordinate space, and each overlay host "owns" a part of this space. In [18], the authors have leveraged the scalable structure of CAN to define an application layer multicast scheme, in which hosts maintain O(d) state and the path lengths are O(dN 1=d ) application level hops, where N is the number of hosts in the net- work. Pastry [19] is a self-organizing overlay network of nodes, where logical peer relationships on the overlay are based on matching prefixes of the node identifiers. Scribe [6] is a large-scale event notification infrastructure that leverages the Pastry system to create groups and build efficient application layer multicast paths to the group members for dissemination of events. Being based on Pas- try, it has similar overlay properties, namely at members, and O(log 2 b N) application level hops between members 6 . Bayeux[24] in another architecture for application layer mul- ticast, where the end-hosts are organized into a hierarchy as defined by the Tapestry overlay location and routing system [23]. A level of the hierarchy is defined by a set of hosts that share a common suffix in their host IDs. Such a technique was proposed by Plaxton et.al. [16] for locating and routing to named objects in a net- work. Therefore, hosts in Bayeux maintain O(b log b N) state and end-to-end overlay paths have O(log b N) application level hops. As discussed in Section 2.3, our proposed NICE protocol incurs an amortized O(k) state at members and the end-to-end paths between members have O(log k N) application level hops. Like Pastry and Tapestry, NICE also chooses overlay peers based on network locality which leads to low stretch end-to-end paths. We summarize the above as follows: For both NICE and CAN- 6 b is a small constant. multicast, members maintain constant state for other members, and consequentlyexchangea constantamount of periodic refreshes mes- sages. This overhead is logarithmic for Scribe and Bayeux. The overlay paths for NICE, Scribe, andBayeuxhavea logarithmic number of application level hops, and path lengths in CAN-multicast asymptotically havea larger number of application level hops. Both NICE and CAN-multicast use a single well-known host (the RP, in our nomenclature) to bootstrap the join procedure of members. The join procedure, therefore, incurs a higher overhead at the RP and the higher layers of the hierarchy than the lower layers. Scribe and Bayeux assume members are able find different "nearby" members on the overlay through out-of-band mechanisms,from which to bootstrap the join procedure. Using this assumption, the join overheads for a large number of joining members can be amortized over the different such "nearby" bootstrap members in these schemes. 8. CONCLUSIONS In this paper, we have presented a new protocol for application-layer multicast. Our main contribution is an extremely low overhead hierarchical control structure over which different data distribution paths canbe built. Our results show that it is possible to build and maintain application-layer multicast trees with very little over- head. While the focus of this paper has been low-bandwidth data stream applications, our scheme is generalizable to different applications by appropriately choosing data paths and metrics used to construct the overlays. We believe that the results of this paper are a significant first step towards constructing large wide-area applications over application-layer multicast. 9. ACKNOWLEDGMENTS We thank Srinivas Parthasarathy for implementing a part of the Narada protocol used in our simulation experiments. We also thank Kevin Almeroth, Lixin Gao, Jorg Liebeherr, Steven Low, Martin Reisslein and Malathi Veeraraghavan for providing us with user accounts at the different sites for our wide-area experiments and we thank Peter Druschel for shepherding the submission of the final version of this paper. 10. --R Resilient overlay networks. Based Trees (CBT): An Architecture for Scalable Multicast Routing. Scalable Secure Group Communication over IP Mulitcast. Scalable application layer multicast. How to Model an Internetwork. SCRIBE: A large-scale and decentralized application-level multicast infrastructure Scattercast: An Architecture for Internet Broadcast Distribution as an Infrastructure Service. RMX: Reliable Multicast for Heterogeneous Networks. Enabling Conferencing Applications on the Internet using an Overlay Multicast Architecture. A Case for End System Multicast. Multicast Routing in Datagram Internetworks and Extended LANs. 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Delis, Distributed middleware architectures for scalable media services, Journal of Network and Computer Applications, v.30 n.1, p.209-243, January 2007 Eli Brosh , Asaf Levin , Yuval Shavitt, Approximation and heuristic algorithms for minimum-delay application-layer multicast trees, IEEE/ACM Transactions on Networking (TON), v.15 n.2, p.473-484, April 2007 Algorithms and Trade-Offs in Multicast Service Overlay Design, Simulation, v.82 n.6, p.369-381, June 2006 Beichuan Zhang , Wenjie Wang , Sugih Jamin , Daniel Massey , Lixia Zhang, Universal IP multicast delivery, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.50 n.6, p.781-806, 13 April 2006 Mohammad S. 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Eugster , Pascal A. Felber , Rachid Guerraoui , Anne-Marie Kermarrec, The many faces of publish/subscribe, ACM Computing Surveys (CSUR), v.35 n.2, p.114-131, June K. K. To , Jack Y. B. Lee, Parallel overlays for high data-rate multicast data transfer, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.51 n.1, p.31-42, 17 January 2007 Hao Yin , Chuang Lin , Feng Qiu , Xuening Liu , Dapeng Wu, TrustStream: a novel secure and scalable media streaming architecture, Proceedings of the 13th annual ACM international conference on Multimedia, November 06-11, 2005, Hilton, Singapore Ananth Rao , Ion Stoica, An overlay MAC layer for 802.11 networks, Proceedings of the 3rd international conference on Mobile systems, applications, and services, June 06-08, 2005, Seattle, Washington H. Saito , K. Taura , T. Chikayama, Collective Operations for Wide-Area Message Passing Systems Using Adaptive Spanning Trees, Proceedings of the 6th IEEE/ACM International Workshop on Grid Computing, p.40-48, November 13-14, 2005 Yongjun Li , James Z. Wang, Cost analysis and optimization for IP multicast group management, Computer Communications, v.30 n.8, p.1721-1730, June, 2007 Suman Banerjee , Christopher Kommareddy , Koushik Kar , Bobby Bhattacharjee , Samir Khuller, OMNI: an efficient overlay multicast infrastructure for real-time applications, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.50 n.6, p.826-841, 13 April 2006 E. W. Biersack , D. Carra , R. Lo Cigno , P. Rodriguez , P. Felber, Overlay architectures for file distribution: Fundamental performance analysis for homogeneous and heterogeneous cases, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.51 n.3, p.901-917, February, 2007 Chae Y. Lee , Ho Dong Kim, Reliable overlay multicast trees for private Internet broadcasting with multiple sessions, Computers and Operations Research, v.34 n.9, p.2849-2864, September, 2007 Tetsuya Kusumoto , Yohei Kunichika , Jiro Katto , Sakae Okubo, Tree-based application layer multicast using proactive route maintenance and its implementation, Proceedings of the ACM workshop on Advances in peer-to-peer multimedia streaming, November 11-11, 2005, Hilton, Singapore Yang Guo , Kyoungwon Suh , Jim Kurose , Don Towsley, P2Cast: peer-to-peer patching scheme for VoD service, Proceedings of the 12th international conference on World Wide Web, May 20-24, 2003, Budapest, Hungary Panayotis Fouliras , Spiros Xanthos , Nikolaos Tsantalis , Athanasios Manitsaris, LEMP: Lightweight Efficient Multicast Protocol for video on demand, Proceedings of the 2004 ACM symposium on Applied computing, March 14-17, 2004, Nicosia, Cyprus Zongming Fei , Mengkun Yang, A proactive tree recovery mechanism for resilient overlay multicast, IEEE/ACM Transactions on Networking (TON), v.15 n.1, p.173-186, February 2007 Yi Cui , Baochun Li , Klara Nahrstedt, On achieving optimized capacity utilization in application overlay networks with multiple competing sessions, Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures, June 27-30, 2004, Barcelona, Spain Yair Amir , Claudiu Danilov , Stuart Goose , David Hedqvist , Andreas Terzis, 1-800-OVERLAYS: using overlay networks to improve VoIP quality, Proceedings of the international workshop on Network and operating systems support for digital audio and video, June 13-14, 2005, Stevenson, Washington, USA Jian-Guang Lou , Hua Cai , Jiang Li, Interactive multiview video delivery based on IP multicast, Advances in Multimedia, v.2007 n.1, p.13-13, January 2007 Jiantao Kong , Karsten Schwan, KStreams: kernel support for efficient data streaming in proxy servers, Proceedings of the international workshop on Network and operating systems support for digital audio and video, June 13-14, 2005, Stevenson, Washington, USA Yuval Shavitt , Tomer Tankel, Hyperbolic embedding of internet graph for distance estimation and overlay construction, IEEE/ACM Transactions on Networking (TON), v.16 n.1, p.25-36, February 2008 Praveen Rao , Justin Cappos , Varun Khare , Bongki Moon , Beichuan Zhang, Net-: unified data-centric internet services, Proceedings of the 3rd USENIX international workshop on Networking meets databases, p.1-6, April 10, 2007, Cambridge, MA Andrea Passarella , Franca Delmastro, Usability of legacy p2p multicast in multihop ad hoc networks: an experimental study, EURASIP Journal on Wireless Communications and Networking, v.2007 n.1, p.38-38, January 2007 Zhang , T. S. Eugene Ng , Animesh Nandi , Rudolf Riedi , Peter Druschel , Guohui Wang, Measurement based analysis, modeling, and synthesis of the internet delay space, Proceedings of the 6th ACM SIGCOMM on Internet measurement, October 25-27, 2006, Rio de Janeriro, Brazil Glen MacLarty , Michael Fry, Towards a platform for wide-area overlay network deployment and management, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.51 n.8, p.2144-2162, June, 2007

scalability;overlay networks;hierarchy;application layer multicast;peer-to-peer systems

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On the characteristics and origins of internet flow rates.

This paper considers the distribution of the rates at which flows transmit data, and the causes of these rates. First, using packet level traces from several Internet links, and summary flow statistics from an ISP backbone, we examine Internet flow rates and the relationship between the rate and other flow characteristics such as size and duration. We find, as have others, that while the distribution of flow rates is skewed, it is not as highly skewed as the distribution of flow sizes. We also find that for large flows the size and rate are highly correlated. Second, we attempt to determine the cause of the rates at which flows transmit data by developing a tool, T-RAT, to analyze packet-level TCP dynamics. In our traces, the most frequent causes appear to be network congestion and receiver window limits.

INTRODUCTION Researchers have investigated many aspects of Internet tra#c, including characteristics of aggregate tra#c [8, 16], the sizes of files transferred, tra#c of particular applications [4] and routing stability [7, 17], to name a few. One area that has received comparatively little attention is the rate at which applications or flows transmit data in the Inter- net. This rate can be a#ected by any of a number of fac- tors, including, for example, application limits on the rate at which data is generated, bottleneck link bandwidth, net-work congestion, the total amount of data the application Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGCOMM'02, August 19-23, 2002, Pittsburgh, Pennsylvania, USA. has to transmit, whether or not the application uses congestion control, and host bu#er limitations. An Internet link may well contain tra#c aggregated from many flows limited by di#erent factors elsewhere in the network. While each of these factors is well understood in isolation, we have very little knowledge about their prevalence and e#ect in the current Internet. In particular, we don't have a good understanding of the rates typically achieved by flows, nor are we aware of the dominant limiting factors. A better understanding of the nature and origin of flow rates in the Internet is important for several reasons. First, to understand the extent to which application performance would be improved by increased transmission rates, we must first know what is limiting their transmission rate. Flows limited by network congestion are in need of drastically different attention than flows limited by host bu#er sizes. Fur- ther, many router algorithms to control per-flow bandwidth algorithms have been proposed, and the performance and scalability of some of these algorithm depends on the nature of the flow rates seen at routers [9, 10, 14]. Thus, knowing more about these rates may inform the design of such algorithms. Finally, knowledge about the rates and their causes may lead to better models of Internet tra#c. Such models could be useful in generating simulation workloads and studying a variety of network problems. In this paper we use data from packet traces and summary flow level statistics collected on backbone routers and access links to study the characteristics and origins of flow rates in the Internet. Specifically, we examine the distribution of flow rates seen on Internet links, and investigate the relationship between flow rates and other characteristics of flows such as their size and duration. Given these macroscopic statistics, we then attempt to understand the causes behind these flow rates. We have developed a tool, called T-RAT, which analyzes traces of TCP connections and infers which causes among several possibilities limited the transmission rates of the flows. Among our significant findings are the following. First, confirming what has been observed previously, the distribution of flow rates is skewed, but not as highly skewed as flow sizes. Second, we find, somewhat surprisingly, that flow rates strongly correlated with flow sizes. This is strong evidence that user behavior, as evidenced by the amount of data they transfer, is not intrinsically determined, but rather, is a function of the speed at which files can be down- loaded. Finally, using our analysis tool on several packet traces, we find that the dominant rate limiting factors appear to be congestion and receiver window limits. We then Trace Date Length # Packets Sampled Bidirectional Access1a Jan. 16, 2001 2 hours 22 million - Yes Access1c Jan. 3, 2002 1 hour Peering1 Jan. 24, 2001 45 minutes 34 million - No Regional1a Jan. 2, 2002 1 hour 1.2 million 1 in 256 No Regional1b Jan. 3, 2002 2 hours 2.3 million 1 in 256 No Regional2 Jan. 3, 2002 2 hour 5 million 1 in 256 No Table 1: Characteristics of 8 packet traces examine the distribution of flow rates among flows in the same causal class (i.e., flows whose rate is limited by the same factor). While we believe our study is the first of its kind to examine the causes of Internet flow rates and relate these causes to other flow characteristics, it is by no means the last word in this area. This paper raises the question, but it leaves many issues unaddressed. However, the value in our work is a new tool that allows for further investigation of this problem, and an initial look at the answers it can provide. Also, while we address flow rates from a somewhat di#er- ent angle, our paper is not the first to study Internet flow rates. A preliminary look at Internet flow rates in a small number of packet traces found the distribution of rates to be skewed, but not as highly skewed as the flow size distribution [14]. This result was consistent with observation in [10] that a small number of flows accounted for a significant number of the total bytes. In recent work, Sarvotham et al [20] found that a single high rate flow usually accounts for the burstiness in aggregate tra#c. In [2], the authors look at the distribution of throughput across connections between hosts and a web server and find that the rates are often consistent with a log-normal distribution. These papers have all made important observations. In this paper, we aim to go beyond this previous work, looking at flow rates making up aggregate tra#c and attempting to understand their causes. The rest of this paper is organized as follows. In the next section we describe the data sets and methodology used in this study. In Section 3 we present various statistics concerning flow rates and related measures. We then describe our rate analyzing tool in Section 4, describe our e#orts to validate its performance in Section 5, and present results of applying it to packet traces in Section 6. We present some conclusions in Section 7. 2. DATASETS AND METHODOLOGY We used data from two sources in our study. The first set of data consisted of 8 packet traces collected over a 14 month period. The traces were collected at high speed access links connecting two sites to the Internet; a peering link between two Tier 1 providers; and two sites on a backbone network. The latter 3 traces were sampled pseudo-randomly (using a hash on the packet header fields) at a rate of 1/256. Sampling was on a per-flow basis, so that all packets from a sampled flow were captured. The packet monitors at the access links saw all tra#c going between the monitored sites and the Internet, so both directions of connections were included in the traces. For the other traces, because of asymmetric routing often only one direction of a connection is visible. The finite duration of the traces (30 minutes to 2 hours) introduces a bias against the largest and most long-lived flows. However, the e#ect of truncation on flow rates, the statistic in which we are most interested, should not be significant. The characteristics of the traces are summarized in Table 1. We supplemented the packet level traces with summary flow level statistics from 19 backbone routers in a Tier 1 provider. Data was collected for 24 hours from the 19 routers on each of 4 days between July, 2000 and November, 2001, yielding 76 sets of data. Because this data was collected concurrently from routers in the same backbone provider, a single flow can be present in more than one of the datasets. We do not know how often this occurred. However, the routers represent a relatively small fraction of the provider's routers, so we expect that each dataset contains a relatively unique set of flows. Records in these datasets contain the IP addresses of the endpoints, port numbers, higher layer protocol, the start time and end time for the flow, the total number of packets and the total number of bytes. Since these datasets lack packet level details, we cannot use them for the trace analysis in Section 4. However, they provide a useful supplement to our results in Section 3, greatly broadening the scope of the data beyond the limited number of packet traces. Each of the 4 days of summary statistics represents between 4 and 6 billion packets and between 1.5 and 2.5 terabytes of data. Flows can be defined by either their source and destination addresses, or by addresses, port numbers and protocol. The appropriateness of a definition depends in part on what one is studying. For instance, when studying router definitions that do per-flow processing, the former definition may be appropriate. When examining the characteristics of individual transport layer connections the latter is preferred. For the results reported in this paper, we used the 5-tuple of IP addresses, port numbers, and protocol number. We also generated results defining flows by source and destination IP addresses only. Those results are not qualitatively di#erent. Also, for the results presented here, we used a second timeout to decide that an idle flow has termi- nated. Repeating the tests with a 15 second timeout again did not qualitatively a#ect the results. In the analysis that follows we report on some basic per-flow statistics, including flow size, duration and rate. Size is merely the aggregate number of bytes transferred in the flow (including headers), and duration is the time elapsed between the first and last packets of a flow. Flow rate is also straightforward (size divided by duration) with the exception that determining a flow rate for very short flows is Cumulative Fraction Flow Rate (bits/sec) Peering1 Regional1a Regional1b Regional2 Figure 1: Complementary distribution of flow Cumulative Fraction Flow Size (bytes) Peering1 Regional1a Regional1b Regional2 Figure 2: Complementary distribution of flow Cumulative Fraction Flow Duration (sec) Peering1 Regional1a Regional1b Regional2 Figure 3: Complementary distribution of flow dura- tion problematic. In particular, rate is not well-defined for single packet flows whose duration by definition is zero. Similarly, flows of very short (but non-zero) duration also present a problem. It does not seem reasonable to say that a 2-packet flow that sends these packets back-to-back has an average rate equal to the line rate. In general, since we are most interested in the rate at which applications transmit data, when calculating rates we ignore flows of duration less than 100 msec, since the timing of these flows' packets may be determined as much by queueing delays inside the network as by actual transmission times at the source. 3. CHARACTERISTICS In this section we examine the characteristics of Internet flows. We begin by looking at the distributions of rate, size and duration, before turning to the question of relationships among them. Throughout, we start with data from the packet traces, and then supplement this with the summary flow data. 3.1 Rate Distribution Figure 1 plots the complementary distribution of flow rates, for flows lasting longer than 100 msec, in the 8 packet traces. The distributions show that average rates vary over several orders of magnitude. Most flows are relatively slow, with average rates less than 10kbps. However, the fastest flows in each trace transmit at rates above 1Mbps; in some traces the top speed is over 10Mbps. For comparison, we also show the complementary distributions of flow size and duration in Figures 2 and 3, respectively. The striking difference here is the longer tail evident in the distributions of flow sizes for the packet traces. One possible explanation of this di#erence is that file sizes are potentially unbounded while flow rates are constrained by link bandwidths. A previous study of rate distributions at a web server suggested that the rate distributions were well described by a log-normal distribution [2]. To test that hypothesis, we use the quantile-quantile plot (Q-Q plot) [3] to compare the flow rate distribution with analytical models. The Q-Q plot determines whether a data set has a particular theoretical distribution by plotting the quantiles of the data set against the quantiles of the theoretical distribution. If the data comes from a population with the given theoretical distri- bution, then the resulting scatter plot will be approximately a straight line. The Q-Q plots in Figures 4 and 5 compare the log of the rate distribution to the normal distribution for two of the traces (Access1c and Regional2). The fit between the two is visually good. As in Reference [2], we further assess the goodness-of-fit using the Shapiro-Wilk normality test [5]. For Access1c (Figure 4), we can not reject the null hypothesis that the log of rate comes from normal distribution at 25% significance level; for Regional2 (Figure 5), we can not reject normality at any level of significance. This suggests the fit for a normal distribution is indeed very good. Applying the Shapiro-Wilk test on all the packet traces and flow summary data, we find that for 60% of the data sets we can not reject normality at 5% significance level. These results give evidence that the flow rates can often be described with a log-normal distribution. The next question we address is how important the fast flows are. In particular, how much of the total bytes transferred are accounted for by the fastest flows? Note that a skewed rate distribution need not imply that fast flows account for a large fraction of the bytes. This will depend on the size of fast flows. Figure 6 plots the fraction of bytes accounted for in a given percentage of the fastest flows for the 8 packet traces. We see that in general, the 10% fastest flows account for between 30% and 90% of all bytes trans- ferred, and the 20% fast flows account for between 55% and 95%. This indicates that while most flows are not fast, these fast flows do account for a significant fraction of all tra#c. Figure 7 shows results for the summary flow data. This . log(Rate) -22 Figure 4: Q-Q plot for Access1c trace log(Rate) -22 Figure 5: Q-Q plot for Regional2 trace0.10.30.50.70.90 Fraction of Bytes Fraction of Fast Flows (D >= 100 msec) Peering1 Regional1a Regional1b Regional2 Figure Fraction of bytes in fastest flows0.10.30.50.70.90 Fraction of Bytes Fraction of Datasets top 20% fast flows top 10% fast flows Figure 7: Distribution of the fraction of bytes in the 10% and 20% fastest flows for summary flow data. figure plots the distribution of the percentage of bytes accounted for by the 10% and 20% fastest flows across the 76 sets of data. The skewed distributions exhibited in the traces are evident here as well. For example, in over 80% of the datasets, the fastest 10% of the flows account for at least 50% of the bytes transferred. Similarly, the fastest 20% of the flows account for over 65% of the bytes in 80% of the datasets. For comparison, the fraction of bytes in the largest flows (not shown) is even greater. We now characterize flows along two dimensions: big or small, and fast or slow. We chose 100 KByte as a cuto# on the size dimension and 10 KByte/sec on the rate dimension. These thresholds are arbitrary, but they provide a way to characterize flows in a two-by-two taxonomy. Table 2 shows the fraction of flows and bytes in each of the 4 categories for the packet traces. Flows that are small and slow, the largest group in each trace, account for between 44% and 63% of flows. However, they account for a relatively small fraction of the total bytes (10% or less.) There are also a significant number of flows in the small-fast category (between 30% and 40%) but these too represent a modest fraction of the total bytes (less than 10% in all but one trace.) On the other hand, there are a small number of flows that are both big and fast (generally less than 10%). These flows account for the bulk of the bytes transferred-at least 60% in all of the traces, and over 80% in many of them. The big-slow category is sparsely populated and these flows account for less than 10% of the bytes. Data for the 76 sets of summary flow statistics (not shown here) are generally consistent with the packet trace results. One question about Internet dynamics is the degree to which tra#c is dominated (in di#erent ways) by small flows. In terms of the number of flows, there is little doubt that the vast majority are indeed small. More than 84% of the flows in all of our traces (and over 90% in some of them) meet our (arbitrary) definition of small. However, before we conclude that the Internet is dominated by these small flows and that future designs should be geared towards dealing with them, we should remember that a very large share of the bytes are in big and fast flows. In 6 of the 8 traces we examined, these flows comprised over 80% of the bytes. Thus, when designing mechanism to control congestion or otherwise deal with tra#c arriving at a router, these big and fast flows are an important (and sometimes dominant) factor. 3.2 Correlations We next examine the relationship between the flow characteristics of interest. Table 3 shows 3 pairs of correlations- duration and rate, size and rate, and duration and size-for the 8 packet traces. We computed correlations of the log of these data because of the large range and uneven distribu- tion. We restricted the correlations to flows with durations longer than 5 seconds. Results for the other flow definitions are similar. The correlations are fairly consistent across traces, and show a negative correlation between duration and rate, a Small-Slow Small-Fast Big-Slow Big-Fast Trace flows bytes flows bytes flows bytes flows bytes Table 2: Fraction of flows and bytes in Small/Slow, Small/Fast, Big/Slow and Big/Fast flows. Trace logD,logR logS,logR logD,logS Peering1 -0.319 0.847 0.235 Regional1a -0.453 0.842 0.100 Regional1b -0.432 0.835 0.136 Regional2 -0.209 0.877 0.287 Table 3: Correlations of size, rate and duration in 8 packet traces slight positive correlation between size and duration and a strong correlation between the size and rate. The correlation between rate and size is also evident in other subsets of flows. For flows longer than 1 second, the correlations range from .65 to .77. For flows lasting longer than seconds, the correlations range from .90 to .95. Figure 8 shows CDFs of the 3 correlations taken across each of our datasets (packet traces and summary flow level statistics). This figure shows that the general trend exhibited in the packet traces was also evident in the summary flow data we examined. The most striking result here is the correlation between size and rate. If users first decided how much data they wanted to transfer (e.g., the size of a file) independent of the network conditions, and then sent it over the network, there would be little correlation between size and rate, 1 a strong correlation between size and duration, and a strongly negative correlation between rate and duration. This is not what we see; the negative correlation between rate and duration is fairly weak, the correlation between size and duration is very weak, and the correlation between size and rate is very strong. Thus, users appear to choose the size of their transfer based, strongly, on the available bandwidth. While some adjustment of user behavior was to be expected, we were surprised at the extent of the correlation between size and rate. could cause some correlation between rate and size. In order to assess the impact of slow-start on the correlations we observed, we eliminated the first 1 second of all flows and recomputed the correlations. For flows lasting longer than 5 seconds, the resulting correlations between size and rate in the 8 traces ranged from .87 to .92, eliminating slow-start as a significant cause of the correlation.0.20.61 Cumulative Fraction Correlation Coefficient corr(logD,logR) corr(logS,logR) Figure 8: CDF of correlations of size, rate and duration across all datasets 4. TCP RATE ANALYSIS TOOL In the previous section we looked at flow rates and their relationship to other flow characteristics. We now turn our attention to understanding the origins of the rates of flows we observed. We restrict our attention to TCP flows for two reasons. First, TCP is used by most tra#c in the Internet [21]. Second, the congestion and flow control mechanisms in TCP give us the opportunity to understand and explain the reasons behind the resulting transmission rates. In this section we describe a tool we built, called T-RAT (for TCP Rate Analysis Tool) that examines TCP-level dynamics in a packet trace and attempts to determine the factor that limits each flow's transmission rate. T-RAT leverages the principles underlying TCP. In partic- ular, it uses knowledge about TCP to determine the number of packets in each flight and to make a rate limit determination based on the dynamics of successive flights. However, as will become evident from the discussion below, principles alone are not su#cient to accomplish this goal. By necessity T-RAT makes use of many heuristics which through experience have been found to be useful. Before describing how T-RAT works, we first review the requirements that motivate its design. These include the range of behavior it needs to identify as well as the environment in which we want to use it. The rate at which a TCP connection transmits data can be determined by any of several factors. We characterize the possible rate limiting factors as follows: . Opportunity limited: the application has a limited amount of data to send and never leaves slow-start. This places an upper bound on how fast it can transmit data. . Congestion limited: the sender's congestion window is adjusted according to TCP's congestion control algorithm in response to detecting packet loss. . Transport limited: the sender is doing congestion avoid- ance, but doesn't experience any loss. . Receiver window limited: the sending rate is limited by the receiver's maximum advertised window. . Sender window limited: the sending rate is constrained by bu#er space at the sender, which limits the amount of unacknowledged data that can be outstanding at any time. . Bandwidth limited: the sender fully utilizes, and is limited by, the bandwidth on the bottleneck link. The sender may experience loss in this case. However, it is di#erent from congestion limited in that the sender is not competing with any other flows on the bottleneck link. An example would be a connection constrained by an access modem. . Application limited: the application does not produce data fast enough to be limited by either the transport layer or by network bandwidth. We had the following requirements in designing T-RAT. First, we do not require that an entire TCP connection, or even its beginning, be observed. This prevents any bias against long-lived flows in a trace of limited duration. Sec- ond, we would like the tool to work on traces recorded at arbitrary places in the network. Thus, the analyzer may only see one side of a connection, and it needs to work even if it was not captured near either the sender or receiver. Fi- nally, to work with large traces, our tool must work in a streaming fashion to avoid having to read the entire trace into memory. T-RAT works by grouping packets into flights and then determining a rate limiting factor based on the behavior of groups of adjacent flights. This entails three main compo- nents: (i) estimating the Maximum Segment Size (MSS) for the connection, (ii) estimating the round trip time, and (iii) analyzing the limit on the rate achieved by the connection. We now describe these components in more detail. As mentioned above, T-RAT works with either the data stream, acknowledgment stream, or both. In what follows, we identify those cases when the algorithm is by necessity di#erent for the data and the acknowledgment streams. 4.1 MSS Estimator The analysis requires that we have an estimate of the MSS for a connection. When the trace contains data packets, we set the MSS to the largest packet size observed. When the trace contains only acknowledgments, estimating the MSS is more subtle, since there need not be a 1-to-1 correspondence between data and acknowledgment packets. In this case, we estimate the MSS by looking for the most frequent common divisor. This is similar to the greatest common divisor, however, we apply heuristics to avoid looking for divisors of numbers of bytes acknowledged that are not multiples of the MSS. 4.2 Round Trip Time Estimator In this section, we present a general algorithm for estimating RTT based on packet-level TCP traces. RTT estimation is not our primary goal but, rather, a necessary component of our rate analyzer. As such, we ultimately judge the algorithm not by how accurately it estimates RTT (though we do care about that) but by whether it is good enough to allow the rate analyzer to make correct decisions. There are three basic steps to the RTT estimation algo- rithm. First, we generate a set of candidate RTTs. Then for each candidate RTT we assess how good an estimate of the actual RTT it is. We do this by grouping packets into flights based on the candidate RTT and then determining how consistent the behavior of groups of consecutive flights is with identifiable TCP behavior. Then we choose the candidate RTT that is most consistent with TCP. We expand on each of these steps below. We generate 27 candidates, 3 sec, where This covers the range of round trip times we would normally expect anywhere beyond the local network. Assume we have a stream of packets, P i , each with arrival time T i and an inter-arrival interval #P . For a candidate RTT, we group packets into flights as follows. Given the first packet, P0 , in a flight, we determine the first packet in the next flight by examining #P i for all packets with arrival times between T0 where fac is a factor to accommodate variation in the round time. We identify the packet P1 with the largest inter-arrival time in this interval. We also examine P2 , the first packet that arrives after T0 +fac -RTT . If #P2 # 2-#P1 , we choose P2 as the first packet of the next flight. Otherwise, we choose P1 . There is an obvious tradeo# in the choice of fac. We need fac to be large enough to cover the variation of RTT. However, setting fac too large will introduce too much noise, thereby reducing the accuracy of the algorithm. Currently, we set fac to 1.7, which is empirically optimal among 1.1, 1.2, ., 2.0. Once a set of flights, F i (i # 0) has been identified for a candidate RTT, we evaluate it by attempting to match its behavior to that of TCP. Specifically, we see whether the behavior of successive flights is consistent with slow-start, congestion avoidance, or response to loss. We elaborate on how we identify each of these three behaviors. Testing for Packet Loss: When the trace contains data packets, we infer packet loss by looking for retransmissions. Let seqB be the largest sequence number seen before flight F . We can conclude that F has packet loss recovery (and a prior flight experienced loss) if and only if we see at least one data packet in F with upper sequence number less than or equal to seqB . For the acknowledgment stream, we infer packet loss by looking for duplicate acknowledgments. Like TCP, we report a packet loss whenever we see three duplicate acknowledgments. In addition, if a flight has no more than 4 acknowledgment packets, we report a packet loss whenever we see a single duplicate. The latter helps to detect loss when the congestion window is small, which often leads to timeouts and significantly alters the timing characteristics. These tests are robust to packet reordering as long as it does not span flight boundaries or cause 3 duplicate acknowledgments. Testing for Congestion Avoidance: Given a flight F , define its flight size, SF , in terms of the number of MSS packets it contains: MSS where seq - F is the largest sequence number seen before F , F is the largest sequence number seen before the next flight. We define a flight's duration DF as the lag between the arrival of the first packet of F and the first packet in the subsequent flight. Testing whether four consecutive flights 2 F i are consistent with congestion avoidance requires determining whether the flight sizes, SF i , exhibit an additive increase pattern. The test is trivial when the receiver acknowledges every packet. In this case, we only need to test whether 2. The test is considerably more complex with delayed ac- knowledgments. In this case, the sizes of successive flights need not increase by 1. Because only every other packet is acknowledged, the sender's congestion window increases by 1 on average every second flight. Further, because the flight size is equal to the sender's window minus unacknowledged packets, the size of successive flights may decrease when the acknowledgment for last packet in the prior flight is delayed. Hence, sequences of flight sizes like the following are common: | {z } | {z } In our algorithm, we consider flights F i be consistent with congestion avoidance if and only if the following three conditions are met: 1. -2 # SF i - predicted predicted ) is the predicted number of segments in flight F i . 2. The flight sizes are not too small and have an overall non-decreasing pattern. Specifically, we apply the following three tests. (i) SF 3. The flight durations are not too di#erent. More specif- ically, The first condition above captures additive increase patterns with and without delayed acknowledgments. The second and third conditions are sanity checks. Testing for Slow-Start: As was the case with congestion avoidance, TCP dynamics di#er substantially during slow-start with and without delayed acknowledgments. We apply di#erent tests for each of the two cases and classify the behavior as consistent with slow-start if either test is passed. To capture slow-start behavior without delayed acknowl- edgments, we only need to test whether SF 2. The following test captures slow-start dynamics when delayed acknowledgments are used. We consider flights F i 2 As the discussion below on delayed acknowledgments in- dicates, and as confirmed by experience, four consecutive flights is the smallest number that in most cases allows us to identify the behavior correctly. to be consistent with slow-start behavior if and only if the following two conditions are met: 1. -3 # SF i - predicted predicted is the predicted number of segments in flight F i , ACKF i-1 is the estimated number of non-duplicate acknowledgment packets in flight F i-1 . (For an acknowledgment stream, can be counted directly. For a data stream, we estimate ACKF i-1 as #SF i-1 /2#.) 2. The flight sizes are not too small and have an over-all non-decreasing pattern. Specifically, we apply the following tests: (i) SF i # SF i-1 The first condition captures the behavior of slow-start with and without delayed acknowledgments. The second and third are sanity checks. Analyzing TCP Dynamics: Having described how we identify slow-start and congestion avoidance, and how we detect loss, we now present our algorithm for assessing how good a set of flights, F i , generated for a candidate RTT is. Let c be the index of the current flight to be examined. Let s be the state of the current flight: one of CA, SS or UN, for congestion avoidance, slow-start and unknown, respectively. Initially, . For a given flight, Fc , we determine the state by examining Fc , Fc+1 , Fc+2 and Fc+3 and applying the following state transitions. . - If there is loss in at least one of the 4 flights, then s transitions to UN. - If the 4 flights show additive increase behavior as described above then we remain in state CA. - Similarly, we also remain in state CA even if we don't recognize the behavior. As with TCP, we only leave CA if there is packet loss. . - If there is loss in at least one of the 4 flights, then s transitions to UN. - If the 4 flights are consistent with multiplicative increase, then s remains SS. Otherwise, s transitions to UN. Note that we can leave state SS when there is packet loss or there is some flight we do not understand. . - If there is loss in at least one flight, s remains UN. - If the four flights are consistent with the multiplicative increase behavior then s transitions to SS. - If the four flights are consistent with additive in- crease, s transitions to CA. Otherwise, s remains UN. As we analyze the set of flights, we sum up the number of packets in flights that are either in CA or SS. We assign this number as the score for a candidate RTT and select the candidate with the highest score. We have made several refinements to the algorithms described above, which we briefly mention here but do not describe further. First, when testing for slow-start or congestion avoidance behavior, if an initial test of a set of flights fails, we see whether splitting a flight into two flights or coalescing two adjacent flights yields a set of flights that matches the behavior in question. If so, we adjust the flight boundaries. Second, to accommodate variations in RTT, we continually update the RTT estimate using an exponentially weighted moving average of the durations of successive flights. Third, in cases where several candidate RTTs yield similar scores, we enhance the algorithm to disambiguate these candidates and eliminate those with very large or very small RTTs. This also allows us to reduce the number of candidate RTTs we examine. 4.3 Rate Limit Analysis Using the chosen RTT, we apply our rate limit analysis to determine the factor limiting a flow's transmission rate. Since conditions can change over the lifetime of a flow, we continually monitor the behavior of successive flights. We periodically check the number of packets seen for a flow. Every time we see 256 packets, or when no packets are seen for a 15 second interval, we make a rate limit determination. We now describe the specific tests we apply to determine the rate limiting factor. Bandwidth Limited: A flow is considered bandwidth limited if it satisfies either of the following two tests. The first is that it repeatedly achieves the same amount of data in flight prior to loss. Specifically, this is the case if: (i) there were at least 3 flights with retransmissions; and (ii) the maximum and minimum flight sizes before the loss occurs di#er by no more than the MSS. The second test classifies a flow as bandwidth limited if it sustains the link bandwidth on its bottleneck link. Rather than attempting to estimate the link bandwidth, we look for flows in which packets are nearly equally-spaced. Specif- ically, a flow is considered bandwidth limited if T hi < 2#T lo , where T lo is the 5 th percentile of the inter-packet times 3 and T hi is the P th percentile. We set #flights #packets )). P must be a function of the flight size. Other- wise, we risk classifying sender and receiver window limited flows that have large flight sizes as bandwidth limited. Congestion Limited: A flow is considered congestion limited if it experienced loss and it does not satisfy the first test for bandwidth limited. Receiver Window Limited: We can only determine that a flow is receiver window limited when the trace contains acknowledgments since they indicate the receiver's advertised window. We determine a flow to be receiver window limited if we find 3 consecutive flights F i flight sizes S i - MSS > awndmax - 3 - MSS, where awndmax is the largest receiver advertised window size. The di#er- 3 In fact, because delayed acknowledgments can cause what would otherwise be evenly spaced packets to be transmitted in bursts of 2, we cannot use the inter-packet times directly in this calculation. For data packets, instead of using the inter-arrival distribution, #P i , directly, we use ence of 3 # MSS is a heuristic that accommodates variations due to delayed acknowledgments and assumes that the MSS need not divide the advertised window evenly. Sender Window Limited: Let SF med and SF 80 be the median and the 80 th percentile of the flight sizes. A flow is considered sender window limited if the following three conditions are met. First, the flow is not receiver window limited, congestion limited, or bandwidth limited. Second, med 3. Finally, there are four consecutive flights with flight sizes between SF Opportunity Limited: A flow is deemed opportunity limited if the total number of bytes transferred is less than # MSS or if it never exits slow-start. The limit of 13 is needed because it is di#cult to recognize slow-start behavior with fewer than 13 packets. Application Limited: A flow is application limited if a packet smaller than the MSS was transmitted followed by a lull greater than the RTT, followed by additional data. Transport Limited: A flow is transport limited if the sender has entered congestion avoidance, does not experience any loss, and the flight size continues to grow. T-RAT is not able to identify unambiguously the rate limiting behaviors in all cases. Therefore, the tool reports two additional conditions. Host Window Limited: The connection is determined to be limited by either the sender window or the receiver window, but the tool cannot determine which. When acknowledgments are not present and the flow passes the sender window limited test above, it is classified as host window limited. Unknown Limited: The tool is unable to match the connection to any of the specified behaviors. 5. VALIDATION Before using T-RAT to analyze the rate limiting factors for TCP flows in our packet traces, we first validated it against measurement data as well as simulations. Specifi- cally, we compared T-RAT's round trip time estimation to estimates provided by tcpanaly [15] over the NPD N2 [18] dataset. 4 Accurate RTT estimation is a fundamental component of the tool since making a rate-limit determination is in most cases not possible without being able to group packets into flights. Once we validated the RTT estimation, we then needed to determine whether the rate analyzer returned the right answer. Validating the results against actual network tra#c is problematic. After all, that is the problem we intend to solve with this tool. Thus, we validated T-RAT against packet traces produced by network simulations and by controlled network experiments in which we could determine the specific factors that limited each flow's transmission rate. 5.1 RTT validation The NPD N2 dataset contains packet traces for over 18, 000 connections. We used 17, 248 of these in which packets were captured at both ends of the connections, so the dataset contains data and acknowledgment packets recorded at both the sender and receiver. We ran tcpanaly over this data and 4 tcpanaly requires traces of both directions of a connection. Therefore, we can use it to validate our tool using 2-way traces, but it cannot address the RTT estimation problem when only a single direction of the connection is available. Cumulative Fraction Accurate within a factor of X Data-based sender-side estimation Data-based receiver-side estimation Ack-based sender-side estimation Ack-based receiver-side estimation estimation Figure 9: RTT validation against NPD N2 data recorded for each connection the median of the RTT estimates it produced. We used these medians to compare to the performance of the RTT estimation of T-RAT. Even though the NPD data includes both directions of connections, we tested our RTT estimation using only a single direction at a time (since the algorithm is designed to work in such cases.) Hence, we consider separately the cases in which the tool sees the data packets at the sender, acknowledgment packets at the sender, data packets at the receiver and acknowledgment packets at the receiver. For each RTT estimate computed by T-RAT we measure its accuracy by comparing it to the value produced by tcpanaly. The results of the RTT validation are shown in Figure 9, which plots the CDF of the ratio between the two values for each of the 4 cases. The figure shows that with access to the data packets at either the sender or the receiver, for over 90% of the traces the estimated RTT is accurate within a factor of 1.15, and for over 95% of the traces the estimated RTT is accurate within a factor of 1.3. Accuracy of RTT estimates based on the acknowledgment stream, while still encouraging, is not as good as data stream analysis. In particular, with ack-based analysis at the receiver, 90% of estimates are accurate within a factor of 1.3 and 95% of traces are accurate within a factor of 1.6. Using the sender-side acknowledgment stream, estimates are accurate within a factor of 1.6 about 90% of the time. We suspect that delayed acknowledgments may be in part responsible for the estimation using the acknowledgment stream. By reducing the number of packets observable per RTT and perturbing the timing of some packets, they may make the job of RTT estimation more di#cult. Further, we speculate that the sender side performance with acknowledgments also su#ers because the acknowledgments at the sender have traversed an extra queue and are therefore subject to additional variation in the network delays they experience. Previous studies have used the round trip time for the initial TCP SYN-ACK handshake as an estimate of per-connection round trip time [6, 11]. We also compared this value to the median value produce by tcpanaly. As shown in Figure this estimate is significantly worse than the others. In general, the SYN-ACK handshake tends to underestimate the actual round trip time. The overall results produced by our tool are encourag- ing. They show that RTT estimation works reasonably well in most cases. The real question, however, is how the rate analyzer works. Are the errors in RTT estimation small enough to allow the tool to properly determine a rate limiting factor, or do the errors prevent accurate analysis? We now turn to the question of the validity of the rate limiting factors. 5.2 Rate Limit Validation We validated the rate limit results of T-RAT using both simulations and experiments in a controlled testbed. In our simulations, we used the ns simulator [13]. By controlling the simulated network and endpoint parameters we created TCP connections that exhibited various rate limiting behav- iors. For example, congested limited behavior was simulated using several infinite source FTP connections traversing a shared bottleneck link, and application limited behavior was simulated using long-lived Telnet sessions. Our simulations included approximately 400 connections and 340,000 pack- ets. T-RAT correctly identified the proper rate limiting behavior for over 99% of the connections. While these simulations provided positive results about the performance of T-RAT, they su#ered from several weak- nesses. First, we were not able to validate all of the rate limiting behaviors that T-RAT was designed to identify. In particular, the TCP implementation in ns does not include the advertised window in TCP packets, preventing experiments that exhibited receiver window limited behavior. Sec- ond, the simulations varied some relevant parameters, but they did not explore the parameter space in a systematic way. This left us with little knowledge about the limits of the tool. Finally, simulations abstract away many details of actual operating system and protocol performance, leaving questions about how the tool would perform on real systems. To further validate the performance of T-RAT we conducted experiments in a testbed consisting of PCs running the FreeBSD 4.3 operating system. In these experiments, two PCs acting as routers were connected by a bottleneck link. Each of these routers was also connected to a high speed LAN. Hosts on these LANs sent and received tra#c across the bottleneck link. We used the dummynet [19] facility in the FreeBSD kernel to emulate di#erent bandwidths, propagation delays and bu#er sizes on the bottleneck link. We devised a series of experiments intended to elicit various rate limiting behaviors, captured packet traces from the TCP connections in these experiments using tcpdump, analyzed these traces using T-RAT, and validated the results reported by T-RAT against the expected behavior. Unless otherwise noted, the bandwidth, propagation delay, and bu#er size on the emulated link were 1.5 Mbps, 25 msec, and KBytes, respectively. We used an MTU of 540 bytes on all interfaces, allowing us to explore a wider range of window sizes (in terms of packets) than would be a#orded with a larger MTU. For some of the rate limiting behaviors, we captured TCP connections on both unloaded and loaded links. In order to produce background load, we generated bursts of UDP traffic at exponentially distributed intervals. The burst size was varied from 1 to 4 packets across experiments, and the average inter-burst interval was generating 10%, 20%, 30% and 40% load on the link. This was not intended to model realistic tra#c. Rather the intention was to perturb the timing of the TCP packets and assess the e#ect of this perturbation on the ability of T-RAT to identify correctly the rate limiting behavior in question. Experiments were repeated with and without delayed ac- knowledgments. All TCP packets were captured at both endpoints of the connection. We tested T-RAT using only a single direction of a connection at a time (either data or acknowledgment) to emulate the more challenging scenario of only observing one direction of a connection. Thus, for each connection we made four independent assessments using data packets at the source, data packets at the destina- tion, acknowledgment packets at the source, and acknowledgment packets at the destination. For each behavior we varied parameters in order to assess how well T-RAT works under a range of conditions. Our exploration of the relevant parameter space is by no means exhaustive, but the extensive experiments we conducted give us confidence about the operation of the tool. In the vast majority of cases T-RAT correctly identified the dominant rate limiting factor. That is, for a given con- nection, the majority of periodic determinations made by T-RAT were correct. Further, for many connections, all of the periodic determinations were correct. In what follows, we summarize the experiments and their results, focusing on those cases that were most interesting or problematic. 5 Receiver Window Limited: In these experiments, the maximum advertised receiver window was varied (by adjusting the receiver's socket bu#er) for each connection, while the sender's window was larger than the bandwidth delay product of the link (and hence did not impact the sender's window.) The parameters of the bottleneck link were such that a window size of packets saturated the link. We tested window sizes between 2 and 20 packets with no background load. Even when the link was saturated, there was su#cient bu#ering to prevent packet loss. With background load, we only tested window sizes up to 10 packets to avoid loss due to congestion. A 5 MByte file was transferred for each connection. T-RAT successfully identified these connections as receiver window limited (using the acknowledgement stream) and host window limited (using the data stream) in most cases. Using the data stream, it did not correctly identify window sizes of 2 packets as receiver window limited. It is not possible to disambiguate this case from a bandwidth limited connection captured upstream of the bottleneck link when delayed acknowledgments are present. In both cases, the trace shows periodic transmission of a burst of 2 packets followed by an idle period. We would not expect receiver window limits to result in flight sizes of 2 packets, so we are not concerned about this failure mode. T-RAT was able to identify a wide range of window sizes as receiver window limited (or host window limited using data packets.) As the number of packets in flight approaches the saturation point of the link, and as a consequence the time between successive flights approaches the inter-packet time, identifying flight boundaries becomes more di#cult. When the tool had access to the data stream, it correctly identified the window limit until the link utilization approached 80%-90% of the link bandwidth. Beyond that it identified the connection as bandwidth limited. With access to the acknowledgment stream, the tool correctly identified the behavior as receiver window limited until the link was fully saturated. As we applied background tra#c to the link, the dominant cause identified for each connection was still receiver 5 More detailed information about the results is available at http://www.research.att.com/projects/T-RAT/. window limited for acknowledgement and host window limited for data packets. However, for each connection T-RAT sometimes identified a minority of the periodic determinations as transport limited when it had access to the data packets. With access to the acknowledgment packets, virtually all of the periodic determinations were receiver window limited. Thus, the advertised window information available in acknowledgments made T-RAT's job easier. Sender Window Limited: These experiments were identical to the previous ones with the exception that in this case it was the sender's maximum window that was adjusted while the receiver window was larger than the bandwidth delay product of the bottleneck link. The results were very similar to the those in the receiver window limited experiments. The tool was again unable to identify flight sizes of 2 packets as sender window limited (which in practice should not be a common occurrence.) RAT was able to identify window sizes as large as 80-90% of the link bandwidth as sender window limited. Beyond that it had trouble di#erentiating the behavior from band-width limited. Finally, as background load was applied to the link, the tool still correctly identified the most common rate limiting factor for each connection, though it sometimes confused the behavior with transport limited. Transport Limited: To test transport limited behavior, in which the connection does congestion avoidance while not experiencing loss, we set the bottleneck link bandwidth to Mbps and the one-way propagation delay to 40 msec, allowing a window size of more than 180 packets (recall we used a 540 byte MTU). In addition, we set the initial value of ssthresh to 2000 bytes, so that connections transitioned from slow-start to congestion avoidance very quickly. With no background tra#c, each connection transferred a 4 MByte file. Without delayed acknowledgments, the window size reached about 140 packets (utilizing 75% of the link) before the connection terminated. When we tested this behavior in the presence of background load, each connection transferred a 2.5 MByte file and achieved a maximum window size of approximately 100 packets (without delayed ac- knowledgments). The smaller file size was chosen to prevent packet loss during the experiments. The experiments were repeated 10 times for each set of parameters. T-RAT successfully identified transport limited as the dominant rate limiting cause for each connection. It made errors in some of the periodic determinations, with the errors becoming more prevalent as the burst size of the background tra#c increased. Whenever T-RAT was unable to determine the correct rate limiting behavior, its estimate of the RTT was incorrect. However, correct RTT estimation is not always necessary. In some cases, the tool was robust enough to overcome errors in the RTT estimation and still determine the proper rate limiting behavior. In assessing transport limited behavior, T-RAT was more successful using data packets than acknowledgment packets, particularly when delayed acknowledgments were used. In contrast to the receiver window limited case above, the acknowledgment packets provide no additional information, and by acknowledging only half of the packets, T-RAT has less information with which to work. Bandwidth Limited: In these experiments, a 10 MByte file was transferred across the bottleneck link with no competing tra#c. The router bu#er was large enough to avoid packet loss, and the sender and receiver windows were large enough to allow connections to saturate the link. We tested bottleneck link bandwidths of 500 Kbps, 1.5 Mbps, and 10 Mbps, with and without delayed acknowledgments. Each experiment was repeated 10 times. In the vast majority of cases, T-RAT properly identified the rate limiting behavior. There are two points to make about these results. First, the RTT estimation produced by the tool was often incorrect. For a connection that fully saturates a bottleneck link, and is competing with no other tra#c on that link, the resulting packet trace consists of stream of evenly spaced packets. There is, therefore, little or no timing information with which to accurately estimate RTT. Nonetheless, the test for bandwidth limiting behavior depends primarily on the distribution of inter-packet times and not on proper estimation of the flight size, so the tool still functions properly. The second observation about these experiments is that the connections were not exclusively bandwidth limited. Rather, they started in congestion avoidance (ssthresh was again set to 2000 bytes) and opened the congestion window, eventually saturating the link. The tool identified the connections as initially transport limited, and then as bandwidth limited once the bottleneck link was saturated. Visual inspection of the traces revealed that the tool made the transition at the appropriate time. In a few instances, the tool was unable to make a rate limiting determination during the single interval in which the connection transitioned states, and deemed the rate limiting behavior to be unknown. Congestion Limited: Congestion limited behavior was tested by transferring 5 MByte files across the bottleneck link with random packet loss induced by dummynet. Tests were repeated with both 2% and 5% loss on the link in a single direction and in both directions. As with our other experiments, we repeated tests with and without delayed acknowledgments, and we repeated 5 transfers in each con- figuration. 6 In nearly all cases, T-RAT identified these connections as congestion limited across all loss rates, acknowledgment strategies, and directionality of loss. For a very small number of the periodic assessments, connections were deemed transport limited. However, a connection that does not experience any loss over some interval will be in congestion avoidance mode and will be appropriately deemed transport limited. Visual inspection of a sample of these instances showed that this was indeed the case. Opportunity Limited: In these experiments, we varied the amount of data transferred by each connection from 1 to 100 packets. The connection sizes and link parameters were such that the sources never left slow-start. However, at the larger connection sizes, the congestion window was large enough to saturate the link. Hence, while the source remained in slow-start, this was not always obvious when examining packet traces. We first review the results without delayed acknowledg- 6 We also performed the more obvious experiment in which multiple TCP connections were started simultaneously with loss induced by the competing TCPs. However, an apparent bug in the version of TCP we used sometimes prevented a connection from ever opening its congestion window after experiencing packet loss. Validating these results was more di#cult since the TCP connections experienced a range of rate limiting factors (congestion, host window, transport.) Nonetheless, visual inspection of those results also indicated that the tool was properly identifying cases of congestion. ments. Using the trace of data packets at the source, T-RAT correctly identified all of the connections as opportunity lim- ited. In the other 3 traces, T-RAT identified between 83 and 88 of the connections as opportunity limited. Most of the failures occurred at connection sizes greater than 80 pack- ets, with a few occurring between 40 and packets. None occurred for connection sizes less than 40 packets. When it failed, T-RAT deemed the connections either transport or bandwidth limited. These cases are not particularly trou- bling, as the window sizes are larger than we would expect to see with regularity in actual traces. With delayed ac- knowledgments, T-RAT reached the right conclusion in 399 out of 400 cases, failing only for a single connection size with acknowledgments at the receiver. Application Limited: Characterizing and identifying application limited tra#c is perhaps more challenging than the other behaviors we study. The test T-RAT uses for application limited tra#c is based on heuristics about packet sizes and inter-packet gaps. However, there are certainly scenarios that will cause the tool to fail. For example, an application that sends constant bit rate tra#c in MSS-sized packets will likely be identified as bandwidth limited. Fur- ther, since this tra#c is by definition limited by the application our tool needs to recognize a potentially wider range of behaviors than with the other limiting factors. Understanding the range of application limited tra#c in the Internet remains a subject for future study. In our e#ort to validate the current tests for application limited tra#c in T-RAT we had the application generate application data units (ADUs) at intervals separated by a random idle times chosen from an exponential distribution. We tested connections with average idle times of 1, 2, 3, 10, 20, 30, 50 and 100 msec. Furthermore, rather than generating MSS-sized ADUs as in our other experiments, we chose the size of the ADUs from a uniform distribution between 333 and 500 bytes, the latter being the MSS in our experiments. The resulting application layer data generation rates would have been between 3.3 Mbps (1 msec average idle time) and kbps (100 msec idle time) without any network limits. In our case (1.5 Mbps bottleneck bandwidth) the highest rates would certainly run into network limits. Since we did not use MSS-sized packets, the resulting network layer tra#c depended on whether or not the TCP Nagle algorithm [12], which coalesces smaller ADUs into MSS-sized packets, is employed. Hence, in addition to repeating experiments with and without delayed acknowledgments, we also repeated the experiments with and without the Nagle algorithm turned on. Assessing the results of these experiments was di#cult. Given that we used a stochastic data generation process, and that one cannot know a priori how this random process will interact with the transport layer, we could not know what the resulting network tra#c would look like. Without a detailed packet-by-packet examination, the best we can do is to make qualitative characterizations about the results. With the Nagle algorithm turned on, T-RAT characterized the two fastest data generation rates (3.3 Mbps and 1.65 Mbps) as a combination of congestion and bandwidth limited. This is what one would expect given the bottleneck link bandwidth. At the lowest data rates (33 Kbps and 66 Kbps) T-RAT deemed the tra#c to be application limited. This is again consistent with intuition. In between, (from 110 Kbps to 1.1 Mbps) the tra#c was characterized vari- 0Access1a Access1b Access1c Access2 Peering1 Regional1a Regional1b Regional2 Percentage of Bytes Congestion Host/Sndr/Rcvr Opportunity Application Transport Unknown Figure 10: Fraction of bytes for each rate limiting factor1030507090 Percentage of Flows Congestion Host/Sndr/Rcvr Opportunity Application Transport Unknown Figure 11: Fraction of flows for each rate limiting factor ously as transport, host window, application, or unknown limited. With the Nagle algorithm disabled, the fastest generation rates were again characterized as congestion and bandwidth limited. At all the lower rates (1.1 Mbps down to 33 Kbps), T-RAT deemed the connections as exclusively application limited when using the data stream, and a combination of application and transport limited when using the acknowledgment stream. Thus, application limited behavior is easier to discern when the Nagle algorithm is turned o#. 6. RESULTS The results of applying T-RAT to the 8 packet traces are shown in Figure 10. For each trace, the plot shows the percentage of bytes limited by each factor. The 4 traces taken from access links are able to di#erentiate between sender and receiver limited flows since they see data and acknowledgment packets for all connections. The peering and regional traces, on the other hand, often only see one direction of a connection and are therefore not always able to di#erentiate between these two causes. We have aggregated the 3 categories identified by T-RAT-sender, receiver and host window limited-into a single category labeled "Host/Sndr/Rcvr" limited in the graph. As shown in Figure 10, the most common rate limiting factor is congestion. It accounts for between 22% and 43% of the bytes in each trace, and is either the first or second most frequent cause in each trace. The aggregate category that includes sender, receiver and host window limited is the second most common cause of rate limits accounting for between 8% and 48% of the bytes across traces. When we were able to make a distinction between sender and receiver window limited flow (i.e., when the trace captured the acknowledgment stream), receiver window limited was a much more prevalent cause than sender window limited, by ratios between 2:1 and 10:1. Other causes-opportunity Cumulative Fraction Flow Rate (Byte/sec) opportunity limited application limited congestion limited transport limited receiver limited Figure 12: Rate distribution by rate limiting fac- tor, Access1b trace0.20.61 Cumulative Fraction Flow Size (bytes) opportunity limited application limited congestion limited transport limited receiver limited Figure 13: Size distribution by rate limiting fac- tor, Access1b trace0.20.61 Cumulative Fraction Flow Duration (sec) opportunity limited application limited congestion limited transport limited receiver limited Figure 14: Duration distribution by rate limiting trace limited, application limited and transport limited-usually accounted for less than 20% of the bytes. Bandwidth limited flows accounted for less than 2% of the bytes in all traces (and are not shown in the plot). For most traces, the unknown category accounted for a very small percentage of the bytes. We examined the 2 traces in which the rate limiting cause for more than 5% of the bytes was unknown and identified 3 factors that prevented the tool from making a rate limiting determination. First, T-RAT cannot accurately estimate round trips on the order of 3 msec or less, and therefore, cannot determine a rate limiting factor for these connections. Second, when the traces were missing packets in the middle of a connection (which may have resulted either from loss at the packet filter or from multi-path routing) estimating the round trip time and the rate limiting cause becomes di#cult. Finally, multiple web transfers across a persistent TCP connection also presented problems. When one HTTP transfer uses the congestion window remaining from the end of the previous transfer a moderate size file may not be opportunity limited (because it is larger than 13 packets and it never enters slow-start) and it may not have enough flights (because the initial flight size is large) for T-RAT to make a rate limit determination. Not surprisingly, when we look at rate limiting factors by flows rather than bytes, the results are very di#erent. Recall that we continuously update the reason a flow is limited, and a single flow may have multiple limiting factors throughout its lifetime. For example, it may be congestion limited for one interval, and after congestion dissipates, become window limited. In those cases when a flow experienced multiple causes, we classified it by the factor that most often limited its transmission rate. Figure 11 shows the percentage of flows constrained by each rate limiting factor for the 8 traces. The most common per-flow factors are opportunity and application limited. Collectively, they account for over 90% of the flows in each of the 8 traces, with opportunity limited accounting for more than 60% and application limited accounting for between 11% and 34%. No other cause accounted for more than 4% of the flows in any trace. These results are consistent with the results reported in Section 3. Namely, most flows are small and slow. Small flows are likely opportunity limited (they don't have enough packets to test bu#er or network limits), and slow flows are likely application limited (not sending fast enough to test bu#er or network limits.) A general trend is evident when comparing the traces taken at access links to those taken at the peering and regional links. The former tend to have a higher percentage of bytes that are window limited. The access links are high speed links connecting a site to the Internet. As such, they support a population with good connectivity to the Internet. The other links are likely seeing a more general cross section of Internet users, some of whom are well-connected and others of whom are not. Since window limits are reached when the bandwidth delay product exceeds bu#er resources, the well-connected users are more likely to reach these limits. This di#erence between the two kinds of traces was evident in Figure 1. That graph shows that the distribution of rates has a longer tail for the access links than for the regional and peering links. We next ask whether these di#erent rate limiting factors can be associated with di#erent performance for users. Figure 12 plots the CDF of the rates for each of the rate limiting factors for the Access1b trace. The graph shows very distinct di#erences between subgroups. Overall, receiver limited and transport limited flows have the largest average rates, followed by congestion limited, application limited and opportunity limited. This same trend was exhibited across the other 7 traces. Figures 13 and 14 plot the distributions of size and duration for each rate limiting factor in the Access1b trace. Receiver limited flows have the largest size distribution, followed by transport and congestion lim- ited. In the duration distribution, congestion limited flows have the longest duration, which is consistent with the observation that flows experiencing congestion will take longer to transmit their data than flows not experiencing congestion 7. CONCLUSION The rates at which flows transmit data is an important and not well-understood phenomenon. The rate of a flow can have a major impact on user experience, and the rates of flows traversing the Internet can have a significant e#ect on network control algorithms. We had two goals in this paper. First, we wanted to better understand the characteristics of flow rates in the Internet. Using packet traces and summary flow statistics we examined the rates of flows and the relationship between flow rates and other flow char- acteristics. We found that fast flows are responsible for most of the bytes transmitted in the Internet, so understanding their behavior is important. We also found a strong correlation between flow rate and size, suggesting an interaction between bandwidth available to a user and what the user does with that bandwidth. Our second goal was to provide an explanation of the reasons why flows transmit at the rates they do. This was an ambitious goal, and we have by no means completely answered this question. We have seen, for a set of Internet packet traces, the reasons that flows are limited in their transmission rates and have looked at di#erences among different categories of flows. We believe our main contribution, however, is to open up an area of investigation that can lead to valuable future research. The tool we developed to study rate limiting behavior provides a level of analysis of TCP connections that can answer previously unanswerable ques- tions. Thus, our tool has applicability beyond the set of results we have obtained with it thus far. Acknowledgments We would like to thank Rui Zhang for her work with us on flow characterization that helped launch this project. We would also like to express our thanks to the anonymous reviewers for many helpful comments on this paper. 8. --R "A Web Server's View of the Transport Layer," "Analyzing Stability in Wide-Area Network Performance," "Graphical Methods for Data Analysis," "Self-similarity in World Wide Web Tra#c: Evidence and Possible Causes," "Goodness-of-Fit Techniques," "Passive Estimation of TCP Round-Trip Times," "Internet Routing Instability," "On the Self-Similar Nature of Ethernet Tra#c (Extended Version)," "Dynamics of Random Early Detection," "Controlling High-Bandwidth Flows at the Congested Router," "Analysis of Internet Delay Times," "Congestion Control in IP/TCP Internetworks" "Approximate Fairness through Di#erential Dropping," "Automated Packet Trace Analysis of TCP Implementations," "Wide-Area Tra#c: The Failure of Poisson Modeling," "End-to-End Routing Behavior in the Internet," "End-to-End Internet Packet Dynamics," "Dummynet: A Simple Approach to the Evaluation of Network Protocols" "Connection-level Analysis and Modeling of Network Tra#c," "Wide Area Internet Tra#c Patterns and Characteristics," --TR area traffic Dummynet Automated packet trace analysis of TCP implementations End-to-end routing behavior in the Internet Self-similarity in World Wide Web traffic Internet routing instability End-to-end internet packet dynamics Connection-level analysis and modeling of network traffic A web server''s view of the transport layer Passive estimation of TCP round-trip times Controlling High-Bandwidth Flows at the Congested Router --CTR Jrg Wallerich , Holger Dreger , Anja Feldmann , Balachander Krishnamurthy , Walter Willinger, A methodology for studying persistency aspects of internet flows, ACM SIGCOMM Computer Communication Review, v.35 n.2, April 2005 David Watson , G. 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Kortebi , L. Muscariello , S. Oueslati , J. Roberts, Evaluating the number of active flows in a scheduler realizing fair statistical bandwidth sharing, ACM SIGMETRICS Performance Evaluation Review, v.33 n.1, June 2005 Tatsuya Mori , Masato Uchida , Ryoichi Kawahara , Jianping Pan , Shigeki Goto, Identifying elephant flows through periodically sampled packets, Proceedings of the 4th ACM SIGCOMM conference on Internet measurement, October 25-27, 2004, Taormina, Sicily, Italy Lukas Kencl , Christian Schwarzer, Traffic-Adaptive Packet Filtering of Denial of Service Attacks, Proceedings of the 2006 International Symposium on on World of Wireless, Mobile and Multimedia Networks, p.485-489, June 26-29, 2006 M. Siekkinen , G. Urvoy-Keller , E. W. Biersack , T. En-Najjary, Root cause analysis for long-lived TCP connections, Proceedings of the 2005 ACM conference on Emerging network experiment and technology, October 24-27, 2005, Toulouse, France Guohan Lu , Xing Li, On the correspondency between TCP acknowledgment packet and data packet, Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, October 27-29, 2003, Miami Beach, FL, USA Arifler , Gustavo de Veciana , Brian L. Evans, A factor analytic approach to inferring congestion sharing based on flow level measurements, IEEE/ACM Transactions on Networking (TON), v.15 n.1, p.67-79, February 2007 Abdesselem Kortebi , Luca Muscariello , Sara Oueslati , James Roberts, Minimizing the overhead in implementing flow-aware networking, Proceedings of the 2005 symposium on Architecture for networking and communications systems, October 26-28, 2005, Princeton, NJ, USA Salvatore Gaglio , Luca Gatani , Giuseppe Re , Alfonso Urso, A Logical Architecture for Active Network Management, Journal of Network and Systems Management, v.14 n.1, p.127-146, March 2006 Allen B. 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Chase, Circus: Opportunistic Block Reordering for Scalable Content Servers, Proceedings of the 3rd USENIX Conference on File and Storage Technologies, March 31-31, 2004, San Francisco, CA Rob Sherwood , Neil Spring, Touring the internet in a TCP sidecar, Proceedings of the 6th ACM SIGCOMM on Internet measurement, October 25-27, 2006, Rio de Janeriro, Brazil Kun-chan Lan , John Heidemann, A measurement study of correlations of internet flow characteristics, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.50 n.1, p.46-62, January 2006 Srikanth Kandula , Dina Katabi , Shantanu Sinha , Arthur Berger, Dynamic load balancing without packet reordering, ACM SIGCOMM Computer Communication Review, v.37 n.2, April 2007 Aditya Akella , Srinivasan Seshan , Anees Shaikh, An empirical evaluation of wide-area internet bottlenecks, Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, October 27-29, 2003, Miami Beach, FL, USA Jayanthkumar Kannan , Jaeyeon Jung , Vern Paxson , Can Emre Koksal, Semi-automated discovery of application session structure, Proceedings of the 6th ACM SIGCOMM on Internet measurement, October 25-27, 2006, Rio de Janeriro, Brazil Atul Adya , Paramvir Bahl , Ranveer Chandra , Lili Qiu, Architecture and techniques for diagnosing faults in IEEE 802.11 infrastructure networks, Proceedings of the 10th annual international conference on Mobile computing and networking, September 26-October 01, 2004, Philadelphia, PA, USA Lukas Kencl , Jean-Yves Le Boudec, Adaptive load sharing for network processors, IEEE/ACM Transactions on Networking (TON), v.16 n.2, p.293-306, April 2008 Kashi Venkatesh Vishwanath , Amin Vahdat, Realistic and responsive network traffic generation, ACM SIGCOMM Computer Communication Review, v.36 n.4, October 2006 Eddie Kohler , Mark Handley , Sally Floyd, Designing DCCP: congestion control without reliability, ACM SIGCOMM Computer Communication Review, v.36 n.4, October 2006 Mahajan , Neil Spring , David Wetherall , Thomas Anderson, User-level internet path diagnosis, Proceedings of the nineteenth ACM symposium on Operating systems principles, October 19-22, 2003, Bolton Landing, NY, USA

flow rates;TCP;network measurement

633558

Quantum communication and complexity.

In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications.

Introduction The area of communication complexity deals with the following type of prob- lem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their goal. In this model, individual computation is free, but communication is expensive and has to be minimized. How many bits do they need to communicate between them in order to solve this? Clearly, Alice can just send her complete input to Bob, but sometimes more efficient schemes are possible. This model was introduced by Yao [52] and has been studied extensively, both for its applications (like lower bounds on VLSI and circuits) and for its own sake. We refer to [38,32] for definitions and results. An interesting variant of the above is quantum communication complexity: suppose that Alice and Bob each have a quantum computer at their disposal and are allowed to exchange quantum bits (qubits) and/or to make use Partially supported by the EU fifth framework project QAIP, IST-1999-11234. Preprint submitted to Elsevier Preprint 21 October 2000 of the quantum correlations given by shared EPR-pairs (entangled pairs of qubits named after Einstein, Podolsky, and Rosen [27]). Can Alice and Bob now compute f with less communication than in the classical case? Quantum communication complexity was first considered by Yao [53] for the model with qubit communication and no prior EPR-pairs, and it was shown later that for some problems the amount of communication required in the quantum world is indeed considerably less than the amount of classical communication. In this survey, we first give brief explanations of quantum computation and communication, and then cover the main results of quantum communication complexity: upper bounds (Section 5), lower bounds (Section 6), and applications (Section 7). We include proofs of some of the central results and references to others. Some other recent surveys of quantum communication complexity are [48,15,35], and a more popular account can be found in [47]. Our survey differs from these in being a bit more extensive and up to date. Quantum Computation In this section we briefly give the relevant background from quantum compu- tation, referring to the book of Nielsen and Chuang [44] for more details. 2.1 States and operations The classical unit of computation is a bit, which can take on the values 0 or 1. In the quantum case, the unit of computation is a qubit which is a linear combination or superposition of the two classical values: More generally, an m-qubit state jOEi is a superposition of all 2 m different classical m-bit strings: The classical state jii is called a basis state. The coefficient ff i is a complex number, which is called the amplitude of jii. The amplitudes form a dimensional complex vector, which we require to have norm 1 (i.e. 1). If some system is in state jOEi and some other is in state j/i, then their joint state is the tensor product We can basically do two things to a quantum state: measure it or perform a unitary operation to it. If we measure jOEi, then we will see a basis state; we will see jii with probability jff . Since the numbers jff induce a probability distribution on the set of basis states they must sum to 1, which they indeed do because jOEi has norm 1. A measurement "collapses" the measured state to the measurement outcome: if we see jii, then jOEi has collapsed to jii, and all other information in jOEi is gone. Apart from measuring, we can also transform the state, i.e., change the am- plitudes. Quantum mechanics stipulates that this transformation U must be a linear transformation on the 2 m -dimensional vector of amplitudes: ff 0:::0 Since the new vector of amplitudes fi i must also have norm 1, it follows that the linear transformation U must be norm-preserving and hence unitary. 2 This in turn implies that U has an inverse (in fact equal to its conjugate transpose U ), hence non-measuring quantum operations are reversible. 2.2 Quantum algorithms We describe quantum algorithms in the quantum circuit model [25,53], rather than the somewhat more cumbersome quantum Turing machine model [24,12]. A classical Boolean circuit is a directed acyclic graph of elementary Boolean gates (usually AND, OR, and NOT), only acting on one or two bits at a time. It transforms an initial vector of bits (containing the input) into the output. A quantum circuit is similar, except that the classical Boolean gates now become elementary quantum gates. Such a gate is a unitary transformation acting only on one or two qubits, and implicitly acting as the identity on the other qubits of the state. A simple example of a 1-qubit gate is the Hadamard transform, which maps basis state jbi to 1 In matrix form, this is An example of a 2-qubit gate is the controlled-NOT (CNOT) gate, which negates the second bit of the state depending on the first bit: jc; bi ! jc; b \Phi ci. Both quantum measurements and quantum operations allow for a somewhat more general description than given here (POVMs and superoperators, respectively, see [44]), but the above definitions suffice for our purposes. In matrix form, this is It is known that the set of gates consisting of CNOT and all 1-qubit gates is universal, meaning that any other unitary transformation can be written as a product of gates from this set. We refer to [4,44] for more details. The product of all elementary gates in a quantum circuit is a big unitary transformation which transforms the initial state (usually a classical bitstring containing the input x) into a final superposition. The output of the circuit is then the outcome of measuring some dedicated part of the final state. We say that a quantum circuit computes some function f : f0; 1g n ! Z exactly if it always outputs the right value f(x) on input x. The circuit computes f with bounded error if it outputs f(x) with probability at least 2=3, for all x. Notice that a quantum circuit involves only one measurement; this is without loss of generality, since it is known that measurements can always be pushed to the end at the cost of a moderate amount of extra memory. The complexity of a quantum circuit is usually measured by the number of elementary gates it contains. A circuit is deemed efficient if its complexity is at most polynomial in the length n of the input. The most spectacular instance of an efficient quantum circuit (rather, a uniform family of such circuits, one for each n) is still Shor's 1994 efficient algorithm for finding factors of large integers. It finds a factor of arbitrary n-bit numbers with high probability using only n 2 polylog(n) elementary gates. This compromises the security of modern public-key cryptographic systems like RSA, which are based on the assumed hardness of factoring. 2.3 Query algorithms A type of quantum algorithms that we will refer to later are the query algo- rithms. In fact, most existing quantum algorithms are of this type. Here the input is not part of the initial state, but encoded in a special "black box" quantum gate. The black box maps basis state ji; bi to ji; b \Phi x i i, thus giving access to the bits x i of the input. Note that a quantum algorithm can run the black box on a superposition of basis states, gaining access to several input bits x i at the same time. One such application of the black box is called a query. The complexity of a quantum circuit for computing some function f is now the number of queries we need on the worst-case input; we don't count the complexity of other operations in this model. In the classical world, this query complexity is known as the decision tree complexity of f . A simple but illustrative example is the Deutsch-Jozsa algorithm [26,23]: suppose we get the promise that the input x 2 f0; 1g n is either or has exactly n=2 0s and n=2 1s. Define in the first case and in the second. It is easy to see that a deterministic classical computer needs queries for this (if the computer has queried n=2 bits and they are all 0, the function value is still undetermined). On the other hand, here is a 1-query quantum algorithm for this problem: (1) Start in a basis state zeroes followed by a 1 (2) Apply a Hadamard transform to each of the (3) Query the black box once Apply a Hadamard transform to the first n qubits (5) Measure the first n qubits, output 1 if the observed state is output 0 otherwise By following the state through these steps, it may be verified that the algorithm outputs 1 if the input is the input is balanced. Another important quantum query algorithm is Grover's search algorithm [30], which finds an i such that x such an i exists in the n-bit input. It has error probability 1=3 on each input and uses O( n) queries, which is optimal [10,13,54]. Note that the algorithm can also be viewed as computing the OR-function: it can determine whether at least one of the input bits is 1. 3 Quantum Communication The area of quantum information theory deals with the properties of quantum information and its communication between different parties. We refer to [11,44] for general surveys, and will here restrict ourselves to explaining two important primitives: teleportation [8] and superdense coding [9]. These pre-date quantum communication complexity and show some of the power of quantum communication. We first show how teleporting a qubit works. Alice has a qubit ff 0 that she wants to send to Bob via a classical channel. Without further resources this would be impossible, but Alice also shares an EPR-pair 1 j11i) with Bob. Initially, their joint state is The first two qubits belong to Alice, the third to Bob. Alice performs a CNOT on her two qubits and then a Hadamard transform on her first qubit. Their joint state can now be written as2 Alice -z Alice then measures her two qubits and sends the result (2 random classical bits) to Bob, who now knows which transformation he must do on his qubit in order to regain the qubit ff 0 j1i. For instance, if Alice sent 11 then Bob knows that his qubit is ff 0 j0i. A bit-flip (jbi ! j1 \Gamma bi) followed by a phase-flip (jbi ! (\Gamma1) b jbi) will give him Alice's original qubit ff 0 j1i. Note that the qubit on Alice's side has been destroyed: teleporting moves a qubit from A to B, rather than copying it. In fact, copying an unknown qubit is impossible [50], which can be seen as follows. Suppose C were a 1-qubit copier, i.e. for every qubit jOEi. In particular would not copy p(j0i+j1i) correctly, since by linearity In teleportation, Alice uses 2 classical bits and 1 EPR-pair to send 1 qubit to Bob. Superdense coding achieves the opposite: using 1 qubit and 1 EPR-pair, Alice can send 2 classical bits b 2 to Bob. It works as follows. Initially they share an EPR-pair 1 Alice applies a phase- flip to her half of the pair. Second, if b 2 applies a bit-flip. Third, she sends her half of the EPR-pair to Bob, who now has one of 4 states jOE b2 b1 i: Since these states are orthogonal, Bob can apply a unitary transformation i and thus learns b 2 Suppose Alice wants to send n classical bits of information to Bob and they do not share any prior entanglement. Alice can just send her n bits to Bob, but, alternatively, Bob can also first send n=2 halves of EPR-pairs to Alice and then Alice can send n bits in n=2 qubits using dense coding. In either case, n qubits are exchanged between them. If Alice and Bob already share n=2 prior EPR-pairs, then n=2 qubits suffice by superdense coding. The following result shows that this is optimal. We will refer to it as Holevo's theorem, because the first part is an immediate consequence of a result of [31] (the second part was derived in [22]). Theorem 1 (Holevo) If Alice wants to send n bits of information to Bob via a qubit channel, and they don't share prior entanglement, then they have to exchange at least n qubits. If they do share unlimited prior entanglement, then Alice has to send at least n=2 qubits to Bob, no matter how many qubits Bob sends to Alice. A somewhat stronger and more subtle variant of this lower bound was recently derived by Nayak [40], improving upon [1]. Suppose that Alice doesn't want to send Bob all of her n bits, but just wants to send a message which allows Bob to learn one of her bits x i , where Bob can choose i after the message has been sent. Even for this weaker form of communication, Alice has to send an n)-qubit message. 4 Quantum Communication Complexity: The Model First we sketch the setting for classical communication complexity, referring to [38,32] for more details. Alice and Bob want to compute some function . If the domain D equals X \Theta Y then f is called a total function, otherwise it is a promise function. Alice receives input x 2 X, Bob receives input y 2 Y , with (x; y) 2 D. As the value f(x; y) will generally depend on both x and y, some communication between Alice and Bob is required in order for them to be able to compute f(x; y). We are interested in the minimal amount of communication they need. A communication protocol is a distributed algorithm where first Alice does some individual computation, and then sends a message (of one or more bits) to Bob, then Bob does some computation and sends a message to Alice, etc. Each message is called a round. After one or more rounds the protocol terminates and outputs some value, which must be known to both players. The cost of a protocol is the total number of bits communicated on the worst-case input. A deterministic protocol for f always has to output the right value f(x; y) for all (x; y) 2 D. In a bounded-error protocol, Alice and Bob may flip coins and the protocol has to output the right value f(x; y) with probability 2=3 for all (x; y) 2 D. We use D(f) and R 2 (f) to denote the minimal cost of deterministic and bounded-error protocols for f , respectively. The subscript '2' in R 2 (f) stands for 2-sided bounded error. For R 2 (f) we can either allow Alice and Bob to toss coins individually (private coin) or jointly (public coin). This makes not much difference: a public coin can save at most O(log n) bits of communication [42], compared to a protocol with a private coin. Some often studied total functions where ffl Equality: EQ(x; ffl Inner product: IP(x; (for is the ith bit of x and x " y 2 f0; 1g n is the bit-wise AND of x and y) ffl Disjointness: DISJ(x; This function is 1 iff there is no i (viewing x and y as characteristic vectors of sets, the sets are disjoint) It is known that n). However, R 2 (EQ) is only O(1), as follows. Alice and Bob jointly toss a random string r 2 f0; 1g n . Alice sends the bit a to Bob (where '\Delta' is inner product mod 2). Bob computes and compares this with a. If then a 6= b with probability 1/2. Thus Alice and Bob can decide equality with small error using O(n) public coin flips and O(1) communication. Since public coin and private coin protocols are close, this also implies that R 2 n) with a private coin. Now what happens if we give Alice and Bob a quantum computer and allow them to send each other qubits and/or to make use of EPR-pairs which they share at the start of the protocol? Formally speaking, we can model a quantum protocol as follows. The total state consists of 3 parts: Alice's private space, the channel, and Bob's private space. The starting state is jxij0ijyi: Alice gets x, the channel is initially empty, and Bob gets y. Now Alice applies a unitary transformation to her space and the channel. This corresponds to her private computation as well as to putting a message on the channel (the length of this message is the number of channel-qubits affected by Alice's operation). Then Bob applies a unitary transformation to his space and the channel, etc. At the end of the protocol Alice or Bob makes a measurement to determine the output of the protocol. We use Q(f) to denote the minimal communication cost of a quantum protocol that computes f(x; y) exactly (= with error probability 0). This model was introduced by Yao [53]. In the second model, introduced by Cleve and Buhrman [21], Alice and Bob share an unlimited number of EPR-pairs at the start of the protocol, but now they communicate via a classical channel: the channel has to be in a classical state throughout the protocol. We use C (f) for the minimal complexity of an exact protocol for f in this model. Note that we only count the communication, not the number of EPR-pairs used. The third variant combines the strengths of the other two: here Alice and Bob start out with an unlimited number of shared EPR-pairs and they are allowed to communicate qubits. We use Q (f) to denote the communication complexity in this third model. By teleportation, 1 EPR-pair and 2 classical bits can replace 1 qubit of communication, so we have bounded-error quantum protocols. Note that a shared EPR-pair can simulate a public coin toss: if Alice and Bob each measure their half of the pair, they get the same random bit. Before continuing to study this model, we first have to face an important ques- tion: is there anything to be gained here? At first sight, the following argument seems to rule out any significant gain. By definition, in the classical world D(f) bits have to be communicated in order to compute f . Since Holevo's theorem says that k qubits cannot contain more information than k classical bits, it seems that the quantum communication complexity should be roughly D(f) qubits as well (maybe D(f)=2 to account for superdense coding, but not less). Fortunately and surprisingly, this argument is false, and quantum communication can sometimes be much less than classical communication complexity. The information-theoretic argument via Holevo's theorem fails, because Alice and Bob do not need to communicate the information in the D(f) bits of the classical protocol; they are only interested in the value f(x; y), which is just bit. Below we will survey the main examples that have so far been found of gaps between quantum and classical communication complexity. 5 Quantum Communication Complexity: Upper bounds 5.1 Initial steps Quantum communication complexity was introduced by Yao [53] and studied by Kremer [37], but neither showed any advantages of quantum over classical communication. Cleve and Buhrman [21] introduced the variant with classical communication and shared EPR-pairs, and exhibited the first quantum protocol provably better than any classical protocol. It uses quantum entanglement to save 1 bit of classical communication. This gap was extended by Buhrman, Cleve, and van Dam [16] and, for more than 2 parties, by Buhrman, van Dam, Hyer, and Tapp [19]. 5.2 Buhrman, Cleve, Wigderson The first impressively large gaps between quantum and classical communication complexity were exhibited by Buhrman, Cleve, and Wigderson [17]. Their protocols are distributed versions of known quantum query algorithms, like the Deutsch-Jozsa and Grover algorithms. The following lemma shows how a query algorithm induces a communication protocol: binary connective (for instance \Phi or "). If there is a T -query quantum algorithm for g, then there is a protocol for f that communicates T (2 log n+4) qubits (and uses no prior entanglement) and that has the same error probability as the query algorithm. Proof The quantum protocol consists of Alice's simulating the quantum query algorithm A on input x?y. Every query in A will correspond to 2 rounds of communication. Namely, suppose Alice at some point wants to apply a query to the state i;b ff ib ji; bi (for simplicity we omit Alice's workspace). Then she adds a j0i-qubit to the state, applies the unitary mapping ji; b; 0i ! sends the resulting state to Bob. Bob now applies the unitary mapping ji; b; x sends the result back to Alice. Alice applies ji; b; x takes off the last qubit, and ends up with the state which is exactly the result of applying an x ? y-query to jOEi. Thus every query to x ? y can be simulated using 2 log of communication. The final quantum protocol will have T (2 log n of communication and computes f(x; y) with the same error probability as A has on input x ? y. 2 Now consider the disjointness function: DISJ(x; Grover's algorithm can compute the NOR of n variables with O( n) queries with bounded-error, the previous lemma implies a bounded-error protocol for disjointness with O( log n) qubits. On the other hand, the linear lower bound for disjointness is a well-known result of classical communication complexity [33,46]. Thus we obtain the following near-quadratic separation: Theorem 2 (Buhrman, Cleve, Wigderson) log n) and R 2 Another separation is given by a distributed version of the Deutsch-Jozsa problem of Section 2.3: define EQ 0 (x; This is a promise version of equality, where the promise is that x and y are either equal or are at Hamming distance n=2. Since there is an exact 1-query quantum algorithm for In contrast, Buhrman, Cleve, and Wigderson use a combinatorial result of Frankl and Rodl [29] to prove the classical lower bound D(EQ 0 n). Thus we have the following exponential separation for exact protocols: Theorem 3 (Buhrman, Cleve, Wigderson) 5.3 Raz Notice the contrast between the two separations of the previous section. For the Deutsch-Jozsa problem we get an exponential quantum-classical separa- tion, but the separation only holds if we force the classical protocol to be exact; it is easy to see that O(log n) bits are sufficient if we allow some error (the classical protocol can just try a few random positions i and check if x not). On the other hand, the gap for the disjointness function is only quadratic, but it holds even if we allow classical protocols to have some error probability. Ran Raz [45] has exhibited a function where the quantum-classical separation has both features: the quantum protocol is exponentially better than the classical protocol, even if the latter is allowed some error probability. Consider the following promise problem P: Alice receives a unit vector v 2 R m and a decomposition of the corresponding space in two orthogonal subspaces H (0) and H (1) . Bob receives an m \Theta m unitary transformation U . Promise: Uv is either "close" to H (0) or to H (1) . Question: which of the two? As stated, this is a problem with continuous input, but it can be discretized in a natural way by approximating each real number by O(log m) bits. Alice and Bob's input is now log m) bits long. There is a simple yet efficient 2-round quantum protocol for this problem: Alice views v as a log m- qubit vector and sends this to Bob. Bob applies U and sends back the result. Alice then measures in which subspace H (i) the vector Uv lies and outputs the resulting i. This takes only qubits of communication. The efficiency of this protocol comes from the fact that an m-dimensional vector can be "compressed" or "represented" as a log m-qubit state. Similar compression is not possible with classical bits, which suggests that any classical protocol for P will have to send the vector v more or less literally and hence will require a lot of communication. This turns out to be true but the proof (given in [45]) is surprisingly hard. The result is the first exponential gap between and R 2 Theorem 4 (Raz) Q 2 n) and R 2 6 Quantum Communication Complexity: Lower Bounds In the previous section we exhibited some of the power of quantum communication complexity. Here we will look at its limitations, first for exact protocols and then for the bounded-error case. 6.1 Lower bounds on exact protocols Quite good lower bounds are known for exact quantum protocols for total functions. For a total function the communication matrix of f . This is an jXj \Theta jY j Boolean matrix which completely describes f . Let rank(f) denote the rank of M f over the reals. Mehlhorn and Schmidt [39] proved that D(f) log rank(f ), which is the main source of lower bounds on D(f ). For Q(f) a similar lower bound follows from techniques of Yao and Kremer [53,37], as first observed in [17]. This bound was later extended to the case where Alice and Bob share unlimited prior entanglement by Buhrman and de Wolf [20]. Their result turned out to be equivalent to a result in Nielsen's thesis [43, Section 6.4.2]. The result is: Theorem 5 Q (f) log rank(f)=2 and C (f) log rank(f). Hence quantum communication complexity in the exact model is maximal whenever M f has full rank, which it does for almost all functions, including equality, (the complement of) inner product, and disjointness. How tight is the log rank(f) lower bound? It has been conjectured that D(f) (log rank(f)) O(1) for all total functions, in which case log rank(f) would characterize D(f) up to polynomial factors. If this log-rank conjecture is true, then Theorem 5 implies that Q (f) and D(f) are polynomially close for all total f , since then Q (f) D(f) (log rank(f)) O(1) (2Q (f)) O(1) . Some small classes of functions where this provably holds are identified in [20]. It should be noted that, in fact, no total f is known where Q (f) is more than a factor of 2 smaller than D(f) (the factor of 2 can be achieved by superdense coding). 6.2 Lower bounds on bounded-error protocols The previous section showed some strong lower bounds for exact quantum protocols. The situation is much worse in the case of bounded-error protocols, for which very few good lower bounds are known. One of the few general lower bound techniques known to hold for bounded-error quantum complexity (without prior entanglement), is the so-called "discrepancy method". This was shown by Kremer [37], who used it to derive an \Omega\Gamma n) lower bound for Q 2 (IP). Cleve, van Dam, Nielsen, and Tapp [22] later independently proved such a lower bound for Q (IP). We will sketch the very elegant proof of [22] here for the case of exact protocols; for bounded-error protocols it is similar but more technical. The proof uses the IP-protocol to communicate Alice's n-bit input to Bob, and then invokes Holevo's theorem to conclude that many qubits must have been communicated in order to achieve this. Suppose Alice and Bob have some protocol for IP. They can use this to compute the following mapping: Now suppose Alice starts with an arbitrary n-bit state jxi and Bob starts with the uniform superposition 1 they apply the above mapping, the final state becomes x\Deltay jyi: If Bob now applies a Hadamard transform to each of his n qubits, then he obtains the basis state jxi, so Alice's n classical bits have been communicated to Bob. Theorem 1 now implies that the IP-protocol must qubits, even if Alice and Bob share unlimited prior entanglement. The above proof works for IP, but does not easily yield good bounds in general. Neither does the discrepancy method, or an approximate version of the rank lower bound which was noted in [17]. New lower bound techniques for quantum communication are required. Of particular interest is whether the upper bound log n) of [17] is tight. Because disjointness can be reduced to many other problems (it is in fact "coNP-complete" [3]), a good lower bound for disjointness would imply many other lower bounds as well. The best known lower bound is only \Omega\Gammaly/ n), which was proven for (DISJ) in [2] (this also follows from discrepancy) and for Q (DISJ) in [20]. Buhrman and de Wolf [20] have a translation of the approximate rank lower bound to properties of polynomials which might prove an \Omega\Gamma n) bound for disjointness, but a crucial link is still missing in their approach. 7 Quantum Communication Complexity: Applications The main applications of classical communication complexity have been in proving lower bounds for various models like VLSI, Boolean circuits, formula size, Turing machine complexity, data structures, automata size etc. We refer to [38,32] for many examples. Typically one proceeds by showing that a communication complexity problem f is "embedded" in the computational problem P of interest, and then uses communication complexity lower bounds on f to establish lower bounds on P . Similarly, quantum communication complexity has been used to establish lower bounds in various models of quantum computation, though such applications have received relatively little attention so far. We will briefly mention some. Yao [53] initially introduced quantum communication complexity as a tool for proving a superlinear lower bound on the quantum formula size of the majority function (a "formula" is a circuit of restricted form). More recently, Klauck [34] used one-way quantum communication complexity lower bounds to prove lower bounds on the size of quantum formulae. Since upper bounds on query complexity give upper bounds on communication complexity (Lemma 1), lower bounds on communication complexity give lower bounds on query complexity. For instance, IP(x; so the \Omega\Gamma n) bound for IP (Section 6.2) implies an \Omega\Gamma n= log n) lower bound for the quantum query complexity of the parity function, as observed by Buhrman, Cleve, and Wigderson [17]. This lower bound was later strengthened to n=2 in [5,28]. Furthermore, as in the classical case, lower bounds on (one-way) communication complexity imply lower bounds on the size of finite automata. This was used by Klauck [34] to show that Las Vegas quantum finite automata cannot be much smaller than classical deterministic finite automata. Finally, Ben-Or [7] has recently applied the lower bounds for IP in a new proof of the security of quantum key distribution. 8 Other Developments and Open Problems Here we mention some other results in quantum communication complexity or related models: ffl Quantum sampling. For the sampling problem, Alice and Bob do not want to compute some f(x; y), but instead want to sample an (x; y)-pair according to some known joint probability distribution, using as little communication as possible. Ambainis et.al. [2] give a tight algebraic characterization of quantum sampling complexity, and exhibit an exponential gap between the quantum and classical communication required for a sampling problem related to disjointness. ffl Spooky communication. Brassard, Cleve, and Tapp [14] exhibit tasks which can be achieved in the quantum world with entanglement and no communication, but which would require communication in the classical world. They also give upper and lower bounds on the amount of classical communication needed to "simulate" EPR-pairs. Their results may be viewed as quantitative extensions of the famous Bell inequalities [6]. ffl Las Vegas protocols. In this paper we just considered two modes of com- putation: exact and bounded-error. An intermediate type of protocols are zero-error or Las Vegas protocols. These never output an incorrect answer, but may claim ignorance with probability at most 1/2. Some quantum- classical separations for zero-error protocols may be found in [18,34]. One-way communication. Suppose the communication is one-way: Alice just sends qubits to Bob. Klauck [34] showed for all total functions that quantum communication is not significantly better than classical communication in this case. ffl Rounds. It is well known in classical communication complexity that allowing Alice and Bob k+1 rounds of communication instead of k can reduce the required communication exponentially. An analogous result has recently been shown for quantum communication [41] (see also [36]). ffl Non-deterministic communication complexity. A non-deterministic protocol has positive acceptance probability on input (x; y) iff f(x; Classically, the non-deterministic communication complexity is characterized by the logarithm of the cover number of the communication matrix M f . Recently, de Wolf [49] showed that the quantum non-deterministic communication complexity is characterized (up to a factor of 2) by the logarithm of the rank of a non-deterministic version of M f . Finally, here's a list of interesting open problems in quantum communication complexity: ffl Raz's exponential gap only holds for a promise problem. Are D(f) and polynomially related for all total f? As we showed in Section 6.1, a positive answer to this question would be implied by the classical log-rank conjecture. A similar question can be posed for the relation between R 2 and Q (f ). ffl Does entanglement add much power to qubit communication? That is, what are the biggest gaps between Q(f) and Q (f ), and between Q 2 (f) and ffl Develop good lower bound techniques for bounded-error quantum protocols. In particular one that gives a good lower bound for disjointness. ffl Classically, Yao [51] used the minimax theorem from game theory to show an equivalence between deterministic protocols with a probability distribution on the inputs, and bounded-error protocols. Is some relation like this true in the quantum case as well? If so, lower bound techniques for exact quantum protocols can be used to deal with the previous question. --R Quantum dense coding and a lower bound for 1-way quantum finite automata The quantum communication complexity of sampling. Complexity classes in communication complexity theory. Elementary gates for quantum computation. Quantum lower bounds by polynomials. On the Einstein-Podolsky-Rosen paradox Security of quantum key distribution. Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states Strengths and weaknesses of quantum computing. Quantum information theory. Quantum complexity theory. Tight bounds on quantum searching. The cost of exactly simulating quantum entanglement with classical communication. Quantum computing and communication complexity. Quantum entanglement and communication complexity. Quantum vs. classical communication and computation (preliminary version). Bounds for small-error and zero-error quantum algorithms Multiparty quantum communication complexity. Communication complexity lower bounds by polynomials. Substituting quantum entanglement for communication. Quantum entanglement and the communication complexity of the inner product function. Quantum algorithms revisited. Quantum theory Quantum computational networks. Rapid solution of problems by quantum computation. Can quantummechanical description of physical reality be considered complete? A limit on the speed of quantum computation in determining parity. Forbidden intersections. A fast quantum mechanical algorithm for database search. Bounds for the quantity of information transmitted by a quantum communication channel. The probabilistic communication complexity of set intersection. On quantum and probabilistic communication: Las Vegas and one-way protocols Quantum communication complexity. On rounds in quantum communication. Quantum communication. Communication Complexity. Las Vegas is better than determinism in VLSI and distributed computing. Optimal lower bounds for quantum automata and random access codes. Interaction in quantum communication complexity. Private vs. common random bits in communication complexity. Quantum Information Theory. Quantum Computation and Quantum Information. Exponential separation of quantum and classical communication complexity. On the distributional complexity of disjointness. Physicists triumph at Guess my Number. Classical versus quantum communication complexity. Characterization of non-deterministic quantum query and quantum communication complexity A single quantum cannot be copied. Probabilistic computations: Toward a unified measure of complexity. Some complexity questions related to distributive computing. Quantum circuit complexity. Grover's quantum searching algorithm is optimal. --TR Private vs. common random bits in communication complexity The probabilistic communication complexity of set intersection On the distributional complexity of disjointness A fast quantum mechanical algorithm for database search Public vs. private coin flips in one round communication games (extended abstract) Quantum Complexity Theory Strengths and Weaknesses of Quantum Computing Communication complexity and parallel computing Communication complexity Quantum vs. classical communication and computation Exponential separation of quantum and classical communication complexity Dense quantum coding and a lower bound for 1-way quantum automata Classical versus quantum communication complexity On quantum and probabilistic communication Interaction in quantum communication and the complexity of set disjointness On communication over an entanglement-assisted quantum channel Quantum Entanglement and Communication Complexity Quantum Entanglement and the Communication Complexity of the Inner Product Function Lower Bounds in the Quantum Cell Probe Model Improved Quantum Communication Complexity Bounds for Disjointness and Equality On Quantum Versions of the Yao Principle Randomized Simultaneous Messages Characterization of Non-Deterministic Quantum Query and Quantum Communication Complexity The Quantum Communication Complexity of Sampling Quantum Lower Bounds by Polynomials Optimal Lower Bounds for Quantum Automata and Random Access Codes Bounds for Small-Error and Zero-Error Quantum Algorithms Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract) Some complexity questions related to distributive computing(Preliminary Report) Communication Complexity Lower Bounds by Polynomials Lower Bounds for Quantum Communication Complexity --CTR Franois Le Gall, Exponential separation of quantum and classical online space complexity, Proceedings of the eighteenth annual ACM symposium on Parallelism in algorithms and architectures, July 30-August 02, 2006, Cambridge, Massachusetts, USA Dmitry Gavinsky , Julia Kempe , Oded Regev , Ronald de Wolf, Bounded-error quantum state identification and exponential separations in communication complexity, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Dmitry Gavinsky , Julia Kempe , Iordanis Kerenidis , Ran Raz , Ronald de Wolf, Exponential separations for one-way quantum communication complexity, with applications to cryptography, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11-13, 2007, San Diego, California, USA

communication complexity;quantum computing

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Logical foundations of cafeOBJ.

This paper surveys the logical and mathematical foundations of CafeOBJ, which is a successor of the famous algebraic specification language OBJ but adds to it several new primitive paradigms such as behavioural concurrent specification and rewriting logic.We first give a concise overview of CafeOBJ. Then we focus on the actual logical foundations of the language at two different levels: basic specification and structured specification, including also the definition of the CafeOBJ institution. We survey some novel or more classical theoretical concepts supporting the logical foundations of CafeOBJ, pointing out the main results but without giving proofs and without discussing all mathematical details. Novel theoretical concepts include the coherent hidden algebra formalism and its combination with rewriting logic, and Grothendieck (or fibred) institutions. However, for proofs and for some of the mathematical details not discussed here we give pointers to relevant publications.The logical foundations of CafeOBJ are structured by the concept of institution. Moreover, the design of CafeOBJ emerged from its logical foundations, and institution concepts played a crucial rle in structuring the language design.

Introduction CafeOBJ is an executable industrial strength algebraic specification language which is a modern successor of OBJ and incorporating several new algebraic specification paradigms. Its definition is given in [12]. CafeOBJ is intended to be mainly used for system specification, formal verification of specifications, rapid prototyping, programming, etc. Here is a brief overview of its most important features. On leave from the Institute of Mathematics of the Romanian Academy. Preprint submitted to Elsevier Preprint 26 February 2000 Equational Specification and Programming. This is inherited from OBJ [27, 17] and constitutes the basis of the language, the other features being somehow built on top of it. As with OBJ, CafeOBJ is executable (by term rewriting), which gives an elegant declarative way of functional programming, often referred as algebraic programming. 3 As with OBJ, CafeOBJ also permits equational specification modulo several equational theories such as associativity, commutativity, identity, idempotence, and combinations between all these. This feature is reflected at the execution level by term rewriting modulo such equational theories. Behavioural Specification. Behavioural specification [21, 22, 13, 29] provides a novel generalisation of ordinary algebraic specification. Behavioural specification characterises how objects (and systems) behave, not how they are implemented. This new form of abstraction can be very powerful in the specification and verification of software systems since it naturally embeds other useful paradigms such as concurrency, object-orientation, constraints, nondeterminism, etc. (see [22] for details). Behavioural abstraction is achieved by using specification with hidden sorts and a behavioural concept of satisfaction based on the idea of indistinguishability of states that are observationally the same, which also generalises process algebra and transition systems (see [22]). CafeOBJ behavioural specification paradigm is based on coherent hidden algebra (abbreviated 'CHA') of [13], which is both an simplification and extension of classical hidden algebra of [22] in several directions, most notably allowing operations with multiple hidden sorts in the arity. Coherent hidden algebra comes very close to the "observational logic" of Bidoit and Hennicker [29]. CafeOBJ directly supports behavioural specification and its proof theory through special language constructs, such as hidden sorts (for states of systems), behavioural operations (for direct "actions" and "observations" on states of sys- tems), behavioural coherence declarations for (non-behavioural) operations (which may be either derived (indirect) "observations" or "constructors" on states of sys- tems), and behavioural axioms (stating behavioural satisfaction). The advanced coinduction proof method receives support in CafeOBJ via a default coinduction relation (denoted =*=). In CafeOBJ, coinduction can be used either in the classical hidden algebra sense [22] for proving behavioural Please notice that although this paradigm may be used as programming, this aspect is still secondary to its specification side. equivalence of states of objects, or for proving behavioural transitions (which appear when applying behavioural abstraction to rewriting logic). 4 Besides language constructs, CafeOBJ supports behavioural specification and verification by several methodologies. 5 CafeOBJ currently highlights a methodology for concurrent object composition which features high reusability not only of specification code but also of verifications [12, 30]. Behavioural specification in CafeOBJ may also be effectively used as an object-oriented (state-oriented) alternative for classical data-oriented specifications. Experiments seem to indicate that an object-oriented style of specification even of basic data types (such as sets, lists, etc.) may lead to higher simplicity of code and drastic simplification of verification process [12]. Behavioural specification is reflected at the execution level by the concept of behavioural rewriting [12, 13] which refines ordinary rewriting with a condition ensuring the correctness of the use of behavioural equations in proving strict equalities Rewriting Logic Specification. Rewriting logic specification in CafeOBJ is based on a simplified version of Mese- guer's rewriting logic (abbreviated as `RWL') [32] specification framework for concurrent systems which gives a non-trivial extension of traditional algebraic specification towards concurrency. RWL incorporates many different models of concurrency in a natural, simple, and elegant way, thus giving CafeOBJ a wide range of applications. Unlike Maude [2], the current CafeOBJ design does not fully support labelled RWL which permits full reasoning about multiple transitions between states (or system configurations), but provides proof support for reasoning about the existence of transitions between states (or configurations) of concurrent systems via a built-in predicate (denoted ==>) with dynamic definition encoding both the proof theory of RWL and the user defined transitions (rules) into equational logic. From a methodological perspective, CafeOBJ develops the use of RWL transitions for specifying and verifying the properties of declarative encoding of algorithms (see [12]) as well as for specifying and verifying transition systems. 4 However, until the time this paper was written, the latter has not been yet explored suffi- ciently, especially practically. 5 This is still an open research topic, the current methodologies may be developed further and new methodologies may be added in the future. Module System. The principles of the CafeOBJ module system are inherited from OBJ which builds on ideas first realized in the language Clear [1], most notably institutions [19, 15]. CafeOBJ module system features several kinds of imports, sharing for multiple imports, parameterised programming allowing multiple parameters, views for parameter instantiation, integration of CafeOBJ specifications with executable code in a lower level language module expressions. However, the concrete design of the language revise the OBJ view on importation modes and parameters [12]. Type System and Partiality. CafeOBJ has a type system that allows subtypes based on order sorted algebra (abbreviated 'OSA') [25, 20]. This provides a mathematically rigorous form of runtime type checking and error handling, giving CafeOBJ a syntactic flexibility comparable to that of untyped languages, while preserving all the advantages of strong typing. At this moment the concrete order sortedness formalism is still open at least at the level of the language definition. CafeOBJ does not directly do partial operations but rather handles them by using error sorts and a sort membership predicate in the style of membership equational logic (abbreviated 'MEL') [33]. The semantics of specifications with partial operations is given by MEL. Logical semantics. CafeOBJ is a declarative language with firm mathematical and logical foundations in the same way as other OBJ-family languages (OBJ, Eqlog [23, 4], FOOPS [24], Maude [32]) are. The mathematical semantics of CafeOBJ is based on state-of-the- art algebraic specification concepts and results, and is strongly based on category theory and the theory of institutions [19, 11, 9, 15]. The following are the principles governing the logical and mathematical foundations of CafeOBJ: P1. there is an underlying logic 6 in which all basic constructs and features of 6 Here "logic" should be understood in the modern relativistic sense of "institution" which the language can be rigorously explained. P2. provide an integrated, cohesive, and unitary approach to the semantics of specification in-the-small and in-the-large. P3. develop all ingredients (concepts, results, etc.) at the highest appropriate level of abstraction. CafeOBJ is a multi-paradigm language. Each of the main paradigms implemented in CafeOBJ is rigorously based on some underlying logic; the paradigms resulting from various combinations are based on the combination of logics. The structure of these logics is shown by the following CafeOBJ cube, where the full arrows mean embedding between the logics, which correspond to institution embeddings (i.e., a strong form of institution morphisms of [19, 15]) (the orientation of arrows goes from "more complex" to "less complex" logics). HA MSA RWL OSRWL HOSA OSA =many rewriting logic The mathematical structure represented by this cube is that of an indexed institution [11]. The CafeOBJ institution is a Grothendieck (or fibred) institution [11] obtained by applying a Grothendieck construction to this cube (i.e., the indexed institution). Note that by employing other logical-based paradigms the CafeOBJ cube may be thought as a hyper-cube (see [12] for details). 1.1 Summary of the paper The first part of this paper is dedicated to the foundations of basic specifications. The main topic of this part is the definition of HOSRWL, the hidden order sorted rewriting logic institution, which embeds all other institutions of the CafeOBJ cube. In this way, the HOSRWL institution contains the mathematical foundations for all basic specification CafeOBJ constructs. The second part of the paper presents the novel concept of Grothendieck institution provides a mathematical definition for a logic (see [19]) rather than in the more classical sense. (developed by [11]) which constructs the CafeOBJ institution from the CafeOBJ cube. The last section contains the definitions of the main mathematical concepts for structuring specification in CafeOBJ. The main concepts of the logical foundations of CafeOBJ are illustrated with several examples, including CafeOBJ code. We assume familiarity with CafeOBJ including its syntax and semantics (see [12] or several papers such as [14]). Terminology and Notations This work assumes some familiarity with basic general algebra (in its many-sorted and order-sorted form) and category theory. Relevant background in general algebra can be found in [18, 26, 34] for the many-sorted version, and in [25, 20] for the order-sorted version. For category theory we generally use the same notations and terminology as Mac Lane [31], except that composition is denoted by ";" and written in the diagrammatic order. The application of functions (functors) to arguments may be written either normally using parentheses, or else in diagrammatic order without parentheses, or, more rarely, by using sub-scripts or super-scripts. The category of sets is denoted as Set, and the category of categories 7 as C at . The opposite of a category C is denoted by C op . The class of objects of a category C is denoted by jC j; also the set of arrows in C having the object a as source and the object b as target is denoted as C (a;b). Indexed categories [35] play an important r"ole in for this work. [36] constitutes a good reference for indexed categories and their applications to algebraic specifica- tion. An indexed category [36] is a sometimes we denote for an index i 2 jIj and B(u) as B u for an index morphism u 2 I. The following 'flattening' construction providing the canonical fibration associated to an indexed category is known under the name of the Grothendieck construc- tion) and plays an important r"ole in mathematics and in particular in this paper. Given an indexed category be the Grothendieck category having objects and hu; ji : arrows. The composition of arrows in B ] is defined by hu; ji;hu 7 We steer clear of any foundational problem related to the "category of all categories"; several solutions can be found in the literature, see, for example [31]. 2 Foundations of Basic Specifications At this level, semantics of CafeOBJ is concerned with the semantics of collections of specification statements. CafeOBJ modules can be flattened to such basic specifications by an obvious induction process on the module composition structure. In CafeOBJ we can have several kinds of specifications, the basic kinds corresponding to the basic CafeOBJ specification/programming paradigms: equational specifications, rewriting specifications, behavioural specifications, and behavioural rewriting specifications. The membership of a basic specification to a certain class is determined by the CafeOBJ convention that each basic specification should be regarded as implementing the simplest possible combination of paradigms resulting from its syntactic content. 2.1 Loose and Tight Denotation The key concept of specification in-the-small is the satisfaction relation between the models and the sentences of a given specification, which is also the key notion of the abstract concept of institution. Each kind of specification has its own concept of satisfaction, and Section 2.2 surveys them briefly. Each class of basic specifications has an underlying logic in the CafeOBJ cube. Specifications can be regarded as finite sets of sentences in the underlying logic. This enables us to formulate the principle of semantics of CafeOBJ specification in-the-small: Each basic specification determines a theory in the corresponding institution. The denotation [[SP]] of a basic specification SP is the class of models MOD(T SP ) of its corresponding theory T SP if loose, and it is the initial model 0 T SP of the theory, if tight. A basic specification can have either loose or initial denotation, and this can be directly specified by the user. CafeOBJ does not directly implement final semantics, however final models play an important r"ole for the loose semantics of behavioural specifications (see [13, 8]). Initial model semantics applies only to non-behavioural specification, and is supported by the following result: Theorem 1 Let T be a theory in either MSA, OSA, RWL, or OSRWL. Then the initial model 0 T exists. This very important result appears in various variants and can be regarded as a classic of algebraic specification theory. The reader may wish to consult [26] for MSA, [25, 20] for OSA, [32] for RWL, and although, up to our knowledge, the result has not yet been published, it is also valid for OSRWL. Because of the importance of the construction of the initial model we briefly recall it here. Let S be the signature of the theory consisting of a set S of sorts (which is a partial order in the order-sorted case) and a ranked (by S ) set of operation symbols (possibly overloaded). The S-sorted set T S of S-terms is the least S-sorted set closed under: - each constant is a S-term (denoted S [];s ng. The operations in S can be interpreted on T S in the obvious manner, thus making it into a S-algebra 0 S . If T is equational, then its ground part is a congruence T on 0 S . Then 0 T is the quotient whose carriers are equivalence classes of S-terms under T . If T is a pure rewriting theory then 0 T is a rewriting logic model whose carriers (0 T ) s are categories with S-terms as objects and concurrent rewrite sequences (using the rules of T ) as arrows. Finally, rewrite theories including equations require the combination between the above two constructions. Example 2 Consider the following CafeOBJ specification of non-deterministic natural numbers: mod! NNAT { protecting(NAT) trans M:Nat | N:Nat => M . trans M:Nat | N:Nat => N . The denotation of NNat is initial and consists [of the isomorphism class] of one model, 0 NNat , the initial model. The main carrier of 0 NNat is a category which has non-empty lists of natural numbers as objects and deletion sequences as arrows. | gets interpreted as a functor which concatenates lists of numbers on objects, and compose in parallel ("horizontally") deletion sequences. 2.2 Hidden Order Sorted Rewriting Logic Institution We devote this section to the definition of the HOSRWL institution (defined for the first time in [8] in the many sorted version) which embeds all CafeOBJ cube institutions. We recall here that the behavioural specification part of HOSRWL is based on the 'coherent hidden algebra' of [13]. The deep understanding of HOS- RWL requires further reading on its main components ([32] for RWL and [13] for CHA) as well as their integration [8]. Signatures Definition 3 A HOSRWL signature is a tuple (H;V;;S;S b ), where (H;) and (V;) are disjoint partial ordered sets of hidden sorts and visible respectively, S is a (H [V;)-order-sorted signature, is a subset of behavioural operations such that s 2 S b w;s has exactly one hidden sort in w. Notice that we may simplify the notation (H;V;;S;S b ) to just (H;V;;S) , or just S, when no confusion is possible. From a methodological perspective, the operations in S b have object-oriented mean- w;s is thought as an action (or "method" in a more classical jargon) on the space (type) of states if s is hidden, and thought as observation (or "attribute" in a more classical jargon) if s is visible. The last condition says that the actions and observations act on (states of) single objects. is an order-sorted signature morphism (H such that hidden sorts h;h These conditions say that hidden sorted signature morphisms preserve visibility and invisibility for both sorts and operations, and the S 0 b F(S b ) inclusion together with (M3) expresses the encapsulation of classes (in the sense that no new actions (methods) or observations (attributes) can be defined on an imported class) 8 . How- ever, these conditions apply only to the case when signature morphisms are used as 8 Without it the Satisfaction Condition fails, for more details on the logical and computational relevance of this condition see [21]. module imports (the so-called horizontal signature morphisms); when they model specification refinement this condition might be dropped (this case is called vertical signature morphism). Proposition 5 HOSRWL signatures and signature morphisms (with the obvious composition) form a category denoted as Sign HOSRWL . Sentences In HOSRWL there are several kinds of sentences inherited from the various CafeOBJ cube institutions. Definition 6 Consider a HOSRWL signature (H;V;;S;S b ). Then a (strict) equation is a sentence of the form where X is a (H [V )-sorted set of variables, t; t 0 are S-terms with variables X , and C is a Boolean(-sorted) S-term, a behavioural equation is a sentence of the form a (strict) transition is a sentence of the form and a behavioural transition is a sentence of the form have the same meaning as for strict equations. All these sentences are here defined in the conditional form. If the condition is missing (which is equivalent to saying it is always true), then we get the unconditional versions of sentences. Notice also that our approach to conditional sentences is slightly different from the literature in the sense that the condition is a Boolean term rather than a finite conjunction of formul. Our approach is more faithful to the concrete level of CafeOBJ and is also more general. This means that a finite conjunction of formul can be translated to a Boolean term by using some special semantic predicates (such as == for semantics equality and ==> for the semantic transition relation, in CafeOBJ). We do not discuss here the full details of this ap- proach, we only further mention that the full rigorous treatment of such conditions can be achieved within the so-called constraint logic [10], which can however be regarded as a special case of an abstract categorical form of plain equational logic [5, 4, Equational attributes such as associativity (A), commutativity (C), identity (I), or idempotence (Z) are just special cases of strict equations. However, the behavioural part of HOSRWL has another special attribute called behavioural coherence [12, 13] which is regarded as a sentence: Definition 7 Let (H;V;;S;S b ) be a signature. Then s coherent is a behavioural coherence declaration for s, where s is any operation S. Definition 8 Given a signature morphism the translation of sentences is defined by replacing all operation symbols from S with the corresponding symbols (via F) from S 0 and by re-arranging the sort of the variables involved accordingly to the sort mapping given by F. Fact 9 If we denote the set of sentences of a signature (H;V;;S;S b ) by Sen HOSRWL (H;V;;S;S b ) and the sentence translation corresponding to a signature morphism F by Sen HOSRWL (F), then this gets a sentence functor Models Models of HOSRWL are rewrite models [32] which are (algebraic) interpretations of the signatures into C at (the category of small categories) rather than in Set (the category of small sets) as in the case of ordinary algebras. Thus, ordinary algebras can be regarded as a special case of rewrite models with discrete carriers. Given a HOSRWL signature (H;V;;S;S b ), a HOSRWL model M interprets: each sort s as a small category M s and each subsort relation s < s 0 as sub-category relation , and each operation s 2 S w;s as a functor Notice that each S-term by evaluating it for each assignment of the variables occurring in t with objects or arrows from the corresponding carriers of M. Model homomorphisms in HOSRWL follow an idea of [29] by refining the ordinary concept of model morphism and reforming the hidden algebra [21, 22] homomorphisms by taking adequate care of the behavioural structure of models. We need first to define the concept of behavioural equivalence. Definition 11 Recall that a S-context c[z] is any S-term c with a marked variable z occurring only once in c. A context c[z] is behavioural iff all operations above z are behavioural. Given a model M, two elements (of the same sort s; they can be either both objects or both arrows) a and a 0 are called behaviourally equivalent, denoted a s a 0 (or just a a 0 ) iff 9 for all visible behavioural contexts c. Remark that the behavioural equivalence is a (H [V )-sorted equivalence relation, and on the visible sorts the behavioural equivalence coincides with the (strict) equality relation. Now we are ready to give the definition of model homomorphism in HOSRWL. between models of a signature (H;V; between the carriers such that (for each sort s): for all a 2 M s there exists a s (a and a 0 can be either both arrows or both elements) such that a h s a 0 , for all a 2 M s , if a h a 0 then (a h b 0 if and only if a 0 b 0 ), for all a;b 2 M s and a s , if a h a 0 and a b then b h a 0 , and for each operation s2S w;s , for all a 2M w and a 0 2M 0 w , a h w a 0 implies M s (a) h w M 0 (component-wise). Notice that when there are no hidden sorts (i.e., we are in some non-behavioural part of HOSRWL), this concept of model homomorphism coincides with the rewriting model homomorphism. For a given signature (H;V;;S), we denote its category of models by MOD HOSRWL . Notice that any signature morphism a model reduct in the usual way (by renaming the sorts of the carriers and the interpretations of the operations accordingly to the mapping of sorts and operations given by F). Therefore we have a contravariant model functor at op . 9 Notice that this equality means an equality between functions M , where This is a relation between the sets of objects together with a relation between the sets of arrows, such that this couple of relations commute with the domain functions, codomain functions, and arrow composition functions. Satisfaction The satisfaction relation between sentences and models is the crucial concept of an institution (see Definition 19). Consider a model M of a signature (H;V;;S;S b ). Then M satisfies an equation, i.e., M only if for all valuations (Notice that this applies both for objects and arrows, since valuations may map variables either to objects or arrows.) M satisfies a behavioural equation, i.e., M only if for all valuations M satisfies a transition, i.e., M only if for each "object there is an arrow a M C (q) is true such that for all "arrow we have a a q M satisfies a behavioural transition, i.e., M only if for each appropriate visible behavioural context c and for each "object valuation" there is an arrow a c true such that for all "arrow we have a c Finally, M satisfies a coherence declaration, i.e., M coherent ), if and only if s preserves the behavioural equivalence on M, i.e., for all a;a 0 2 Mw . 11 I.e. a is a natural transformation. Notice that the behavioural coherence of both the behavioural operations and of operations of a visible rank is trivially satisfied. Example 14 Consider the following CafeOBJ behavioural specification of non-deterministic naturals: mod* NNAT-HSA { protecting(NAT) *[ NNat ]* vars vars or S2 -> N . Notice that for all models M of NNAT-HSA, This situation when the operations which are neither behavioural or a data type operations (i.e. with visible rank) are automatically coherent is rather natural and occurs very often in practice, and this corresponds to the so-called coherence conservative methodology of [13]. The definition of the satisfaction relation between sentences and models completes the construction of the HOSRWL institution: Theorem 15 (Sign HOSRWL ; Sen HOSRWL ; MOD HOSRWL ; j=) is an institution. For the definition of institution see Definition 19 given below. We omit here the proof of this result which is rather long and tedious and follows the same pattern as proofs of similar results, also reusing some of them (such as the proof that RWL is an institution). At the end of the presentation of the HOSRWL institution we give a brief example of a CafeOBJ specification in HOSRWL: Example Consider a behavioural specification of sets of non-determinstic naturals mod* SETS { protecting(NNAT) vars vars S or (E in S) . in S1) or (E in S2) . in S1) and (E in S2) . not (E in S1) . where NNAT is the RWL specification of non-determinstic naturals of Example 2. Notice that each model of SETS satisfies the usual set theory rules (such as commutativity and associativity of union and intersection, De Morgan laws, etc.) only behaviourally, not necessarily in the strict sense. For example, the following De Morgan behavioural rule is a consequence of the specification SETS. Also, the following behavioural tran- sition btrans add(M | N, S) => add(M, S) . is a consequence of SETS too. Specifications in full HOSRWL naturally occurs in the case of a behavioural specification using concurrent (RWL) data types. However the practical significance of full HOSRWL is still little understood. The real importance of the HOSRWL institution is its initiality in the CafeOBJ cube. We will see below that the existence of all possible combinations between the main logics/institutions of CafeOBJ is crucial for the good properties of the CafeOBJ institution. 2.3 Operational vs. Logical Semantics The operational semantics underlies the execution of specifications or programs. As with OBJ, the CafeOBJ operational semantics is based on rewriting, which in the case of proofs is used without directly involving the user defined transitions (rules) but rather involving them via the built-in semantic transition predicate ==>. For executions of concurrent systems specified in rewriting logic, CafeOBJ uses both the user-defined transitions and equations. Since rewriting is a very well know topic in algebraic specification, we do not insist here on the standard aspects of rewriting. However, the operational semantics of behavioural specification requires a more sophisticated notion of rewriting which takes special care of the use of behavioural sentences during the rewriting process, which we call behavioural rewriting [12, 13]: Definition 17 Given a HOSRWL signature S and a S-algebra A, a behaviourally coherent context for A is any S-context c[z] such that all operations above 12 the marked variable z are either behavioural or behaviourally coherent for A. Notice that any behavioural context is also behaviourally coherent. The following Proposition from [13] ensures the soundness of behavioural rewriting Proposition Consider a HOSRWL signature S, a set E of S-sentences regarded as a TRS (i.e. term rewriting system), and a S-algebra A satisfying the sentences in E. If t 0 is a ground term and for any rewrite step t which uses a behavioural equation from E, the rewrite context has a visible behaviourally coherent sub-context for A, then A If the rewrite context is behaviourally coherent for A, then A The completeness of the operational semantics with respect to the logical semantics is a two-layer completeness going via the important intermediate level of the proof calculi. Denotational Proof Calculus Operational The completeness of the proof calculus is one of the most important class of results in algebraic specification, for equational logic we refer to [25], and for rewriting logic to [32]. In the case of rewriting logic the relationship between the proof calculus and rewriting is very intimate, but for equational logic the completeness of rewriting can be found, among other many places, in [18, 7]. Notice that hidden logics of the CafeOBJ cube do not admit a complete (finitary) proof calculus. However, advanced proof techniques support the verification process in the case of behavioural specifications, most notably the hidden coinduction method (see [22] for the original definition, [12, 13] for its realization in CafeOBJ, and [6] for the details for the case of proving behavioural transitions). 12 Meaning that z is in the subterm determined by the operation. 3 The CafeOBJ Institution In this section we define the CafeOBJ institution, which is a Grothendieck construction on the CafeOBJ cube. The Grothendieck construction for institutions was first introduced by [11] and generalises the famous Grothendieck construction for categories [28]. The essence of this Grothendieck construction is that it constructs a 'disjoint sum' of all institutions of the CafeOBJ cube, also introducing theory morphisms across the institution embeddings of the CafeOBJ cube. Such extra theory morphisms were first studied in [9]. However, one advantage of the Grothendieck institutions is that they treat the extra theory morphisms as ordinary theory morphisms, thus leading to a conceptual simplification with respect to [9]. The reader might wonder why one cannot live with HOSRWL only (which embeds all the CafeOBJ cube institutions) and we still need a Grothendieck construction on the CafeOBJ cube. The reason for this is that the combination of logics/institutions realized by HOSRWL collapses crucial semantic information, therefore a more refined construction which preserves the identity of each of the CafeOBJ cube institutions, but yet allowing a concept of theory morphism across the institution embeddings, is necessary. For example, in the case of specifications with loose semantics without a RWL component, the carriers of the models of these specifications should be sets rather than categories, which is not possible in HOSRWL. Therefore, such specifications should be given semantics within the appropriate institution of the CafeOBJ cube rather than in HOSRWL. Example 36 illustrates this argument. 3.1 Institutions We now recall from [19] the definitions of the main institution concepts: Definition 19 An institution consists of (1) a category Sign, whose objects are called signatures, (2) a functor Sen : Sign!Set, giving for each signature a set whose elements are called sentences over that signature, (3) a functor MOD : Sign op ! C at giving for each signature S a category whose objects are called S-models, and whose arrows are called S-(model) mor- phisms, and a relation for each S2 jSignj, called S-satisfaction, such that for each morphism j in Sign, the satisfaction condition holds for each m 0 2 jMOD(S 0 )j and e 2 Sen(S). We may denote the reduct functor MOD(j) by j and the sentence translation Sen(j) by j( ). be an institution. For any signature S the closure of a set E of S-sentences is . (S;E) is a theory if and only if E is closed, i.e., A theory morphism that j(E) E 0 . Let Th() denote the category of all theories in . For any institution , the model functor MOD extends from the category of its signatures Sign to the category of its theories Th(), by mapping a theory (S;E) to the full subcategory MOD(S;E) of MOD(S) formed by the S-models which satisfy liberal if and only if the reduct functor The institution is liberal if and only if each theory morphism is liberal. Definition 22 An institution exact if and only if the model functor MOD : Sign op ! C at preserves finite limits. is semi-exact if and only if MOD preserves only pullbacks. Definition 23 Let and 0 be institutions. Then an institution homomorphism consists of (1) a (2) a natural transformation a (3) a natural transformation such that the following satisfaction condition holds for any S 0 -model m 0 from 0 and any S 0 F-sentence e from . Fact 24 Institutions and institution homomorphisms form a category denoted as Ins. The following properties of institution homomorphisms were defined in [11] and play an important r"ole for Grothendieck institutions: Definition 25 An institution homomorphism means that M S-model M that satisfies all sentences in E . an embedding iff F admits a left-adjoint F (with unit z); an institution embedding is denoted as liberal iff b S 0 has a left-adjoint b S 0 for each S 0 2 jSign 0 j. An institution embedding exact if and only if the square below is a pullback MOD(SFF) MOD(Sz) (jF) signature morphism in . 3.2 Indexed and Grothendieck Institutions The following definition from [11] generalises the concept of indexed category [36] to institutions. Definition 26 An indexed institution is a functor : I op ! Ins. The CafeOBJ cube is an indexed institution where the index category I is the 8-element lattice corresponding to the cube (i.e., the elements of the lattice correspond to the nodes of the cube and the partial order is given by the arrows of the cube). Definition 27 The Grothendieck institution ] of an indexed institution : I op ! Ins has (1) the Grothendieck category Sign ] as its category of signatures, where Sign: I op ! C at is the indexed category of signatures of the indexed institution , C at as its model functor, where for each index i 2 jIj and signature S 2 jSign i j, and its sentence functor, where for each index i 2 jIj and signature S 2 jSign i j, and for each S e for each index For the category minded readers we mention that [11] gives a higher level characterisation of the Grothendieck institution as a lax colimit in the 2-category Ins (with institutions as objects, institution homomorphisms as 1-cells, and institution modifications as 2-cells; see [11] for details) of the corresponding indexed institution. This means that Grothendieck institutions are internal Grothendieck objects 14 in Ins in the same way as Grothendieck categories are Grothendieck objects in C at . For the fibred category minded readers, in [11] we also introduce the alternative formulation of fibred institution and show that there is a natural equivalence between split fibred institutions and Grothendieck institutions. We would also like to mention that the concept of extra theory morphism [9] across an institution homomorphism 0 ! (with all its subsequent concepts) is recuper- ated as an ordinary theory morphism in the Grothendieck institution of the indexed institution given by the homomorphism 0 ! (i.e., which has ! as its index category). Now we are ready to define the institution of CafeOBJ: Definition 28 The CafeOBJ institution is the Grothendieck institution of the CafeOBJ cube. 3.3 Properties of the CafeOBJ Institution In this section, we briefly study the most important institutional properties of the CafeOBJ institution: existence of theory colimits, liberality (i.e. free construc- tions), and exactness (i.e. model amalgamation). The institution homomorphisms of the CafeOBJ cube are all embeddings; this makes the CafeOBJ cube an embedding-indexed institution (cf. [11]). As we will see below, this property of the CafeOBJ cube plays an important r"ole for the properties of the CafeOBJ institution. Theory Colimits. The existence of theory colimits is crucial for any module system in the Clear-OBJ tradition. Let us recall the following result from [11]: 14 From [11], a Grothendieck object in a 2-category is a lax colimit of a 1-functor to that 2-category. Theorem 29 Let : I op ! Ins be an embedding-indexed institution such that I is J-cocomplete for a small category J. Then the category of theories Th( ) of the Grothendieck institution ] has J-colimits if and only if the category of signatures Sign i is J-cocomplete for each index i 2 jIj. Corollary The category of theories of the CafeOBJ institution is small cocomplete Notice that the fact that the lattice of institutions of the CafeOBJ cube is complete (as a lattice) means exactly that the index category of the CafeOBJ cube is (small) cocomplete, which is a precondition for the existence of theory colimits in the CafeOBJ institution. In the absence of the combinations of logics/institution of the CafeOBJ cube (such as HOSRWL), the possibility of theory colimits in the CafeOBJ institution would have been lost. Liberality. Liberality is a desirable property in relation to initial denotations for structured specifications. In the case of loose denotations liberality is not necessary. Since the behavioural specification paradigm involves only loose denotations, in the case of the CafeOBJ institution, we are therefore interested in liberality only for the non-behavioural theories. Recall the following result from [11]: Theorem 31 The Grothendieck institution ] of an indexed institution : I op ! Ins is liberal if and only if i is liberal for each index i 2 jIj and each institution homomorphism u is liberal for each index morphism u 2 I. Corollary In the CafeOBJ institution, each theory morphism between non- behavioural theories is liberal. Notice that this corollary is obtained from the theorem above by restricting the index category to the non-behavioural square of the CafeOBJ cube, and from the corresponding liberality results for equational and rewriting logics. Exactness. Firstly, let us extend the usual exactness results for equational and rewriting logics to the CafeOBJ cube: Proposition 33 All institutions of the CafeOBJ cube are semi-exact. As shown in [9] and [11], in practice exactness is a property hardly achieved at the global level by the Grothendieck institutions. In [11] we give a necessary and sufficient set of conditions for (semi-)exactness of Grothendieck institutions. One of them is the exactness of the institution embeddings, which fails for the embeddings from the non-RWL institutions into the RWL institutions of the CafeOBJ cube. In the absence of a desired global exactness property for the CafeOBJ institution, we need a set of sufficient conditions for exactness for practically significant particular cases. In [9] we formulate a set of such sufficient conditions, but this problem is still open. 4 Foundations of Structured Specifications In this section we survey the mathematical foundations of the CafeOBJ module composition system. CafeOBJ module composition system follows the principles of the OBJ module system which are inherited from earlier work on Clear [1]. Consequently, CafeOBJ module system is institution-independent (i.e., can be developed at the abstract level of institutions) in the style of [15]. In the actual case of CafeOBJ, the institution-independent semantics is instantiated to the CafeOBJ institution. The following principle governs the semantics of programming in-the- large in CafeOBJ: (L) For each structured specification we consider the theory corresponding to its flattening to a basic specification. The structuring constructs are modelled as theory morphisms between these corresponding theories. The denotation [[SP]] of a structured specification is determined from the denotations of the components recursively via the structuring constructs involved. The general structuring mechanism is constituted by module expressions, which are iterations of several basic structuring operations, such as (multiple) imports, parameters, instantiation of parameters by views, translations, etc. 4.1 Module Imports Module imports constitute the most primitive structuring construct in any module composition system. The concept of module import in the institution-independent semantics of CafeOBJ is based on the mathematical notion of inclusion system. Module imports are modeled as inclusion theory morphisms between the theories corresponding to flattening the imported and the importing modules Inclusion systems where first defined by [15] for the institution-independent study of structuring specifications. Weak inclusion systems were introduced in [3], and they constitute a simplification of the original definition of inclusion systems of [15]. We recall the definition of inclusion systems: Definition 34 hI ; Ei is a weak inclusion system for a category C if I and E are two sub-categories with jI (1) I is a partial order, and (2) every arrow f in C can be factored uniquely as The arrows of I are called inclusions, and the arrows of E are called tions. 15 The domain (source) of the inclusion i in the factorisation of f is called the image of f and denoted as Im( f ). An injection is a composition between an inclusion and an isomorphism. A weak inclusion system hI ; Ei is an inclusion system iff I has finite least upper bounds (denoted +) and all surjections are epics (see [15]). The inclusion system for the category of theories of the CafeOBJ institution is obtained by lifting the inclusion system for its category of signatures (see [15, 3]). The inclusion system for the category of signatures is obtained from the canonical inclusion systems of the categories of signatures of the CafeOBJ cube institutions by using the following result from [11] (which appeared previously in a slightly different form in [9]): Theorem C at be an indexed category such that I has a weak inclusion system hI I has a weak inclusion system hI preserves inclusions for each inclusion index morphism u 2 I I , and preserves inclusions and surjections and lifts inclusions uniquely for each surjection index morphism Then, the Grothendieck category B ] has an inclusion system hI B is inclusion iff both u and j are inclusions, and surjection iff both u and j are surjections. In the case of the CafeOBJ institution, this result is applied for the indexed category of signatures of the CafeOBJ cube. Example 36 Consider the following module import: mod* TRIV { [ Elt 15 Surjections of some weak inclusion systems need not necessarily be surjective in the ordinary sense. mod* NTRIV { protecting(TRIV) trans M:Elt | N:Elt => M . trans M:Elt | N:Elt => N . Module TRIV gets a MSA loose theory, which has all sets as its denotation. Module NTRIV gets a RWL loose theory, which has as denotations categories with an interpretation of j as an associative binary functor, and which satisfies the couple of choice transitions of NTRIV. The module import TRIV ! NTRIV corresponds to an injective extra theory morphism T across the forgetful institution MSA. More formally, the inclusion signature morphism underlying T TRIV ! T NTRIV can be represented as hu; ji where u is the institution morphism RWL ! MSA and j is the signature inclusion S TRIV ! u(S NTRIV ) (where S TRIV is the MSA signature of TRIV, S NTRIV is the RWL signature of NTRIV, and u(S NTRIV ) is the reduct of S NTRIV to a MSA signature). Notice that u is an inclusion since the CafeOBJ cube admits a trivial inclusion system in which all arrows are inclusions, that the reduct from RWL signatures to MSA signatures is an identity, and that S TRIV ! S NTRIV is an inclusion of MSA signatures. An interesting aspect of this example is given by its model theory. The denotation of this module import is the model reduct functor MOD(T NTRIV in the CafeOBJ institution. From Definition 27, this means b u which means a two level reduction. The first level, b u , means getting rid of the arrows of the carrier (i.e. making the carrier discrete) of the model and regarding the interpretation of j as a function rather than functor. The second level, MOD MSA (j), is a reduction internal to MSA which forgets the interpretation of j . It is very important to notice that the correct denotation for this module import can be achieved only in the framework of the CafeOBJ institution, the fact that this is a Grothendieck institution being crucial. None of the institutions of the CafeOBJ cube (such as RWL for example) would have been appropriate to give the denotation of this example. We denote the partial order of module imports by . By following the OBJ tradi- tion, we can distinguish between three basic kinds of imports, protecting, extend- ing, and using. At the level of the language, these should be treated just as semantic declarations which determine the denotation of the importing module from the denotation of the imported module. Definition 37 Given a theory morphism model M of T , an expansion of M along j is a model M 0 of T 0 satisfying the following properties: the expansion is protecting, there is an injective 16 model homomorphism M ,! M 0 j iff the expansion is extending, there is an arbitrary model homomorphism M!M 0 j iff the expansion is using, and with respect to j iff the expansion is free. Definition 38 Fix an import SPSP 0 and let T and T 0 be the theories corresponding to SP and SP 0 , respectively. Then is an expansion of the same kind as the importation mode involved of some model M 2 (and in addition free if SP 0 is initial) g. Multiple imports are handled by a lattice structure on imports. The (finite) least upper bounds (called sums in [15]) of module imports corresponds to the weak inclusion system of theory morphisms being a proper inclusion system. In [16] we lift sums from inclusion systems for ordinary theory morphisms to extra theory morphisms; this result can be easily translated to the conceptual framework of Grothendieck institutions. The (finite) greatest lower bounds (called intersections) are defined as the pullback of the sums. The details of this construction for the inclusion system of extra theory morphisms are given in [16]; also this construction can be easily translated to the conceptual framework of Grothendieck institutions. In practice, one of the important properties of the sum-intersection square is to be a pushout besides being a pullback square. This result for the inclusion system of extra theory morphisms is given in [16]; again this can be easily translated to Grothendieck institutions. Under a suitable concept of 'injectivity'. 17 Which means that M 0 is the free object over M with respect to the model reduct functor This relies the construction of finite limits in Grothendieck (fibred) categories. 4.2 Parameterisation Parameterised specification and programming is an important feature of all module systems of modern specification or programming languages. In CafeOBJ The mathematical concept of parameterised modules is based on injections (in the sense of Definition 34) in the category of theories of the CafeOBJ institution: Parameterised specifications SP(X :: P) are modelled as injective theory morphisms from the theory corresponding to the parameter P to the theory corresponding to the body SP. Views are modelled as theory morphisms The denotation [[SP]] of the body is determined from the denotation of the parameter accordingly to the parameterisation mode involved as in the case of module imports (Definition 38). We distinguish two opposite approaches on parameters: a shared and a non-shared one. In the 'non-shared' approach, the multiple parameters are mutually disjoint (i.e., for X and X 0 two different parameters) and they are also disjoint from any module imports T 0 T (i.e., 0). In the 'shared' approach this principle is relaxed to being disjoint outside common imports, i.e., two different parameters and . The 'non-shared' approach has the potentiality of a much more powerful module system, while the 'shared' approach seems to be more convenient to implement (see [12] for details). The CafeOBJ definition gives the possibility of the whole range of situations between these two extremes by giving the user the possibility to control the sharing. Example 39 This is an example adapted from [12]. Consider the (double param- eterised) specification of a 'power' operation on monoids, where powers are elements of another (abstract) monoid rather than natural numbers. mod* MON { mod* MON-POW (POWER :: MON, M :: MON) { vars vars The diagram defining MON-POW is MON-POW POWER r r r r r r r r r r r r where MON-POW consists of two copies of MON (labelled by M and, POWER) re- spectively, plus the power operation together with the 3 axioms defining its action. This means TRIV is not shared, since the power monoid and the base monoid are allowed to have different carriers. The denotation consists of all protecting expansions (with interpretations of " ) to MON-POW of non-shared amalgamations of monoids corresponding to the two parameters. In the 'shared' approach, the parameterisation diagram is MON z z z z z z z z z z z z z z z z MON POWER In this case, the denotation consists of all two different monoid structures on the same set, plus an interpretation of " satisfying the 'power' equations. In CafeOBJ such sharing can be achieved by the user by the command share which has the effect of enforcing that the modules declared as shared are included rather than 'injected' in the body specification. In this case we have just to specify share(TRIV) The following defines parameter instantiation by pushout technique for the case of single parameters. This definition can be naturally extended to the case of multiple parameters (for details about instantiation of multiple parameters in CafeOBJ see [12]). be a parameterised module and let T its representation as theory morphism. Let be a view. Then the instantiation T SP (v) is given by the following pushout of theory morphisms in the CafeOBJ institution: in the 'non-shared' approach, and by the following co-limit in the 'shared' approach. The semantics of parameter instantiation relies on preservation properties of conservative extensions by pushouts of theory morphisms. Recall the concept of conservative theory morphism from [15]: Definition 41 A theory morphism conservative iff any model M of T has a protecting expansion along j. Preservation of conservative extensions in Grothendieck institutions is a significantly harder problem than in ordinary institutions. Such technical results for Grothendieck institutions have been obtained in [16] but within the conceptual frame-work of extra theory morphisms. 5 Conclusions and Future Work We surveyed the logical foundations of CafeOBJ which constitute the origin of the concrete definition of the language [12]. Some of its main features are: simplicity and effectiveness via appropriate abstractness, cohesiveness, flexibility, provides support for multi-paradigm integration, provides support for the development of specification methodologies, and uses state-of-art methods in algebraic specification research. We defined the CafeOBJ institution, overviewed its main properties, and presented the main mathematical concepts and result underlying basic and structured specification in CafeOBJ. Besides theoretical developments, future work on CafeOBJ will mainly concentrate on specification and verification methodologies, especially the object-oriented ones emerging from the behavioural specification paradigm. This includes refining the existing object composition methodology based on projection operations [30, 14, 12] but also the development of new methodologies and careful identification of the application domains most suitable to certain specification and verification methodologies. The development of CafeOBJ has been an interplay process among language de- sign, language and system implementation, and methodology development. Although the language design is based on solid and firm mathematical foundations, it has been greatly helped by the existence of a running system, which gave the possibility to run various relevant examples, thus giving important feedback at the level of concrete language constructs and execution commands. The parallel development of methodologies gave special insight on the relationship between the various paradigms co-existing in CafeOBJ with consequences at the level of design of the language constructs. We think that the interplay among mathematical semantic design of CafeOBJ, the system implementation, and the methodology development has been the most important feature of CafeOBJ design process. We believe this promises the sound and reasonable development of a practical formal specification method around CafeOBJ. --R The semantics of Clear Principles of Maude. Virgil Emil C Principles of OBJ2. Theorem Proving and Algebra. Institutions: Abstract model theory for specification and programming. A hidden agenda. An initial algebra approach to the specification Observational logic. Categories for the Working Mathematician. Some fundamental algebraic tools for the semantics of computation --TR Initiality, induction, and computability Unifying functional, object-oriented and relational programming with logical semantics Some fundamental algebraic tools for the semantics of computation, part 3 Conditional rewriting logic as a unified model of concurrency Order-sorted algebra I Institutions: abstract model theory for specification and programming Logical support for modularisation Principles of OBJ2 Membership algebra as a logical framework for equational specification Towards an Algebraic Semantics for the Object Paradigm Observational Logic The Semantics of CLEAR, A Specification Language Component-Based Algebraic Specification and Verification in CafeOBJ --CTR Rzvan Diaconescu, Herbrand theorems in arbitrary institutions, Information Processing Letters, v.90 n.1, p.29-37, 15 April 2004 Rzvan Diaconescu, Behavioural specification for hierarchical object composition, Theoretical Computer Science, v.343 n.3, p.305-331, 17 October 2005 Miguel Palomino, A comparison between two logical formalisms for rewriting, Theory and Practice of Logic Programming, v.7 n.1-2, p.183-213, January 2007 Mauricio Ayala-Rincn , Ricardo P. Jacobi , Luis G. A. Carvalho , Carlos H. Llanos , Reiner W. Hartenstein, Modeling and prototyping dynamically reconfigurable systems for efficient computation of dynamic programming methods by rewriting-logic, Proceedings of the 17th symposium on Integrated circuits and system design, September 07-11, 2004, Pernambuco, Brazil Rzvan Diaconescu, Institution-independent Ultraproducts, Fundamenta Informaticae, v.55 n.3-4, p.321-348, August Razvan Diaconescu, Institution-independent ultraproducts, Fundamenta Informaticae, v.55 n.3-4, p.321-348, June Rzvan Diaconescu, Interpolation in Grothendieck institutions, Theoretical Computer Science, v.311 n.1-3, p.439-461, 23 January 2004 M. Ayala-Rincn , C. H. Llanos , R. P. Jacobi , R. W. Hartenstein, Prototyping time- and space-efficient computations of algebraic operations over dynamically reconfigurable systems modeled by rewriting-logic, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.11 n.2, p.251-281, April 2006 Francisco Durn , Jos Meseguer, Maude's module algebra, Science of Computer Programming, v.66 n.2, p.125-153, April, 2007 Narciso Mart-Oliet , Jos Meseguer, Rewriting logic: roadmap and bibliography, Theoretical Computer Science, v.285 n.2, p.121-154, 28 August 2002

behavioural specification;institutions;CafeOBJ;algebraic specification

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Lower bounds for the rate of convergence in nonparametric pattern recognition.

We show that there exist individual lower bounds corresponding to the upper bounds for the rate of convergence of nonparametric pattern recognition which are arbitrarily close to Yang's minimax lower bounds, for certain "cubic" classes of regression functions used by Stone and others. The rates are equal to the ones of the corresponding regression function estimation problem. Thus for these classes classification is not easier than regression function estimation.

Introduction ),. be independent identically distributed R d f0; 1g-valued random variables. In pattern recognition (or classication) one wishes to decide whether the value of Y (the label) is 0 or 1 given the (d-dimensional) value of X (the observation), that is, one wants to nd a decision function g dened on the range of X taking values 0 or 1 so that g(X) equals to Y with high probability. Assume that the main aim of the analysis is to minimize the probability of error : xg be the a posteriori probability (or regression) function. Introduce the Bayes-decision and let be the Bayes-error. Denote the distribution of X by . Introduce kfk q R d) 1=q . It is well-known (see Devroye, Gyor and Lugosi [8]), that for each measurable function the relation Z 1 I fg 6=g g d (2) holds, where I A denotes the indicator function of the event A. Therefore the function g achieves the minimum in (1) and the minimum is L . In the classication problem we consider here, the distribution of (X; Y ) (and therefore also and g ) is unknown. Given only the independent sample D of the distribution of (X; Y ), one wants to construct a decision rule (R d f0; 1g) n 7! f0; 1g such that is close to L . In this paper we study asymptotic properties of EL n L . If we have an estimate n of the regression function and we derive a plug-in rule g n from n quite naturally by then from (2) we get easily (see [8]). This shows that if k in the same sense, and the latter has at least the same rate, that is, in a sense, classication is not more complex than regression function estimation. It is well-known, that there exist regression function estimates, and so classication rules, which are universally consistent, that is, which satisfy for all distributions of (X; Y ). This was rst shown in Stone [14] for nearest neighbor estimates (see also [8] for a list of references). Classication is actually easier than regression function estimation in the sense that if then for the plug-in rule (see [8] Chapter 6), that is, the relative expected error of g n decreases faster than the expected error of n . Moreover, if k then for the plug-in rule (see Antos [1]), that is, the relation also holds for strong consistency. However the value of the ratio above cannot be universally bounded, the convergence can be arbitrary slow. It depends on the behavior of near 1=2 and the rate of convergence of f n g. Unfortunately, there do not exist rules for which EL n L tends to zero with a guaranteed rate of convergence for all distributions of (X; Y ). Theorem 7.2 and Problem 7.2 in Devroye et al. [8] imply the following slow-rate-of-convergence result: Let fa n g be a positive sequence converging to zero with 1=16 a 1 a :. For every sequence fg n g of decision rules, there exists a distribution of (X; Y ), such that X is uniformly distributed on [0; 1], 2 f0; 1g for all n. Therefore, in order to obtain nontrivial rate-of-convergence results, one has to restrict the class of distributions. Then it is natural to ask what the fastest achievable rate is for a given class of distributions. This is usually done by considering minimax rate-of-convergence results, where one derives lower bounds according to the following denition. Denition 1 A positive sequence fa n g is called lower rate of convergence for a class D of distributions of (X; Y ) if lim sup sup (X;Y )2D a n Remark. In many cases the limit superior in this denition could be replaced by limit inferior or inmum, because the lower bound for the minimax loss holds for all (su-ciently large) n. Since and determine the distribution of (X; Y ), the class D of distributions is often given as a product of a class H of allowed distributions of X and a class F of allowed regression functions. For example, H may consist of all absolute continuous distributions, or all distributions (distribution-free approach), or one particular distribution (distribution- sensitive approach). In this paper we give lower bounds for some classes D in the last (and strongest) for- mat, when H contains one (uniform) distribution, and class of functions (with a parameter ), dened later. For minimax lower-rate results on other types of distribution classes (e.g., Vapnik-Chervonenkis classes), see Devroye et al. [8] and the references therein. For related results on the general minimax theory of statistical estimates see Ibragimov and Khasmiskii [9], [10], [11], Korostelev and Tsybakov [12]. Yang [16] points out that while (3) holds for every xed distribution for which f n g is consistent, the optimal rate of convergence for many usual classes is the same in classication and regression function estimation. He shows many examples and some counterexamples to this phenomenon with rates of convergence in terms of metric entropy. Classication seems to have the same complexity as regression function estimation for classes which are rich near 1=2. (See also Mammen and Tsybakov [13].) For example, it was shown in Yang [16] that for the distribution classes D above, the optimal rate of convergence is fn 2+d g, the same as for regression function estimation (see also Stone [15]). In some sense, such lower bounds are not satisfactory. They do not tell us anything about the way the probability of error decreases as the sample size is increased for a given classi- cation problem. These bounds, for each n, give information about the maximal probability of error within the class, but not about the behavior of the probability of error for a single xed distribution as the sample size n increases. In other words, the \bad" distribution, causing the largest probability of error for a decision rule, may be dierent for each n. For example, the previous lower bounds for the classes D does not exclude the possibility that there exists a sequence fg n g such that for every distribution in D , the expected probability of error EL n L decreases at an exponential rate in n. In this paper, we are interested also in \individual" minimax lower bounds that describe the behavior of the probability of error for a xed distribution (X; Y ) as the sample size n grows. Denition 2 A positive sequence fa n g is called individual lower rate of convergence for a class D of distributions of (X; Y ) if fgng sup (X;Y )2D lim sup a n where the inmum is taken over all sequences fg n g of decision rules. The concept of individual lower rate has been introduced in Birge [7] concerning density estimation. For individual lower-rate results concerning pattern recognition see Antos and We will show that for every sequence fb n g tending to zero, fb n n 2+d g is an individual lower rate of convergence for the classes D . Hence there exist individual lower rates of these classes, which are arbitrarily close to the optimal lower rates. These rates are the same as the individual lower rates for the expected L 2 () error of regression function estimation for these classes (see also Antos, Gyor and Kohler [4] and Antos [3]). Both for regression function estimation and pattern recognition, the individual lower rates are optimal, hence we extend Yang's observation for the individual rates for these classes. Our results also imply that the ratio ELn L can tend to zero arbitrary slowly (even for a xed sequence f n g). Next we give the denitions of the function classes, for which we derive lower rates of convergence. Let and denote the L q norm regarding to the Lebesgue-measure on R d by k k Lq () . Denition 3 For given 1 q 1, R 2 be the class of functions f such that for R d ) with kD moreover, for every such with kD where D denotes the partial derivative with respect to For q < 1, we assume R 1. For all the rate-of-convergence results below hold if some of M 0 ,. ,MR 1 are innite, that is, if we omit some of the conditions of the rst kind. These classes are generalizations of the Lipschitz classes of Example 2 in Yang [16], Antos, Gyor and Kohler [4], Birge [7] and Stone [15], and also generalizations of a special case of Example 4 in Yang [16] for polinomial modulus of continuity. Denition 4 For given ~ and M > 0, let V M) be the class of functions f such that kfk1 M and for - > 0 where R R r and e i is the i th unit vector in R d . These classes are just Example 5 in Yang [16]. We assume that M;M 0 > 1=2. Denition 5 Denote one of the classes Lip ;d and V ;d by F . Let D be the class of distributions of (X; Y ) such that (i) X is uniformly distributed on [0; 1] d , It is well-known, that there exist regression function estimates f n g, which satisfy lim sup sup (X;Y )2D +d (see, e.g., [16] and Barron, Birge and Massart [6]). (This remains true replacing condition (i) in Denition 5, e.g., with the assumption that is in a class of distributions with uniformly bounded density funtions. Note that the rate depends only on and d.) Thus for the plug-in rules fg n g lim sup sup (X;Y )2D 2+d that is, sup (X;Y )2D (EL n L 2+d . (The rate n 2+d might be formulated as d=, the exponent of dimension is just the exponent of 1= in the -metric entropy of the classes, see [7].) To handle the two types of classes together, let ~ for the classes Lip ;d . Thus holds in all cases. (We may call as the exponent of global smoothness following [7].) Now we give conditions for a general function class F , which assure lower rates of convergence for the corresponding distribution class D . A subclass of regression functions will be indexed by the vectors of +1 or 1 components, where are introduced below. Denote the set of all such vectors by C. Let be the uniform distribution on [0; 1] d . Assumption 1 With some ~ and satisfying (6), for any probability distribution fp j g for every j 2 N , there is a function with support on the brick I , such that these are disjoint even for dierent j and j 0 , and for every c 2 C, (c) (which are independent of fp j g and j). Note that by (6). Theorem 1 If Assumption 1 holds for a function class F , then the sequence a +d is a lower rate of convergence for the corresponding distribution class D . Assumption 1 is similar to A'2(k) in Birge [7]. In this form, it will be required only for the individual lower rates. It seems from the proof (choice of fp j g) that for the minimax rates above, we only use the weaker form: Assumption 2 With some ~ and satisfying (6), for any 2 (0; 1], there is a function with support on the brick I = I in [0; 1] d , such that for every (c) r Also km k 2 This is almost the same as Assumption 3 in Yang [16] and A2(k) in Birge [7] in case of polinomial metric entropy, and Theorem 1 is analogous to Yang's Theorem 2. Both Yang's and our theorems give as a special case: Corollary 1 If F is Lip ;d or V ;d , then the sequence a 2+d is a lower rate of convergence for the class D . Our main result is the following extention of Theorem 1 to individual lower rates (see Antos [2] for Lip ;d if Theorem 2 Let fb n g be an arbitrary positive sequence tending to zero. If Assumption 1 holds for a function class F , then the sequence b n a 2+d is an individual lower rate of convergence for the corresponding distribution class D . Remark 1. Applying for the sequence f b n g, Theorem 2 implies that for all fg n g there is such that lim sup b n a n Remark 2. Certainly Theorems 1 and 2 hold if we increase the class by leaving condition (i) from Denition 5. Focusing again on the classes Lip ;d and Corollary 2 Let fb n g be an arbitrary positive sequence tending to zero. If F is Lip ;d or then the sequence b n a 2+d is an individual lower rate of convergence for the class D . Call a sequence fc n g an upper rate of convergence for a class D, if there exist rules fg n g which satisfy lim sup sup (X;Y )2D that is, sup (X;Y )2D (EL n L it an individual upper rate of convergence for a class D, if there exist rules fg n g which satisfy sup (X;Y )2D lim sup This implies only that for every distribution in D, EL n L possibly with dierent constants. Then (5) implies that n 2+d is an upper rate of convergence, and thus also an individual upper rate of convergence for D . While Theorem 1 shows only that there is no upper rate of convergence for D better than n , it follows from Theorem 2 that n d is even the optimal individual upper rate for D in the sense, that there doesn't exist an individual upper rate c n of convergence for D , which satises lim Moreover (3) and (4) imply fgng sup (X;Y )2D lim sup which shows that Theorem 2 cannot be improved by dropping b n . This shows the strange nature of individual lower bounds, that while every sequence tending to zero faster than 2+d is an individual lower rate for D , n 2+d itself is not that. 3 Proofs The proofs of the theorems apply the following lemma: l-dimensional real vector taking values in [ 1=4; 1=4] l , let C be a zero mean random variable taking values in f1;+1g, and let Y l be inde- pedent binary variables given C with Then for the error probability of the Bayes decision for C based on ~ Proof. The Bayes decision is 1 if Y Y (see [8]). One can verify that where Y I fY i =1g +2 I fY i =0g Y where For arbitrary 0 < q < 1=2, q if and only if j log T j log 1 q By Markov's inequality log 1 q Moreover because of j log T , we get log T Using the inequality for 1=4 x 1=4 on the one hand and on the other hand so Hence log T jg Thus Efg q@ 1 Efj log T jg log 1 q qA q@ 1q P log 1 q By choosing Proof of Theorem 1. The method of this proof diers from that of Yang, it can be easily modied to individual lower bound in Theorem 2. Assumption 1 and 0 (c) 1 imply that each distribution (X; Y ) with X Unif[0; 1] d , Y 2 f0; 1g and EfY for all x 2 [0; 1] d for some c 2 C is contained in D , which implies lim sup gn sup (X;Y )2D a n lim sup sup (X;Y ):XUnif[0;1] d ;EfY jX=xg= (c) (x);c2C a n Let g n be an arbitrary rule. By denition, fI A j;k =2 : j; kg is an orthogonal system in for the measure R A therefore the projection ^ of g is given by I A j;k (x) where R (g n 1=2)I A j;k =2 d R I A j;k =2 d R A j;k R A j;k R A j;k R A j;k R A j;k R A j;k (Note that ^ arbitrary. Note that g = 1+ I A j;k. Then by (2) Z (c) 1 I fgn 6=g g d Z d Z d KX Let ~ c n;j;k be otherwise. Because of j^c n;j;k c j;k j I f~c n;j;k 6=c j;k g , we get I f~c n;j;k 6=c j;k g p +d This proves where Equations (8) and imply lim sup gn sup (X;Y )2D (;M) a n K gn sup a n To bound the last term, we x the rules fg n g and choose c 2 C randomly. Let be a sequence of independent identically distributed random variables independent of X 1 , . , which satisfy PfC 1=2. Next we derive a lower bound for The sign ~ c n;j;k can be interpreted as a decision on C j;k using D n . Its error probability is minimal for the Bayes decision C n;j;k , which is 1 if PfC therefore l be those X i 2 A j;k . Then given distributed as in the conditions of Lemma 1 with u depends only on C n fC j;k g and on X r 's with r 62 fi therefore is independent of Now conditioning on X the error of the conditional Bayes decision for C j;k based on (Y depends only on (Y Pf By Jensen-inequality Pf 1e 10E K1np 2+d independently of k. Thus if np 2+d Pf and where 2+d for j n 1 2+d , 2+d cn +1 so lim sup sup a n lim sup a n This together with (11) implies the assertion. 2 Proof of Theorem 2. We use the notations and results of the proof of Theorem 1. Now we have by fgng sup (X;Y )2D lim sup b n a n K fgng sup lim sup b n a n In this case we have to choose fp j g independently from n. Since b n and a n tend to zero, we can take a subsequence fn t g t2N of fng n2N with b n t 2 t and a 1= that a 1= and choose fp j g as where q t is repeated 2 t =q t times. So t:d2 t a 1= a 1= t:an t an a 1= a 1= by a 1= specially for ER ns (C) K 3 ts b ns a ns : (15) We nish the proof in the spirit of Lemma 1 in Antos and Lugosi [5]. Using (15) one gets fgng sup lim sup b n a n fgng sup lim sup R ns (c) b ns a ns K 3 fgng sup lim sup R ns (c) ER ns (C) K 3 fgng lim sup R ns (C) ER ns (C) Because of (12) and the fact that for all c 2 C the sequence fR ns (C)=ER ns (C)g is uniformly bounded, so we can apply Fatou's lemma to get fgng sup lim sup b n a n K 3 fgng lim sup R ns (C) ER ns (C) This together with (14) implies the assertion. 2 Proof of Corollary 1 and 2. We must prove that the classes Lip ;d and V ;d satisfy Assumption 1. The parameters ~ and are given as in the text satisfying (6). For any fp j g, we give the required functions m j and sets A j;k . First we pack the disjoint sets A j;k into [0; 1] d in the following way: Assume for simplicity, that not, then the index of the minimal i takes the role of the rst dimension in the construction below.) For a given fp j g, let fB j g be a partition of [0; 1] such that B j is an interval of length p j . We pack disjoint translates of I j into the brick This gives Y and since, for x 1, bxc x=2 and p j d Y d Y where Let be the uniform distribution on [0; 1] d and Choose a function 1=4] such that (I) the support of m is a subset of [0; 1] d , denotes a diagonal matrix. Thus m j is a contraction of m from [0; 1] d to I j . Now we have km 2 and We only have to check that because of (III), for every c 2 C, (c) Case F Note that the functions m j;k have disjoint support, and this holds also for their derivatives. If q < 1, then for R q thus For with kD (c) k q c j;k D kD Moreover, for every with kD (c) (c) (x)k Lq c j;k - D c j;k - D c j;k - D We can choose m such that its support is in [1=3; 2=3] d . Now if k-k < p j =3 then the support of - D m j;k is in A j;k , hence they are disjoint. Thus for the rst term, with M c j;k - D For the second term, c j;k - D c j;k D c j;k D c j;k D c j;k D c j;k D kD This gives kD (c) If . For R with P d kD c j;k D kD Moreover, for every - 2 R d , introduce x 0 and x 00 as a function of x the following way: If x and x+ - fall in the same A j;k , then let x dierent bricks, then consider the segment x(x + -), and let x 0 and x 00 be the intersections of this segment and the borders of the bricks x and x vanishing outside A j;k , any of its partial derivatives (up to order R) is zero on the border of A j;k , thus in both case D (c) (c) with kD (c) (c) (x)k 1 (c) (c) (c) (c) (x)k 1 kD (c) (c) (x 00 (c) (c) Using the fact that x and x 0 are in the same brick, and so the support of D j;k (x) is in A j;k , for the second term kD (c) (c) c j;k (D kp R (D sup We get the same bound similarly for the rst term, which gives kD (c) (See also [15], p. 1045.) Case F Now k (c) k1 c j;k R i c j;k R i c j;k R i We can choose m such that its support is in [1=2; 1] d . Now, if R i - < then the support of R i i;- m j;k is in A j;k , hence they are disjoint. For the rst term, with M c j;k R i For the second term, c j;k R i c j;k r r (R r (R r (R This gives --R Lower bounds on the rate of convergence of nonparametric pattern recog- nition Performance limits of nonparametric estimators. Strong minimax lower bounds for learning. On nonparametric estimation of regression. Statistical Estimation: Asymptotic Theory. On the bounds for quality of nonparametric Minimax Theory of Image Reconstruction. Smooth discrimination analysis. Consistent nonparametric regression. Optimal global rates of convergence for nonparametric regression. Minimax nonparametric classi --TR Characterizing rational versus exponential learning curves Strong Minimax Lower Bounds for Learning MiniMax Methods for Image Reconstruction Lower Bounds on the Rate of Convergence of Nonparametric Pattern Recognition

nonparametric pattern recognition;individual rates of convergence

633577

A geometric approach to leveraging weak learners.

AdaBoost is a popular and effective leveraging procedure for improving the hypotheses generated by weak learning algorithms. AdaBoost and many other leveraging algorithms can be viewed as performing a constrained gradient descent over a potential function. At each iteration the distribution over the sample given to the weak learner is proportional to the direction of steepest descent. We introduce a new leveraging algorithm based on a natural potential function. For this potential function, the direction of steepest descent can have negative components. Therefore, we provide two techniques for obtaining suitable distributions from these directions of steepest descent. The resulting algorithms have bounds that are incomparable to AdaBoost's. The analysis suggests that our algorithm is likely to perform better than AdaBoost on noisy data and with weak learners returning low confidence hypotheses. Modest experiments confirm that our algorithm can perform better than AdaBoost in these situations.

Introduction Algorithms like AdaBoost [7] that are able to improve the hypotheses generated by weak learning methods have great potential and practical benefits. We call any such algorithm a leveraging algorithm, as it leverages the weak learning method. Other examples of leveraging algorithms include bagging [3], arc-x4 [5], and LogitBoost [8]. One class of leveraging algorithms follows the following template to construct master hypotheses from a given sample The leveraging algorithm begins with a default master hypothesis H 0 and then for - Constructs a distribution D t over the sample (as a function of the sample and the current master hypothesis H possibly t). - Trains a weak learner using distribution D t over the sample to obtain a weak hypothesis h t . Picks ff t and creates the new master hypothesis, authors were supported by NSF Grant CCR 9700201. This is essentially the Arcing paradigm introduced by Breiman [5, 4] and the skeleton of AdaBoost and other boost-by-resampling algorithms [6, 7]. Although leveraging algorithms include arcing algorithms following this template, leveraging algorithms are more general. In Section 2, we introduce the GeoLev algorithm that changes the examples in the sample as well as the distribution over them. In this paper we consider 2-class classification problems where each y +1g. However, following Schapire and Singer [15], we allow the weak learner's hypotheses to be "confidence rated," mapping the domain X to the real num- bers. The sign of these numbers gives the predicted label, and the magnitude is a measure of confidence. The master hypotheses produced by the above template are interpreted in the same way. Although the underlying goal is to produce hypotheses that generalize well, we focus on how quickly the leveraging algorithm decreases the sample error. There are a variety of results bounding the generalization error in terms of the performance on the sample [16]. Given a sample the margin of a hypothesis h on instance x i is y i h(x i ) and the margin of h on the entire sample is the vector hypothesis that correctly labels the sample has a margin vector whose components are all positive. Focusing on these margin vectors provides a geometric intuition about the leveraging problem. In particular, a potential function on margin space can be used to guide the choices of D t and ff t . The distribution D t is the direction of steepest descent and ff t is the value that minimizes the potential of H Leveraging algorithms that can be viewed in this way perform a feasible direction descent on the potential function. An amortized analysis using this potential function can often be used to bound the number of iterations required to achieve zero sample error. These potential functions give insight into the strengths and weaknesses of various leveraging algorithms. Boosting algorithms have the property that they can convert weak PAC learning algorithms into strong PAC learning algorithms. Although the theory behind the Adaboost algorithm is very elegant, it leads to the somewhat intriguing result that minimizing the normalization factor of a distribution will reduce the training error [14, 15]. Our search for a better understanding of how AdaBoost reduces the sample error led to our geometric algorithms, GeoLev and GeoArc. Although the performance bounds for these algorithms are too poor to show that they have the boosting property, these bounds are incomparable to AdaBoost's in that they are better when the weak hypotheses contain mostly low-confidence predictions. The main contributions of this paper are as follows: We use a natural potential function to derive a new algorithm for leveraging learners, called GeoLev (for Geometric Leveraging algorithm). We highlight the relationship between AdaBoost, Arcing and feasible direction linear programming [10]. - We use our geometric interpretation to prove convergence bounds on the algorithm GeoLev. These bound the number of iterations taken by GeoLev to achieve ffl classification error on the training set. We provide a general transformation from GeoLev to an arcing algorithm GeoArc, for which the same bounds hold. We summarize some preliminary experiments with GeoLev and GeoArc. We motivate a novel algorithm, GeoLev, by considering the geometry of "margin space." Since many empirical and analytical results show that good margins on the sample lead to small generalization error [2, 16], it is natural to seek a master hypothesis with large margins. One heuristic is to seek a margin vector with uniformly large margins, i.e. a vector parallel to 1). This indicates that the master hypothesis is correct and equally confident on every instance in the sample. The GeoLev algorithm exploits this heuristic by attempting to find hypotheses whose margin vectors are as close as possible to the 1 direction. We now focus on a single iteration of the leveraging process, dropping the time subscripts. Margin vectors will be printed in bold face and often normalized to have Euclidean length one. Thus H is the margin vector of the master hypothesis H , whose i th component is Let the goal vector, p m), be 1 normalized to length one. Recall that m is the sample size, so all margin vectors lie in ! m , and normalized margin vectors lie on the m dimensional unit sphere. Note that it is easy to re-scale the confidences - multiplying the predictions of any hypothesis H by a constant does not change the direction of H's margin vector. Therefore we can assume the appropriate normalization without loss of generality. The first decision taken by the leverager is what distribution D to place on the sample. Since distribution D has m components, it can also be viewed as a (non-negative) vector in ! m . The situation in margin-space at the start of the iteration is shown in Figure 1. In order to decrease the angle ' between H and g we must move the head of H towards g. All vectors at angle ' to the goal vector g lie on a cone, and their normalizations lie on the "rim" shown in the figure. If h, the weak hypothesis's margin vector (which need not have unit length), is parallel to H or tangent to the "rim", then no addition of h to H can decrease the angle to g. On the other hand, if the line H cuts through the cone, then the angle to the goal vector g can be reduced by adding some multiple of h to H. The only time the angle to g cannot be decreased is when the h vector lies in the plane P which is tangent to the cone and contains the vector H, as shown in Figure 2. theta Fig. 1. Situation in margin space at the start of an iteration. If the weak learner learns at all, then its hypothesis h is better than random guessing, so the learners "edge", E i-D (y i h(x i )), will be positive. This means that D \Delta h is positive, and if distribution D (viewed as a margin vector) is perpendicular to plane P then h lies above P . Therefore the leverager is able to use h to reduce the angle between H and g. As suggested by the figures, the appropriate direction for D is In general neither jjDjj If all components of D are positive, it can be normalized to yield a distribution on the sample for the weak learner. However, it is possible for some components of D to be negative. In this case things are more complicated 1 . If a component of D is negative, then we flip both the sign of that component and the sign of the corresponding label in the sample. This creates a new direction D 0 which can be normalized to a distribution D 0 and a new sample S 0 with the same x i 's but (possibly) new labels y 0 . The modified sample S 0 and distribution D 0 are then used to generate a new weak hypothesis, h. Let h 0 be the margins of h on the modified sample S 0 , so h 0 In fact it is this complication which differentiates GeoLev from Arcing algorithms. Arcing algorithms are not permitted to change the sample in this way. A second transformation avoiding the label flipping is discussed in section 5. Fig. 2. The direction D for the distribution used by GeoLev. as the sign flips cancel. The second decision taken by the algorithm is how to incorporate the weak hypothesis h into its master hypothesis H . Any weak hypothesis with an "edge" on the distribution D described above can be used to decrease '. Our goal is to find the coefficient ff so that H jjH+ffhjj2 decreases this angle as much as possible. Taking derivatives shows that ' is minimized when From this discussion we can see that GeoLev performs a kind of gradient descent. If we consider the angle between g and the current H as a potential on margin space, then D is the direction of steepest descent. Moving in a direction that approximates this gradient takes us towards the goal vector. Since we have only little control over the hypotheses returned by the weak learner, an approximation to this direction is the best we can do. The step size is chosen adaptively to make as much use of the weak hypothesis as possible. The GeoLev Algorithm is summarized in Figure 3. 3 Relation to Previous Work Breiman [5, 4] defines arcing algorithms using potential functions that can be expressed as component-wise functions of the margins having the form 2 Breiman allows the component-wise potential f to depend on the sum of the ff i 's in some arcing algorithms. Input: A sample a weak learning algorithm. Initialize master hypothesis H to predict 0 everywhere m) Repeat: do if add else add do call weak learner with distribution D over S 0 , obtaining hypothesis h Fig. 3. The GeoLev Algorithm. Breiman shows that, under certain conditions on f , arcing algorithms converge to good hypotheses in the limit. Furthermore, he shows that AdaBoost is an arcing algorithm with is an arcing algorithm with polynomial f(x). For completeness, we describe the AdaBoost algorithm and show in our notation how it is performing feasible direction gradient descent on the potential function AdaBoost fits the template outlined in the introduction, choosing the distribution Z where Z is the normalizing factor so that D sums to 1. The master hypothesis is updated by adding in a multiple of the new weak hypothesis. The coefficient ff is chosen to minimize exp the next iteration's Z value. Unlike GeoBoost, the margin vectors of AdaBoost's hypotheses are not normalized. We now show that AdaBoost can be viewed as minimizing the potential by approximate gradient descent. The direction of steepest descent (w.r.t. the components of the margin vector) is proportional to (5), the distribution AdaBoost gives to the weak learner. Continuing the analogy, the coefficient ff given to the new hypothesis should minimize the potential, X of the updated master hypothesis, which is identical to (6). Thus AdaBoost's behavior is approximate gradient descent of the function defined in (7), where the direction of descent is the weak learner's hypothesis. Furthermore, the bounds on AdaBoost's performance proven by Schapire and Singer are implicitly performing an amortized analysis over the potential function (8). Arc-x4 also fits the template outlined in the introduction, keeping an unnormalized master hypothesis. In our notation the distribution chosen at trial t is proportional to This algorithm can also be viewed as a gradient descent on the potential function at the t th iteration. Rather than computing the coefficient ff as a function of the weak hypothesis, arc-x4 always chooses ff = 1. Thus each h t has weight 1=t in the master hypothesis, as in many gradient descent methods. Unfortunately, the dependence of the potential function on t makes it difficult to use in an amortized analysis. This connection to gradient descent was hinted at by Freund [6] and noted by Breiman and others [4, 8, 13]. Our interpretation generalizes the previous work by relaxing the constraints on the potential function. In particular, we show how to construct algorithms from potential functions where the direction of steepest descent can have negative components. The potential function view of leveraging algorithms shows their relationship to feasible descent linear programming, and this relationship provides insight into the role of the weak learner. Feasible direction methods try to move in the direction of steepest descent. However, they must remain in the feasible region described by the constraints. A descent direction is chosen that is closest to the (negative) gradient \Gammarf while satisfying the constraints. For example, in a simplified Zoutendijk method, the chosen direction d satisfies the constraints and maximizes \Gammarf \Delta d . Similarly, the leveraging algorithms discussed are constrained to produce master hypotheses lying in the span of the weak learner's hypothesis class. One can view the role of the weak learner as finding a feasible direction close to the given distribution (or negative gradient). In fact the weak learning assumption used in boosting and in the analysis of GeoLev implies that there is always a feasible direction d such that \Gammarf \Delta d is bounded above zero. The gradient descent framework outlined above provides a method for deriving the corresponding leveraging algorithm from smooth potential functions over margin space. The potential functions used by AdaBoost and arc-x4 have the advantage that all the components of their gradients D are positive, and thus it is easy to convert D into a distribution. On the other hand, the methods outlined in the previous section and section 5 can be used to handle gradients with negative components. The approach used by R-atsch et al. [13] can similarly be interpreted as a potential function of the margins. Recently, Friedman et al. [8] have given a maximum likelihood motivation for AdaBoost, and introduced another leveraging algorithm based on the log-likelihood criteria. They indicate that minimizing the square loss potential, performed less well in experiments than other monotone potentials, and conjecture that its non-monotonicity (penalizing margins greater than 1) is a contributing factor. Our methods described in section 5 may provide a way to ameliorate this problem. Convergence Bound In this section we examine the number of iterations required by GeoLev to achieve classification error ffl on the sample. The key step shows how the sine of the angle between the goal vector g and the master hypothesis H is reduced each iteration. Upper bounding the resulting recurrence gives a bound on how rapidly the training error decreases. We begin by considering a single boosting iteration. The margin space quantities are as previously defined (recall that g and H are 2-normed, while D and h are not). In addition, let H 0 denote the new master hypothesis at the end of the iteration, and ' 0 the angle between H 0 and g. We assume throughout that the sample is finite. (D \Delta h) to be the edge of the weak learner's hypothesis h with respect to the distribution given to the weak learner. Our bound on the decrease in ' will depend on h only through r and jjhjj 2 . Note that r was chosen to maintain consistency with the work of Schapire and Singer [15] and that At the start of the iteration and at the end of the iteration sin(' 0 Recall that H 0 is H+ ffh normalized, and since H already has unit length, Lemma 1. The value cos 2 (' 0 ) is maximized (and sin(' 0 ) minimized) when (g Proof The lemma follows from examination of the first and second derivatives of cos 2 (' 0 ) with respect to ff. 2 Using this value of ff a little algebra shows that Although we desire bounds that hold for all h, we find it convenient to first minimize (15) with respect to (H \Delta h). The remaining dependence on h will be expressed as a function of r and jjhjj 2 in the final bound. Lemma 2. Equation (15) is minimized when (H Proof Again the lemma follows after examining the first and second derivatives with respect to (H \Delta h). 2 This considerably simplifies (15), yielding (D Recall that (D Therefore, We will bound this in two ways, using different bounds on jjDjj 1 . The first of these bounds is derived by noting that jjDjj 1 - jjDjj 2 . Recall that ('). Combining this with (18) and the bound on jjDjj 1 yields sin s jjhjj 2: (20) Repeated application of this bound yields the following theorem. Theorem 1. If r are the edges of the weak learner's hypotheses during the first T iterations, then the sine of the angle between g and the margin vector for the master hypothesis computed at iteration T is at most Y s We can bound jjDjj 1 another way to obtain a a bound which is often better. Note that jjDjj 1 - (D \Delta Substituting this into (18) and continuing as before yields sin Continuing as above results in the following theorem. Theorem 2. Let r be the edges of the weak learner's hypotheses and be the angles between g and the margins of the master hypotheses at the start of the first T iterations. If ' T+1 is the angle between g and the margins of the master hypothesis produced at iteration T then Y s To relate these results to the sample error we use the following lemma. Lemma 3. If sin(') ! is the angle between g and a master hypothesis H, then the sample error of H is less than ffl. Proof Assume sin(') ! R=m, so is 2-normed, this can only hold if H has more than m positive components. Therefore the master hypothesis correctly classifies more examples and the sample error rate is at most (R \Gamma 1)=m. 2 Combining Lemma 3 and Theorem 2 gives the following corollary. Corollary 1. After iteration T , the sample error rate of GeoLev's master hypothesis is bounded by Y The recurrence of Theorem 2 is somewhat difficult to analyze, but we can apply the following lemma from Abe et al. [1]. Lemma 4. Consider a sequence fg t g of non-negative numbers satisfying g t+1 - positive constant. If f c all t 2 N . Given a lower bound r on the r t values and an upper bound H 2 on jjh t jj 2 , then we can apply this lemma to recurrence (22). Setting and H 2m shows that sin 2: (25) This, and the previous results lead to the following theorem. Theorem 3. If the weak learner always returns hypotheses with an edge greater than r and H 2 is an upper bound on jjh t jj 2 , then GeoLev's hypothesis will have at most ffl training error after iterations. Similar bounds have been obtained by Freund and Schapire [7] for AdaBoost. Theorem 4. After T iterations, the sample error rate of AdaBoost's master hypothesis is at most Y s 2: (27) The dependence on jjhjj 1 is implicit in their bounds and and can be removed when h t Comparing Corollary 1 and Theorem 4 leads to the following observations. First, the bound on GeoLev does not contain the square-root. If this were the only difference, then it would correspond to a halving of the number of iterations required to reach error rate ffl on the sample. This effect can be approximated by a factor of 2 on the r 2 terms. A more important difference is the factors multiplying the r 2 terms. With the preceding approximation GeoLev's bound has 2m sin 2 (' t )=jjh t jj 2 aboost's bound has 1/jjh t jj 2 1 . The larger this factor the better the bound. The dependence on sin 2 (' t ) means that GeoLev's progress tapers off as it approaches zero sample error. If the weak hypotheses are equally confident on all examples, then jjh t jj 2 2 is times larger than jjh t jj 2 1 and the difference in factors is simply 2 sin 2 (' t ). At the start of the boosting process ' t is close -=2 and GeoLev's factor is larger. However, sin 2 (' t ) can be as small as 1=m before GeoLev predicts perfectly on the sample. Thus GeoLev does not seem to gain as much from later iterations, and this difficulty prevents us from showing that GeoLev is a boosting algorithm. On the other hand, consider the less likely situation where the weak hypotheses produce a confident prediction for only one sample point, and abstain on the rest. Now jjh t jj 2 1 , and GeoLev's bound has an extra factor of about 2m sin 2 (' t ). GeoLev's bounds are uniformly better 3 than AdaBoost's in this case. 5 Conversion to an Arcing Algorithm The GeoLev algorithm discussed so far does not fit the template for Arcing algorithms because it modifies the labels in the sample given to the weak learner. 3 We must switch to recurrence (20) rather than recurrence (22) when sin 2 (' t ) is very small. This also breaks the boosting paradigm as the weak learner may be required to produce a good hypothesis for data that is not consistent with any concept in the underlying concept class. In this section we describe a generic conversion that produces arcing algorithms from leveraging algorithms of this kind without placing an additional burden on the weak learner. Throughout this section we assume that the weak learner's hypotheses produce values in [\Gamma1; +1]. The conversion introduces an wrapper between the weak learner and leveraging algorithm that replaces the sign-flip trick of section 2. This wrapper takes the weighting D from the leveraging algorithm, and creates the distribution by setting all negative components to zero and re-normalizing. This modified distribution D 0 is then given to the weak learner, which returns a hypothesis h with a margin vector h. The margin vector is modified by the wrapper before being passed on to the leveraging algorithm: if D(x i ) is negative then h i is set to \Gamma1. Thus the leveraging algorithm sees a modified margin vector h 0 which it uses to compute ff and the margins of the new master hypothesis. The intuition is that the leveraging algorithm is being fooled into thinking that the weak hypothesis is wrong on parts of the sample when it is actually correct. Therefore the margins of the master hypothesis are actually better than those tracked by the leveraging algorithm. Furthermore, the apparent "edge" of the weak learner can only be increased by this wrapping transformation. This intuition is formalized in the following theorems. Theorem 5. If is the edge of the weak learner with respect to the distribution it sees, and r is the edge of the modified weak hypothesis with respect to the (signed) weighting D requested by the leveraging algorithm, then r 0 - r. Proof ensures that both D 0 otherwise, and The assumption on h implies r - 1, so r 0 is minimized at Theorem 6. No component of the master margin vector t used by the wrapped leveraging algorithm is ever greater than the actual margins of the master hypothesis Proof The theorem follows immediately by noting that each component of h 0 is no greater than the corresponding component of h t . 2 We call the wrapped version of GeoLev, GeoArc, as it is an Arcing algorithm. It is instructive to examine the potential function associated with GeoArc: ip min This potential has a similar form to the following potential function which is zero on the entire positive orthant: ip The leveraging framework we have described together with this transformation enables the analysis of some undifferentiable potential functions. The full implications of this remain to be explored. 6 Preliminary Experiments We performed experiments comparing GeoLev and GeoArc to AdaBoost on a set of 13 datasets(the 2 class ones used in previous experiments) from the UCI repository. These experiments were run along the same lines as those reported by Quinlan [12]. We ran cross validation on the datasets for two class classification. All leveraging algorithms ran for 25 iterations, and used single node decision trees as implemented in MLC++ [9] for the weak hypotheses. Note that these are \Sigma1 valued hypotheses, with large 2-norms. It was noticed that the splitting criterion used for the single node had a large impact on the results. Therefore, the results reported for each dataset are those for the better of mutual information ratio and gain ratio. We report only a comparison between AdaBoost and GeoLev, GeoArc performed comparably to GeoLev. The results are illustrated in figure 4. This figure is a scatter plot of the generalization error on each of the datasets. These results appear to indicate that the new algorithms are comparable to AdaBoost. Further experiments are clearly warranted and we are especially interested in situations where the weak learner produces hypotheses with small 2-norm. 7 Conclusions and Directions for Further Study We have presented the GeoLev and GeoArc algorithms which attempt to form master hypotheses that are correct and equally confident over the sample. We found it convenient to view these algorithms as performing a feasible direction gradient descent constrained by the hypotheses produced by the weak learner. The potential function used by GeoLev is not monotonic: its gradient can have negative components. Therefore the direction of steepest descent cannot simply be normalized to create a distribution for the weak learner. We described two ways to solve this problem. The first constructing a modified sample by flipping some of the labels. This solution is mildly unsatisfying as it strengthens the requirements on the weak learner - the weak learner must now deal with a broader class of possible targets. Therefore we also presented a second transformation that does not increase the requirements on the weak learner. In fact, using this second transformation can actually improve the efficiency of GeoLev AdaBoost Fig. 4. Generalization error of GeoLev versus AdaBoost after 25 rounds. the leveraging algorithm. One open issue is whether or not this improvement can be exploited to improve GeoArc's performance bounds. A second open issue is to determine the effectiveness of these transformations when applied to other non-monotonic potential functions, such as those considered by Mason et al. [11]. We have upper bounded the sample error rate of the master hypotheses produced by the GeoLev and GeoArc algorithms. These bounds are incomparable with the analogous bounds for AdaBoost. The bounds indicate that Ge- oLev/GeoArc may perform slightly better at the start of the leveraging process and when the weak hypotheses contain many low-confidence predictions. On the other hand, the bounds indicate that GeoLev/GeoArc may not exploit later iterations as well, and may be less effective when the weak learner produces valued hypotheses. These disadvantages make it unlikely that the GeoArc algorithm has the boosting property. One possible explanation is that GeoLev/GeoArc aim at a cone inscribed in the positive orthant in margin space. As the sample size grows, the dimension of the space increases and the volume of the cone becomes a diminishing fraction of the positive orthant. AdaBoost's potential function appears better at navigating into the "corners" of the positive orthant. However, our preliminary tests indicate that after 25 iterations the generalization errors of GeoArc/GeoLev are similar to AdaBoost's on 13 classification datasets from the UCI repository. These comparisons used 1-node decision tree classifiers as the weak learning method. It would be interesting to compare their relative performances when using a weak learner that produces hypotheses with many low-confidence predictions. Acknowledgments We would like to thank Manfred Warmuth,Robert Schapire, Yoav Freund, Arun Jagota, Claudio Gentile and the EuroColt program committee for their useful comments on the preliminary version of this paper. --R Polynomial learnability of probabilistic concepts with respect to the Kullback-Leibler divergence A training algorithm for optimal margin classifiers. Bagging predictors. Arcing the edge. Boosting a weak learning algorithm by majority. A decision-theoretic generalization of on-line learning and an application to boosting Additive logistic re- gression: a statistical view of boosting Data mining using MLC Improved generalization through explicit optimization of margins. Bagging, boosting and c4. margins for adaboost. Boosting the margin: a new explanation for the effectiveness of voting methods. Improved boosting algorithms using confidence-rated predictions Estimation of Dependences Based on Empirical Data. --TR A theory of the learnable What size net gives valid generalization? Polynomial learnability of probabilistic concepts with respect to the Kullback-Leibler divergence Equivalence of models for polynomial learnability A training algorithm for optimal margin classifiers The design and analysis of efficient learning algorithms Learning Boolean formulas An introduction to computational learning theory Boosting a weak learning algorithm by majority Bagging predictors Exponentiated gradient versus gradient descent for linear predictors A decision-theoretic generalization of on-line learning and an application to boosting General convergence results for linear discriminant updates An adaptive version of the boost by majority algorithm Drifting games Additive models, boosting, and inference for generalized divergences Boosting as entropy projection Prediction games and arcing algorithms Improved Boosting Algorithms Using Confidence-rated Predictions An Empirical Comparison of Voting Classification Algorithms Margin Distribution Bounds on Generalization

classification;boosting;learning;gradient descent;ensemble methods

633623

Phase transition for parking blocks, Brownian excursion and coalescence.

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n > m), and places. For a noncomputer science-minded reader, we shall use the metaphore of n cars parking on m places: each car ci chooses a place pi at random, and if pi is occupied, ci tries successively finds an empty place. Pittel [42] proves that when /m goes to some positive limit > 1, the size B1m,1 of the largest block of consecutive cars satisfies 2( converges weakly to an extreme-value distribution. In this paper we examine at which level for n a phase transition occurs between o(m). The intermediate case reveals an interesting behavior of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connected components of the random graph are related to the multiplicative coalescent.

Introduction We consider hashing with linear probing for a hashing table with n places f1; 2; :::; ng, items places. Hashing with linear probing is a fundamental object in analysis of algorithms: its study goes back to the 1960's (Knuth [17], or Konheim & Weiss [19]) and is still active (Pittel [27], Knuth [18], or Flajolet et al. [11]). For a non computer science-minded reader, we shall use, all along the paper, the metaphore of m(n) cars parking on n places, leaving E(n) places empty: each car c i chooses a place p i at random, and if p i is occupied, c i tries successively nds an empty place. We use the convention that place place 1. Under the name of parking function, hashing with linear probing has been and is still studied by combinatorists (Schutzenberger [34], Riordan [30], Foata & Riordan [12], Francon [15] or Stanley 1 Institut Elie Cartan, INRIA, CNRS and Universite Henri Poincare, BP 239, 54 506 Vandoeuvre Cedex, France. [email protected] Universite Libre de Bruxelles, Departement d'Informatique, Campus Plaine, CP 212, Bvd du Triomphe, 1050 Bruxelles, Belgium. [email protected] [35, 36]). There is a nice development on the connections between parking functions and many other combinatorial objects in Section 4 of [11] (see also [35]). In this paper, and also in [20], we use mainly a { maybe less exploited { connection between parking functions and empirical processes of mathematical statistics (see also the recent paper [25]) . Pittel [27] proves that when E(n)=n goes to some positive limit , the size of the largest block of consecutive cars B (1) converges weakly to an extreme-value distribution. This paper is essentially concerned with what we would call the "emergence of a giant block" (see [5, 7] for an historic of the emergence of the giant component of a random graph, and also [2, 14, 16]): Theorem 1.1 For n and m(n) going jointly to+1, we have: in which B (1) belongs almost surely to ]0; 1[. So the threshold phenomenon is less pronounced than in the random graph pro- cess. However, the behaviour during the transition is reminiscent of the random graph process: while Aldous [2] observed a limiting behaviour of connected components for the random graph process related to the multiplicative coalescent, here, it rather seems that the additive coalescent (cf. Aldous & Pitman [4]) comes into play. Theorem 1.3 describes the random variable B 1 (). k1 be the decreasing sequence of sizes of blocks, ended by an innite sequence of 0's. Dene analogously R n;n (R (k) k1 as the sequence of sizes of blocks when the blocks are sorted by increasing date of birth (in increasing order of rst arrival of a car: for instance, on Figure 1, for Theorem 1.2 If lim E(n) The law of X() is characterized by the fact that (X 1 ()+X 2 ()+ ::: +X k ()) k1 is distributed as the sequence k1 in which the N k are standard Gaussian and independent. Figure 1: parking schemes for places. One recognizes the marginal law of the -valued fragmentation process derived from the continuum random tree, introduced by Aldous & Pitman in their study of the standard additive coalescent [4]. In order to describe the limit of Bn n , we dene a family of operators on the space E of continuous nonnegative functions f(x), y Let e be a normalized Brownian excursion, that is a 3-Bessel bridge (see [29] Chap. XI and XII for background). Let k1 be the sequence of widths of excursions of e, sorted in decreasing order. By excursion of a function f , we understand the restriction of the function f to an interval [a; b] in which f does not have any change of sign, and more precisely, such that b[. The Brownian motion is known to have innitely many excursions in the neighborhood of any of its zeros. This property holds true for e, with the exception of 0, that is a.s. an isolated point in the set of zeroes of e. Nevertheless e has, almost surely, innitely many excursions in the interval [0; 1] (it can be seen as a consequence of Theorem 4.1, or more generally, of Cameron- Martin-Girsanov formula). We have: Theorem 1.3 If lim E(n) p Incidentally, the length L() of the excursion of e beginning at 0 is studied by Bertoin [6] in a recent paper: he gives the transition kernel of the Markov process Figure 2: parking schemes for places. Theorem 1.1 is the width of the largest excursion of e. The fact that B 1 () belongs almost surely to ]0; 1[, and has a density, follows from the next Theorem about the Brownian excursion, that was in turn suggested by the more combinatorial in nature Theorem 1.2: Theorem 1.4 The law of the size-biased permutation Y () of B() satises law k1 in which the N k are standard Gaussian and independent. The size-biased permutation of a random probability distribution such as B() is constructed as follows. Consider a sequence of independent, positive, integer-valued random variables distributed according to B(): With probability 1 each positive integer appears at least once in the sequence (I k ) k1 . Erase each repetition after the rst occurence of a given integer in the sequence: remains a random permutation ((k)) k1 of the positive integers. Set: Size-biased permutations of random discrete probabilities have been studied by Aldous [3] and Pitman, [23], [24]. The most celebrated example is the size-biased permutation of the sequence of limit sizes of cycles of a random permutation. While the limit distribution of the sizes of the largest, second largest . cycle have a complicated expression (see Dickman [9], Shepp & Lloyd [32]), the size-biased permutation k1 in which the U k are uniform on [0; 1] and independent. See also the beautiful developments about Poisson-Dirichlet distributions, in [3], [21], [22], [24] and [26]. Actually, Theorem 1.4 gives a implicit description of the law of B(), for instance it proves that almost surely each Y k () is positive, and thus a.s. 1. There exist even formulas, due to Perman [21], giving the joint distribution of the k th largest weights of a random discrete probability in term of the joint densities of its size-biased permutation, in the special case where the random discrete probability comes from the order statistics for jumps of normalised subordinators these formulas do not seem to apply here. Flajolet & Salvy [13] have a direct approach, to the computation of the density of B 1 (), by methods based on Cauchy coe-cient integrals to which the saddle point method is applied: the density is a variant of the Dickman function. Note that Bertoin [6] studies the stochastic process as a tool for the study of the excursions of the re ected Brownian motion with a varying drift. He establishes interesting properties of , for instance Markov property. Finally, set: We have Theorem is the stable subordinator with exponent 1=2, meaning that, for any k and any k-tuple of positive numbers Thus, as a stochastic process, S has the same law as the process of hitting times of the Brownian motion. With Theorem 1.4, this is still another feature that Y () shares with the stochastic additive coalescent. Theorems 1.5 and 1.4 suggest that the process (Y ()) 0 has the same law as the -valued fragmentation process, providing an alternative construction of the stochastic additive coalescent. A formal proof is out of the scope of this paper (however see the concluding remarks). Theorem 1.5 has interest in itself, but it is is relevant to the study of parking schemes only if we are able to prove weak convergence of the process "size of the block containing car c 1 " to the process (Y 1 ()) 0 . The next Theorem lls incompletely this gap. Let (R (1) the sequence of successive widths R (1) n;k of the block containing car c 1 when cars are parked on n places, and k places are still empty. If k n, set R (1) R (1) n;d ne We are able to build on the same space a Brownian excursion e and a sequence of parking schemes of n cars on n places, in such a way that: Theorem 1.6 Pr We would need an almost sure convergence for the Skorohod topology, in order to ll completely the gap. The fact that S() is a pure jump process makes sense in the parking scheme context, since the block of car c 1 is known to increase by O(n) while only O( n) cars arrived: it can only be explained by coalescence with other blocks of size O(n), that is, by instantaneous jumps. The paper is organized as follows. Section 2 analyses the block containing a given car or a given site, leading to a proof of Theorem 1.1 (i),(ii). Section 3 provides a combinatorial proof of Theorem 1.2. Section 4 uses a decomposition of sample paths of e (cf. Theorem 4.1) which, we believe, has interest in itself, to provide a simple proof of Theorems 1.4 and 1.5. Theorem 4.1 is proven at Section 5, with the help of the combinatorial identity (2.2). Proofs of Theorem 1.3 and 1.1 (iii) are also given in Section 5, where we exhibit a close coupling between empirical processes of mathematical statistics and the prole associated with a parking scheme (see previous gures, and for a denition of the prole see Section 5). Section 6 is devoted to the proof of Theorem 1.6 . Section 7 concludes the paper. 2 On the block containing a given car, or a given site In this Section,we give the proof of Theorem 1.1((i) and (ii)). In order to do that, we give a partial proof of Theorem 1.2, concerning the size R (1) n;E(n) of the block containing car c 1 : we have Theorem 2.1 If n 1=2 E(n) ! > 0, R (1) law in which N is standard Gaussian. Proof : The probability that, when parking m cars on n places, the block containing car c 1 has k elements, denoted Pr(R (1) Clearly, the number of parking schemes for m cars on n places is n m . One has to choose the set of k 1 cars that belong to the same block as c 1 , giving the factor the place where this block begins, giving the factor n, the way these k cars are allocated on these k places, giving the factor nally one has to park the m k remaining cars on the n k 2 remaining places, leaving one empty place at the beginning and at the end of the block containing car c 1 , and this gives the factor (n Flajolet et al. (1998) or Knuth (1998) for justication of the third and of the last factor). Note that these computations would hold for any given car instead of c 1 . At the end of this Section, we shall prove that: Lemma 2.2 For any 0 < < 1=2 there exists a constant C() such that, whenever , simultaneously, k , we have: '(n; m; k)n f Proof of Theorem 2.1. Owing to Lemma 2.2 yields, for 0 < a < b < 1, that Pr(an R (1) a Doing Z +1e y=2 dy Thus, is a density of probability, and R (1) n;E(n) =n has for limit law (x)dx. Furthermore, the previous change of variable entails that if some random variable W has the density (x), then 2 W=(1 W ) has a 1=2;1=2 law, that is, 2 W=(1 W ) has the same law as the square of a standard Gaussian random variable. } As a consequence of Theorem 2.1, we prove now (i) and (ii) of Theorem 1.1. Considering (ii), provided that E(n) n R (1) n;d ne : As n), for any > 0 and for n large n;d ne < nx): Due to Theorem 2.1, we obtain that for any > 0 lim sup Clearly, for x < 1, Pr 0: As regards (i), let L n be the length of the block of cars containing place 1 (resp. the length of the largest block) when car c bn p nc arrives. We have, for k > 0, Pr(R (1) and place 1 is empty with probability: ne We have also and thus Assuming p we obtain that for any , when n is large enough, not depending on !, so that: lim sup nally yielding (i). } We prove (iii) of Theorem 1.1 in Section 5, together with Theorem 1.3. Proof of Lemma 2.2. Setting, for brevity, E(n), we can in We obtain n) +O( and nally: exp 3 Proof of Theorem 1.2 We rst provide a useful identity leading to the proof of Theorem 1.2. Set We have Theorem 3.1 Y Proof : The choice of the elements in each of the blocks can be done in Y ways, and they can be arranged inside each of these blocks in Y ways. It is more convenient to argue in terms of conned parking schemes, as in Knuth (1998) or in Flajolet et al. (1998): that is, we can assume the last place to be empty, since rotations does not change the sizes of blocks. The total number of conned parking schemes is n m 1 (n m). We obtain a conned parking scheme with sizes k 1 , . , for the i rst blocks, respectively, by inserting these i blocks successively, with an empty place attached to the right of them, insertion taking place at the front of the conned parking scheme for the remaining cars, or just after one of the empty places of the conned parking scheme for the remaining cars. There are possible insertions for the rst possible insertions for the second block, and so on . Finally, the probability p(k) on the left hand of Theorem 3.1 is given by Y It is not hard to check that this last expression is the same as the right hand of Theorem 3.1. } Proof of Theorem 1.2. Set: Using the same line of proof as in Theorem 2.1, the approximations of Lemma 2.2 for '(n; m; k) and Theorem 3.1 yield the joint density of (X 1 Y f Equivalently, the conditional law of X j has the f in Equivalently again, X j is distributed as in which N j is standard Gaussian and independent of (X 1 We have: and so a straightforward induction gives Theorem 1.2. } 4 On the excursions of e In this Section, we give the proofs of our last two main theorems, Theorem 1.4 and Theorem 1.5. 4.1 Decomposition of paths of e These results are simple consequences of a property of decomposition of sample paths of e that, we believe, has interest in itself: let U 1 be a random variable uniformly distributed on [0; 1] and independent of e and let D (resp. F ) denote the last zero of e before U 1 (resp. the rst zero of e after U 1 ), so that Y 1 We have: Theorem 4.1 We have: has the same distribution as N 2 , in which N is standard Gaussian ; (ii) f is a normalized Brownian excursion, independent of Y 1 (iii) Let W be uniformly distributed on [0; 1] and independent of e. Given (f; V ) distributed as T The introduction of W is a rather unpleasant feature. Actually, the study of h n suggests that, if '(t) denote the local time of r at 0, on the interval [0; t], point (iii) should be replaced by reaches its unique minimum at a point distributed as T If we could prove this last point, a second problem would arise: we do not know uniformly distributed on [0; 1] and independent of (f; Y 1 ()). However, fV + Wg has surely these properties. On the other hand, for the purpose of proving Theorem 1.2, the introduction of W is harmless, as the law of the length of the excursion that straddles U 1 is the same for T Theorem 4.1 is proven at Subsection 5.4. The starting point of the proof is the combinatorial identity: 4.2 Proof of Theorem 1.4 It should not be di-cult, following the line of proof of Theorem 4.1, subsection 5.4, to exhibit a on which there is almost sure convergence ofn (R (1) for each k, yielding Theorem 1.4. We prefer to borrow the nice idea of Section 6.4 in Pitman & Yor [26], that uses the decomposition of sample paths of a Brownian bridge to prove distributional properties of a sample from a Poisson-Dirichlet distribution. We introduce, as in [26], a sequence U being independent of e : with probability 1, U k falls inside some excursion (D of e ; if this excursion has width B j (), we dene I yielding a size-biased permutation of B(), as explained in the introduction. Set: The random variables U T (k) are independent and uniformly distributed on [0; D] [ [F; 1], and there exist a unique number V k 2]0; 1[ such that k1 is a sequence of independent random variables, uniform on [0; 1], and independent of (e; U 1 ). In view of Theorem 4.1, this leads to Lemma 4.2 Given that Y 1 the sequence Y distributed as (1 x)Y ( Actually, but among the (U i ) i2 , only the U T (k) are useful to determine Y (). Actually Y () is a size-biased permutation of the sequence of widths of excursions of r: more precisely it is the size-biased permutation built with the help of the sequence V . As a consequence, it is also the size-biased permutation of the sequence of widths of excursions of q, built with the help of the sequence ~ k1 . This ends the proof of Lemma 4.2, as ~ V is a sequence of independent and uniform random variables, independent of (r; W ). Using Lemma 4.2, we prove, by induction on k, the two following properties has the distribution asserted in Theorem 1.4 ; distributed as (1 s k )Y ( The conditional law of is the conditional law of (1 s k )Y ( Thus, due to Lemma 4.2, it has the same law as in giving point(2) for k + 1. Set: Due to point(2) for k, and to point (i) of Theorem 4.1, given that (Y j or equivalently giving 4.3 Proof of Theorem 1.5 There exists a similar result in a seemingly dierent setting, that is, for the standard additive coalescent (cf. [4]). Given that (D; F distributed as a+xU in which U is uniform on ]0; 1[. On the other hand, it is easy to see that, for t 2 [0; 1], x Thus given (D; F distributed as Since this last distribution does not depend on a, it is also the conditional distribution of (Y Equivalently, by change of variables, the conditional distribution of (S( y, is the same as the unconditional distribution of 1+y This last statement yields (1.1), by induction on k: assuming that property at we see that, given that S( 1 y, Owing to Theorem 4.1, 5 Proof of Theorems 1.3 and 4.1 denote the number of cars that tried to park on place k, successfully or not. The proof of Theorem 1.3 will be in three steps: in Subsection 5.2, using Theorems of Doob and Vervaat we shall prove that Theorem 5.1 If lim n weakly in which h is dened, with the help of a uniform random variable U independent of e, by: As a rst result, we shall establish in Subsection 5.1 a close coupling between H k and the empirical processes of mathematical statistics. Theorem 5.1 is a generalization of a similar Theorem established in Marckert & Chassaing, for the case In Subsection 5.3 we shall prove the convergence of the widths of excursions of h n to that of e, using essentially results of Section 2.3 in Aldous (1997). Figure 3: Prole. As only if place k is empty, the width of an excursion of h n turns out to be the length of some block of cars, normalized by 1=n. We shall call h n the prole of the parking scheme. 5.1 Connection between parking and empirical processes Propositions 5.3, 5.4 and 5.5 at the end of this subsection, are the key points for the convergence of blocks' sizes. The model of cars parking on places can be described by a sequence (U k ) k1 of independent uniform random variables, car c k being assumed to park (or to try to on place p i if U k falls in the interval . Let Y k denote the number of cars that tried rst to park on place k. We have: since either place k is occupied by car c i and, among the H k cars that tried place k, only car c i won't visit place k place k is empty and H We understand this equation, when This induction alone does not give the H k 's, since we do not have any starting value. We have thus to nd an additional relation, and this is the purpose of Proposition 5.2, that gives a rst connection between hashing (or parking) and the empirical process. Let V be dened by ng in which m is the empirical process of mathematical statistics (see Shorack & Wellner [33], Csorgo & Revesz [8] or Pollard [28] for background). We just recall that, given a sample (U almost all interesting statistics are functionals of the empirical distribution and that the empirical process m is dened by: The process m gives a measure of the accuracy of the approximation of the true distribution function t by the empirical distribution function Fm (t), and was, as such, extensively studied in mathematical statistics. Proposition 5.2 In the hashing table, place V (n) is empty. Proof of Proposition 5.2. Let: Since we have: clearly: Thus is nonnegative, in which we understand that Y . (while S is not a convention). Obviously some place k is empty if and only if H To end the proof, we shall assume that H V (n) is positive and we shall deduce, by induction on k, that so that no place is empty, in contradiction with m < n. Due to (5.4) for thus, due to (5.3): so that H V (n) 1 2, and that is the starting point of our induction. Now we assume that for any i k, H V (n) i 2. From (5.3) we obtain that for any i k, so that due to (5.4), Now that we know the true value of H k for some point k, namely (n), we can use (5.3) to compute each value of H k . Sizes of blocks of cars will follow, as blocks of cars are in correspondence with blocks of indices k such that H k > 0 (blocks that we call also later excursions of H k ). We nd the following explicit connection between empirical processes and H k , in the same spirit as in Marckert & Chassaing [20]. Proposition 5.3 For any k 2 f1; 2; ::: ; n 1g, that can be rewritten in terms of the empirical process: Proposition 5.4 For any k 2 fV (n) Proof of Proposition 5.3. Set: places in the set fV (n); and: Using (5.3) we obtain: To obtain Proposition 5.3, we only have to prove that R We already have There exist a last index j < k such that Z As a consequence, is the last empty place before including Now R acording to place V (n) being empty or not. But due to (5.3) and the fact that V (n) is the last empty place before place we have H V Two cases arise: either H V In the rst case we have simultaneously R In the second case H V (n)+k 1 entails both R and also W j+1 (= This ends the tedious proof by induction, but these facts will prove useful as Z k will be easier to handle than R k , when dealing with uniform convergence in the next subsection. } We notice, for further use, that we just proved that Proposition 5.5 A record of W k or of Z k means that is an empty place. 5.2 Convergence to e Let us recall that Donsker (1952), following an idea of Doob, proved that: Theorem 5.6 Let be a Brownian bridge. We have: weakly We shall also need the next Theorem to prove Theorem 5.1: Theorem 5.7 (Verwaat, 1979 [37]) Let V be the almost surely unique point such that b(V dened by normalized Brownian excursion, independent of V . Proof of Theorem 5.1. According to the Skorohod representation theorem (cf. Rogers & Williams, (1994) II.86.1) we can assume the existence of a and on this space a sequence n and a Brownian bridge b, such that, for almost any converges uniformly on [0; 1] to b(!). We could in such a way to build for each m a corresponding sequence U independent random vari- ables, and the corresponding random parking scheme. However this would not really be needed for the proof, only for the mental picture. Note that such a sequence U (m) would not necessarily be embedded in U (m+1) . Each m denes a sequence S m;n as in the previous subsection, and denes also the corresponding V (n), as follows: 1kn Finally, let: be the corresponding number of cars that tried, successfully or not, place number k, and set: z Assuming lim n E(n) Lemma 5.8 For almost any !, uniformly Lemma 5.9 For almost any !, V (n; !)=n converges to V (!). As a consequence, we have Lemma 5.10 For almost any !, uniformly and z n (t) uniformly and also: Lemma 5.11 For almost any !, uniformly Taking Theorem 5.1 is a reformulation of Lemma 5.11, since we e(ft (V (n)=n)g) uniformly Due to the estimates in the proof of Lemma 5.8, uniform convergence holds also true, indierently, for continuous and stepwise linear versions of h n , y n , z n or m (bntc=n) (we shall use this remark in the proof of Theorem 4.1). Proof of Lemma 5.8. Let M n denote max 0<kn Y k;n , where Y k;n denotes the number of cars that want to park at place k. We have: and, as Y k;n is binomially(m; 1=n) distributed: nE[exp(KY 1;n )] exp( KC log n) Thus Borel-Cantelli Lemma entails that for a suitable C, with probability 1 the supremum norm of m (bntc=n) m (t) vanishes as quickly as C log n Proof of Lemma 5.9. For this proof and the next one, we consider an ! such that simultaneously m and m (bntc=n) converges uniformly to b, and such that b reaches its minimum only once (we know that the set of such !'s has measure 1). We set: It is then straightforward, >from the continuity property of b, that the rst minimum of m (bntc=n) (i.e. V (n)=n) converges to the only minimum of b (i.e. V clearly Now the minimum of b(t) over the set [0; 1]=[V "; V +"] is b(V )+ for some positive , and thus, if necessarily jV V (n)=nj < " . } Proof of Lemma 5.10. Clearly: Proof of Lemma 5.11.Let z According to Proposition 5.3, or to Proposition 5.4, we have: 0st where 0st 0st Thus Lemma 5.11 follows from the uniform convergence of z n to z in Lemma 5.10. } 5.3 Proof of Theorem 1.3 The widths of excursions of h n (t) above zero are the sizes of the blocks of cars of the corresponding parking scheme, normalized by n. Unfortunately, uniform convergence of h n to h does not entails convergence of sizes of excursions. Using the line of proof of Aldous (1997, Section 2.3), we shall argue that the excursions of h n above 0 are also the excursions of z n above its current minimum: now the uniform convergence of z n to z entails convergence of sizes of excursions of z n above its current minimum to sizes of excursions of z above its current minimum, provided that z does never reach its current minimum two times. This last condition is classically satised for almost each sample path z, so that we have almost sure convergence of sizes of excursions of z n or equivalently of sizes of blocks. Note that excursions of z above its current minimum are also excursions of e above 0. More precisely, we shall apply to z n and z the following weakened form of Lemma 7, p. 824 of [2]: Lemma 5.12 Suppose f : [0; +1[ ! R is continuous. Let E be the set of nonempty intervals I = (l; r) such that: sl Suppose that, for intervals I 1 , I 2 2 E with l 1 < l 2 we have Suppose also that the complement of [ I2E (l; r) has Lebesgue measure 0. Let uniformly on [0; 1]. Suppose (t n;i , i 1) satisfy the following: Write m(n)g. Then (n) ! for the vague topology of measures on [0; 1] (0; 1]. As we are dealing with point processes, such as n or , we have: Proposition 5.13 (n) ! for the vague topology if and only if, for any y > 0 such that ([0; 1] (i) for n large enough, (n) ([0; 1] [y; (ii) for any x 2 [0; 1] [y; 1] such that (fxg) > 0 there is a sequence of points x n , 0, such that x n ! x. As an easy consequence, partly due to the fact that second components add up to 1: Corollary 5.14 If (n) ! for the vague topology, then the sequence of second components of points of (n) , sorted in decreasing order, converge componentwise and in ' 1 to the corresponding sequence for . The Lemmata and Propositions of this subsection are hold to be straightforward by specialists, as well as the stochastic calculus points in the next proof, so we give their proofs in the annex. If (f the sequence of > second components of (n) (resp. of ) is nothing else but 1 Proof of Theorem 1.3. Lemma 5.12 and Corollary 5.14, applied to of Subsection 5.2, entails Theorem 1.3. Hypothesis of Lemma 5.12, concerning a regular sample path of z(t), that is, almost surely, for any l 1 < l 2 , setting r), the Lebesgue measure of O c is 0, are well known to hold true, as e is solution of: du; (see [29], Chp. XI, the whole Section 3, and notably Ex. 3.11). For the t n;i 's in Lemma 5.12, we choose the records of z n (t) so, due to Lemma 5.5, the nt n;i 's are the empty places, counted starting at V (n). Then, not depending on i, we have z n, and hypothesis (iii) of Lemma 5.12 is satised by z n . } This last proof is also a proof of Theorem 1.1 is the width of the widest excursion of e. 5.4 Proof of Theorem 4.1 Theorem 4.1 is basically a consequence of the following identity (m n 2): which is equivalent to relation (2.2). Let U be uniformly distributed and independent of the Brownian excursion e. The decomposition of n m according to the length k of the block containing car c m , in the identity 5.7, yields the desintegration of e according to the value x of Y 1 () asserted in Theorem 4.1: the factor gives the distribution of the shape of the (suitably scaled) excursion ~ f n of h n corresponding to the block that contains car c 1 . Marckert & Chassaing (1999) proved that if if each of the conned parking schemes are equiprobable, then weakly we shall deduce that the shape f of the excursion of e that contains U 1 , being the limiting prole of ~ f n , is independent of its width Y 1 and is distributed as a normalized Brownian excursion. If we make a random rotation with angle d(n k 1)W e of the parking scheme of the m k remaining cars on the n k 1 remaining places, we obtain (n equiprobable parking schemes, with n m ' leading to the fact that r(fW + :g) is distributed as T Let us give a formal proof: Proof of Theorem 4.1. Let C be the space of continuous functions on [0; 1], with the topology of uniform convergence. The triplet of independent random variables denes the random variable (X 1 (); f; q) and its law Q, that is a probability measure on the space [0; 1] C 2 . The normalized Brownian excursion e resp. T denes the probability measure (resp. x ) on C. Theorem 4.1 can be written equivalently: Z 1f(; x) Z Z for any bounded uniformly continuous function on the space [0; 1] C 2 . It is harmless to assume that In order to prove Theorem 4.1, we shall exhibit a probability and, on this space, a sequence of [0; almost surely, in to (X 1 (); f; q), for the product topology of [0; ne; (iii) the conditional law, k , of f n given that X , does not depend on n and satises: weakly (iv) the conditional law, n;k , of q n given that X n;k weakly As a consequence of (i): for any bounded uniformly continuous function . We shall prove now that properties (ii) to (iv) are su-cient to insure that, for any bounded uniformly continuous function satisfying have Z 1f(; x) Z entailing (5.8). We end the proof with the construction of (X Let M be the bound for jj . Set: Z 1f(; x) Z Z a Z Z a Z Z (dnxe=n; Z Z '(n; m; Z Z By dominated convergence, owing to (iii) and (iv), lim n B A. By uniform continuity of f and , lim Finally lim n C n D due to Lemma 2. Construction of (X Assume we are in the setting of subsection 5.2, that is we use the Skorohod Theorem to obtain on some space a sequence m that converges almost surely uniformly to a Brownian bridge We enlarge this space to in order to obtain two uniform random variables U 1 and W independent of of b. We set ne for sake of brevity. We shall build our sequence (X will describe, partly, the parking scheme for m(n) cars on n places. Let us collect some basic facts concerning empirical processes: m has m positive jumps with height 1 m , at places that we call (V (m) Between the jumps m has a negative slope m. The random vector uniformly distributed on the simplex f0 < x 1 < x 2 < ::: < xm < 1g and a random permutation of its components would yield a sequence (U (m) independent uniform random variables on [0; 1], car c k trying to park rst on place l Unfortunately is not provided with the space It means that, given m , we know the number of cars that that will park on each place k, say (Y m;n is the number of jumps of m taking place in the interval n ]), but we do not know which car parks on which place. This is a slight di-culty, since we have in mind to choose nX n as the place where car c 1 parks, so that (ii) follows at once from relation (2.2). In order to circumvent this problem, note that a random permutation can be described by a random bijection from f2; 3; 4; :::; mg to f1; 2; 3; :::; m 1g and a random uniform integer (1), independent of : rst we choose (1), then we choose ::; (m)) at random from the m 1 remaining integers, renumbered >from 1 to m 1. We shall take that is U (m) car c 1 parks at place dnU (m) 1 e. We shall leave undened. In addition to the fact that (1) and U (m) are random uniform, we get that, almost surely: U (m) Incidentally, the remaining jumps are independent of U (m) 1 and uniformly distributed on the simplex f0 < x 1 < x 2 < ::: < xm 1 < 1g. the rst empty place on the left of dnU (m) e (resp. the rst empty place on the right), that is, the beginning (resp. the end) of the block containing car c 1 . Easy considerations on the uniform convergence of z n to z give that, almost surely, Recall that R (1) yielding point (ii) and also a part of point (i): almost surely, Let ~ R (n)h n Uniform convergence of h n to e, uniform continuity of e(t) and (5.10) entails the uniform convergence of ~ f n to f and the uniform convergence of ~ to As regards (iii), relation (5.7) tells us that, given that nX ~ f n is the prole associated to one of the parking schemes of k cars on k places. Accordingly, its conditional law ~ k converges weakly to . Similarly, due to the random rotation d(n R (n) e, given that nX is the prole associated to one of the (n k 1) m k equiprobable parking schemes of m k cars on n k 1 places: its conditional law ~ n;k converges weakly to x under the hypothesis of (iv). Unfortunately, ~ n;k (C), so we have a little bit of additional work: we just replace ~ f n and ~ q n by their corresponding piecewise-linear continuous approx- imations, f n and q n . Relation (5.6) insures that f n (resp. q n ) converges uniformly to f (resp. g), yielding point (i). Relation (5.6) insures also that their laws k (resp. 6 Proof of Theorem 1.6 Once again, we use the setting of subsection 5.2, that is we use the Skorohod Theorem to obtain on some space a sequence n that converges almost surely uniformly to a Brownian bridge We enlarge again the probability space, so that we have a sequence of independent uniform random variables (U k ) k1 , both independent and independent of the sequence ( m does contain the information about the number of cars that tried to park on each of the n places, but it contains no information about the chronology. We shall use (U k ) 2kn to recover the chronology, that is, to pick at random, one jump after the other, the jumps of give the place of car c 1 , through U (n) 1 dened at relation (5.9), and (U k ) 2kn will generate a random permutation of the rank of the remaining order statistics f2; 3; :::; ng. As a consequence, from a given n and (U k ) 2kn , we can recover the whole history of the process of parking n cars on n places, without losing the almost sure uniform convergence of n to a Brownian bridge, needed for the convergence of sizes of blocks. Thus for each n; k we dene a prole h n;k (t) associated with the parking of the rst cars on the n places, and we use the corresponding notation z n;k (t). The prole h n;k (t) is given by Proposition 5.4, based on the empirical process n;k (t) obtained by erasing the k last random choices of order statistics (places of jumps) of n . More precisely, let ~ uniformly distributed on the simplex denote the remaining jumps of n , once U (n) 1 has been erased. Let be the random permutation generated from (U k the rank of U k , once (U k ) 2kn is sorted in increasing order, is (k). We assert that if the rst try of car c 1 is place dnU (n) e, and the rst try of car c k (k 2) is place dn ~ e, the n n parking schemes are equiprobable. The k last random choices ( ~ provide a sample of k independent and uniform random variables, with an associated empirical process ~ n;k (t). We have: n;k (t) The DKW inequality gives Pr(sup n;k (t)j x) 4 thus, using Borel-Cantelli lemma, we obtain easily that, for " > 0, Pr sup sup Accordingly, by a simple glance at the proof of Lemma 5.9, we see that, if V (n; denotes the rst minimum of , the convergence of V (n; k)=n to V , for n, is uniform, almost surely. Compared with Lemma 5.9, we just have to change slightly the denitions of u sup By inspection of relation 5.5, we see that the convergence of Z n (; ne (t) to Z(; t, uniformly for (; holds true almost surely, that is Pr uniformly on Z owing to 6.12 and to the fact that sup ne ne 0: R (k) n;d ne Lemma 5.12 yields Proposition 6.1 Pr Theorem 1.6 follows at once. } 7 Concluding remarks The main pending question, in our opinion, is about does it provide an alternative construction of the stochastic additive coalescent ? This construction would then complete the parallel between the additive coalescent and the multiplicative coalescent, given in the concluding remarks of [4], as the stochastic multiplicative coalescent was identied by Aldous [2] as the limit of the normalized sequence of sizes of connected components in the random graph model, and also as the sequence of widths of excursions of a simple stochastic process. An easy corrolary of Theorems 4.1, 1.4 and 1.5 is that we have the joint law of the sequence of shapes and sizes (or widths) of excursions of e, the sizes being given by Theorem 1.4: an easy induction proves that the shapes are independent, distributed as e, and independent of the sizes. An argument in the proof of Theorem then says that each of the clusters of the fragmentation process B() starts anew a fragmentation process distributed as , given that the cluster has size x. It rises the following questions: is this also a property of the standard additive coalescent ? Is it a charasteristic property of the standard additive coalescent Which additional properties are eventually needed to characterize the standard additive coalescent ? The scaling factor x in is easily understood, but the time change is less clear. However it is explained asymptotically by a property of parking schemes: the time unit for the discrete fragmentation process associated with parking n cars on n places, is the departure of p cars. Due to the law of large numbers, during one time unit, a given block of cars with size xn loses approximately x x xn cars, meaning that, for the internal clock of this block, p Acknowledgements The starting point for this paper was the talk of Philippe Flajolet at the meeting ALEA in February 1998 at Asnelles (concerning his paper with Viola & Poblete), and a discussion that Philippe and the authors had in a small cafe of the French Riviera after the SMAI meeting of September 1998. Some discussions of the rst author with Marc Yor, and also with Uwe Rossler were quite fruitful. We thank Philippe Laurencot for calling our attention to the works of Aldous, Pitman & Evans on coalescence models. The papers of Perman, Pitman, & Yor about random probability measures and Poisson-Dirichlet processes were also of a great help. --R The continuum random tree. critical random graphs and the multiplicative coalescent. Exchangeability and related topics. The standard additive coalescent. The probabilistic method. A fragmentation process connected with Brownian motion. On the frequency of numbers containing prime factors of a certain relative magnitude. On the Analysis of Linear Probing Hashing Mappings of acyclic and The birth of the giant compo- nent The art of computer programming. Linear Probing and Graphs Order statistics for jumps of normalised subordinators. Exchangeable and partially exchangeable random partitions. Random discrete distributions invariant under size-biased permuta- tion A polytope related to empirical distribu- tions The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator the probable largest search time grows logarithmically with the number of records. Convergence of stochastic processes. Ballots and trees. Ordered cycle lengths in a random permutation. Empirical processes with applications to statis- tics On an enumeration problem. in Mathematical Essays in Honor of Gian-Carlo Rota (B A relation between Brownian bridge and Brownian excursion --TR Linear probing: the probable largest search time grows logarithmically with the number of records The first cycles in an evolving graph The art of computer programming, volume 3 --CTR Philippe Chassaing , Guy Louchard, Reflected Brownian Bridge area conditioned on its local time at the origin, Journal of Algorithms, v.44 n.1, p.29-51, July 2002 Svante Janson, Individual displacements for linear probing hashing with different insertion policies, ACM Transactions on Algorithms (TALG), v.1 n.2, p.177-213, October 2005 Jean Bertoin, Random covering of an interval and a variation of Kingman's coalescent, Random Structures & Algorithms, v.25 n.3, p.277-292, October 2004

brownian excursion;hashing with linear probing;coalescence;empirical processes;parking

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Feature selection with neural networks.

We present a neural network based approach for identifying salient features for classification in feedforward neural networks. Our approach involves neural network training with an augmented cross-entropy error function. The augmented error function forces the neural network to keep low derivatives of the transfer functions of neurons when learning a classification task. Such an approach reduces output sensitivity to the input changes. Feature selection is based on the reaction of the cross-validation data set classification error due to the removal of the individual features. We demonstrate the usefulness of the proposed approach on one artificial and three real-world classification problems. We compared the approach with five other feature selection methods, each of which banks on a different concept. The algorithm developed outperformed the other methods by achieving higher classification accuracy on all the problems tested.

Introduction Learning systems primary source of information is data. For numerical systems like Neural Networks (NNs), data are usually represented as vectors in a subspace of R k whose components - or features - may correspond for example to measurements performed on a physical system or to information gathered from the observation of a phenomenon. Usually all features are not equally informative: some of them may be noisy, meaningless, correlated or irrelevant for the task. Feature selection aims at selecting a subset of the features which is relevant for a given problem. It is most often an important issue: the amount of data to gather or process may be reduced, training may be easier, better estimates will be obtained when using relevant features in the case of small data sets, more sophisticated processing methods may be used on smaller dimensional spaces than on the original measure space, performances may increase when non relevant information do not interfere, etc. Feature selection has been the subject of intensive researches in statistics and in application domains like pattern recognition, process identification, time series modelling or econometrics. It has recently began to be investigated in the machine learning community which has developed its own methods. Whatever the domain is, feature selection remains a difficult problem. Most of the time this is a non monotonous problem, i.e. the best subset of p variables does not always contain the best subset of q variables (q < p). Also, the best subset of variables depends on the model which will be further used to process the data - usually, the two steps are treated sequentially. Most methods for variable selection rely on heuristics which perform a limited exploration on the whole set of variable combinations. In the field of NNs, feature selection has been studied for the last ten years and classical as well as original methods have been employed. We discuss here the problem of feature selection specifically for NNs and review original methods which have been developed in this field. We will certainly not be exhaustive since the literature in the domain is already important, but the main ideas which have been proposed are described. We describe in sections 2 and 3 the basic ingredients of feature selection methods and the notations. We then briefly present, in section 4, statistical methods used in regression and classification. They will be used as baseline techniques. We describe, in section 5, families of methods which have been developed specifically for neural networks and may be easily implemented either for regression or classification tasks. Representative methods are then compared on different test problems in section 6. . Basic ingredients of feature selection methods. A feature selection technique typically requires the following ingredients: . a feature evaluation criterion to compare variable subsets, it will be used to select one of these subsets, . a search procedure, to explore a (sub)space of possible variable combinations, . a stop criterion or a model selection strategy. 2.1.1 Feature evaluation Depending on the task (e.g. prediction or classification) and on the model (linear, logistic, neural networks.), several evaluation criteria, based either on statistical grounds or heuristics, have been proposed for measuring the importance of a variable subset. For classification, classical criteria use probabilistic distances or entropy measures, often replaced in practice by simple interclass distance measures. For regression, classical candidates are prediction error measures. A survey of classical statistical methods may be found in (Thomson 1978) for regression and (McLachlan 1992) for classification. Some methods rely only on the data for computing relevant variables and do not take into consideration the model which will then be used for processing these data after the selection step. They may rely on hypothesis about the data distribution (parametric methods) or not (non parametric methods). Other methods take into account simultaneously the model and the data - this is usually the case for NN variable selection. 2.1.2 Search In general, since evaluation criteria are non monotonous, comparison of feature subsets amounts to a combinatorial problem (there are 2 k -1 possible subsets for k variables), which rapidly becomes computationally unfeasible, even for moderate input size. Branch and Bound exploration (Narendra and Fukunaga 1977) allows to reduce the search for monotonous criteria, however the complexity of these procedures is still prohibitive in most cases. Due to these limitations, most algorithms are based upon heuristic performance measures for the evaluation and sub-optimal search. Most sub-optimal search methods follow one of the following sequential search techniques (see e.g. Kittler, . start with an empty set of variables and add variables to the already selected variable set (forward methods) . start with the full set of variables and eliminate variables from the selected variable set (backward methods) . start with an empty set and alternate forward and backward steps (stepwise methods). The Plus l - Take away r algorithm is a generalisation of the basic stepwise method which alternates l forward selections and r backward deletions. 2.1.3 Subset selection - Stopping criterion Let be given a feature subset evaluation criterion and a search procedure. Several methods examine all the subsets provided by the search (e.g. 2 k -1 for an exhaustive search or k for a simple backward search) and select the most relevant according to the evaluation criterion. When the empirical distribution of the evaluation measure or of related statistics is known, tests may be performed for the (ir)relevance hypothesis of an input variable. Classical sequential selection procedures use a stop criterion: they examine the variables sequentially and stop as soon as a variable is found irrelevant according to a statistical test. For classical parametric methods, distribution characteristics (e.g. estimates of the evaluation measure variance) are easily derived (see sections 4.1 and 4.2). For non parametric or flexible methods like NNs, these distributions are more difficult to obtain. Confidence intervals which would allow to perform significance testing might be computed via monte carlo simulations or bootstrapping. This is extremely prohibitive and of no practical use except for very particular cases (e.g. Baxt and White 1996). Hypothesis testing is thus seldom used with these models. Many authors use instead heuristic stop criteria. A better methodology, whose complexity is still reasonable in most applications, is to compute for the successive variable subsets provided by the search algorithm an estimate of the generalization error (or prediction risk) obtained with this subset. The selected variables will be those giving the best performances. The generalization error estimate may be computed using a validation set or cross-validation or algebraic methods although the latter are not easy to obtain with non linear models. Note that this strategy involves retraining a NN for each subset. 3 . Notations We will denote ( , ) k g the realization of a random variable pair (X,Y) with probability distribution P. x i will be the i th component of x and x l the l th pattern in a given data set D of cardinality N. In the following, we will restrict ourselves to one hidden layer NNs, the number of input and output units will be denoted respectively by k and g. The transfer function of the network will be denoted f. Training will be performed here according to a Mean Squared Error criterion (MSE) although this is not restrictive. We will consider, in the selection methods for classification and regression tasks. 4 . Model independent Feature Selection We introduce below some methods which perform the selection and the classification or regression steps sequentially, i.e. which do not take into account the classification or regression model during selection. These methods are not NN oriented and are used here for the experimental comparison with NN specific selection techniques (section 6). The first two are basic statistical techniques aimed respectively at regression and classification. These methods are not well fitted for NNs since the hypothesis they rely on do not correspond to situations where NNs might be useful. However since most NN specific methods are heuristics they should be used for a baseline comparison. The third one has been developed more recently and is a general selection technique which is data hypothesis free and might be used for any system either for regression or classification. It is based on a probabilistic dependence measure between two sets of variables. 4.1 Feature selection for linear regression We will consider only linear regression, but the approach described below may be trivially extended for multiple regression. Let x 1 , x 2 , . x k and y be real variables which will be supposed centered. Let us denote: the current approximation of y with p selected variables (the x i are renumbered so that the p first selected variables correspond to numbers 1 to p). The residuals x are assumed identically and independently distributed. Let us denote: l l =1 For forward selection, the choice of the p th variable is usually based on R p 2 , the partial correlation coefficient (table 1) between y and regressor f (p) , or on an adjusted coefficient 1 . This coefficient represents the proportion of y total variance explained by the regressor f (p) . The p th variable to select is the one for which f (p) maximizes this coefficient. The importance of a new variable is usually measured via a Fisher test (Thompson, 1978) which compares the models with p-1 and p variables (Fs(p) forward in table 1). Selection is stopped if 1 The adjusted coefficient R is often used instead of R p . Fs(p) forward < F(1,N-p,a) the Fisher statistics with (1,N-p) degrees of freedom for a confidence level of a. Choice Stop (y l ) 2N Backward SSR l =1 Table 1: Choice and Stop criteria used with statistical forward and backward methods. Note that F S could also be used in place of R p 2 as a choice criterion: forward When p-1 variables have already been selected, R p-1 2 has a constant value in [0,1] and maximizing F s is similar to maximizing R p 2 . Equation (4.1.3) selects variables in the same order as R p 2 does. For backward elimination the variable eliminated from the remaining p is the less significant in terms of the Fisher test i.e. it is the one with the smallest value of SSR p-1 or equivalently of Fs(p) backward (table 1). Selection is stopped if backward > F(1,N-p,a). 4.2 Feature Selection For Classification For classification, we shall select the variable subset which allows the best separation of the data. Variable selection is usually performed by considering a class separation criterion for the choice criterion and an associated F-test as stopping criterion. As for regression, forward, backward or stepwise methods may be used. Data separation is usually computed through an inter-class distance measure (Kittler, 1986). The most frequent discriminating measure is the Wilks lambda (Wilks, 1963) L sV p defined as follows: where W is the intra-class matrix dispersion corresponding to the selected variable set SV p , B the corresponding inter-class matrix 2 and |M| the determinant of matrix M. The determinant of a covariance matrix being a measure of the volume occupied by the data, |W| measures the mean volume of the different classes and |W+B| the volume of the whole data set. These quantities are computed for the selected variables so that a good discriminating power corresponds to a small value of L sv p : the different classes are represented by compact clusters and are well separated. This criterion is well suited in the case of multinormal distributions with equal covariance for each class, it is meaningless for e.g. multimodal distributions. This is clearly a very restrictive hypothesis. With this measurement the statistic F s , defined below, has a F(g-1,N-g-p+1,a) distribution (McLachlan 1992): We can then use the Wilks lambda both for estimating the discriminating power of a variable and for stopping the selection in forward, backward (Habbema and Hermans, 1977) or stepwise methods. For the comparisons in section 6, we used Stepdisc, a stepwise method based on (4.2.2) with a 95% confidence level. 4.3 Mutual Information When data are considered as realization of a random process, probabilistic information measures may be used in order to compute the relevance of a set of these two quantities are defined as : x l -class j with g the number of classes, n j the number of samples in class j, - j the mean of class j and - the global mean. variables with respect to other variables. Mutual information is such a measure which is defined as: a,b where a and b are two variables with probability density P(a) and P(b). Mutual information is independent from any inversible and differentiable transformation of the variables. It measures the "uncertainty reduction" on b when a is known. It is also known as the Kullbak-Leibler distance between the joint distribution P(a,b) and the marginal distribution product P(a)*P(b). The method described below does not make use of restrictive assumptions on the data and is therefore more general and attractive than the ones described in sections 4.1 and 4.2, especially when these hypothesis do not correspond to the data processing model, which is usually the case for NNs. It may be used either for regression or discrimination. On the other hand such non parametric methods are computationally intensive. The main practical difficulty here is the estimation of the joint density P(a,b) and of the marginal densities P(a) and P(b). Non parametric density estimation methods are costly in high dimensions and necessitate a large amount of data. The algorithm presented below uses the Shannon entropy (denoted H(.)) to compute the mutual information It is possible to use other entropy measures like quadratic or cubic entropies (Kittler, 1986). Battiti (1994) proposed to use mutual information with a forward selection algorithm called MIFS (Mutual Information based Feature Selection). P(a,b) is estimated by Fraser algorithm (Fraser and Swinney, 1986), which recursively partitions the space using c 2 tests on the data distribution. This algorithm can only compute the mutual information between two variables. In order to compute the mutual information between x p and the selected variable set SV p-1 does not belong to SV p-1 ), Battiti uses simplifying assumptions. Moreover, the number of variables to select is fixed before the selection. This algorithms uses forward search and variable x p is the one which maximises the value : where SV p-1 is the set of p - 1 already selected variables. Bonnlander and Weigend (1994) use Epanechnikov kernels for density estimation (H-rdle, 1990) and a Branch&Bound (B&B) algorithm for the search (Narendra and Fukunaga, 1977). B&B warrants an optimal search if the criterion used is monotonous and it is less computationally intensive than exhaustive search. For the search algorithm, one can also consider the suboptimal floating search techniques proposed by Pudil et al. (1994) which offer a good compromise between the sequential methods simplicity and the relative computational cost of the Branch&Bound algorithm. For the comparisons in section 6, we have used Epanechnikov kernels for density estimation in (4.3.2), a forward search, and the selection is stopped when the MI increase falls below a fixed threshold (0.99). 5 . Model dependent feature selection for Neural Networks Model dependent feature selection attempts to perform simultaneously the selection and the processing of the data: the feature selection process is part of the training process and features are sought for optimizing a model selection criterion. This "global optimization" looks more attractive than model-independent selection where the adequacy of the two steps is up to the user. However, since the value of the choice criterion depends on the model parameters, it might be necessary to train the NN with different sets of variables: some selection procedures alternate between variable selection and retraining of the model parameters. This forbids the use of sophisticated search strategies which would be computationally prohibitive. Some specificities of NNs should also be taken into consideration when deriving feature selection algorithms: . NNs are usually non linear models. Since many parametric model-independent techniques are based on the hypothesis that input-output variables dependency is linear or that input variables redundancy is well measured by linear correlation between these variables, such methods are clearly ill fitted for NNs. . The search space has usually many local minima, and relevance measures will depend on the minimum the NN will have converged to. These measures should be averaged over several runs. For most applications this is prohibitive and has not been considered here. . Except for (White 1989) who derives results on the weight distribution there is no work in the NN community which might be used for hypothesis testing. For NN feature selection algorithms, choice criteria are mainly based on heuristic individual feature evaluation functions. Several of them have been proposed in the literature, we have made an attempt to classify them according to their similarity. We will distinguish between: . zero order methods which use only the network parameter values. . first order methods which use the first derivatives of network parameters. . second order methods which use second derivatives of network parameters. Most feature evaluation criteria only allow to rank variables at a given time, the value of the criterion by itself being non informative. However, we will see that most of these methods work reasonably well. Feature selection methods with neural networks use mostly backward search although some forward methods have also been proposed (Moody 1994, Goutte 1997). Several methods use individual evaluation of the features for ranking them and do not take into consideration their dependencies or their correlations. This may be problematic for selecting minimal relevant sets of variables. Using the correlation as a simple dependence measure is not enough since NNs capture non linear relationships between variables, on the other hand, measuring non linear dependencies is not trivial. While some authors simply ignore this problem, others propose to select only one variable at a time and to retrain the network with the new selected set before evaluating the relevance of remaining variables. This allows to take into account some of the dependencies the network has discovered among the variables. More critical is the difficulty for defining a sound stop criterion or model choice. Many methods use very crude techniques for stopping the selection, e.g. a threshold on the choice criterion value, some rank the different subsets using an estimation of the generalization error. The latter is the expected error performed on future data and is defined as: where in our case, r(x,y) is the euclidean error between desired and computed outputs. Estimates can be computed using a validation set, cross-validation or algebraic approximations of this risk like the Final Prediction 1970). Several estimates have been proposed in the statistical (Gustafson and Hajlmarsson 1995) and NN (Moody 1991, Larsen and Hansen 1994) literature. For the comparison in section 6, we have used a simple threshold when the authors gave no indication for the stop criterion and a validation set approximation of the risk otherwise. 5.1 Zero Order Methods For linear regression models, the partial correlation coefficient can be expressed as a simple function of the weights. Although this is not sound for non linear models, there have been some attempts for using the input weight values in the computation of variable relevance. This has been observed to be an inefficient heuristic: weights cannot be easily interpreted in these models. A more sophisticated heuristic has been proposed by Yacoub and Bennani (1997), it exploits both the weight value and the network structure of a multilayer perceptron. They derived the following criterion: I O where I, H, O denote respectively the input, hidden and output layer. For a better understanding of this measure, let us suppose that each hidden and output unit incoming weight vector has a unitary L 1 norm, the above equation can be written as: In (5.1.2), the inner term is the product of the weights from input i to hidden unit j and from j to output o. The importance of variable i for output o is the sum of the absolute values of these products over all the paths -in the NN- from unit i to unit o. The importance of variable i is then defined as the sum of these values over all the outputs. Denominators in (5.1.1) operate as normalizing factors, this is important when using squashing functions, since these functions limit the effect of weight magnitude. Note that this measure will depend on the magnitude of the input, the different variables should then be in a similar range. The two weight layers do have different role in a MLP which is not reflected in (5.1.1), for example, if the outputs are linear, the normalization should be suppressed in the inner summation of (5.1.1). They used a backward search and the NN is retrained after each variable deletion, the stop criterion is based on the evolution of the performances on a validation set, elimination is stopped as soon as performances decrease. 5.2 First Order Methods Several methods propose to evaluate the relevance of a variable by the derivative of the error or of the output with respect to this variable. These evaluation criteria are easy to compute, most of them lead to very similar results. These derivatives measure the local change in the outputs wrt a given input, the other inputs being fixed. Since these derivatives are not constant like in linear models, they must be averaged over the training set. For these measures to be fully meaningful, inputs should be independent and since these measures average local sensitivity values, the training set should be representative of the input space. 5.2.1 Saliency Based Pruning (SBP) This backward method (Moody and Utans 1992) uses as evaluation criterion the variation of the learning error when a variable x i is replaced by its empirical mean here since variables are assumed centered): where MSE x l l l l l This is a direct measure of the usefulness of the variable for computing the output. For large values of N, computing S i is costly, and a linear approximation may be used: f y x l l l l l x Variables are eliminated in the increasing order of S i . For each feature set, a NN is trained and an estimate of the generalization error - a generalization of the Final Prediction Error criterion - is computed. The model with minimum generalization error is selected. Changes in MSE is not ambiguous only when inputs are not correlated. Variable relevance being computed once here, this method does not take into account possible correlations between variables. Relevance could be computed from the successive NNs in the sequence at a computational extra-cost (O(k 2 computations instead of O(k) in the present method). 5.2.2 Methods using computation of output derivatives For a linear model the output derivative wrt any input is a constant, which is not the case for non linear NNs. Several authors have proposed to measure the sensitivity of the network transfer function with respect to input x i by computing the mean value of outputs derivative with respect to x i over the whole training set. In the case of multilayer perceptrons, this derivative can be computed progressively during learning (Hashem, 1992). Since these derivatives may take both positive and negative values, they may compensate and produce an average near zero. Most measures use average squared or absolute derivatives. Tenth of measures based on derivatives have been proposed, and many others could be defined, we thus give below only a representative sample of these measures. The sum of the derivative absolute values has been used e.g. in Ruck et al. l =1 For classification Priddy et al. (1993) remark that since the error for decision j x may be estimated by 1 - f j (x) , (5.2.3) may be interpreted as the absolute value of the error probability derivative averaged over all decisions (outputs) and data. Squared derivatives may be used instead of the absolute values, Refenes et al. (1996) for example proposed for regression a normalized sum: x f y f x l l x x where var holds for variance. They also proposed a series of related criteria, among - a normalized standard deviation of the derivatives: f x f x f x l l l l x - a weighted average of the derivatives absolute values where the weights reflect the relative magnitude of x and f(x): f x x f l i l l x x All these measures being very sensitive to the input space representativeness of the sample set, several authors have proposed to use a subset of the sample in order to increase the significance of their relevance measure. In order to obtain robust methods, "non-pathological" training examples should be discarded. For regression and radial basis function networks, Dorizzi et al. (1996) propose to use the 95% percentile of the derivative absolute value: -f Aberrant points being eliminated, this contributes to the robustness of the measure. Note that the same idea could be used with other relevance measures proposed in this paper. Following the same line, Czernichow (1996) proposed a heuristic criterion for regression, estimated on a set of non pathological examples whose cardinality is N'. The proposed choice criterion is: -f -l -f -l For classification, Rossi (1996), following a proposition made by Priddy et al. (1993), considers only the patterns which are near the class frontiers. He proposes the following relevance measure: f x f x l frontier l x x x The frontier is defined as the set of point for which - ( ) > x l f x e where e is a fixed threshold. Several authors have also considered relative contribution of partial derivatives to the gradient as in (5.2.9). All these methods use a simple backward search. For the stopping criteria, all these authors use heuristic rules, except for Refenes et al. (1996) who define statistical tests for their relevance measures. For non linear NNs, this necessitates an estimation of the relevance measure distribution, which is very costly and in our opinion usually prohibits this approach, even if it looks attractive. 5.2.3 Links between these methods All these methods use simple relevance measures which depend upon the gradient of network outputs with respect to input variables. It is difficult to rank the different criteria, all that can be said is that it is wise to use some reasonable rules like discarding aberrant points for robustness, or retraining the NN after discarding each variable and computing new relevance measures for each NN in the sequence, in order to take into account dependencies between variables. In practice, all these methods give very similar results as will be shown in section 6. We summarize below in table 2 the main characteristics of relevance measures for the different methods. Derivative used Task C/R Data used (Moody (5.2.1)) -f C/R All (Refenes (5.2.5)) -f C/R All (Dorizzi (5.2.7)) -f C/R Non pathological data (Refenes (5.2.6)) -f C/R All (Czernichow (5.2.8)) -f -C/R Non pathological data (Refenes (5.2.4)) -f (Ruck (Rossi C Frontier between classes Table 2. Computation of the relevance of a variable by different methods using the derivative of the network function. C /R denote respectively Classification and Regression tasks. 5.3 Second Order Methods Several methods propose to evaluate the relevance of a variable by computing weight pruning criteria for the set of weights of each input node. We present below three methods. The first one is a Bayesian approach for computing the weight variance. The other two use the hessian of the cost function for computing the cost function dependence upon input unit weights. 5.3.1 Automatic Relevance Determination (ARD) This method was proposed by MacKay (1994) in the framework of Bayesian learning. In this approach, weight are considered as random variables and regularization terms taking into account each input are included into the cost function. Assuming that the prior probability distribution of the group of weights for the i th input is gaussian, the input posterior variance s i 2 is estimated (with the help of the hessian matrix). ARD has been successful for time serie prediction, learning with regularization terms improved the prediction performances. However ARD has not really been used as a feature selection method since variables were not pruned during training. 5.3.2 Optimal Cell Damage Several neural selection methods have been inspired by weight pruning techniques. For the latter, the decision of pruning a weight is made according to a relevance criterion often named the weight saliency, the weight being pruned if its saliency is low. Similarly, the saliency for an input cell is usually defined as the sum of its weights saliencies. where fan-out(i) is the set of weights of input i. Optimal Cell Damage (OCD) has been proposed by Cibas et al. (1994a, 1996) similar method has also been proposed by Mao et al., 1994). This feature selection method is inspired from the Optimal Brain Damage (OBD) weight pruning technique developed by LeCun (1990). In OBD, the connection saliency is defined by : which is an order two Taylor expansion of MSE variation around a local minimum. The Hessian matrix H can be easily computed using gradient descent but this may be computationally intensive for large networks. For OBD, the authors use a diagonal approximation for the hessian which can then be computed in O(N). The saliency of an input variable is defined accordingly as: Cibas et al. (1994) proposed to use (5.3.5) as a choice criterion for eliminating variables. The NN is trained so as to reach a local minimum, variables whose saliency is below a given threshold are eliminated. The threshold value is fixed by cross validation. This process is then repeated until no variable is found below the threshold. This method has been tested on several problems and gave satisfying results. Once again, the difficulty lies in selecting an adequate threshold. Furthermore, since several variables can be eliminated simultaneously whereas only individual variable pertinence measures are used, significant sets of dependent variables may be eliminated. For stopping, the generalization performances of the NN sequence are estimated via a validation set and the variable set corresponding to the NN with the best performances is chosen. The hessian diagonal approximation has been questioned by several authors, Hassibi and Stork (1993), for example, proposed a weight pruning algorithm, Optimal Brain Surgeon (OBS) which is similar to OBD, but uses the whole hessian for computing weight saliencies. Stahlberger and Riedmiller (1997) proposed a feature selection method similar to OCD except that it takes into account non diagonal terms in the hessian. For all these methods, saliency is computed using for performance measure the error variation on the training set. Weight estimation and model selection both use the same data set, which is not optimal. Pedersen et al. (1996) propose two weight pruning methods gOBD and gOBS that compute weight saliency according to an estimate of the generalization error: the Final Prediction Error (Akaike 1970). Similarly to OBD and OBS, these methods could be also transformed into feature selection methods. 5.3.3 Early Cell Damage (ECD) Using a second order Taylor expansion, as in the OBD family of methods, is justified only when a local minimum is reached and the cost is locally quadratic in this minimum. Both hypothesis are barely met in practice. Tresp et al. (1997) propose two weight pruning techniques from the same family, coined EBD (Early Brain Damage) and EBS (Early Brain Surgeon). They use a heuristic justification to take into account early stopping by adding a new term in the saliency computation. These methods can be extended for feature ranking, we will call ECD (Early Cell Damage) the EBD extension. For ECD, the saliency of input i is defined as: The algorithm we propose is slightly different from OCD: only one variable is eliminated at a time, and the NN is retrained after each deletion. For choosing the "best" set of variables, we have used a variation of the "selection according to an estimate of the generalization error" method. This estimate is computed using a validation set. Since the performances may oscillate and be not significantly different, several subsets may have the same performances (see e.g. figure 1). Using a Fisher test we compare any model performances with those of the best model, we then select the set of networks whose performances are similar to the best ones and choose among these networks the one with the smallest number of input variables. 6 . Experimental comparison We now present comparative performances of different feature selection methods. Comparing these methods is a difficult task: there is not a unique measure which characterizes the importance of each input, the selection accuracy also depends on the search technique and on the variable subset choice criterion. In the case of NNs, these different steps rely on heuristics which could be exchanged from one method to the other. The NNs used are multilayer perceptrons with one hidden layer of 10 neurons. The comparison we provide here is not intended for a definite ranking of the different methods but for illustrating the general behavior of some of the methods which have been described before. We have used two synthetic classification problems which illustrate different difficulties of variable selection. In the first one the frontiers are "nearly" linear and there are dependent variables as well as pure noise variables. The second problem has non linear frontiers and variables can be chosen independent or correlated. The first problem has been originally proposed by Breiman et al. (1984). It is a three class waveforms classification problem with 19 noisy dependent features. We have also used a variation of this problem where 21 pure noise variables are added to the 19 initial variables (there are 40 inputs for this variant). The training set has 300 patterns and the test set 4300. A description of this problem is provided in the appendix. The performances of the optimal Bayes classifier estimated on this test set are 86% correct classification. A performance comparison appears in tables 3 and 4 for these two instances. Method p * Selected Variables Perf . Stepdisc (4.2.2) 14 000110111111111011100 0000000000000000000 (Bonnlander (4.3.2)) 12 000011101111111110000 0000000000000000000 (Moody (Ruck (5.2.3)) (Dorizzi (5.2.7)) (Czernichow (5.2.8)) 17 010111111111111111100 0000000000000000000 (Cibas (5.3.5)) 9 000001111110111000000 0000000000000000000 82.26 % (Leray (5.3.6)) 11 000001111111111100000 0000000000000000000 Table 3. Performance comparison of different variable selection on the noisy wave problem. For the noisy problem, all methods do eliminate pure noise variables. Except for the two methods at the bottom of table 3 which give slightly lower performances and select fewer variables , all give similar values around 85% correct. Stepdisc also gives good performances since in this problem data have a unimodal distribution and the frontiers are nearly linear. For the non noisy problem, performances and methods ordering change. The two techniques at the bottom of table 4 give now slightly better performances. Method p * Selected Variables Perf . None 21 111111111111111111111 Stepdisc (4.2.2) 14 001110101111111011100 (Bonnlander (4.3.2)) 8 000001100111101010000 (Moody (5.2.1)) (Ruck (5.2.3)) (Dorizzi (Czernichow (Cibas (5.3.5)) 15 001011111111111110100 (Leray Table 4. Performance comparison of different variable selection methods on the original wave problem. Number of remaining variables ECD OCD Figure 1. Performance comparison of two variable selection methods (OCD and ECD) according to the number of remaining variables for the noisy wave problem. Figure shows performance curves for two methods, OCD and ECD, estimated on a validation set. Since we have used a single validation set, there are small fluctuations in the performances. Some form of cross validation should be used in order to get better estimates, the test strategy proposed for ECD looks also attractive in this case. It can be seen that for this problem, performances are more or less similar during the backward elimination (they slightly rise) until they quickly drop when relevant variables are removed.842060100 None Yacoub Moody Cibas Leray Ruck Stepdisc Bonnlander Czernichow Figure 2. Performance comparison of different variable selection methods vs. percentage of selected variables on the original wave problem. x axis: percentage of variables selected, y axis: percentage of correct classification. Figure 2 gives the repartition of different variable selection methods for the original wave problem according to their performances (y axis) and the percentage of selected variables (x axis). The best methods are those with the best performances and the lower number of variables. In this problem, "Leray" is satisfying (see figure 2). "Yacoub" does not delete enough variables while "Bonnlander" deletes too much variables. The second problem is a two class problem in a 20 dimensional space. The classes are distributed according two gaussians with respectively - 1 =(0,.,0), (a is chosen so that ||- 1 - 2 In this problem, variable relevance is ordered according to their index: x 1 is useless, x i+1 is more relevant than x i . Method p * Selected Variables Perf . Stepdisc (4.2.2) 17 10001111111111111111 (Bonnlander (4.3.2)) 5 00010000000000011011 90.60 % 94.86 % (Moody (5.2.1)) 9 01000100011000110111 92.94 % (Ruck 94.86 % (Dorizzi (5.2.7)) 11 00000000101111111111 (Czernichow (5.2.8)) 9 00000000001101111111 (Cibas (Leray (5.3.6)) 15 01011011101110111111 Table 5. Performance comparison of different variable selection methods on the two gaussian problem with uncorrelated variables.9193954080% None Yacoub Stepdisc Leray Cibas Dorizzi Ruck Czernichow Moody Bonnlander Figure 3. Performance comparison of different variable selection methods vs. percentage of selected variables on the two gaussian problem with uncorrelated variables. x axis: percentage of variables selected, y axis: percentage of correct classification. Table 5 shows that Stepdisc is not adapted for this non linear frontier: it is the only method that selects x 1 which is useless for this problem. We can remark on figure 3 that Bonnlander's method deletes too many variables whereas Yacoub's stop criterion is too rough and does not delete enough variables. In an other experiment, we replaced the I matrix in S 1 and S 2 by a block diagonal matrix. Each block is 5x5 so that there are four groups of five successive correlated variables in the new problem. Method p * Selected Variables Perf . 90.58 % Stepdisc (4.2.2) 11 00001101011010110111 (Bonnlander (4.3.2)) 5 00001001010000100001 (Ruck 91.06 % (Leray 90.72 % Table 6. Performance comparison of different variable selection methods on the two gaussian problem with correlated variables. Table 6 gives the results of some representative methods for this problem: . Stepdisc still gives a model with good performances but selects many correlated variables, . Bonnlander's method selects only 5 variables and gives significantly lower results, . Ruck's method obtains good performances but selects some correlated variables, . Leray's method, thanks to the retraining after each variable deletion, find models with good performances and few variables (7 compared to 10 and 11 for Ruck and Stepdisc). 7 . Conclusion We have reviewed variable selection methods developed in the field of Neural Networks. The main difficulty here is that NNs are non linear systems which do not make use of explicit parametric hypothesis. As a consequence, selection methods rely heavily on heuristics for the three steps of variable selection : relevance criterion, search procedure - NN variable selection use mainly backward search - and choice of the final model. We first discussed the main difficulties for developing each of these steps. We then introduced different families of methods and discussed their strengths and weaknesses. We believe that a variable selection method must remain computationally feasible for being useful, and we have not considered techniques which rely on computer intensive methods like e.g. bootstrap at each step of the selection . Instead, we have proposed a series of rules which could be used in order to enhance several of the methods which have been described, at a reasonable extra computational cost, e.g. retraining each NN in the sequence and computing the relevance for each of these NN allows to take into account some correlations between variables, simple estimates of the generalization error may be used for the evaluation of a variable subset, simple tests on these estimates, allow to choose minimal variable sets (section 5.3.3). Finally we performed a comparison of representative NN selection techniques on synthetic problems. --R Statistical Predictor Identification Using Mutual Information for Selecting Features in Supervised Neural Net Learning Bootstrapping confidence intervals for clinical input variable effects in a network trained to identify the presence of acute myocardial infraction Selecting Input Variables Using Mutual Information and Nonparametric Density Evaluation Classification and Regression Trees. Fogelman Souli- Fogelman Souli- Architecture Selection through Statistical Sensitivity Analysis. Independent Coordinates for Strange Attractors from Mutual Information Extracting the Relevant Decays in Time Series Modelling Gustafson and Hajlmarsson Selection of Variables in Discriminant Analysis by F-statistic and Error Rate Applied Nonparametric Regression. Econometric Society Monograph n. Sensitivity Analysis for Feedforward Artificial Neural Networks with Differentiable Activation Functions. Second Order Derivatives for Network Pruning Feature Selection and Extraction Generalized performances of regularized neural networks models. Bayesian Non-linear Modelling for the Energy Prediction Competition Discriminant Analysis and Statistical Pattern Recognition Note on generlization Principled Architecture Selection for Neural Networks: Application to Corporate Bond Rating Prediction. Prediction Risk and Architecture Selection for Neural Networks in From Statistics to Neural Networks - Theory and Pattern Recognition Applications A Branch and Bound Algorithm for Feature Subset Selection. Floating search methods in feature selection. Pattern Recognition Letters Attribute Suppression with Multi-Layer Perceptron Fast Network Pruning and Feature Extraction Using the Unit-OBS Algorithm Selection of Variables in Multiple Regression. Learning in Artificial Neural Networks Mathematical Statistics HVS: A Heuristic for Variable Selection in Multilayer Artificial Neural Network Classifier. --TR Feature selection for automatic classification of non-Gaussian data Image enhancement and thresholding by optimization of fuzzy compactness Introduction to statistical pattern recognition (2nd ed.) Floating search methods in feature selection Merging back-propagation and Hebbian learning rules for robust classifications Feature Selection Neural Networks for Pattern Recognition --CTR E. Gasca , J. S. Snchez , R. Alonso, Rapid and brief communication: Eliminating redundancy and irrelevance using a new MLP-based feature selection method, Pattern Recognition, v.39 n.2, p.313-315, February, 2006 Rajen B. Bhatt , M. Gopal, On fuzzy-rough sets approach to feature selection, Pattern Recognition Letters, v.26 n.7, p.965-975, 15 May 2005 M. Bacauskiene , A. Verikas, Selecting salient features for classification based on neural network committees, Pattern Recognition Letters, v.25 n.16, p.1879-1891, December 2004 Chao-Ton Su , Long-Sheng Chen , Tai-Lin Chiang, A neural network based information granulation approach to shorten the cellular phone test process, Computers in Industry, v.57 n.5, p.412-423, June 2006 D. Franois , F. Rossi , V. Wertz , M. Verleysen, Resampling methods for parameter-free and robust feature selection with mutual information, Neurocomputing, v.70 n.7-9, p.1276-1288, March, 2007 R. E. Abdel-Aal, GMDH-based feature ranking and selection for improved classification of medical data, Journal of Biomedical Informatics, v.38 n.6, p.456-468, December 2005

regularization;classification;feature selection;neural network

63382

Stochastic Petri Net Analysis of a Replicated File System.

The authors present a stochastic Petri net model of a replicated file system in a distributed environment where replicated files reside on different hosts and a voting algorithm is used to maintain consistency. Witnesses, which simply record the status of the file but contain no data, can be used in addition to or in place of files to reduce overhead. A model sufficiently detailed to include file status (current or out-of-date) as well as failure and repair of hosts where copies or witnesses reside, is presented. The number of copies and witnesses is not fixed, but is a parameter of the model. Two different majority protocols are examined.

Introduction Users of distributed systems often replicate important files on different hosts, to protect them from a subset of host failures. The consistency of the files across the database is difficult to maintain manually, and so a data abstraction, the replicated file, has been introduced to automate the update procedure [Ellis83, Popek81, Stonebraker79]. The consistency of the files is most often maintained by assigning votes to each copy of the file and by automatically assembling a quorum (majority) of votes for a file access. Requiring a majority for each update insures that at most one write set can exist at any time, and that a quorum automatically includes at least one of the most recently updated copies. Such a system tolerates host failures to the extent that a minority of votes may be unavailable at any time, but a file update will still be permitted. P-aris suggests replacing some copies with witnesses, which contain no data but which can testify to the current state of the copies. Witnesses have low storage costs and are simple to update [Paris86a, Paris86]. In [Paris86] it is believed that the replacement of some copies with witnesses has a minor impact on the availability of the file system. When an availability analysis of such a replicated file system is performed, a Markov chain is often used [Jajodia87, Paris86a, Paris86]. If the Markov chain is constructed manually, the analysis is often limited to small systems (perhaps three copies) modeled at the highest level. Without the proper tools, it is difficult to develop a Markov model of a large system detailed enough to include both failure and repair of hosts, and voting and updating procedures. A higher level "language" is needed for the description of the system to allow the automatic generation of the Markov chain. The stochastic Petri net model provides such a language. We present a stochastic Petri net model for a distributed file system, whose structure is independent of the number of copies and witnesses. This model is automatically converted into a Markov chain and solved numerically for the state probabilities (the generated Markov chains have up to 1746 states). After describing the replicated file system being considered and defining the stochastic Petri net model, we incrementally develop a model that includes copies and witnesses, failure and repair, requests and voting. We then solve the model for the availability of the filesystem under static and adaptive voting algorithms. Then we examine the performance/reliability tradeoffs associated with preferring copies vs. preferring witnesses to participate in a quorum, when more than a majority are available. Replicated File System Mutual consistency of the various representatives of a replicated file can be maintained in at least three ways [Ellis83]: writing to all copies at each update, designating a 'primary' copy as leader [Parker83, Stonebraker79], or using various weighted voting algorithms [Davcev85, Ellis83, Gifford79, Jajodia87, Paris86a, Paris86, Thomas79]. In a weighted voting algorithm, each copy is assigned a number of votes; a set of copies representing a majority of votes (a quorum) must be assembled for each update. Copies may be assigned different weights (number of votes), including none, and different quorums can be defined for read and write operations. Consistency is guaranteed as long as the quorums are high enough to disallow simultaneous read and write operations on disjoint sets of copies. The simplest quorum policies require a majority of copies to participate in any read or write, by assigning one vote to each. However, the system administrator can alter the performance by manipulating the voting structure of a replicated file [Gifford79]. Associated with each representative is a timestamp, or version number, which is increased each on each update. At any time, the copies representing a majority of votes have the same (highest) version number and the assemblage of a majority is guaranteed to contain a representative with the highest version number. Copies with the same version number are identical. Each time a quorum of copies is gathered, any out-of-date copy participating in the quorum is brought up-to-date before the transaction is processed. Copies not participating in the quorum become obsolete, since they will no longer have the highest version number. P-aris suggests replacing some of the copies by witnesses to decrease the overhead of maintaining multiple copies of a file. Witnesses contain no data, but can vote to confirm the current state of copies. They can be created and maintained easily; they simply contain version numbers reflecting the most recent write observed. A quorum can be gathered counting witnesses as copies, but at least one current copy must be included. If there is no copy with the same version number as the highest witness' version number, then an update cannot be performed. P-aris further suggests transforming witnesses into copies and vice versa, as needed. Assumptions In the model developed, we assume for simplicity that each representative (copy or witness) is assigned one vote. This vote assignment is called the uniform assignment in [Barbara87], where it is shown to optimize reliability for fully connected, homogeneous systems with perfect links. We also assume that copies and witnesses are not transformed into the other. Further, we assume the following scenario for the gathering of a quorum. When a request is received, a status request message is sent to each site known to contain a representative. Each representative residing on a functioning host computer answers, communicating its status. If more than a majority of representatives are available, a subset is chosen to participate in the update. The selection process of the members in the participating subset tries to minimize the time needed to service the request by including as many current representatives as possible, preferably witnesses since they are fast to update. A quorum is thus formed by first choosing a current copy, then, as needed, current witnesses, more current copies, outdated witnesses, and outdated copies, in that order. All current copies and witnesses participating in the quorum are updated, while the remaining ones become outdated, since they will not have the most recent timestamp. The host computers can fail and be repaired, thus not all representatives are available at all times. A representative on a machine that has failed is assumed to be out-of-date when the machine is subsequently repaired. This is a conservative, but not unrealistic, assumption. It is possible to exactly model whether a representative has gone out-of-date while the host was down, but we chose not to do so because the model would be more complex. If a quorum cannot be formed when a request is received, a manual reconstruction is initiated, to restore the system to its original state. Again, to keep the model from becoming too complex, we only consider write requests and we assume that there is a time lapse between requests. We also assume that hosts do not fail while a quorum is being gathered. Hosts may fail during the update procedure. All times between events are assumed exponentially distributed, to allow a Markov chain analysis. If the times have a general distribution, semi-Markov analysis or simulation would be needed [Bechta84a]. 4 Stochastic Petri Net A Petri net is a graphical model useful for modeling systems exhibiting concurrent, asynchronous or nondeterministic behavior [Peterson81]. The nodes of a Petri net are places (drawn as circles), representing conditions, and transitions (drawn as bars), representing events. Tokens (drawn as small filled circles) are moved from place to place when the transitions fire, and are used to denote the conditions holding at any given time. As an event is usually enabled by a combination of conditions, a transition is enabled by a combination of tokens in places. An arc is drawn from a place to a transition or from a transition to a place. Arcs are used to signify which combination of conditions must hold for the event to occur and which combination of conditions holds after the event occurs. If there is an arc from place p to transition t, p is an input place for t, if there is an arc from t to p, p is an output place for t. (These conditions are not exclusive.) A transition is enabled if each input place contains at least one token; an enabled transition fires by removing a token from each input place and depositing a token in each output place. Stochastic Petri nets (SPN) were defined by associating an exponentially distributed firing time with each transition [Molloy81, Natkin80]. A SPN can be analyzed by considering all possible markings (enumerations of the tokens in each place) and solving the resulting reachability graph as a Markov chain. Generalized stochastic Petri nets (GSPN) allow immediate (zero firing time) and timed (exponentially distributed firing time) transitions; immediate transitions are drawn as thin bars, timed transitions are drawn as thick bars. GSPN are solved as Markov chains as well [Ajmone84]. Extended stochastic Petri nets (ESPN) [Bechta84a, Bechta85] allow transition times to be generally distributed. In some cases an ESPN can be solved as a Markov chain or as a semi-Markov process, otherwise it can be simulated. The model used in this paper is basically the GSPN with variations from the original (or the current) definition, so we will use the term stochastic Petri net (SPN) with the generic meaning of "Petri net with stochastic timing". Three additional features to control the enabling of transition are included in our SPN model: inhibitor arcs, transition priorities, and enabling functions. An inhibitor arc [Peterson81] from a place to a transition disables the transition if the corresponding input place is not empty. If several transitions with different priorities are simultaneously enabled in a marking, only the ones with the highest priority are chosen to fire, while the others are disabled. An enabling function is a logical function defined on the marking, if it evaluates to false, it disables the transition. More specifically, a transition t is enabled if and only if (1) there is at least one token in each of its input places, (2) there is no token in any of its inhibiting places, (3) its enabling funtion evaluates to true, and (4) no other transition u with priority over t and satisfying (1), (2), and (3) exists. In the description, each transition is labeled with a tuple ("name","priority","enabling function"), where the enabling function is identically equal to true if absent. The priority is indicated by an a lower number indicates a higher priority. If several enabled transitions are scheduled to fire at the same instant, a probability distribution (possibly marking-dependent) is defined across them to determine which one(s) will fire. In the SPN we present, only immediate transitions require the specification of these probabilities, since the probability of contemporary firing for continuous (exponential) distributions is null. 5 Model of Quorum Consider the SPN shown in figure 1 (part of a larger one considered later). At a certain moment in time, the SPN contains a token in the place labeled START, and a number of tokens in the places labeled CC (current copies), CW (current witnesses), OW (outdated witnesses), and OC (outdated copies), representing the number of representatives in the corresponding state at the beginning of the observation. The remaining places are empty. Assume that it is possible to reach a quorum, that is, there is a token in the CC place and that a minority of the hosts are down. Transition T16 is enabled and can fire after an exponentially distributed amount of time signifying the time needed to send status request messages and receive responses. After T16 fires, the assembly of the quorum begins. It is likely that more than a majority of representatives respond; if so, a subset of them must be selected to participate in the update. As mentioned earlier, we would like to minimize the time needed to perform the update, so we prefer to include as many witnesses as possible. We must always include a current copy in the quorum, so a token is removed from the CC place and deposited in the QC (quorum copies) place when T16 fires. A token is also deposited in the DRIVE place, to start the gathering of other representatives. Transitions T 17, T 18, T19 and T20 represent the inclusion of current witnesses, current copies, outdated witnesses and outdated copies in the quorum, respectively. We want these transitions to fire in the order T 17, T 18, T19 and T 20, so they are assigned priorities of 2, 3, 4, and 5 respectively. We want these transitions to fire only until a majority is reached, so we associate an enabling function, q with them. The logical function q evaluates to true if a majority of representatives has been selected to participate in the update transaction; q is the logical complement of q. Transition T17 will fire once for each current witness, then transition T18 will fire once for each current copy. Then outdated witnesses will be updated, and if necessary, outdated copies will be made current. These last two events requires some time, so transitions T19 and T20 are timed. As soon as q evaluates to true, the gathering will stop, so T 18, T 19, and T20 might not fire as many times as they are enabled. When q evaluates to true, transition T 26, with priority 6, is enabled. At this point, the tokens in places QC and QW (quorum witnesses) denote the participants in the update. The sum of the tokens in these two places represents a majority of the representatives in the file system. We want to embed this SPN describing the gathering of a quorum within a larger SPN including failures, repairs, and the update procedure. We will consider the quorum gathering model as a subnet activated when a token is deposited in the START place and exited when a token is deposited in the DONE place. The CC, CW, OW, and OC places will be shared with the larger net. The shared places are drawn with a double-circle in all the figures. The numbers we used for the transitions correspond to one of the two instances of the subnet. The numbers for the second instance are T 21, T 22, T 23, T 24, T 25, T 27, respectively, instead of T 16, T 17, T 18, T 19, T 20, T 26. 6 The File System Model The overall model for the distributed file system is shown in figure 2, where the contents of each of the boxes labeled "Form Quorum" is the SPN shown in figure 1, with the exception of the CC, CW, OC, and OW places, which are shared with the overall model. The left portion of figure 2 models the failure and repair of hosts where copies reside; the right portion models the failure and repair of hosts where witnesses reside; the center portion models an update transaction. We will describe each portion separately. Places CW, DW, and OW (on the far right in the figure) represent the number of witnesses that are current, down (host has failed), and out-of-date. Transition T2 represents the failure of the host (we assume that all hosts are identical), transition T4 represents the repair of the host, and transition T40 represents failure of the host of an out-of-date witness. Since we assume that a witness goes out-of-date when the host fails, the output place for transition T4 is place OW. A similar structure for the copies consists of transitions T 1, T3 and T 39, and places CC, DC, and OC. We assume that a copy on a host that has failed and is subsequently repaired will check for the existence of a quorum in the file system, possibly bringing itself up-to-date. This action is represented by the box labeled "Form Quorum 2". When a host on which a copy resides is repaired, its corresponding token is deposited in the OC place, but a token is also deposited in the TRY place to represent the attempt to form a quorum. It is then possible that the out-of-date copy needs to be brought up-to-date before reaching a quorum (transition T41 insures that no more than one token exists in place TRY). When a token is deposited in place TRY, if it is possible to reach a quorum, then transition T15 fires and a token is deposited in the START.2 place in the SPN of figure 1. Also, a token is deposited in place INQ.2, which is simply used to signify that a quorum is being formed. The QC, DONE, and QW places in the quorum gathering SPN are the same as those shown in the overall SPN. The CC, CW, OC and OW places in the quorum gathering SPN are the same as the respective places in the overall SPN. Thus, there are arcs from these places into the boxes that are not shown explicitly in figure 2; these arcs are shown explicitly in figure 1. After a quorum is gathered and the out-of-date copy is updated (transition T 13), the representatives are returned to their respective places (transitions T12 and T 14). Transition T13 has higher priority than transitions T12 or T 14, so that it fires before them. The center section of the SPN models the normal update process. Transition T0 models the time lapse between update requests. When a request is received, a token is deposited in place REQ. If a quorum can be formed (function f evaluates to true), transition T5 will fire and deposit a token in the START.1 place. The box labeled "Form Quorum 1" is another instance of the quorum gathering SPN, with the QC.1, QW.1, and DONE.1 places in common with the overall SPN. Once the quorum gathering is completed, the remaining current representatives become outdated (transitions T8 and T 10). When these two transitions have moved all the tokens from CC and CW to OC and OW respectively, T9 fires, allowing the update to occur. At this point, the representatives may be updated (transitions T6 and T 7), or the hosts on which they reside may fail (transitions T28 and T 29), after which we await the next request. 7 Manual Reconstruction of File Configuration The SPN just defined produces a Markov chain with absorbing states, which correspond to situations where a quorum cannot be formed because more than a majority of hosts are down. Since we wanted to analyze the steady-state availability of the system, we introduce a manual intervention to reconstruct the system when a quorum is not possible. Figure 3 shows the SPN used to model the manual reconstruction of the file structure. When a request is received and it is not possible to gather a quorum (function f evaluates to false), transition T30 (in figure 2) fires and deposits a token in the START place of figure 3, which we now describe. The firings of transitions T31 and gather together the outdated representatives and deposit the corresponding tokens in places C and W. The failed hosts must be repaired before their representatives can be gathered; the repair is represented by timed transitions T36 and T 37. When all the representatives have been gathered, transitions T33 and T34 may then begin firing, to bring representatives up-to-date. After these two transitions have emptied places C and W, transitions T38 can fire, putting a token in DONE. Transition T35 (in figure 2) can then remove this token and the token residing in INREC, restoring the initial configuration. 8 Model Specification The model is specified to the solution package in CSPL (C-based Stochastic Petri net Language) based on the C programming language. A set of predefined functions available for the definitions of SPN entities distinguishes CSPL from C. Some of these predefined functions are place, trans, iarc, oarc, and harc, used for defining places, transitions, input arcs, output arcs and inhibitor arcs, respectively. There are also built-in functions for debugging the SPN and for specifying the solution method and desired output measures. The enabling, distribution, and probability functions for each transition are defined by the user as C functions; marking-dependency can be obtained using other built-in functions such as mark("place"), returning the number of tokens in the specified place. The enabling functions q and f , used respectively in the quorum gathering subnet (figure 1) and to trigger the reconstruction subnet (in figure 2) are logical functions defined on the current marking of the SPN. To keep these functions simple, we defined three places in the overall SPN, INQ.1, INQ.2, and INREC. There is a token in place INQ.1 if a quorum is being formed for a normal update; a token is in place INQ.2 if an out-of-date copy is checking to see if a valid quorum is possible; place INREC holds a token during manual reconstruction of the file system. The CSPL code describing q and f respectively is the following (in the C programming language "&&" means "logical and" and ' '!" means "logical not"): int q() f else Thus, q evaluates to true iff more than half of the total number of representatives is in the places QW and QC (at least one of them is a copy, given the structure of our SPN). int f() f if (mark(CC) && !mark(INQ.1) && !mark(INQ.2) && else Thus, if there is a current copy (a token in place CC), and if no other quorum or reconstruction is in progress (no token in any of places INQ.1, INQ.2 or INREC), and if more representatives are up than are down, the transition is enabled. The firing rates for some timed transitions depend on the number of tokens in the input place. As an example, transition T1 fires at a rate equal to the failure rate of an individual host multiplied by the number of operational hosts (tokens in place CC). float where "phi" is a floating point constant representing the failure rate of a single host. The marking- dependent firing rates for the other transitions representing host failures are defined similarly. 9 Model Experimentation For two different quorum definitions, we varied the mean time to repair failed hosts, the number of copies, and the number of witnesses. For both sets of experiments, we assumed the parameters listed in table 1 and considered the mean time to repair to be 1 hour, 2 hours, or 8 hours. In the first set of experiments, we defined a quorum to be the majority of all representatives (a quorum must include a current copy) as previously discussed. The second set of experiments relaxed this definition by requiring a quorum to consist of a majority of representatives on operational hosts. For example, suppose that a total of 3 copies and 2 witnesses were created, but only one of each is now available and current. Given the requirement that a majority of all representatives must participate in a quorum, an update cannot be serviced now. However, if we require that a quorum consist of a majority of representatives on operational hosts, a quorum can be formed using the available copy and witness. Under the second assumption, updates can continue until the last copy fails, then a manual reconstruction must be performed. This idea was concurrently investigated by Jajodia and Mutchler in [Jajodia87], where it was termed dynamic voting. A similar approach is the adaptive voting used in the design of fault-tolerant hardware systems [Siewiorek82]. To model adaptive voting instead of static voting, only the definitions of q and f need to be changed, to eliminate the dependency on places DW and DC. We analyzed the availability of the system by observing the probability that a token is in place INREC. This measure represents the steady-state probability that the system is undergoing manual reconstruction of the file system. The availability of the file system, the complement of this probability, is shown under the two quorum definitions in table 2 for hours. The model of the system with one copy had 11 states, while the model of the system with 4 copies and 3 witnesses had 1746 states. When hours, we can see how the adaptive voting technique increases the availability if the total number of representatives is small. This technique may decrease availability if there are many representatives (see for example the configuration with 5 copies and 2 witnesses). The availability of the system decreases as copies are replaced by witnesses for both voting techniques up to a certain point, then it increases again. This behavior can be explained considering the factors affecting the overall availability. First of all, replacing copies with witnesses decreases the total number of copies, so the probability of having all the copies down (regardless of the state of the witnesses) is higher, with a negative effect on the availability. An especially sharp decrease in the availability is experienced going from dn=2e + 1 to dn=2e, when n representatives are available (we present only the case n odd, see [Jajodia87] for a discussion of the case n even), the reason is the quorum policy that we are assuming. Most of the time no host is down, so, assuming 7, the quorum will contain two copies and two witnesses in the (5,2) case, and one copy and three witnesses in the (4,3) case. Having only one copy in the quorum implies that whenever that copy becomes unavailable, the other copies will be useless, because they will be out-of-date. The increase of availability after the point dn=2e is less intuitive, the main cause is probably the quorum policy itself (this suggests that further investigations on the quorum policies and their effect on the availability are needed). When hours, the adaptive voting technique always increases availability although the improvement decreases as the number of witnesses increases. With 5 representatives, allowing one of the representatives to be a witness increases the availability under static voting, but decreases it under adaptive voting. With 7 representatives, the assignment of two of them to witness status maximizes availability under static voting. However, the adaptive algorithm still performs better. Under the assumptions stated, the addition of witnesses to a fixed number of copies does not increase the availability. This can be seen by comparing the results for 3 copies and 0 witnesses (0.99908), 3 copies and 2 witnesses (0.95992), and 3 copies and 4 witnesses (0.95663) (see table adaptive voting). The reason for the decrease in availability when adding extra witnesses can be explained by examining the procedure used for the selection of the participants in a quorum. The desire to speed the quorum gathering process gives preference to witnesses in the quorum, since the update procedure for a witness is fast. However, this preference increases the probability of copies being out-of-date. Since a current copy is necessary for a quorum and redundant copies are often out-of-date, a decrease in availability results. In trying to improve the availability without considering performance, the quorum gathering procedure would give preference to copies as participants. This can be easily reflected in the SPN model under consideration. Only the priorities associated with the transitions in figure 1 need to be changed, to include current copies, outdated copies, current witnesses and then outdated witnesses, in that order. Table 3 compares the availability with 3 copies and 0, 2, or 4 witnesses with the two different preferences in the formation of the quorum using the adaptive voting algorithm. In this case, the addition of witnesses favorably affects the availability. The preference for copies in the quorum increases the time needed to perform an update, since it takes longer to write a copy than a witness. In table 4, we compare the steady-state probability of the system being in the "update" state. We can estimate this probability from the SPN by looking at the probability that a token is in place INQ.1 (figure 2): a token is more likely to be in place INQ.1 when preference in forming a quorum is given to copies over witnesses. Conclusions We have presented a detailed stochastic Petri net model of a replicated file system for availability analysis. The model included failure and repair of hosts, file updates, quorum formation, and manual reconstruction of the file system. The total number of representatives of a file was varied between one and seven; the composition of the representatives (number of copies and witnesses) was varied as well. Using this model, we investigated two different voting algorithms used to maintain file consis- tency. The first algorithm, static voting, required a majority of all representatives to participate in an update. The second algorithm, adaptive voting, required a majority of representatives on operational units to participate in an update. In most cases, the adaptive voting algorithm resulted in a higher availability of the file system. The effect was more pronounced when the mean time to repair a host was higher, and thus a smaller number of hosts was available at a given time. We then examined the process by which actual participants in an update are selected, given that more than a majority are available. A desire for a fast update process leads to a preference for witnesses in a quorum, with the effect of decreasing the availability of the file system. This decrease in availability can be attributed to the fact that representative not partecipants in the quorum become outdated, and so copies are frequently out-of-date. To increase the availability of the file system, preference is given to copies as participants in the update, with an unfavorable effect on performance. Since it takes longer to perform an update on a copy than on a witness, it takes longer to service each request if more copies are quorum participants. Future modeling efforts will look at the effect of converting copies to witnesses and vice versa and at the effect of using different quorum policies on the availability of the file system. We will also investigate heuristics for determining when to perform the conversion. We espect to be able to explore a wide number of policies by making minor changes or additions to the SPN we have presented. --R A class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor The Reliability of Voting Mechanisms. Extended Stochastic Petri Nets: Applications and Analysis. The Design of a Unified Package for the Solution of Stochastic Petri Net Models. Consistency and Recovery Control for Replicated Files. The Roe File System. Weighted Voting for Replicated Data. Dynamic voting. On the Integration of Delay and Throughput Measures in Distributed Processing Models. Reseaux de Petri Stochastiques. Voting with Witnesses: A Consistency Scheme for Replicated Files. Voting with a Variable Number of Copies. Detection of Mutual Inconsistency in Distributed Systems. Petri Net Theory and the Modeling of Systems. LOCUS: A Network Transparent The Theory and Practice of Reliable System Design. Concurrency Control and Consistency of Multiple Copies of Data in Distributed INGRES. A Majority Consensus Approach for Concurrency Control for Multiple Copy databases. --TR A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems The reliability of voting mechanisms Dynamic voting A Majority consensus approach to concurrency control for multiple copy databases Consistency and recovery control for replicated files Petri Net Theory and the Modeling of Systems Extended Stochastic Petri Nets The Design of a Unified Package for the Solution of Stochastic Petri Net Models Weighted voting for replicated data LOCUS a network transparent, high reliability distributed system --CTR G. Chiola , M. A. Marsan , G. Balbo , G. Conte, Generalized Stochastic Petri Nets: A Definition at the Net Level and its Implications, IEEE Transactions on Software Engineering, v.19 n.2, p.89-107, February 1993 Yiannis E. Papelis , Thomas L. Casavant, Specification and analysis of parallel/distributed software and systems by Petri nets with transition enabling functions, IEEE Transactions on Software Engineering, v.18 n.3, p.252-261, March 1992 Yuan-Bao Shieh , Dipak Ghosal , Prasad R. Chintamaneni , Satish K. Tripathi, Modeling of Hierarchical Distributed Systems with Fault-Tolerance, IEEE Transactions on Software Engineering, v.16 n.4, p.444-457, April 1990 Changsik Park , John J. Metzner, Efficient Location of Discrepancies in Multiple Replicated Large Files, IEEE Transactions on Parallel and Distributed Systems, v.13 n.6, p.597-610, June 2002

performance reliability tradeoffs;concurrency control;majority protocols;distributed databases;fault tolerant computing;file status;voting algorithm;witnesses;stochastic Petri net model;replicated file system;distributed environment;petri nets

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Recurrence equations and their classical orthogonal polynomial solutions.

The classical orthogonal polynomials are given as the polynomial solutions pn(x) of the differential equation (x)y''(x) is a polynomial of at most second degree and (x) is a polynomial of first degree.In this paper a general method to express the coefficients An, Bn and Cn of the recurrence equation in terms of the given polynomials (x) and (x) is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three-term recurrence equation, i.e., a homogeneous linear three-term recurrence equation with polynomial coefficients.In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equation (x)y(x) denote the forward and backward difference operators, respectively, and a similar approach applies to classical q-orthogonal polynomials, being solutions of the q-difference equation denotes the q-difference operator.

Introduction Families of orthogonal polynomials p n (x) (corresponding to a positive-definite measure) satisfy a three-term recurrence equation of the form with C n A n A e.g. [6, p. 20]. Moreover, Favard's theorem states that the converse is also true. On the other hand, in practice one is often interested in an explicit solution of a given recurrence equation. Therefore it is an interesting question to ask whether a given recurrence equation has classical orthogonal polynomial solutions. In this paper an algorithm is developed which answers this question for a large class of classical orthogonal polynomial systems. Furthermore, we present results of our corresponding Maple implementation retode and compare these with the Maple implementation rec2ortho of Koornwinder and Swarttouw [13]. These programs overlap, but rec2ortho does not cover Bessel, Hahn and q-polynomials, whereas retode does not include the Meixner- case. Classical Orthogonal Polynomials A family of polynomials of degree exactly n is a family of classical continuous orthogonal polynomials if it is the solution of a differential equation of the type is a polynomial of at most second order and is a polynomial of first order ([4], [14]). Since one demands that p n (x) has exact degree n, by equating the coefficients of x n in (3) one gets Similarly, a family p n (x) of polynomials of degree exactly n, given by (2), is a family of classical discrete orthogonal polynomials if it is the solution of a difference equation of the type where denote the forward and backward difference operators, respectively, and and are again polynomials of at most second and of first order, respectively, see e.g. [19]. Again, (4) follows. Finally, a family p n (x) of polynomials of degree exactly n, given by (2), is a family of classical q-orthogonal polynomials if it is the solution of a q-difference equation of the type where denotes the q-difference operator [7], and are again polynomials of at most second and of first order, respectively. By equating the coefficients of x n in (6) one gets where the abbreviation denotes the so-called q-brackets. Note that lim q!1 It can be shown (see e.g. [15]) that any solution p n (x) of either (3), (5) or (6) satisfies a recurrence equation (1). The following is a general procedure to find the coefficients of the recurrence equation (as well as of similar structural formulas for classical orthogonal polynomials, see [11]) in terms of the coefficients a; b; c; d and e of oe(x) and -(x): 1. Substitute p n in the differential equation (3), in the difference equation (5) or in the q-difference equation (6), respectively. 2. Equating the coefficients of x n yields - n , given by (4) and (7), respectively. 3. Equating the coefficients of x respectively, as rational multiples of k n . 4. Substitute p n (x) in the proposed equation, and equate again the three highest coeffi- cients. In the case of the recurrence equation (1), this yields and e by linear algebra. 5. Substituting the values of k 0 n given in step (3) in these equations yields the three unknowns in terms of a; b; c; d; e; n; k With regard to the recurrence equation coefficients, we collect these results in Theorem 1 Let p n family of polynomial solutions of the differential equation (3). Then the recurrence equation (1) is valid with and in terms of the coefficients a; b; c; d and e of the given differential equation. family of polynomial solutions of the difference equation (5). Then the recurrence equation (1) is valid with and in terms of the coefficients a; b; c; d and e of the given difference equation. family of polynomial solutions of the q-difference equation (6). Then the recurrence equation (1) is valid with a a and in terms of the coefficients a; b; c; d and e of the given q-difference equation. 2 3 The Inverse Characterization Problem It is well-known ([4], see also [5], [14]) that polynomial solutions of (3) can be classified according to the zeros of oe(x), leading to the normal forms of Table 1 besides linear transformations x 7! Ax+B. The type of differential equation that we consider is invariant under such a transformation. family 1. 2. 1 \Gamma2x H n (x) Hermite polynomials 3. x \Gammax 5. Table 1: Normal Forms of Polynomial Solutions This shows that the only orthogonal polynomial solutions are linear transforms of the Hermite, Laguerre, Bessel and Jacobi polynomials (for details see e.g. [11]), hence using a mathematical dictionary one can always deduce the recurrence equation. Note, however, that this approach except than being tedious may require the work with radicals, namely the zeros of the quadratic polynomial oe(x), whereas our approach is completely rational: Given k n+1 =k n 2 Q(n), the recurrence equation is given rationally by Theorem 1. Moreover, Theorem 1 represents the recurrence equation by a unique formula. It is valid also in the cases of Table 1:1 and 4a, with the trivial solution p In both cases we have the recurrence equation p n+1 Now, we will use the fact that these equations are given explicitly to solve an inverse problem. Assume one knows that a polynomial system satisfies a differential equation (3). Then by the classification of Table 1 it is easy to identify the system. On the other hand, given an arbitrary holonomic three-term recurrence equation it is less obvious to find out whether there is a polynomial system satisfying (15), being a linear transform of one of the classical systems (Hermite, Laguerre, Jacobi, Bessel), and to identify the system in the affirmative case. In this section we present an algorithm for this purpose. Note that Koornwinder and Swarttouw [13] have also considered this question and in their Maple implementation rec2ortho propose a solution based on the careful ad hoc analysis of the input polynomials q n ; r n ; and s n . Their Maple implementation deals with the following families: Hermite, Charlier, Laguerre, Meixner-Pollaczek, Meixner, Krawtchouk, and Jacobi. Let us start with a recurrence equation of type (15). Without loss of generality we assume that neither q has a nonnegative integer zero w.r.t. n. Otherwise, a suitable shift can be applied, see Algorithm 1 and Example 1. Therefore, in the sequel we assume that the recurrence equation is valid, but neither q nonnegative integer zeros. We search for solutions Next, we divide (16) by q n (x), and replace n by n \Gamma 1. This brings (16) into the form For being a linear transform of a classical orthogonal system, there is a recurrence equation (1) therefore (18) and (19) must agree. We would like to conclude that t n This follows if we can show that p n (x)=p x). For a proof of this assertion, see [10]. Therefore we can conclude that t n Hence if (18) does not have this form, i.e., if either t n (x) is not linear in x or u n (x) is not a constant with respect to x, we see that p n (x) cannot be a linear transform of a classical orthogonal polynomial system. In the positive case, we can assume the form (19). Since we propose solutions (17), equating the coefficients of x n+1 in (19) we get (v Hence the given A generates the term ratio k n+1 =k n . In particular k n turns out to be a hypergeometric term, (i.e., k n+1 =k n is rational,) and is uniquely determined by (20) up to a normalization constant k the zeros of w n are a subset of the zeros of q is defined by (20) for all n 2 N from k 0 . In the next step we can eliminate the dependency of k n by generating a recurrence equation for the corresponding monic polynomials e . For e p n (x), we get by e A n e e e with e A n Then our formulas (9)-(10) read in terms of e e d) and e \Gamman d) (a and these are independent of k n by construction. Now we would like to deduce a; b; c; d and e from (21)-(22). Note that as soon as we have found these five values, we can apply a linear transform (according to the zeros of oe(x)) to bring the differential equation in one of the normal forms of Table 1 which finally gives us the desired information. We can assume that e C n are in lowest terms. If the degree of either the numerator or the denominator of e B n is larger than 2, then by (21) p n (x) is not a classical system. Similarly, if the degree of either the numerator or the denominator of e C n is larger than 4, by (22) the same conclusion follows. Otherwise we can multiply (21) and (22) by their common denominators, and bring them therefore in polynomial form. Both resulting equations must be polynomial identities in the variable n, hence all of their coefficients must vanish. This gives a nonlinear system of equations for the unknowns a; b; c; d and e. Any solution of this system with not both a and d being zero yields a differential equation (3), and hence given such a solution one can characterize it via Table 1. Therefore our question can be resolved in this case. In particular, if one of the cases Table 1.1 or 1.4a applies, then there are no orthogonal polynomial solutions. If the nonlinear system does not have such a solution, we deduce that no such values a; b; c; d and e exist, hence no such differential equation is satisfied by p n (x), implying that the system is not a linear transformation of a classical orthogonal polynomial system. Hence the whole question boils down to decide whether the given nonlinear system has nontrivial solutions, and to find these solutions in the affirmative case. As a matter of fact, with Gr-obner bases methods, this question can be decided algorithmically ([16]-[18]). Such an algorithm is implemented, e.g., in the computer algebra system REDUCE [17], and Maple's solve command can also solve such a system. Note that the solution of the nonlinear system is not necessarily unique. For example, the Chebyshev polynomials of the first and second kind T n (x) and U n (x) satisfy the same recurrence equation, but a different differential equation. We will consider this example in more detail later. If we apply this algorithm to the recurrence equation p n+2 of the power generates the complete solution set, given by Table 1:1 and 1:4a. The following statement summarizes the above considerations. Algorithm 1 This algorithm decides whether a given holonomic three-term recurrence equation has shifted, linear transforms of classical orthogonal polynomial solutions, and returns their data if applicable. 1. Input: a holonomic three-term recurrence equation 2. Shift: Shift by nonnegative integer zero is a zero of q 3. Rewriting: Rewrite the recurrence equation in the form If either t n (x) is not a polynomial of degree one in x or u n (x) is not constant with respect to x, then return "no orthogonal polynomial solution exists"; exit. 4. Standardization: Given now A define (v according to (20). 5. Make monic: Set e A n and bring these rational functions in lowest terms. If the degree of either the numerator or the denominator of e B n is larger than 2, or if the degree of either the numerator or the denominator of e C n is larger than 4, return "no classical orthogonal polynomial solution exists"; exit. 6. Polynomial e d) and e d) (a using the as yet unknowns a; b; c; d and e. Multiply these identities by their common denom- inators, and bring them therefore in polynomial form. 7. Equating Coefficients: Equate the coefficients of the powers of n in the two resulting equations. This results in a nonlinear system in the unknowns a; b; c; d and e. Solve this system by Gr-obner bases methods. If the system has no solution or only one with then return "no classical orthogonal polynomial solution exists"; exit. 8. Output: Return the classical orthogonal polynomial solutions of the differential equations (3) given by the solution vectors (a; b; c; d; e) of the last step, according to the classification of Table 1, together with the information about the standardization given by (20). This information includes the density ae(x) e R -(x) (see e.g. [14]), and the supporting interval through the zeros of oe(x). 1 2 Remark Assume that a given recurrence equation contains parameters. Then our implementation determines for which values of the parameters there are orthogonal polynomial solutions, by solving not only for a; b; c; d and e, but moreover for those parameters. Example 1 As a first example, we consider the recurrence equation 1 If the zeros of oe(x) are not real, then these orthogonal polynomials are not positive-definite. The Bessel system is never positive-definite [2]. we see that the shift p n (x) := P n+1 (x) is necessary, i.e. (23). For we have the recurrence equation In the first steps this recurrence equation is brought into the form hence and therefore Moreover, for monic e e hence e 1. The polynomial identities concerning e C n of step 5 of the algorithm yield At this point we have already determined Hence possible classical orthogonal polynomial solutions of (24) are defined in the interval In the first of the above cases, i.e. for and the differential equation corresponding to the density e R -(x) The corresponding orthogonal polynomials are multiples of translated Chebyshev polynomials of the first kind (see e.g. [1], Table 22.2, and (22.5.11); C n (x) are monic, but C hence finally In the second of the above cases, i.e. for one gets the equation a with two possible solutions that give the differential equations and They correspond to the densities r and r respectively, hence the orthogonal polynomials are multiples of the Jacobi polynomials Finally, in the third of the above cases, i.e. for corresponding to the density e R -(x) The corresponding orthogonal polynomials are multiples of translated Chebyshev polynomials of the second kind (see e.g. [1], Table 22.2, and (22.5.13); S n (x) are monic, see also Table 22.8), hence U We see that the recurrence equation (24) has four different (shifted) linearly transformed classical orthogonal polynomial solutions! Using our implementation, these results are obtained by strict:=true: Warning: several solutions found @ @x @ @x @ @x @x which gives the corresponding differential equations, the intervals and densities, as well as the the term ratio k n+1 =k With Koornwinder-Swarttouw's rec2ortho, these results are obtained by the statements rec2ortho((n+2)/(n+1),0,n/(n+1)), rec2ortho((n+2)/(n+1),0,n/(n+1),4,0), rec2ortho((n+2)/(n+1),0,n/(n+1),2,-1), and rec2ortho((n+2)/(n+1),0,n/(n+1),2,1), respectively. Note that here the user must know the initial values to determine possible orthogonal polynomial solutions, whereas our approach finds all possible solutions at once. Example 2 As a second example, we consider the recurrence equation depending on the parameter ff 2 R. Here obviously the question arises whether or not there are any instances of this parameter for which there are classical orthogonal polynomial solutions. In step 6 of Algorithm 1 we therefore solve also for this unknown parameter. This gives a slightly more complicated nonlinear system, with the unique solution ae oe Hence the only possible value for ff with classical orthogonal polynomial solutions is in which case one gets the differential equation with density in the interval [\Gamma1=2; 1], corresponding to linearly transformed Laguerre polynomials. Using our implementation, these results are obtained by strict:=false: Warning : parameters have the values; [2 @ @x With Koornwinder-Swarttouw's rec2ortho, this result can also be obtained. On the other hand, the Bessel polynomials are not accessible with Koornwinder-Swarttouw's rec2ortho. 4 Classical Discrete Orthogonal Polynomials In this section, we give similar results for classical orthogonal polynomials of a discrete variable (see Chapter 2 of [19]). The classical discrete orthogonal polynomials are given by a difference equation (5). family 1. 1 ff x translated Charlier 2a. x 0 x n falling factorial 3. x - (fl 4. x p pols. 5. pols. Table 2: Normal Forms of Discrete Polynomials These polynomials can be classified similarly as in the continuous case according to the functions oe(x) and -(x); up to linear transformations the classical discrete orthogonal polynomials are classified according to Table 2 (compare [19], Chapter 2). In particular, case (2a) corresponds to the non-orthogonal solution x n in Table 1. Similarly as for the powers d dx the falling factorials x n := It turns out that they are connected with the Charlier polynomials by the limiting process lim \Gamman where we used the hypergeometric representation given in [19, (2.7.9)]. Note, however, that other than in the differential equation case the above type of difference equation is not invariant under general linear transformations, but only under integer shifts. We will have to take this under consideration. The classical discrete orthogonal polynomials satisfy a recurrence equation (1) with A given by Theorem 1. Similarly as in the continuous case, this information can be used to generate an algorithm to test whether or not a given holonomic recurrence equation has classical discrete orthogonal polynomial solutions. Obviously the first three steps of this algorithm agree with those given in Algorithm 1. Algorithm 2 This algorithm decides whether a given holonomic three-term recurrence equation has classical discrete orthogonal polynomial solutions, and returns their data if applicable. 1. Input: a holonomic three-term recurrence equation 2. Shift: Shift by maxfn 2 N 0 jn is zero of either q necessary. 3. Rewriting: Rewrite the recurrence equation in the form If either t n (x) is not a polynomial of degree one in x or u n (x) is not constant with respect to return "no orthogonal polynomial solution exists"; exit. 4. Linear Transformation: Rewrite the recurrence equation by the linear transformation x 7! x\Gammag f with (as yet) unknowns f and g. 5. Standardization: Given now A define (v according to (8). 6. Make monic: Set e A n and bring these rational functions in lowest terms. If the degree of either the numerator or the denominator of e B n is larger than 2, if the degree of the numerator of e C n is larger than 6, or if the degree of the denominator of e C n is larger than 4, then return "no classical discrete orthogonal polynomial solution exists"; exit. 7. Polynomial Identities: Set e according to (11), and e according to (12), in terms of the unknowns a; b; c; d; e; f and g. Multiply these identities by their common denominators, and bring them therefore in polynomial form. 8. Equating Coefficients: Equate the coefficients of the powers of n in the two resulting equations. This results in a nonlinear system in the unknowns a; b; c; d; e; f and g. Solve this system by Gr-obner bases methods. If the system has no solution, then return "no classical discrete orthogonal polynomial solution exists"; exit. 9. Output: Return the classical orthogonal polynomial solutions of the difference equations (5) given by the solution vectors (a; b; c; d; e; f; g) of the last step, according to the classification given in Table 2, together with the information about the standardization given by (8). This information includes the necessary linear transformation g, as well as the discrete weight function ae(x) given by ae(x) (see e.g. [19]). Proof: The proof is an obvious modification of Algorithm 1. The only difference is that we have to take a possible linear transformation fx into consideration since the difference equation (5) is not invariant under those transformations. This leads to step 4 of the algorithm. 2 Note that an application of Algorithm 2 to the recurrence equation p n+2 which is valid for the falling factorial p n generates the difference equation x\Deltarp n of Table 2:2a. Example 3 We consider again the recurrence equation (31) depending on the parameter ff 2 R. This time, we are interested in classical discrete orthogonal polynomial solutions. According to step 4 of Algorithm 2, we rewrite (31) using the linear transformation x 7! x\Gammag f with as yet unknowns f and g. Step 5 yields the standardization =f In step 8, we solve the resulting nonlinear system for the variables fa; b; c; d; e; f; resulting in ae d d d oe This is a rational representation of the solution. However, since we assume ff to be arbitrary, we solve the last equation for b. This yields which cannot be represented without radicals. Substituting this into (32) yields the solution ae d oe d and e being arbitrary. It turns out that for ff ! 1=4 this corresponds to Meixner or Krawtchouk polynomials. With Koornwinder-Swarttouw's rec2ortho, this result can be also obtained. Moreover, rec2ortho determines that for ff ? 1=4 one gets Meixner-Pollaczek polynomials. These polynomials are not accessible by our approach. Example 4 Here we want to discuss the possibility that a given recurrence equation might have several classical discrete orthogonal solutions. Whereas the recurrence equation of the Hahn polynomials h (ff;fi) (x; N) has (besides several linear transformations) only this single classical discrete orthogonal solution, the case results in two essentially different solutions. Here one has the recurrence equation An application of Algorithm 2 shows that this recurrence equation corresponds to the two different difference equations and Using our implementation, these results are obtained by strict:=true: Warning : parameters have the values; ag Warning : parameters have the values; ag Warning: several solutions found Note that Hyperterm(upper,lower,z,x) denotes the hypergeometric term (= summand) of the hypergeometric function hypergeom(upper,lower,z) with summation variable x, see [9]. Hahn polynomials are not accessible with Koornwinder-Swarttouw's rec2ortho. 5 Classical q-Orthogonal Polynomials In this section, we consider the same problem for classical q-orthogonal polynomials ([7], [12], see e.g. [8]). The classical q-orthogonal polynomials are given by a q-difference equation (6). These polynomials can be classified similarly as in the continuous and discrete cases according to the functions oe(x) and -(x); up to linear transformations the classical q-orthogonal polynomials are classified according to Table 3. family 1. 2. 3. pols. 4. 1 a+1\Gammax (a) pols. 5. x xq+a+q 7. x pols. 8. 9. pols. alternative q-Charlier pols. 11. pols. 12. pols. 13. U (a) pols. 14. 15. (q Table 3: Normal Forms of q-Polynomials For the sake of completeness we have included all families from [8], Chapter 3, although they overlap in several instances. The non-orthogonal polynomial solutions are the powers x n and the q-Pochhammer functions The classical q-orthogonal polynomials satisfy a recurrence equation (1) with A given by Theorem 1. Similarly as in the continuous and discrete cases, this information can be used to generate an algorithm to test whether or not a given holonomic recurrence equation has classical q- orthogonal polynomial solutions. Algorithm 3 This algorithm decides whether a given holonomic three-term recurrence equation has classical q-orthogonal polynomial solutions, and returns their data if applicable 1. Input: a holonomic three-term recurrence equation 2. Shift: Shift by maxfn 2 N 0 jn is zero of either q necessary. 3. Rewriting: Rewrite the recurrence equation in the form If either t n (x) is not a polynomial of degree one in x or u n (x) is not constant with respect to return "no q-orthogonal polynomial solution exists"; exit. 4. Linear Transformation: Rewrite the recurrence equation by the linear transformation x 7! x\Gammag f with (as yet) unknowns f and g. 5. Standardization: Given now A define (v 6. Make monic: Set e A n and bring these rational functions in lowest terms. If the degree (w.r.t N := q n ) of the numerator of e B n is larger than 3, the degree of the denominator of e B n is larger than 4, the degree of the numerator of e C n is larger than 7, or the degree of the denominator of e C n is larger than 8, then return "no classical q-orthogonal polynomial solution exists"; exit. 7. Polynomial Identities: Set e according to (13), and e according to (14), in terms of the unknowns a; b; c; d; e; f and g. Multiply these identities by their common denominators, and bring them therefore in polynomial form. 8. Equating Coefficients: Equate the coefficients of the powers of in the two resulting equations. This results in a nonlinear system in the unknowns a; b; c; d; e; f and g. Solve this system by Gr-obner bases methods. If the system has no solution, then return "no classical q-orthogonal polynomial solution exists"; exit. 9. Output: Return the q-classical orthogonal polynomial solutions of the q-difference equations given by the solution vectors (a; b; c; d; e; f; g) of the last step, according to the classification given in Table 3, together with the information about the standardization given by (8). This information includes the necessary linear transformation as well as the q-discrete weight function ae(x) given by ae(qx) ae(x) Proof: The proof is an obvious modification of Algorithms 1 and 2. 2 Example 5 We consider the recurrence equation depending on the parameter ff 2 R. This time, we are interested in classical q-orthogonal polynomial solutions. According to step 4 of Algorithm 3, we rewrite (31) using the linear transformation x 7! x\Gammag f with as yet unknowns f and g. Step 5 yields the standardization =f In step 8, we solve the resulting nonlinear system for the variables fa; b; c; d; e; f; resulting in the following nontrivial solution that corresponds-for 1-to the q-difference equation x Hence for every ff 2 R and every scale factor f there is a q-classical solution that corresponds to q-Hermite I polynomials, see Table 3, which have real support for ff ! 0. Using our implementation, these results are obtained by Warning : parameters have the values; ae(x) Note that q-polynomials are not accessible with Koornwinder-Swarttouw's rec2ortho. Note The Maple implementation retode, and a worksheet retode.mws with the examples of this article can be obtained from http://www.imn.htwk-leipzig.de/~koepf/research.html. Acknowledgment The first named author thanks Tom Koornwinder and Ren'e Swarttouw for helpful discussions on their implementation rec2ortho [13]. Examples 2 and 4 given by recurrence equation were provided by them. Thanks to the support of their institutions I had a very pleasant and interesting visit at the Amsterdam universities in August 1996. --R Handbook of Mathematical Functions. The Bessel polynomials. A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols On polynomial solutions of a class of linear differential equations of the second order. An Introduction to Orthogonal Polynomials. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Hypergeometric Summation. Algorithms for classical orthogonal polynomials. Representations of orthogonal polynomials. Compact quantum groups and q-special functions rec2ortho: an algorithm for identifying orthogonal polynomials given by their three-term recurrence relation as special functions Die Charakterisierung der klassischen orthogonalen Polynome durch Sturm- Liouvillesche Differentialgleichungen Solving polynomial equation systems by Groebner type methods. Algebraic solution of nonlinear equation systems in REDUCE. On decomposing systems of polynomial equations with finitely many solutions. Classical orthogonal polynomials of a discrete variable. --TR Representations of orthogonal polynomials

q-difference equation;structure formula;computer algebra;maple;differential equation

634745

Monotonicity and Collective Quantification.

This article studies the monotonicity behavior of plural determiners that quantify over collections. Following previous work, we describe the collective interpretation of determiners such as all, some and most using generalized quantifiers of a higher type that are obtained systematically by applying a type shifting operator to the standard meanings of determiners in Generalized Quantifier Theory. Two processes of counting and existential quantification that appear with plural quantifiers are unified into a single determiner fitting operator, which, unlike previous proposals, both captures existential quantification with plural determiners and respects their monotonicity properties. However, some previously unnoticed facts indicate that monotonicity of plural determiners is not always preserved when they apply to collective predicates. We show that the proposed operator describes this behavior correctly, and characterize the monotonicity of the collective determiners it derives. It is proved that determiner fitting always preserves monotonicity properties of determiners in their second argument, but monotonicity in the first argument of a determiner is preserved if and only if it is monotonic in the same direction in the second argument. We argue that this asymmetry follows from the conservativity of generalized quantifiers in natural language.

Introduction Traditional logical studies of quantification in natural language concentrated on the interactions between quantifiers and distributive predicates - predicates that describe properties of atomic entities. Generalized Quantifier Theory (GQT), as was applied to natural language semantics in the influential works of Barwise and Cooper (1981), Benthem (1984) and Keenan and Stavi (1986), followed this tradition and concentrated on 'atomic' quantification. The framework that emerges from these works provides a general treatment of sentences such as the following. (1) All the students are happy. Some girls arrived. No pilot is hungry. Most teachers are Republican. Exactly five boys smiled. Not all the children sneezed. In these sentences, the denotations of both the nominal (e.g. students, girls, etc.) and the verb phrase (e.g. be happy, arrived, etc.) are traditionally treated as distributive predicates, which correspond to subsets of a domain of (arbitrary) atomic entities. Standard GQT assigns determiners such as all, some and most denotations that are relations between such sets of atomic entities. While this general treatment is well-motivated, it does not account for the interactions between quantifiers and collective predicates. Consider for instance the following sentences. (2) All the colleagues cooperated. Some girls sang together. No pilots dispersed. Most of the sisters saw each other. Exactly five friends met at the restaurant. Not all the children gathered. According to most theories of plurals, nominals such as colleagues, sisters and friends and verb phrases such as cooperated, gathered and saw each other do not denote sets of atomic entities, but rather sets of collections of such entities. There are various theories about the algebraic structure of such collections, but for our purposes in this article it is sufficient to assume that collections are sets of atomic entities. Thus, we assume that collective predicates denote sets of sets of atomic entities. Consequently, the standard denotation of determiners in GQT as relations between sets of atoms is not directly applicable to sentences with collective predicates Early contributions to the study of collective quantification in natural language, most notably Scha (1981), propose that meanings of 'collective statements' as in (2) are derived using 'collective' denotations of determiners. 1 More recent works, including among others Van der Does (1992,1993), Dalrymple et al. (1998) and Winter (1998,2001), , propose to derive such collective meanings of determiners from their standard distributive denotations in GQT using general mappings that apply to these distributive meanings. In the works of Van der Does and Winter, type shifting operators apply to a standard determiner denotation D, which ranges over atomic entities, and derives a determiner of a higher type O(D), which ranges over sets of atomic entities. 2 The study of collective quantification as in (2) is reduced in these theories to the study of the available O mapping(s) from standard determiners to determiners over collections. We follow Winter (2001) and adopt one general type shifting principle for collective quantification that unifies Scha's and Van der Does' `neutral' and 'existential' liftings of determiners into one operator. This operator is referred to as determiner fitting (dfit). 1 For earlier works on plural quantification within Montague Grammar see Bennett (1974) and Hauser (1974). As we shall see below, the bounded composition operator that Dalrymple et al. propose can also be cast to a type shifting operator on determiners. The type shifting approach establishes a connection between standard GQT and linguistic theories of plurality. A natural question that arises in this context is: what are the relations between semantic properties of standard quantifiers in GQT and properties of their 'collectivized' version? In this article we concentrate on the monotonicity properties of determiners, which, as far as standard distributive quantification is concerned, are one of the best studied aspects of quantification in natural language. 3 Consider for instance the simple valid entailments (denoted by ')') in (3), with the determiner all. a. All the students are happy ) All the rich students are happy. b. All the students are very happy ) All the students are happy. Intuitively, the entailments in (3a-b) show that, in simple sentences, the determiner all licenses a replacement of its first argument (students) by any subset of this argument (e.g. rich students), and licenses a replacement of its second argument (very happy) by any superset of this argument (e.g. happy). Thus, the determiner all is classified as downward monotone in its first argument but upward monotone in its second argument. The starting point for the investigations in this work is the observation that such monotonicity entailments are not always preserved when the determiner quantifies over collections. Consider for instance the contrast between the sound entailment in (3a) and the invalid entailment in (4) below. (4) All the students drank a whole glass of beer together All the rich students drank a whole glass of beer together. In a situation where the students are s 1 and s 3 and the rich students are s 1 and assume that the group fs drank a whole glass of beer together, but no other group did. In this situation, the antecedent in (4) is obviously true, but the consequent is false. However, as we shall see, many other plural determiners do not lose their monotonicity properties when they apply to collective predicates. This variation calls for a systematic account of the monotonicity properties of determiners in their collective usage, in relation to their monotonicity properties in standard GQT. The aim of this work is to study these relations in detail. We will prove that, under the adopted determiner fitting operator, 'monotonicity loss' with all is strongly 3 See Ladusaw (1979), Fauconnier (1978) and much recent work on the linguistic centrality of monotonicity for describing the distribution of negative polarity items like any or ever. For instance, in correlation with the monotonicity properties of all as reflected in (3), the negative polarity item ever can appear in the nominal argument of all (e.g. in (i) below), in which it is downward monotone, but not in the verb phrase argument (e.g. in (ii)), where all is upward monotone: (i) All the [students who have ever visited Haifa][came to the meeting]. (ii) \LambdaAll the [students who came to the meeting][have ever visited Haifa]. connected to the fact that the monotonicity properties of this determiner are different in its two arguments. We show that determiner fitting preserves the monotonicity properties of determiners in their second argument, and it further preserves monotonicity properties of determiners which have the same monotonicity properties in both arguments. However, with determiners such as all, not all, some but not all, and either all or none (of the), which are monotone in their first argument, but have a different monotonicity property in the second argument, monotonicity in the first argument is not preserved under determiner fitting. We claim that the origin of these (empirically welcome) results is in the 'neutral' process that Scha proposed for collective quantification, and that the combination of this treatment with an 'ex- istential' lifting, which is empirically well-motivated, has no effects whatsoever on the (non-)preservation of (non-)monotonicity with collective quantifiers. The structure of the rest of this article is as follows. Section 2 reviews some familiar notions from GQT that are used in subsequent sections. Section 3 describes previous treatments of collective quantification and the uniform type shifting strategy that is adopted in this paper. Section 4 establishes the facts pertaining to (non-)preservation of (non-)monotonicity under type shifting with all possible monotonicity properties of determiners in GQT. 2 Notions from generalized quantifier theory This section reviews some familiar notions from standard GQT that are important for the developments in subsequent sections. For an exhaustive survey of standard GQT see Keenan and Westerst-ahl (1996). The main property of quantifiers that is studied in this article is monotonicity, which is a general concept that describes 'order preserving' properties of functions over partially ordered domains. partially ordered sets, and let f be a function from A 1 to B. The function f is called upward (downward) monotone in its i-th argument iff for all a 1 2 A We say that f is monotone in its i-th argument iff f is either upward or downward monotone in its i-th argument. Extensional denotations are given relative to an arbitrary non-empty finite set E, to which we refer as the domain of atomic entities, or simply atoms. Given a non-empty domain E, a determiner over E is a function from -(E) \Theta -(E) to 1g. 4 Hence, a determiner is a relation between subsets of E. The set -(E), the power set of E, is ordered by set inclusion. The set f0; 1g, the domain of truth values, is ordered by implication, which is simply the numerical '-' order on 4 Later in the paper, we refer to such determiners as Atom-Atom determiners, since both their arguments are sets of atomic entities. Determiner Denotation: for all A; B ' E: Monotonicity all all 0 not all (:all 0 some some 0 no no most most 0 exactly five exactly 5 0 Table 1: standard denotations of some determiners 1g. Since a determiner is a two-place function, we use the terms left monotonicity and right monotonicity for referring to its monotonicity in the first and second arguments respectively. We use the following notation: ffl "MON, #MON and -MON for determiners that are upward left-monotone, downward left-monotone and not left-monotone, respectively. ffl MON", MON# and MON- for determiners that are upward right-mono- tone, downward right-monotone and not right-monotone, respectively. We combine these two notations, and say for instance that the determiner all is #MON" according to its definition as the subset relation. The standard denotations that are assumed for some determiners are given in table 1, together with their monotonicity properties. The well-known conservativity property of determiners reflects the observation that the truth value that determiners in natural language assign to any pair of sets A and B is identical to the truth value they assign to A and A " B. This is observed in (seemingly obvious) equivalences such as the following. All the/some/no/most of the/exactly five cars are blue , All the/some/no/most of the/exactly five cars are blue cars. Formally, conservativity of determiner functions is defined as follows. determiner D over E is conservative (CONS) iff for all Thus, in order to evaluate the truth value of a sentence D students are hungry, with a conservative determiner D, we do not have to know the set of all hungry entities, but only the set of hungry students. Let us mention a useful constancy aspect of the meaning of most natural language determiners. 5 Van Benthem (1984) defines the permutation invariant (PI) determiners as follows. 5 The exceptions to PI are determiners such as my, her or this, which we henceforth ignore. Definition 3 (permutation invariance) A determiner D over E is permutation invariant (PI) iff for all permutations - of E and for all Roughly speaking, when a determiner D is PI, this means that it is not sensitive to the identity of the members in its arguments, but only to set-theoretical relations between its arguments. It is easy to verify that the determiners in table 1 are all PI. The following two definitions characterize two trivial classes of determiners. Definition 4 (left/right triviality) A determiner D over E is called left (right) trivial iff for all A; B; C ' E: respectively). Intuitively, an LTRIV (RTRIV) determiner is insensitive to the identity of its left (right) argument. For instance, the determiners less than zero and at least zero, which are both LTRIV and RTRIV, assign the same truth value (0 and 1 respec- tively) to all possible arguments. We occasionally restrict our attention to determiners that are not right trivial. This is because non-right-triviality is a stronger restriction on conservative determiners than non-left-triviality - provably, all conservative LTRIV determiners are RTRIV. However, a conservative RTRIV determiner is not necessarily LTRIV. For instance, the determiner D s.t. iff A 6= ; is conservative and RTRIV but is not LTRIV. In this article we study the monotonicity properties of non-right-trivial conservative determiners that satisfy permutation invariance, which are also in the main focus of the general theory of quantification in natural language. 3 Collective determiners and type shifting principles The type shifting account of plural determiners that was initiated by Scha (1981) is motivated by sentences such as those in (2), which involve collective predicates. In this section we review previous treatments of collective quantification, and concentrate on the proposal of Winter (1998,2001) that aims to solve some of the empirical problems for Scha's and Van der Does' proposals. As an example that illustrates many aspects of the interpretation of collective determiners, consider the following sentence. Exactly five students drank a whole glass of beer together. The denotation of the collective predicate drank a whole glass of beer together is assumed to be an element of -(E)) - a set of sets of atomic elements. To sentence (6), the meaning of the determiner exactly five from table 1 is shifted so that it can combine with this collective predicate. This section deals with the proper way(s) to define such a shifting operator. 3.1 The type shifting operators of Scha and Van der Does Scha (1981) proposes an extension of standard GQT to the treatment of collectivity phenomena as in sentence (6). The work in Van der Does (1992,1993) contains a systematic reformulation of Scha's approach using type shifting operators within contemporary GQT. In the systems of Scha and Van der Does (henceforth S&D), both distributive and collective verbal predicates (e.g. smile, meet) denote elements of -(E)). However, nominal predicates such as students standardly denote sub-sets of E. Accordingly, in S&D's proposal, collective determiners are functions from -(E) \Theta -(E)) to f0; 1g. We distinguish such Atom-Set determiners, where the first argument is a set of atoms and the second argument is a set of sets of atoms, from the standard Atom-Atom determiners of GQT as in table 1, where both arguments are sets of atoms. Van der Does follows Van Benthem (1991:68) and proposes that the Atom-Set determiners that are necessary for the interpretation of plural sentences can be obtained systematically from Atom-Atom determiners. There are two collective shifts that are proposed for sentences like (6) in S&D's works. 6 One collective operator is called Existential operator). For sentence this operator generates a statement that claims that there is a set of exactly five students and that this set drank a whole glass of beer together. In general, for any Atom-Atom determiner D over a domain E, applying the E operator leads to the Atom-Set determiner E(D) which is defined for all A ' E and B ' -(E) by: Scha proposes another collective analysis of plural determiners, which Van der Does refers to as neutral. In sentence (6), for instance, this neutral analysis counts all the individual students who participated in sets of students that drank a whole glass of beer together, and requires that the total number of these students is exactly five. For any Atom-Atom determiner D over E, the corresponding Atom- determiner N(D) (Van der Does' N 2 is defined for all A ' E and B ' -(E) by: Note that the set [(B " -(A)) contains x if and only if x is an element of a subset of A that belongs to B. The N operator involves a mapping of the left argument A of the determiner, which is a set of atoms, into the power set of A, which is a set of sets of atoms. Such a mapping from sets to sets of sets is useful in S&D's strategy, as in most other theories of plurals, since it makes a connection between distributive predicates and collective predicates. This is required whenever a predicate over atoms semantically interacts with other elements that range over collections (e.g. sets). In the theory of plurals, such a mapping is often referred to as a distributivity op- erator. The power set operator - is sufficient as a distributivity operator for our 6 S&D also assume a distributive shift, which is irrelevant for our purposes here. In addition, Van der Does (1992) considers a third collective shift but (inconclusively) dismisses it in his 1993 article. purposes in this paper. 7 We say that a set A ' -(E) is a distributed set of atoms set of atoms A ' E. In sentences where the second argument of the determiner is an ordinary distributive predicate, its meaning under S&D's treatment can be defined as a distributed set of atoms rather than a set of atoms. This is needed in order to match the type of the second argument of the lifted Atom-Set determiner. For instance, the standard meaning of sentence (7) below is captured using the N operator as in (8), and not simply by directly applying the Atom-Atom denotation of the determiner exactly five to two sets of atoms, as in standard GQT. Exactly five students sang. (N(exactly 5 0 ))(student 0 )(-(sing 0 )), where student sing 0 It is easy to verify that this analysis is equivalent to the standard analysis of (7). More generally, observe the following fact. Fact 1 For every conservative Atom-Atom determiner D over E, for all 3.2 Problems for S&D's strategies One empirical problem for S&D's type shifting analysis follows from a warning in Van Benthem (1986:52-53), and is accordingly referred to as the Van Benthem problem for plural quantification. Van Benthem mentions that any general existential lifting such as the E operator is problematic, because it turns any Atom-Atom determiner into an Atom-Set determiner that is upward right-monotone. Quite expectedly, this property of the E operator is empirically problematic with many Atom-Atom determiners that are not upward right monotone. For in- stance, using the E operator, sentence (9) below, with the MON# determiner no, gets the interpretation in (10). met yesterday at the coffee shop. This analysis of sentence (9) makes the strange claim that an empty set met yesterday at the coffee shop, which is clearly not what sentence claims. 8 The 7 A more common version of a distributivity operator is the - + operator, which maps each set to its power set minus the empty set: f;g. Using the operator would not change the results in this paper, and therefore we use the simpler power set operator. For arguments in favor of a distributivity operator that is a more sophisticated than - see Schwarzschild (1996). For counterarguments see Winter (2000). 8 Using - + instead of - in the definition of the E operator (which is what S&D do), sentence is analyzed as a contradiction. Obviously, this is not the correct analysis of the sentence either. existential analysis reverses the monotonicity properties of the determiner no, so that E(no 0 ) is "MON". However, the determiner no remains #MON# in this case even though it is used for quantification over collections. For instance, sentence sentence (11a) below and does not entail sentence (11b). students met yesterday evening at the coffee shop. b. No people (ever) met at the coffee shop. The problem is manifested even more dramatically when the second argument of the determiner is a distributive predicate (distributed by -). For instance, sentence (12) below is analyzed as in (13), which is a tautology (for choose Less than five students smiled. Although the Van Benthem problem indicates that the existential operator is inadequate, this operator still captures one effect that the N operator by itself does not handle. To see that, reconsider sentence (6), restated in (14) below, and its analyses using the N and E operators. Exactly five students drank a whole glass of beer together. a. (N(exactly 5 0 ))(student 0 )(drink beer 0 b. (E(exactly 5 0 ))(student 0 )(drink beer 0 The analysis in (14a) requires that the total number of students in sets of students that drank a glass of beer together is five. However, in addition, sentence (14) also requires that there was a set of five students who drank a whole glass of beer together. The E operator in the analysis in (14b) imposes this requirement, but fails to take into account the total number of students who drank a glass of beer, and therefore leads to the Van Benthem problem. A similar dilemma arises with upward monotone determiners, as in the following example. More than five students drank a whole glass of beer together. In this case too, the N operator imposes a requirement only on the total number of students involved in beer drinking events, whereas what we need in this case is an existential reading, requiring that there actually was a set with more than five students who drank a whole glass of beer together. In S&D's systems there 9 If we replace - by - sentence (12) is analyzed as being equivalent to at least one student smiled, which is bad enough. is no clear specification of how to capture both aspects of collective quantification without generating undesired truth conditions. 10 Another problem for S&D's strategy is in the type of collective determiners it assumes. In S&D's proposal, any collective determiner is an Atom-Set determiner. However, in many cases, a collective predicate may also appear in the left argument of a determiner. For instance, reconsider the following example from (2). All the colleagues cooperated. In this case, the plural noun colleagues is collective: to say that a and b are colleagues is not the same as saying that a is a colleague and b is a colleague. Other collective nouns like brothers, sisters, friends etc. lead to similar problems. Other cases where the first argument of a determiner is collective appear when a distributive noun is modified by a collective predicate. For instance, consider the following examples. Exactly four similar students smiled. (18) Most of the students who saw each other played chess. In these cases, the interpretation of the first argument involves intersection of a distributive predicate (distributed by -) with a collective predicate. For instance, the denotation of the nominal similar students in (17) is obtained by intersecting the set of sets of students with the set of sets of similar entities. These examples indicate that collective determiners should allow collective predicates in both arguments, and not only in the right argument as in S&D's lifting strategies. 3.3 Dalrymple et al.'s bounded composition operator Dalrymple et al. (1998) concentrate on the semantics of reciprocal expressions (each other, one another) in sentences with simple plural NPs such as the children and Mary and John. However, they also address the problem of interpreting reciprocal expressions in the following sentences, where a collective reciprocal predicate combines with a quantifier of more complex monotonicity properties. a. Many people are familiar to one another. b. Most couples in the apartment complex babysit for each other. c. At most five men hit each other. Dalrymple et al. observe the existential requirement in sentences (19a-b), and their treatment of such sentences, with MON" determiners, is accordingly a reformulation of the E operator of Scha and Van der Does. However, to overcome the problems that the existential requirement creates in sentences such as (19c), with tries to overcome this problem by proposing a syntactic mechanism of feature propagation that is designed to rule out some of the undesired effects of his semantic system. non-MON" determiners, Dalrymple et al. use a different analysis for such sen- tences. In example (19c), their analysis requires that each set of men who hit each other contains at most five men. It seems quite likely that for the case of sentence (19c), this analysis reflects a possible reading. 11 Dalrymple et al. combine the two processes they assume for MON" and non- MON" determiners into one general operator that they call Bounded Composition. This operator can be cast as a lifting operator of determiners, so that for any Atom-Atom determiner D over E, the corresponding Atom-Set determiner BC(D) is defined for all A ' E and B ' -(E) by: 12 The first conjunct in this definition reflects a counting process, parallel to S&D's N operator. 13 The second conjunct adds to this process an existential requirement, similar to S&D's E operator. However, there are two modifications in the usage of these two processes, compared to S&D's strategies: 1. Unlike the N operator, the counting process within BC does not require the total union of to be in the generalized quantifier D(A), but only requires that each set of maximal cardinality within is in D(A). 2. The existential requirement overcomes Van Benthem's problem, due to the disjunct properly weakens the E operator with determiners that satisfy The motivation for the introduction of the bounded composition operator is to treat collective readings of GQs with reciprocal predicates. However, we believe that the combination of counting and existential processes is a promising aspect of Dalrymple et al.'s proposal also for other cases of collectivity. On the other hand, the empirical adequacy of the counting process as implemented within the BC operator is not completely clear to us. Some speakers we consulted accept Dalrymple et al.'s assumption that sentences such as (19c), with a downward monotone quan- tifier, can be true even though the 'total' set of people who participated in sets of Dalrymple et al.'s intuitions about the meaning of (19c) seem to be similar to those of Schein (1993), who proposes an event semantics of plurals. For some remarks on the empirical question concerning the generality of this analysis see our discussion below. 12 The operator that Dalrymple et al. propose is defined as a 4-ary relation between a determiner, a set of atoms, a binary relation and the meaning of the reciprocal expression. For instance, in sentence (19a), these are (respectively) the meanings of the expressions many, people, familiar to and one another. For our purposes it is sufficient to consider the determiner alone, because the compositional interpretation of reciprocal predicates such as familiar to one another is not in the focus of this article. 13 The requirement jA n Xj - jA n Y j in this conjunct is needed only when we assume infinite domains. Over finite domains it follows from the requirement jXj - jY j. people who hit each other is not in the quantifier (i.e. in this case - includes more than five members). However, these judgments did not seem to be highly robust and they vary considerably when the determiner at most five is replaced by other non-MON" determiners such as less than five, exactly five or between five and ten. For example, consider the following sentence. Exactly five students hit each other. Assume that there was a set of exactly five students who hit each other, and that in addition there was only one other set of students A who hit each other. In case there are four students in A, then the BC operator renders sentence (20) true. However, if A contains six students then the BC operator takes sentence (20) to be false. We did not trace such a difference in our informants' intuitions about the sentence. As a general operation for deriving collective readings of GQs, the BC operator shows some undesired effects when the quantifier it derives interacts with a so-called 'mixed' predicate. These predicates (unlike predicates formed with re- ciprocals) can also be true of singleton sets, in addition to sets with two or more elements. Consider for example the following sentence. (21) At most five students drank a whole glass of beer (together or separately). In a situation where there are ten students and each student drank a whole glass of beer on her own, sentence (21) is clearly false. However, if we assume that no students shared any glass of beer between them, the BC operator makes sentence (21) true, because there is no relevant set of students with more than five members: all the relevant sets are singletons. For these reasons, in the proposal below we choose to study the counting process of the N operator. We leave for further research the empirical study of the exact interpretation of sentences such as (19c), as well as the formal study of the BC operator that is motivated by their interpretation. 3.4 Determiner fitting and the witness condition To overcome the two problems of S&D's mechanism that were pointed out in sub-section 3.2, Winter (1998,2001) proposes to reformulate the N and E operators as one operator called dfit (for determiner fitting). This operator, unlike S&D's N and operators and Dalrymple et al.'s BC operator, maps an Atom-Atom determiner into a Set-Set determiner, i.e. a determiner where both arguments can be collective predicates. To define the dfit operator, let us first reformulate N as an operator from Atom-Atom determiners to Set-Set determiners. This reformulation of the N operator is called count, and is defined as follows. Definition 5 (counting operator) Let D be an Atom-Atom determiner over E. The corresponding Set-Set determiner count(D) is defined for all A; B ' -(E) by: By giving a symmetric Set-Set denotation to collective determiners, this definition involves two separate sub-processes within the process of counting members of collections. The first sub-process is the intersection of the right argument with the left argument of the determiner. The second sub-process is the union of the sets in each of the two arguments. The intersection sub-process reflects the conservativity of (distributive/collective) quantification in natural language: the elements of the right argument that need to be considered are only those that also appear in the left argument. This also holds for the Set-Set determiner count(D). 14 The union sub-process is simply a natural 'participation' adjustment of the type of the Atom-Atom determiner's arguments: for any collective predicate A, an atom x is in [A iff x participates in a set in A. The count operator generalizes S&D's N operator in the following sense. Fact 2 For every conservative Atom-Atom determiner D over E, for all A ' E Thus, like the N operator, count respects the semantics of conservative determiners on distributive predicates (cf. fact 1). Corollary 3 For every conservative Atom-Atom determiner D over E, for all As with Dalrymple et al.'s BC operator, a counting process (of the count op- erator) is combined with an existential requirement. In order to do that, a useful notion is the notion of witness set from Barwise and Cooper (1981). Definition 6 (witness set) Let D be an Atom-Atom determiner over E and let A and W be subsets of E. We say that W is a witness set of D and A iff W ' A and D(A)(W For example, the only witness set of the determiner every 0 and the set man 0 is the set man 0 itself. A witness of some 0 and man 0 is any non-empty subset of man 0 . We sometimes sloppily refer to a witness set of a determiner D and a set A as 'witnessing the quantifier D(A).' 15 To the count operator we now add an 'existential' condition that is formalized using a witness operator. 14 Note that we still assume that the Atom-Atom determiner D that is lifted by the count operator is conservative. However, even when D is conservative, lifting it by an alternative operator count does away with the intersection process within count, would not guarantee sound conservativity equivalences such as between (i) and (ii) below. (i) All the students are similar. (ii) All the students are similar students. Using the count 0 operator, sentence (i) would be treated, contrary to intuition, as being true if every student is similar to something else (potentially a non-student). But sentence (ii) would be treated by the count 0 operator as being false in such a situation. Barwise and Cooper define witness sets on quantifiers explicitly, but they reach the argument A indirectly by defining what they call a live on set of the quantifier. This complication is unnecessary for our purposes. Definition 7 (witness operator) Let D be an Atom-Atom determiner over E. The corresponding Set-Set determiner wit(D) is defined for all A; B ' -(E) by: In words: the witness operator maps an Atom-Atom determiner D to a Set-Set determiner that holds of any two sets of sets A; B iff their intersection A " B is empty or contains a witness set of D and [A. A similar strategy for quantification over witness sets is proposed in (1997). While Szabolcsi's witness operation is used only for MON'' determiners, the witness operator that is defined above is designed to be used for all determiners. This is the reason for the disjunction in the definition of the wit operator with an emptiness requirement on A " B. As we shall see below, this will allow us to apply the witness operator as a general strategy, also in cases like (9), without imposing undesired existential requirements as in (10). The general determiner fitting operator that we use is simply a conjunction of the counting operator and the witness operator. Definition 8 (determiner fitting operator) Let D be an Atom-Atom determiner over E. The corresponding Set-Set determiner dfit(D) is defined for all To exemplify the operation of the dfit operator, consider the analysis in (23) below of sentence (14), repeated as (22). In this analysis, the noun students is treated as the distributed set of atoms -(student 0 ). This is needed in order to match the general type of the left argument of the Set-Set determiner that is derived by the dfit operator. Exactly 5 students drank a whole glass of beer together. dfit(exactly 5 0 )(-(student 0 ))(drink beer 0 ) The first conjunct in this formula is derived by the count operation, and guarantees that exactly five students participated in sets of students drinking beer. The second conjunct is a result of the witness condition, and it verifies that there exists at least one such set that is constituted by exactly five students. By combining the counting process and the existential process in this way, the dfit operator captures some properties of collective quantification that seem quite puzzling under S&D's double lifting strategy. On the one hand, as problem indicates, in sentences such as (9) and (12), with MON# determiners, the existential strategy is problematic and only the N operator is needed. In sentences such as (15), with MON" determiners, the existential strategy is needed and the N operator is redundant. Moreover, when the determiner is MON-, as in (14), the existential analysis is needed in combination with the neutral analysis. The dfit operator distinguishes correctly between these cases. As we will presently see, in those cases where a simple existential analysis would be problematic, the witness condition is trivially met due to the counting condition within dfit. We characterize two such cases: cases such as (12), where the two arguments of the determiner are distributed sets of atoms, and cases such as (9), where the determiner is downward right-monotone. In other cases, the witness condition does add a non-trivial requirement to the counting operator. First, let us observe that when the arguments of a Set-Set determiner dfit(D) are two distributed sets of atoms -(A) and -(B), the witness operator adds nothing to the requirement that Fact 4 Let D be a conservative Atom-Atom determiner over E. Then for all Proof. Assume that 1. By conservativity, the witness set the existential requirement in wit. From this fact and corollary 3 it directly follows that the witness operator is redundant in dfit when the arguments of the determiner are both distributed sets of atoms. Corollary 5 Let D be a conservative Atom-Atom determiner over E. Then for all Similarly, when a determiner is downward monotone in its right argument, the witness operator is again redundant in the definition of dfit : Fact 6 Let D be a MON# Atom-Atom determiner over E. Then for all In other cases - that is, when A or B are not DSAs and D is not MON" - (count(D))(A)(B) does not entail (wit(D))(A)(B). Accordingly, an existential requirement is invoked by the sentence. This is illustrated by the entailments from the sentences in (24) to sentence (25): a. More than/exactly five students drank a whole glass of beer together. b. More than five/exactly five students who drank a whole glass of beer together smiled. c. More than/exactly five students who drank a whole glass of beer together hit each other later. There was (at least) one group of more than/exactly five students who drank a whole glass of beer together. These entailments are not captured by the count operator alone, but they follow from the wit requirement within the dfit operator. 3.5 Determiners that are trivial for plurals Before moving on to the next section and to the monotonicity properties of Set-Set determiners under the dfit operator, there is an additional notion that we need to introduce, which refines the right triviality property for plural determiners. An illustration of the point is the behavior of the definite article. In singular sentences such as (26) below, the 'Russellian' interpretation of the definite article in GQT analyses it as a universal determiner that in addition imposes a uniqueness condition on its left argument. The definition of this determiner is given in (27). (26) The student smiled. (27) the 0 By contrast, in a plural sentence such as (28) below, this definition would be inad- equate. The sentence here imposes a plurality requirement on the left argument of the determiner, rather than uniqueness. A possible definition of this determiner is given in (29). (28) The students smiled. (29) the 0 getting into the question of definites, us make one simple point. We do not expect the meaning in (27), which is appropriate for the singular definite article, to be a meaning of any plural determiner. The reason is that such a determiner function, which imposes singularity on its left argument, would contradict the implication, prominent with plurals, that there are at least two elements in the left argument. 17 Consequently, no plural noun could appear with such a determiner without leading to a trivial statement: a contradiction or a tautology. As with singular determiners, we do not expect such trivialities with plural determin- ers. Crucially, note that the determiner in (27) is not RTRIV (or LTRIV). We can therefore assume that plural determiners show a stronger notion of non-triviality than RTRIV, which we call triviality for plurals (PTRIV). Formally: Definition 9 (triviality for plurals) A determiner D over E is called trivial for plurals (PTRIV) iff for all Most contemporary works assume that the definite article does not denote a determiner, but some version of the iota operator. For works that propose a unified definition of the definite article for singular and plural NPs, see Sharvy (1980) and Link (1983). We believe that this view on articles is justified (cf. Winter (2001)), and that (in)definite articles such as a and the should not denote determiners in GQT. Therefore, the dfit operator does not apply to these articles. For one thing, it does not seem possible to derive the collective reading of the definite article from its singular denotation in (27) using the same operator that applies to other determiners. 17 Whether this implication is truth-conditional or presuppositional is irrelevant here. See some discussion of this point in Krifka (1992), Schwarzschild (1996), Chierchia (1998) and Winter (1998, Informally, a PTRIV determiner is indifferent to the identity of its right argument whenever its left argument is a set with two or more entities. The determiner the 0 sg as defined above is PTRIV but not RTRIV. We hypothesize that all plural determiner expressions in natural language (though not necessarily all singular determiner expressions) denote non-PTRIV determiners. This hypothesis about plural determiners will play a role in the next section. 4 Monotonicity properties of collective determiners The facts that were mentioned in the introduction indicate that standard monotonicity properties of determiners are not always preserved when they apply to collective predicates. In this section we show that, using the count operator, monotonicity properties in the right argument of a determiner are always preserved, in agreement with intuition. However, whether or not a determiner preserves its left monotonicity property when it applies to collective predicates, depends on its monotonicity property in the right argument. We observe that the reason for these different results for determiners of different monotonicity properties is the asymmetric conservativity element within the definition of the count operator, which intersects the right argument with the left argument, but not vice versa. Further, we mention without proof that the same results concerning (non-)preservation of monotonicity properties hold for the dfit operator, which is defined using count. The following fact summarizes all the cases where (non-)monotonicity is preserved under count. Fact 7 Let D be a determiner over E. If D belongs to one of the classes "MON", #MON#, MON" or MON#, then the Set-Set determiner count(D) belongs to the same class. If D is conservative and -MON (MON-), then count(D) is also -MON (MON-). Proof sketch. For the monotone cases the claim immediately follows from the definition of count. For the non-monotone cases it is enough to note that if D is -MON, then there are A 0' A 1 ' E, A 2 ' A 0' E and B; C tive, we can apply corollary 3 (page 13) and get: 0, which shows that count(D) is -MON too. The proof is similar when D is MON-. Fact 7 explains why in many cases, as in the following examples, determiners do not show any surprising monotonicity patterns when they appear with collective predicates. (30) Some (rich) students drank a whole glass of (dark) beer together ) Some students drank a whole glass of beer together. (some is "MON") whole glass of beer together students drank a whole glass of (dark) beer together. (no is #MON#) (32) All/most of the students drank a whole glass of dark beer together ) All/most of the students drank a whole glass of beer together. (all and most are MON") Exactly five students drank a whole glass of beer together Exactly five rich students drank a whole glass of (dark) beer together. (exactly five is -MON-) However, as we saw in (4) above, determiners are sometimes more surprising in their monotonicity behavior with collective predicates. To cover all the monotonicity classes of determiners, we also have to consider the left argument of determiners such as all, not all and some but not all, which are monotone in their left argument but have a different monotonicity property in their right argument. The theorem below will show that almost all these determiners lose their left monotonicity under count. The only exceptions are PTRIV determiners that are #MON" or "MON#. Provably, these determiners preserve left monotonicity under count. However, as claimed above, PTRIV determiners are not expected in the class of plural determiners in natural language. Moreover, it is not hard to show that in each of the monotonicity classes #MON" and "MON# there is only one PTRIV determiner that is conservative, PI and not RTRIV. These two determiners are the following determiners - D 0 and its complement Once these two special cases are observed, we can establish the following result. Theorem Let D be a conservative determiner over E that is not RTRIV. If D satisfies one of the conditions (i) and (ii) below, then the Set-Set determiner count(D) is -MON. (i) D is non-PTRIV and is either #MON" or "MON#. (ii) D is PI and is either #MON- or "MON-. To make the proof of this theorem more readable, we first prove the following lemma. Lemma 8 Let D be a conservative determiner over E. If D satisfies one of the conditions (i) and (ii) below then there are X; Y; Y (i) D is non-PTRIV and #MON". (ii) D is PI and #MON-. Proof. Assume first that D is non-PTRIV and #MON". Because D is non- PTRIV and conservative, there are and it follows from Therefore, we can choose If, on the other hand, D is PI and #MON-, then (by #MON- and conserva- tivity) there are B s.t. the following hold: (D is not MON#); (D is not MON"). It follows that both A and A 0 are not empty. If jAj ? 1 then simply choose Let - be a permutation on E s.t. follows from ( ) that ;. Otherwise, A fyg, and from ( ), it follows from Therefore, we can choose Proof of theorem. We first prove the theorem for a determiner D that is either #MON" or #MON-. Because D is not RTRIV, it cannot be "MON. Using the same reasoning as in the proof of the non-monotone cases in fact 7, it is straight-forward to show that in each of the two cases count(D) is not "MON. It is left to be shown that count(D) is also not #MON. By lemma 8, there are X; Y; Y s.t. and the following holds: three subsets of -(E), A, A 0 and B as follows: g. it follows from ( ) that Hence, count(D) is not #MON. Assume now that D is either "MON# or "MON-. Again, it is straightforward to show that count(D) is not #MON. To see that count(D) is also not "MON, consider the negation of D: :D, defined by The determiner :D is either "MON# or "MON-, respectively. Further, :D is non-PTRIV if and only if D is non-PTRIV, and the same holds for the properties CONS, PI and non-RTRIV. Thus, it follows from condition ( ) above that for the same Using the same A, A 0 and B as above we get that count(D) is not "MON. In the proof of the theorem we use one construction of subsets of -(E) - A, A 0 and B - that applies to all four classes of determiners. However, it is not always convenient to apply this construction to sentences in natural language, because the set A 0 is neither DSA nor a purely collective predicate (i.e. it contains singletons in it). To overcome this empirical difficulty, assume that condition ( ) is satisfied for some determiner D that satisfies the conditions in the theorem. This assumption is tenable, with no loss of generality, for all determiners that are #MON". Under this assumption, we can use the following construction. Leave A and B as they are in the proof, and use the following A 00 instead of A g. Now consider the following two sentences: a. All the students drank a whole glass of beer together. b. All the students who've been roommates drank a whole glass of beer together. Assume that and that Y g. Clearly, these sets satisfy condition ( ) with respect to the determiner all. Following the construction Assume now that the denotation of students is A, and that the denotation of students who've been roommates is A 00 , i.e. all the couples of students. Assume further that the denotation of drank a whole glass of beer together is B. In this situation it is clear that sentence (34a) is true. However, since, for instance, s 1 and s 3 were roommates, but did not drink a whole glass of beer together, sentence (34b) is not true in this situation. An example for a "MON- determiner is the determiner some but not all. This determiner is formally defined as follows, for all A; B ' E: some but not all 0 Consider the following two sentences: a. Some but not all of the students who've been roommates drank a whole glass of beer together. b. Some but not all of the students drank a whole glass of beer together. Clearly, the same X , Y and Y 0 from the previous example satisfy condition ( ) with respect to the determiner some but not all. Following the same line, assume that the denotations of students, students who've been roommates and drank a whole glass of beer together are the same as in the previous example. Now, sentence (35a) is true in this situation, since there is a set of students that were roommates and also drank a whole glass of beer together, namely fs 1 but it is not true that all the students who've been roommates drank a whole glass of beer Monotonicity of D Monotonicity of count(D) Example "MON" "MON" some #MON#MON# less than five #MON" -MON" (!) all "MON# -MON# (!) not all -MON-MON- exactly five -MON# -MON# not all and (in fact) less than five (of the) -MON" -MON" most #MON-MON- (!) all or less than five (of the) "MON-MON- (!) some but not all (of the) Table 2: (non-)monotonicity under count together (cf. the argument in the previous example). On the other hand, sentence (35b) is not true in this situation, simply because all the students drank a whole glass of beer together. Table 2 summarizes the (non-)preservation of monotonicity properties under count for the nine classes of determiners according to the monotonicity of their two arguments. Note again that for each of the two classes #MON" and "MON#, there is an exception to the result that is mentioned in the table: the PTRIV deter- miners. The exclamation marks emphasize the cases in which left monotonicity is not preserved. There are two natural extensions to these results, which we state here without proof. First, we note that fact 7 and the 'monotonicity loss' theorem equally hold when count is replaced by dfit. Thus, adding the witness condition to count does not change the monotonicity (non-)preservation results that were established above for count. The proof of this claim is quite laborious, but routine. Linguistically, it implies that existential processes in collective quantification should not lead to problems that are similar to Van Benthem's problem, or to any other change in the monotonicity properties of determiners beyond what was shown above. Another point that worths mentioning is that the results that were shown above equally hold for global determiners: functors from domains E to determiners DE over E. It is important to appeal to global functors because linguistic items such as all, some, five etc. have global definitions and properties, and are not simply defined over a given domain, as assumed throughout this article. We say that a global determiner D is conservative, permutation invariant, (left/right) trivial or (upward/downward left/right) monotone, if the local determiner DE satisfies the respective property for any non-empty domain E. Using this global perspective, all the results that were proven above for local determiners equally hold of global determiners that satisfy the extension property (cf. Van Benthem (1984)). The When we say that a global determiner D satisfies extension, this means that given a domain E and any two sets A;B ' E, the local determiners DE 0 s.t. all agree on the truth value that they assign to A and B. reason for assuming this global property is our results concerning non-monotone determiners. Note that a global determiner D is -MON (MON-) if and only if there is E s.t. DE is not #MON (MON#) and there is is not "MON (MON"). From this it does not yet automatically follow that there is a domain E s.t. DE is -MON (MON-), as required in fact 7 and the main theorem. How- ever, provided that a global determiner D is non-trivial in both its arguments and satisfies extension, the existence of such a domain does follow from the assumption that D is -MON (MON-). Since most works on GQT assume that natural language determiners satisfy extension (see Keenan and Westerst-ahl (1996)), the generalization of our results to global determiners is both linguistically and technically straightforward. 5 Conclusion The formal study of the interactions between quantifiers and collective predicates has to deal with many seemingly conflicting pieces of evidence that threaten to blur the interesting logical questions that these phenomena raise. In this article we have studied the monotonicity properties of collective quantification, which is a central aspect of the problem of collectivity. We showed that to a large extent, the principles that underly monotonicity of collective quantification follow from standard assumptions on quantification in natural language in general. The count opera- tor, which is a straightforward extension of Scha's `neutral' analysis of collective determiners, involves a simple 'conservativity element' - intersection of the right argument with the left argument, and a 'participation element' - union of both set of sets arguments. The conservativity element within the count operator is responsible for the two a priori unexpected asymmetries in the monotonicity behavior of collective determiners: 1. Only determiners with 'mixed' monotonicity properties change their behavior when they quantify over collections. 2. Only the left monotonicity properties of such determiners may change in these cases. We believe that the reduction of certain asymmetries in the domain of collective quantification to the asymmetric conservativity principle is a desirable result that another aspect of the central role that this principle plays in natural language semantics. Two open questions should be mentioned. First, in this article we did not address the 'universal' reading that certain collective determiners show, as treated by Dalrymple et al.'s bounded composition operator. More empirical research is needed into these phenomena, which indicate that there may be more than one strategy of plural quantification. The formal properties of such universal strategies and the linguistic restrictions on their application should be further explored. Sec- ond, although we characterized the logical monotonicity properties of collective determiners, we did not study the linguistic implications that these properties may have for the analysis of negative polarity items. These items normally appear only in downward entailing environments, and it should be checked whether they are sensitive to 'monotonicity loss' under collectivity of all. For instance, a sentence such as the following, where all is not left downward monotone, is expected not to license the negative polarity item any in its left argument. (36) ?All the students who had any time drank a whole glass of beer together. Whether or not this expectation is borne out is not clear to us, and we must leave these and other implications of 'monotonicity loss' to further research. Acknowledgments This research was partly supported by grant no. 1999210 ("Extensions and Implementations of Natural Logic") from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The second author was also partly supported by NWO grants for visiting the UiL OTS of the Utrecht University in the summers of 2000 and 2001. We are indebted to Johan van Benthem for discussions that initiated our interest in questions of monotonicity and collectivity. Thanks also to Nissim Francez, Richard Oehrle, Mori Rimon and Shuly Wintner for their remarks. The main results in this article were presented at the conference Formal Grammar/Mathematics of Language(FGMOL), Helsinki, August 2001. We are grateful to the editor of JoLLI and an anonymous reviewer for thorough and constructive comments that helped us to improve an earlier version of this work. --R Generalized quantifiers and natural language. Some Extensions of a Montague Fragment of English. Plurality of mass nouns and the notion of 'semantic parameter'. Reciprocal expressions and the concept of reciprocity. Implication reversal in a natural language. Quantification in an Extended Montague Grammar. A semantic characterization of natural language determiners. Generalized quantifiers in linguistics and logic. quantifiers. Polarity Sensitivity as Inherent Scope Relations. The logical analysis of plurals and mass terms: a lattice theoretical approach. In Bauerle Plurals and Events. A more general theory of definite descriptions. Strategies for scope taking. Questions about quantifiers. Essays in Logical Semantics. Language in Action: categories Applied Quantifier Logics: collectives Sums and quantifiers. Flexible Boolean Semantics: coordination Distributivity and dependency. Flexibility Principles in Boolean Semantics: coordination --TR

monotonicity;determiner;plural;collectivity;generalized quantifier;type shifting

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Removing Node Overlapping in Graph Layout Using Constrained Optimization.

Although graph drawing has been extensively studied, little attention has been paid to the problem of node overlapping. The problem arises because almost all existing graph layout algorithms assume that nodes are points. In practice, however, nodes may be labelled, and these labels may overlap. Here we investigate how such node overlapping can be removed in a subsequent layout adjustment phase. We propose four different approaches for removing node overlapping, all of which are based on constrained optimization techniques. The first is the simplest. It performs the minimal linear scaling which will remove node-overlapping. The second approach relies on formulating the node overlapping problem as a convex quadratic programming problem, which can then be solved by any quadratic solver. The disadvantage is that, since constraints must be linear, the node overlapping constraints cannot be expressed directly, but must be strengthened to obtain a linear constraint strong enough to ensure no node overlapping. The third and fourth approaches are based on local search methods. The third is an adaptation of the EGENET solver originally designed for solving general constraint satisfaction problems, while the fourth approach is a form of Lagrangian multiplier method, a well-known optimization technique used in operations research. Both the third and fourth method are able to handle the node overlapping constraints directly, and thus may potentially find better solutions. Their disadvantage is that no efficient global optimization methods are available for such problems, and hence we must accept a local minimum. We illustrate all of the above methods on a series of layout adjustment problems.

Introduction Graph drawing has been extensively studied over the last fteen years. However, almost all research has dealt with graph layout in which the nodes are treated as points in the layout of graphs. Unfortunately, treating nodes as points is inadequate for many applications. For example, a textual label is frequently added to each node to explain some important information, as in many illustrative diagrams for engineering designs, the human body and the geography, and even a satisfactory layout for a point-based graph may lead to node overlapping when labels are considered. For this reason, we are interested in layout adjustment which takes an initial graph layout with the sizes of the nodes as inputs, and then modies the original graph layout so that node overlapping is removed. We primarily address the problem of layout adjustment in a dynamic context. This is useful in interactive applications such as graph display in which sub-graphs are enlarged and shrunk or node labels are changed. Here the aim is take an existing graph layout and remove node overlapping while preserving the user's mental map of the graph. Following, Misue et al [17] in order to minimize change to the mental map we preserve the graph's orthogonal ordering, that is to say, the relative ordering of the nodes in both the x and y direction, and attempt to place nodes as closely as possible to their original positions. We also consider layout adjustment in the context of a static graph. In this case we wish to minimize the area of the graph in the new layout. However, since we assume that some sophisticated graph layout algorithm has been used to give an initial layout for the graph, like the dynamic case we still try to preserve the initial layout while removing node overlapping. In this paper we study four dierent proposals for performing graph layout adjust- ment. All approaches model the problem as a constrained optimization problem. An advantage of viewing layout adjustment as a constrained optimization problem is that we can also add constraints which capture the semantics of the diagram if it is not just a simple graph. For example, we can add constraints which specify that certain nodes must be on the boundary of the graph or which specify the relative placement of nodes. Our rst approach to layout adjustment is the simplest. It performs the minimum linear scaling in both the x and y directions which will remove node-overlapping. We give a simple and e-cient algorithm based on dynamic programming for nding this minimum. The disadvantage is that uniform scaling means that nodes are often moved unnecessarily far, and so this approach may not lead to good layout. The second approach is based on formulating the node overlapping problem as a convex quadratic programming problem which can then be solved using a quadratic solver. This approach is attractive because algorithms for convex quadratic optimization are well understood, and global optimization is possible in polynomial- time. The main disadvantage arises from the need to model node overlapping constraints using linear constraints. Unfortunately, the no node-label overlap constraint between nodes u and v is inherently disjunctive, i.e. u is su-ciently to the left of v or u is su-ciently above v or u is su-ciently to the right of v or u is su-ciently below v. This cannot be expressed directly but must be strengthened to obtain a linear constraint strong enough to ensure no node overlapping. Optimal layout with respect to this stronger linear constraint may be sup-optimal with respect to the original no node-label overlap constraints. In contrast the third and fourth approaches allow the disjunctive node overlapping constraint to be expressed directly. Both approaches use local search methods [1, 2, 7, 6, 27]. A local search method starts with a current value for each variable, and by examining the local neighbourhood tries to move to a point which is closer to the optimum. Constraints are handled as penalties to the optimization function. The search proceeds until a local minimum is found. When the local minimum represents a solution (all constraints are satised), the algorithm terminates. Otherwise, it may penalize this local minimum to avoid visiting it again, and then continue the search until it nds a solution. Because for layout adjustment problems we have a starting position \close" to the global optimum, namely the initial layout, in many cases local search can e-ciently nd the optimum. The third approach is a modication of the EGENET [11] solver (an extension of the GENET model [27]) to perform layout adjustment. Modications are required to (a) handle a constraint optimization problem (as opposed to constraint satisfaction for which EGENET is designed) and (b) to handle real ( oating point) variables, rather than variables with a discrete domain. Our work represents the rst attempt, that we know of, to investigate how the EGENET solver can be used to solve constrained optimization problems involving real numbers 1 . Our fourth approach is an adaptation of the Lagrangian multiplier methods (LMMs) [13, 20, 26], a classical optimization technique used in operations research. Lagrangian multiplier methods are a general approach to constrained optimization which have been applied to many di-cult problems successfully. Here we develop a specic Lagrangian algorithm where variables are treated in pairs corresponding to node positions. Because of the non-convexity of the constraints, we repeatedly optimize using the results of the previous optimization to give a new, possibly better, starting position for the search. There has been little work on layout adjustment and mental map preservation. The most closely related papers are that of Eades et al [9] in which mental map was proposed for the rst time and orthogonal ordering, proximity relations and topology were given as three criteria, and that of Misue et al [17], which provides a force-directed algorithm called the force scan algorithm to perform dynamic graph layout adjustment. We demonstrate that our methods, apart from the uniform scaling approach, give better layout than the force scan algorithm, although they are slower. Other related work includes Lyons [15] who tries to improve the distribution of the nodes in the new layout according to some measures of distribution (cluster busting) while simultaneously trying to minimize the dierence between the two layouts according to some measures of dierence (anchored graph drawing). Eades et al [8] try to preserve the mental map in animated graph drawing by using a modied spring algorithm to provide smooth user transitions. Other related work includes: [19], [4], [14], [18] and [21]. A preliminary version of our second proposal, including a proof, appeared in [10]. For a comprehensive survey of graph drawing methods, see [3, 22]. Finally we mention some previous work [24, 25] on adapting the GENET model to solve constraint optimization problems [23]. Guided Local Search (GLS) [25] iteratively calls a local search procedure, based on the GENET variable updating scheme, to modify and minimize an augmented cost function until a predened stopping condition is reached. GLS had been successful in solving Radio Link Frequency Assignment Problems. This paper is organized as follows. In Section 2 we describe how to view the lay-out adjustment problem as a constrained optimization problem and introduce both a static and dynamic version of the problem. In Section 3 we describe the simplest approach to solving the layout adjustment problem, linear scaling. In Section 4 we show how to replace the disjunctive no-node overlapping constraints of the layout problem with linear approximations. This allows us to transform the layout adjustment problem to a quadratic programming problem, which can then be handled by any quadratic solver. In Section 5 we introduce the original EGENET model. Then, we discuss how to modify the EGENET model to handle both the static and dynamic layout problems as continuous constrained optimization problems. In Section 6 we brie y describe some basic concepts in multiplier methods, and then discuss how to adapt these concepts to derive a Lagrangian-based search procedure specialized for layout adjustment problems. We provide an empirical evaluation of our four proposals using a set of arbitrary graph layout problems investigating both their e-ciency and eectiveness in Section 7. We also compare our approaches to the force-scan algorithm of Misue et al. Finally, we summarize and conclude our work in Section 8. 2. Layout Adjustment as Constrained Optimization All our algorithms for layout adjustment are based on translating the problem into a constrained quadratic optimization problem of the form minimize with respect to C where is a quadratic expression and C is a (possibly non-linear) collection of constraints. All are in terms of variables representing the position of each node. We assume that we are given A (possibly directed) graph ng is the set of nodes in the graph, and E V V is the set of edges in the graph, that is to say, there is a edge from node u to node v. A node labelling for G which consists of two vectors wn ) and each node v to the width (w v ) and height (h v ) of its label respectively. And an initial layout for the graph G, which consists of two vectors, x n ) which map each node v to its x or y position respectively. That is to say, node v is placed at Intuitively, as shown in Figure 1, each node has a rectangular bounding box and the display area has origin at its lower left corner with x axis rightwards and y upwards. The variables of the layout adjustment problem are the x and y coordinates of the nodes, which we shall represent using two vectors x and y. The constraints on the problem for layout adjustment must ensure that there is no node overlapping in the resulting solution. These constraints (C no ) can be expressed as: for all u; Y Figure 1. Notation (v to the right of u) to the right of v) or equivalently In fact, we usually want the nodes to be separated by some minimum distance d and not directly abutt each other. This is simply handled by modifying the height and width of each node by adding the distance d, and treating the problem as no overlap of these new larger nodes. As indicated in the introduction, we are primarily concerned with layout adjustment in the context of dynamic graph layout. This is useful in interactive applications such as graph display in which sub-graphs are enlarged and shrunk or node labels are changed. Here the aim is take an existing graph layout and remove node overlapping while preserving the user's mental map of the graph. One heuristic to ensure that the aesthetic criteria remain satised and that the new layout is \similar" to the initial layout is to preserve the orthogonal ordering of the original layout [17]. The idea is to preserve the relative ordering of the nodes in the x and y directions. In our context, linear constraints (C oo ) which ensure the preservation of orthogonal ordering are: for each u; v then x u < x v and if x 0 v then x and similarly for the y-direction. We shall use this heuristic for our rst two approaches|uniform scaling and quadratic programming|but not for the local search based methods. The dynamic layout adjustment problem is to nd a new layout for G, x and y, so that the constraints C are satised and the position of each node is as close as possible to its original position. We encode the dynamic layout adjustment problem as a constrained optimization problem by setting the objective function dyn to be, and we wish to minimize dyn subject to C (1) where we can choose the constraints of the problem C to be either C no or C no ^C oo . The new layout is given by an optimal solution to the above problem. Apart from layout adjustment in a dynamic context, we might also consider layout adjustment in a static context. In this case the underlying assumption is that the initial layout has been generated using some sophisticated graph layout algorithm which captures aesthetic criteria. We do not want to redo this work in the layout adjustment phase. Hence, static layout adjustment should remove node-label overlapping but still preserve the (presumably) aesthetically pleasing node placement of the initial layout. We are also interested in minimizing the overall area of the graph. Preserving the initial layout is very similar to preserving the user's mental model, and so we can use the same techniques: constraints to preserve orthogonal ordering and a term in the objective function to move nodes as little as possible. The main dierence to the dynamic layout adjustment problem is an extra term in the objective function, stat , which allows the area of the new layout to be minimized. More precisely, the static layout adjustment problem is to nd a new layout for G, x and y, by solving the constrained optimization problem subject to C (2) where C is either C no or C no ^ C oo , k 0 is a weighting factor and and towards which the layout is supposed to be shrunk. can be the arithmetical average of all nodes' x (y) coordinates, or median of all nodes' x (y) coordinates. One can also take the position of centroid of a graph or even a predened point as c. Minimization will attempt to place the nodes as close as possible to a predened position, so reducing the overall area of the new layout. Despite its dierent motivation, the formulation of the static layout adjustment problem is very similar to that for the static. Thus, in the following sections, we will focus on techniques for solving the dynamic case since they can also be used with little modication to solve the static case. 3. Using Uniform Scaling We rst consider a very simple approach to layout adjustment in which we uniformly scale the graph to remove overlapping. We nd c x , m x , c y and m y and move each node v ) to (m x x 0 this is a linear transformation, it preserves the graph's orthogonal ordering (indeed, all of the original graph's structure) and the scaling factors m x and m y are chosen so that node overlapping is removed. The disadvantage is that uniform scaling may cause nodes to move unnecessarily. We rst examine how to compute c x and c y for given scaling factors m x and m y . It follows from Equation 1 that we must minimize scale where this is This is minimized when where x and y are the means of the x 0 v 's and y 0 's, respectively. It follows that x and 2 y are the variances of the x 0 v 's and y 0 's, respectively. Now we consider how to compute the scaling factors. Consider node dimensions (w dimensions (w similarly for my ij . By construction mx ij is the minimum amount of scaling in the x direction which will remove the overlap between the two nodes and similarly mx ij is the minimum amount of scaling in the y direction to remove this overlap. Hence we can solve the layout adjustment problem using scaling by nding scale factors sx and sy that solve the constrained optimization problem: minimize scale subject to 81 i < Inspection of Equation 4 reveals that we should choose the minimal scaling factors removes overlapping. This means that m x will be mx ij for some overlapping nodes i and j and similarly for m y . This observation leads to the algorithm shown in Figure 2. It computes c x , which remove all overlapping (if possible) and which minimize layout-scale compute the mean mu x and variance 2 x of the x 0 's compute the mean mu y and variance 2 y of the y 0 's lexicographically into the array a[1:::m] let a[k] have form (mx /* compute possible pairs of scaling factors bmx[k] and bmy[k] */ endfor /* now determine which pair has least cost */ bestcost cost y if (cost < bestcost) then bestcost := cost best := k endif endfor solution else m x := bmx[best] Figure 2. Scaling method for layout adjustment scale . The only subtlety to notice is that, for each i, the scaling factors bmx[i] and bmy[i] remove all overlapping since scaling by bmx[i] in the x-dimension removes the overlapping corresponding to a[1]; :::; a[i] and scaling by bmy[i] in the y-dimension removes the overlapping corresponding to a[i The main part of the algorithm has complexity O(m log m) where m is the number of overlapping nodes. Since the number of overlapping nodes is O(jV j 2 ), the overall complexity is O(jV j 2 log jV j). 4. Using Quadratic Programming Quadratic programming is used to nd the global optimum of a convex quadratic objective function where the constraints are a conjunction of linear arithmetic equalities 9and inequalities. Quadratic programming has been widely studied in operations research and interior point methods provide polynomial time algorithms for solving such problems. In Section 2 we saw how to encode the layout adjustment problem as a constrained optimization problem in which the objective function is a convex quadratic for- mula. Unfortunately, the constraints cannot be expressed as conjunctions of linear arithmetic constraints since the no overlap constraints involve disjunction. To use a quadratic programming based approach, we need to replace these disjunctive constraints by linear approximations which will guarantee that the no overlap constraints hold. Since each individual disjunct is a linear constraint, the straightforward way to do this is to choose which disjunctive possibility must hold. In order to construct the linear approximation of the no-overlap constraints C no with respect to the x direction (C no x ) we dene the \right horizontal neighbours" of nodes. We then constrain each node to have no overlap with its right neighbours. In a sense we hardwire into the constraints which direction the nodes must move in to remove the overlapping. For any node v, its right horizontal neighbour nodes set right(v) is the set containing all nodes u such that in the initial position: (1) u 6= v; (2) u is to the right of v v ); and (3) u and v could overlap if moved only in the x direction (either x 0 or x 0 right(v) is an immediate right horizontal neighbour node of node v if there does not exist a node u such right(v). Similarly we can dene the upper vertical neighbour nodes set, upper(v), and immediate upper vertical neighbour node, u v . It is straightforward to dene the constraints in C no x That is for each node v 2 V , we compute its immediate right horizontal neighbours, and for each of these, r v say, add the constraint into C no x . Similarly we dene C no y . Generation of the orthogonal ordering constraints C oo x and C oo y for the x and y directions, respectively, is conceptually straightforward. For e-ciency it is important to eliminate as many redundant constraints as possible. The precise algorithm is given in [10] We treat the layout adjustment problem as two separate optimization problems, one for the x dimension and one for the y dimension, by breaking the optimization function into two parts and the constraint into two parts. The constraints in the direction C x are given by C no x together with the C oo constraints on x variables x ). We similarly dene C y . For the dynamic layout adjustment problem it follows from the denition of dyn that the optimization problems are to subject to C x (6) quadratic-opt compute C x x := minimize x subject to C x compute C y y := minimize y subject to C y Figure 3. Quadratic programming approach to layout adjustment for the x-direction, and subject to C y (7) for the y-direction. The new layout is given by x and y where x is the solution to and y is the solution to (7). The advantage in separating the problem in this way is twofold. First, it improves e-ciency since it roughly halves the number of constraints considered in each problem. Second, if we solve for the x-direction rst, it allows us to delay the computation of C y to take into account the node overlapping which has been removed by the optimization in the x-direction. The actual layout adjustment algorithm used is given in Figure 3. First the problem in the x dimension is solved. Then x 0 is reset to be x, the x positions discovered by this optimization. Doing this may reduce the number of upper vertical neighbours and so reduce the size of C y and also allow more exibility in node placement in the y-direction. As we previously observed, polynomial time algorithms for solving quadratic programming problems exist. However, for medium sized problems of several hundred to one thousand constraints the preferred method of solution is an active set method. We have used an incremental implementation of the active set method provided by the C++ QOCA constraint solver [5]. The key idea behind the active set method is to solve a sequence of constrained optimization problems O 0 , ., O t . Each problem minimizes f with respect to a set of equality constraints, A, called the active set. The active set consists of the original equality constraints plus those inequality constraints that are \tight," in other words, those inequalities that are currently required to be satised as equalities. The other inequalities are ignored for the moment. 5. Using the EGENET Solver In this section, we investigate how the EGENET solver can be used to solve the layout adjustment problem. In order to use quadratic programming we needed to linearize the no-overlap constraints and implicitly xed choices about whether to remove the overlap in the vertical or the horizontal direction. This means that potentially better solutions to the layout adjustment problem were never considered. EGENET can handle the disjunctive no-overlap constraints directly and hence is attractive as a method for solving the constrained optimization problems associated with layout adjustment. But, because this formulation is non-convex only a local optimum can be found. In the remainder of this section we rst review the original EGENET model for solving discrete constrained satisfaction problems. We then examine how to modify the original EGENET to handle continuous constrained satisfaction problems (without optimization). Lastly, we discuss how the modied EGENET can be used to solve the layout adjustment problem as a continuous constrained optimization problem. To our knowledge, this work represents the rst attempt to investigate how the EGENET solver can be used to solve continuous constrained optimization problems. 5.1. The Original EGENET Solver A constraint satisfaction problem (CSP) [16] is a tuple (U; D;C), where U is a nite set of variables, D denes a nite set D z , called the domain of z, for each z 2 U , and C is a nite set of constraints restricting the combination of values that the variables can take. A solution is an assignment of values from the domains to their respective variables so that all constraints are satised simultaneously. CSPs are well-known to be NP-hard in general. GENET [27] is an articial neural network, based on the min-con ict heuristic (MCH), for solving arbitrary CSPs with binary constraints. The MCH is to assign a value causing the minimum number of constraint con icts to each variable so as to quickly nd the local minima in the search space. Lee et. al. [12] extended GENET to EGENET with a generic representation scheme for handling both binary and non-binary constraints. EGENET has been successfully applied to solve non-binary CSPs such as the car-sequencing problems and cryptarithmetic problems in an e-cient manner [12]. Constraints in EGENET are represented by functions from values of the variables to non-negative numbers. For example the constraint z 1 z 2 can be represented by the function Alternatively we can represent the same constraint just using a Boolean valued function, as A general CSP can be formulated as a a discrete unconstrained optimization problem as follows. min z2D zn is the Cartesian product of the (nite) domains for all the n variables, m is the total number of constraints, and g i (z) denotes the penalty function for each constraint C i in the CSP, and i represents the weight given to the constraint. The penalty g i (z) equals 0 if the variable assignment z returns a positive integer. The weights associated with constraints are all initially 1, but may be modied by the EGENET procedure. In this formulation, the goal is to minimize the output of the cost function f(z) whose value depends on the number of unsatised constraints and the weights associated with these constraints. For each solution z to the original CSP, f(z satises all the constraints. EGENET uses a simple local search rule to minimize the cost function f(z), and a heuristic learning rule to change weights of constraints until it nds a solution z . Initially, a complete and random variable assignment z 0 is generated. Then, the network executes a convergence procedure as follows. Each variable is asynchronously updated in each convergence cycle. The update simply nds the value for each variable which gives lowest total cost without modifying any other vari- ables. When there is no change in any value assigned to the variables, the network is trapped in a local minima. If the local minimum does not represents a solution, a heuristic learning rule, is used to update the weight i for any violated constraint C i in the CSP so as to help the network escape from these local minima. The net-work convergence procedure iterates until a solution is found or a predetermined resource limit is exceeded. 5.2. The Modied EGENET Solver The original EGENET solver only supports constraints over nite domains. There- fore, to solve the layout adjustment problems with the EGENET approach, we have to consider how to modify the original EGENET solver so as to handle problems with continuous domains. To handle such a continuous constrained problem, we have to decide how to represent the real-number domain of each variable in an EGENET network. The domain of each variable in a CSP is nite and is usually represented by a nite set of contiguous integers. For continuous constrained problems, this is inappropriate, since, even if we consider that real variables only take oating point values, there are too many possibilities. Instead we represent the range of possible values of a variable z i simply by a lower and upper bound l i ::u i . For problems with no natural lower and upper bounds on variables we need to generate them. The tighter the bounds used the more e-cient the search will be since it examines a smaller area. But giving initial bounds which are too tight may lead to no solution being found even when one does exist. Since the original EGENET variable updating function assumes a nite number of possible values in the domain of each variable, to update a continuous variable in the modied EGENET network, we need to treat the domain of each continuous variable represented as a range l as a set with nite number of elements bounded by the values l and u only. In other words, we sample the domain of a continuous variable by only trying a nite number of possible values in its range, when updating its value within the EGENET computation. The number of elements we try in each update is called the domain sampling size. The larger the domain sampling size the closer to the true local optimum solution we are likely to get, but the more computation is required at each step. We use a domain sampling size of 10 throughout our experiments, although this could be changed (even within a computation). In order to overcome the coarseness of the search when using a small domain sampling size without increasing the computational overhead too greatly, whenever a solution is found we can revise the bounds of variables inwards towards the solution just found. This has the eect of focusing the search around the solution just found. This approach is used in the algorithm below. This extra exibility makes our algorithm substantially dierent from the case where the domain sizes of the variables remain xed throughout the computation (as used for discrete constrained optimization and satisfaction problems). To handle a continuous constrained optimization problem we simply add a optimization component to the function to be minimized. The augmented cost function is As we shall see in the next section, this is closely related to the Lagrange multiplier methods. Pseudo-code for the general optimization algorithm EGENET-opt is given in Figure 4. The algorithm takes the domain sampling size DSZ, the number n of variables in z and the cost function cost() as input, and returns a tuple consisting of the best solution found z b and its cost best. The basic idea is similar to that of GLS [25]. First, we initialize all the EGENET variables using initialize vars (usually this is just random). Then, we use the var order function to produce a permutation perm which determines the order in which the variables z will be updated. The core of the method is the updating loop where each of the sample values for a variable z v are tried in turn, and the value which gives minimum cost is retained as the new value. This represents a simple local search on the best value of variable z v . Because the cost function includes penalties for violated constraints this will drive the search towards solutions. This loop continues until a local minima is found (where no variable is updated). When the local minimum represents a solution, EGENET-opt invokes the function revise bounds to try to revise the bounds for the domains of the EGENET variables. Then, it computes the solution cost according to the cost function, and updates the best solution x b and best if the cost of the current solution is smaller than best. To modify the augmented cost function, EGENET-opt invokes penalize ctr(z), which is basically the same as the original EGENET heuristic learning mechanism, to penalize any violated constraint with respect to the current variable assignment. Similarly, EGENET-opt invokes penalize opt(z) to modify the form of the optimization function that occurs in the augmented cost function. The algorithm iterates until the function stopping criterion detects that a predened stopping criterion is fullled, and therefore returns true. initialize vars(z) z b := z best := cost(z b ) repeat repeat perm := var order(n) mincost := cost(z) minval := z v ; if cost(z) < mincost mincost := cost(z) minval := z v ; endif endfor z v := minval endfor until (no update for all v 2 z) if (z represents a solution) revise bounds(z) if (cost(z) < best) z b := z best := cost(z b ) endif endif penalize ctr(z) penalize opt(z) until (stopping criterion()) return hz b ; besti Figure 4. A General EGENET-based Optimization Algorithm Clearly, the e-ciency of using EGENET-opt to solve a continuous constrained optimization problem, and the quality 2 of the best solution found by EGENET- opt depends largely on how we dene the augmented cost function, and how we appropriately modify this cost function by the penalize ctr and penalize opt function. 5.3. Handling Layout Adjustment The variables of the layout adjustment problem are x and y, the x and y coordinates of each node. Given the exible representation scheme for any general constraint in the EGENET model, it is straightforward to dene a no-overlap constraint as a disjunctive constraint in the EGENET network, which ensures that there will be no overlapping between the labels i and j, as follows: Figure 5 shows how the constraint is represented in the modied EGENET net- work. It denes a function g ij which measures the amount of overlap in the x and y directions, returning 0 if there is no overlap. A no-overlap constraint is applied to each pair (i; j) of nodes where 1 i < j n. This demonstrates one advantage of using the EGENET approach to solve the layout adjustment problems : it is, in general, simple to represent the constraints involved in these problems or other related graph layout problems in the EGENET network. Indeed arbitrary additional constraints can be added to the problem straightforwardly. fx fy fy Figure 5. The EGENET network for a no-overlap constraint. We need to determine suitable ranges for the variables of the problem, in order to make use of the modied EGENET algorithm above. We would like to ensure that the variables' initial ranges contains a solution to the problem, if it exists. A conservative approach is to nd a solution using the uniform scaling approach and then to use the minimal range containing that position and the original value. For instance, if (x u v ) is the position computed by the uniform scaling approach for node v and the position of node v in the original graph is is then the initial range for x v can be set to [x otherwise, and similarly for y v . Clearly this range is guaranteed to include at least one solution. The problem is that scaling factors tend to be large, so the range is also large. In practice, by making some assumptions about the density of the nodes we can start with smaller ranges, and thus reduce the search. Dene max x to be the maximum overlap in the x direction of any pair of nodes, and dene max y similarly. We use initial ranges of [x 0 for y v . This means no solution is guaranteed, but this is only true for graph where there is a dense overlapping in the initial layout. In practice this is rare. For layout adjustment problems the initial value for the variables x and y returned by initialize vars is given by the initial layout x 0 and y 0 . One of the reasons for investigating local search methods is that this initial layout is usually quite close to the global optimum, and hence local search around the initial layout is likely to nd a good solution. The variable ordering strategy var order used simply updates the variables in one dimension (x) in a random order before or after updating the variables in the other dimension (y). This is eective since the relationship between the x and y variables are weak (only through disjunctive constraints). By evaluating each dimension in turn we get a faster convergence to local minima. Other strategies are, of course, possible. The revise bounds function is dened as follows. When a solution is found at x then for each x i , if x i > x 0 i , we reset the upper bound to be the current value x i and the lower bound is reset to x 0 i we reset the lower bound to be the current value x i and the upper bound is reset to x 0 the bounds are unchanged. Similarly for y. This encodes the strategy that, since we have found a solution on one side of the starting point, we will not look on the other side, and we will not look further away as looking farther away will tend to reduce the optimality of the solution. For the graph layout problem, stopping criterion returns true when a predetermined number of iterations have been tried, or when the cost of the current solution is worse than or equal to the previous solution. The total resource limit is initially set to 1000. Each time a better solution is found, the resource limit is reset to double the amount of resource used to nd the previous solution if this is less than 100. Otherwise, the resource limit is reset to the minimum of 1000 or 120% of the resource used when nding the previous solution. All that remains is to determine the penalize ctr and penalize opt functions. The relative weights of the objective function and the constraint penalties are impor- tant. For example, for the graph layout adjustment problem, if the augmented cost function is biased towards the penalty for constraint violation, that is node over- lapping, then EGENET-opt may take a longer time, or even fail, to return a good solution. 3 On the other hand, if the augmented cost function is biased towards the solution cost, then EGENET-opt may fail to nd a solution satisfying all the no-overlap constraints. Recall the objective function dyn for layout adjustment. As the quantity (usually in the order of 100 or 1000) is always much larger than the penalty value of the constraints (which are Boolean) for constraint violations, we need to normalize this quantity so as to avoid bias towards the solution cost. If the range of x v is [l v ::u v ] then the maximum value of the term of (l v c x . Denote this by omax v . The normalized optimization function for x v is simply norm x Clearly, the value returned by norm x real number in the range of We can similarly dene a normalized optimization function norm y for variables y. Accordingly, we dene the augmented cost function cost(x; y) as as the sum of the penalty for constraint violations, and the sum of normalized optimization function as follows. (norm x where m denotes the total number of constraints in the problem. When a local minimum is found, the penalty function penalize opt changes the normalized optimization functions to take into account the new smaller ranges for some variables. The penalize ctr function simply increases the weight i of a violated constraint by 1 (as in the original EGENET model). Note that in the EGENET approach we have ignored the orthogonal ordering constraints, and simply concentrated on the overlap constraints. Since, in practice, we have found that minimimizing dyn preserves the structure of the original graph, and so tends to preserve orthogonality, even though there are not explicit constraints to do so. 6. Using A Pseudo-Lagrangian Method The EGENET local search method described in the previous section is a modica- tion of a discrete constrained satisfaction algorithm to solve continuous constrained optimization problems. In the end, it looks very similar to a Lagrange multiplier method. This inspired us to directly solve the problem using such an approach. 6.1. Lagrangian Multiplier Methods Lagrange multiplier methods are a general approach to continuous constrained optimization that can tackle non-linear objective functions with non-linear constraints. A general continuous equality-constrained objective function is formulated as follows minimize f(z) subject to where f(z) is the objective function and g(z) is a vector of functions representing the constraint penalties. The Lagrangian function associated with this problem is a weighted sum of the objective function and the constraints. It is dened as : where is a vector of Lagrange multipliers. The Lagrangian function is related to the local extrema of the problem (8) by the following theorem (see e.g. [13]). Theorem 1 Let z be a local extremum of f(z) subject to g(z). Assume that z is a regular point 4 . Then there exists a vector such that r z f(z) Figure 6. Overlap calculation. Based on the above theorem there are a number of method for solving constrained optimization problems. The most widely used is the rst-order method represented as an iterative process: z where k is a step-size parameter. Intuitively the equations represent counteracting forces to achieve a good solution to the optimization problem. When a constraint is violated (11) increases the weight of the constraint, forcing the search towards a solution. In contrast (10) performs descent in the optimization direction once all of the Lagrange multipliers are xed. 6.2. Layout Adjustment with Lagrange Multipliers The Lagrangian approach to layout adjustment performs an iterative process like that dened above. Rather than a synchronous update of all variables at once, each node v is treated in turn, and its two coordinate variables x v and y v are updated together. This allows the non-overlap constraints to be handled in a more meaningful way. When two nodes i and j overlap as illustrated in Figure 6, then the overlap of j on i in eect creates a force on i in the direction shown. If node i moves either a distance dx ij in the x direction or dy ij in the y direction the overlap will disappear. We choose the minimum magnitude of dx ij or dy ij as the resulting constraint violation. In eect, we treat the constraint function g ij as follows: (0; do not overlap Given this denition then r x i otherwise, and similarly r y i The two constraint functions g ij and g ji are symmetric and represent a single underlying overlap constraint. Hence, for each pair we have a Lagrange multiplier ij representing the current weight of the constraint. The basic local search is then simply a Lagrangian optimization using a rst-order stepping. Since the optimization only nds a local minima, one optimization may not nd a very good solution. After a local minima is reached the pseudo-Lagrangian method reduces the step-size by half, and also moves the nodes closer to their positions in the initial layout since this will further decrease the objective function albeit at the risk of introducing overlapping. This lets the search potentially nd better solu- tions. Eventually after some number of such reoptimizations the method nishes, returning the best solution found. For the experiments the initial stepsize was 0.125 and the factor limit 255. The function initialize multiplier simply initialized all the Lagrange multipliers to 1. The resource limit was never exceeded. The algorithm in Figure 7 does not take into account the orthogonal ordering constraints. These could be added using the standard Lagrangian approach of adding new constraint functions, namely and Unfortunately this simple addition can lead to divergent behaviour. One simple approach to handle the equality constraints of the orthogonal ordering is instead to simply force their compliance. Let S be a set of coordinates which must all be equal, e.g. fx g. For each such set we compute the average x2S x)=jSj and set each value within to this average value, e.g. x 1 := This approach while seemingly ad-hoc, has quite a principled justication. If we replaced the set of variables S by a single variable (in eect using the equation constraints as substitutions) then the change in this variable would be determined from the sum of the changes of the variables in the set S. The average is simply a function of the sum of the changes of the variables in S. However, this approach does not handle the inequality constraints of the orthogonal ordering. As for the EGENET approach, in our empirical evaluation of the Langrangian approach we have not included constraints to preserve the original orthogonal or- dering, since again in practice, we have found that minimimizing dyn preserves the structure of the original graph, and so tends to preserve orthogonality. initialize multiplier() best := +1 stepsize := initial step size repeat if (i 6= endif endfor endfor if (g(x; if (x; y) < best best := (x; y) else stepsize := stepsize=2 initialize multiplier() endif endif until (factor > limit or resource limit exceeded) return Figure 7. Pseudo-Lagrangian method for layout adjustment 7. Empirical Evaluation In this section we compare the performance of the Scaling Algorithm (SCALE) presented in Section 3, the Force Scan Algorithm (FSA) of Misue et al [17], the quadratic solver approach (QUAD) presented in Section 4, the modied EGENET solver described in Section 5 and the Psuedo-Lagrangian method (PLM) discussed in Section 6 on a set of nine dynamic graph layout adjustment problems. QUAD is implemented in Borland C++ for Windows Version 4:5. The other solvers are implemented in C, and compiled by the GCC compiler Version 2:7:3 on Linux. All tests were performed on a Pentium PC running at 155Mhz. Table 1. CPU time taken by SCALE, FSA, QUAD, EGENET and PLM for layout adjustment of the example problems. Graph # nodes SCALE FSA QUAD EGENET PLM Table 2. Value of dyn in the adjusted layout using SCALE, FSA, QUAD, EGENET and PLM for layout adjustment on the example problems. Graph # nodes SCALE FSA QUAD EGENET PLM 9 17 38880 127008 77760 5535 3688 A quantitative comparison of these dierent methods on the sample problems is provided in Table 1 and 2. In each table, the rst column gives the identifying number for the graph while the second column gives the number of nodes in the sample graph. Table 1 details the CPU time in seconds taken to nd the adjusted layout for each method while Table 2 details the value of dyn for the adjusted layout given by each method. For local search methods such as EGENET and PLM, the averages of CPU time and cost over 10 successful runs are reported in each case. For all the cases we have tested, both EGENET and PLM can successfully nd an (sub-)optimal nal layout without node overlapping. It should be noted that for both EGENET and PLM, the solvers will halt when the current solution found is worse than the previous one. For EGENET, the solver will also halt after the, possibly reset, resource limit is exceeded. Broadly speaking we nd that the ranking of methods with respect to the CPU time taken to nd the adjusted layout, from fastest to slowest, is SCALE, FSA, QUAD, PLM and EGENET. SCALE and FSA are considerably faster than the other methods, QUAD somewhat faster than PLM and the modied EGENET solver takes much longer than the other solvers. This is probably because the modied EGENET solver can only consider a nite number of points in the domains of each variable for each updating, and slowly revise the domains after each learning. The values of dyn reported in Table 2 provide a simple numerical measure of the quality of the adjusted layout. However, it is also important to look at the actual aesthetic quality of the adjusted layout. For this reason we now look at each of the example graphs and show the resulting layout adjustment with each method. Note that we have used an asterisk ( ) to indicate the method giving the (subjectively) best layout for each example in Table 2. (a) Graph 1, general (b) Graph 1 with box nodes (c) Layout with SCALE (d) Layout with FSA Layout with QUAD (f) Layout with EGENET (g) Layout with PLM Figure 8. Initial and layout adjustment for Graph 1. Figure 8(a) and 8(b) respectively show the initial layout of Graph 1 as an idealized graph with circles as nodes and as a labelled graph. Figure 8(c), 8(d), 8(e), 8(f) and 8(g) give the resulting layout adjustment using SCALE, FSA, QUAD, mod- ied EGENET and PLM respectively. All methods provide reasonable layout ad- justment, although we note that EGENET introduces an edge/node label overlap. PLM gives the best adjustment. Note that, as in all of our gures, we have scaled each graph to have a maximum height or width of one inch. Thus the adjusted layouts may not have the same scale. Smaller (usually better) layouts may be identied by their relatively larger node labels. (a) Graph 2 Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 9. Initial and resulting layouts for Graph 2. Surprisingly for Graph 2 as shown in Figure 9, the simplest method SCALE nds the best layout, while the other methods only nd a local minimum. All methods give reasonable layout adjustment with QUAD giving the worst layout. Figure shows the initial and resulting layouts for Graph 3 which has been chosen as an example of the kind of explanation diagram widely used in Biology or Engineering textbooks. Arguably, QUAD gives the best adjusted layout since it best preserves the original graph's structure. SCALE increases the size by too much while EGENET and to a lesser extent PLM change the orthogonal ordering of the graph and remove its symmetry. Graph 4 as shown in Figure 11(a) is an example of a rooted tree. Such labelled graphs are commonly used to display data structures or organization structures in many real-life applications. All methods give reasonable layout. The worst is by SCALE since it unnecessarily increases the size, while the layout found by QUAD is slightly better than that found by PLM and EGENET. Graph 5 is another example of rooted tree layout. This time PLM gives the best layout, although it has changed the orthogonal ordering, closely followed by FSA which preserves the orthogonal ordering. Both QUAD and EGENET introduce an edge/label crossing. Graph 6 was carefully designed to show that layout adjustment may introduce edge crossings even though the layout adjustment method preserves the graph's orthogonal ordering. Both FSA and QUAD produce the worst layouts since they introduce edge crossing. SCALE does not introduce any edge crossing in the nal a) Graph 3 (b) Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 10. Initial and resulting layouts for Graph 3. (a) Graph 4 (b) Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 11. Graph 4 | tree layout adjustment (1). layout since the initial layout does not contain any edge crossing, but the resulting layout is, as usual, unnecessarily wide. The modied EGENET and PLM methods produce very similar (and good) layouts with that produced by EGENET slightly better than that produced by PLM. As shown in Figure 14(a), Graph 7 is a rather pathological graph with no edges but with lots of node overlapping occurred at dierent horizontal levels. SCALE clearly gives the worst layout adjustment while the modied EGENET gives the best layout which is neatly packed and fairly close to the initial layout. On the other hand, PLM gives a stack-like layout which looks quite dierent from the original graph. a) Graph 5 (b) Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 12. Graph 5 | tree layout adjustment (2). Graph 8 shown in Figure 15(a) is a simplied version of Graph 7 with some nodes removed. Thus, there is less node overlapping in Graph 8. All of the algorithms produce a result closer to the original graph, but in this case PLM gives the most aesthetically pleasing layout. Graph 9 is an X-shaped graph with symmetry about both the x- and y-axis. All layouts are reasonable. The layouts by SCALE, FSA and QUAD retain the symmetry while those produced by the modied EGENET and PLM do not. QUAD produces the best layout. As we can see no method produces uniformly better layout adjustment, even SCALE produces the best layout for one example. However in general, SCALE and FSA produce the worst layout. In general, PLM closely followed by QUAD produce the best layout. However, EGENET and PLM can lose the original structure of the graph and to preserve this structure we would need to add extra constraints into the modied EGENET and PLM solvers. Generally, QUAD and FSA preserve the structure, in part because they preserve the orthogonal ordering, but may still introduce edge overlapping not found in the original graph. SCALE is guaranteed to preserve all of the original structure since it performs a simple uniform scaling. (a) Graph 6 without node labels (b) Graph 6 with node labels (c) Layout with SCALE (d) Layout with FSA (e) Layout with QUAD (f) Layout with EGENET (g) Layout with PLM Figure 13. Initial and resulting layouts for Graph 6. 7.1. Resource-bounded Layout Adjustment Although the PLM solver produces the best layout, Table 1 suggests that it is substantially slower than the SCALE and FSA solvers. A natural question to ask is, given a (small) xed amount of time, which method will give the best layout? This question makes sense because the PLM solver employs local search techniques and at any point in time has a current best solution. Thus we can stop the PLM solver after any time interval and look at the quality of the solution and compare this to that of the SCALE and FSA solvers. It does not make sense to perform the same experiment with QUAD since there is no concept of \the current best solution." The graph in Figure 17 shows the cost of the best solution found so far (as a multiple of the best found eventually) versus time for the graphs 1, 7 and 9 during the execution of the PLM algorithm. The rst non-overlapping solution is found for each graph within 0.02 seconds. Except for Graph 9 the value of dyn for this solution is smaller than that of the solution eventually found by SCALE, FSA or a) Graph 7 (b) Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 14. Initial and resulting layouts for Graph 7. (a) Graph 8 (b) Graph 8 by SCALE (c) Graph 8 by FSA (d) Graph 8 by QUAD (e) Graph 8 by EGENET (f) Graph 8 by PLM Figure 15. Initial and resulting layouts for Graph 8. a) Graph 9 (b) Layout with SCALE (c) Layout with FSA (d) Layout with QUAD (e) Layout with EGENET (f) Layout with PLM Figure 16. Initial and resulting layouts for Graph 9. QUAD (for Graph 9 the rst solution has cost 8108). For each of the graphs PLM nds a solution whose value of dyn is within 20% of the eventual best in less than half the time required to nd the best. For these examples we could safely stop PLM after 0.15 seconds and obtain a solution within 20% of the best found and, in better than any solution found by the other algorithms. 8. Conclusion We have studied the problem of layout adjustment for graphs in which we wish to remove node overlapping while preserving the graph's original structure and hence the user's mental map of the graph. We have given four algorithms to solve this problem, all of which rely on viewing it as a constrained optimization problem. Empirical evaluation of our algorithms shown that they are reasonable fast and give nice layout, which is better than that of the comparable algorithm, the FSA of Misue et al. Generally speaking, the approach based on Lagrangian methods gives the best layout in a reasonable time. However, it may not preserve the orthogonal layout of the original graph. The quadratic programming approach also produces good layout in a reasonable time and does preserve the graph's orthogonal ordering. Uniform scaling gives the fastest and simplest approach to layout adjustment. It may lead to unnecessary enlargement of the graph, but is guaranteed to preserve all of the original structure in the graph. Our results are not only interesting for layout adjustment of graphs: They also suggest techniques for laying out non-overlapping windows and labels in maps. graph 9 graph 7 graph 1 Figure 17. Cost of best solution found so far against time for graphs 1, 7, and 9 9. Acknowledgement We would like to thank Peter Eades for his comments on our work, and Yi Xiao for the quadratic solver. Notes 1. Note that unless the number of pixels is used, real numbers are usually used to denote the positions and sizes of the nodes in a graph layout since it allows greater exibility. 2. Even though the global optimality of the resulting solution cannot be guaranteed by the EGENET approach as a local search method, the experimental results of the related GLS approach reported in [25] show that for a set of the real-life military frequency assignment problems GLS always found better solutions than those found by conventional search methods. 3. Here, we mean a solution with its cost close enough to the globally optimal cost of the optimization problem. 4. A regular point of constraints g is one where rg 1 are linearly independent. --R Boltzmann machines for traveling salesman problems. A discrete stochastic neural network algorithm for constraint satisfaction problems. Algorithms for drawing graphs: an annotated bibliography. Solving linear arithmetic constraints for user interface applications. GENET: A connectionist architecture for solving constraint satisfaction problems by iterative improvement. Solving small and large scale constraint satisfaction problems using a heuristic-based microgenetic algorithm Online animated graph drawing using a modi Preserving the mental map of a diagram. Removing node overlapping using constrained optimization. Extending GENET for Non-Binary CSP's Towards a more e-cient stochastic constraint solver busting in anchored graph drawing. Consistency in networks of relations. Layout adjustment and the mental map. Experimental and theoretical results in interactive orthogonal graph drawing. Edge: an extendible graph Automatic graph drawing and readablity of diagrams. Foundations of Constraint Satisfaction. The tunneling algorithm for partial csps and combinatorial optimization problems. Partial constraint satisfaction problems and guided local search. Methods of Optimization. Solving satisfaction problems using neural-networks --TR --CTR Huang , Wei Lai, Force-transfer: a new approach to removing overlapping nodes in graph layout, Proceedings of the twenty-sixth Australasian conference on Computer science: research and practice in information technology, p.349-358, February 01, 2003, Adelaide, Australia Wanchun Li , Peter Eades , Nikola Nikolov, Using spring algorithms to remove node overlapping, proceedings of the 2005 Asia-Pacific symposium on Information visualisation, p.131-140, January 01, 2005, Sydney, Australia Huang , Wei Lai , A. S. M. Sajeev , Junbin Gao, A new algorithm for removing node overlapping in graph visualization, Information Sciences: an International Journal, v.177 n.14, p.2821-2844, July, 2007 Huang , Peter Eades , Wei Lai, A framework of filtering, clustering and dynamic layout graphs for visualization, Proceedings of the Twenty-eighth Australasian conference on Computer Science, p.87-96, January 01, 2005, Newcastle, Australia

constrained optimization;graph layout;disjunctive constraints

634941

Capillary instability in models for three-phase flow.

Standard models for immiscible three-phase flow in porous media exhibit unusual behavior associated with loss of strict hyperbolicity. Anomalies were at one time thought to be confined to the region of nonhyperbolicity, where the purely convective form of the model is ill-posed. However, recent abstract results have revealed that diffusion terms, which are usually neglected, can have a significant effect. The delicate interplay between convection and diffusion determines a larger region of diffusive linear instability. For artificial and numerical diffusion, these two regions usually coincide, but in general they do not.Accordingly, in this paper, we investigate models of immiscible three-phase flow that account for the physical diffusive effects caused by capillary pressure differences among the phases. Our results indicate that, indeed, the locus of instability is enlarged by the effects of capillarity, which therefore entails complicated behavior even in the region of strict hyperbolicity. More precisely, we demonstrate the following results. (1) For general immiscible three-phase flow models, if there is stability near the boundary of the saturation triangle, then there exists a Dumortier-Roussarie-Sotomayor (DRS) bifurcation point within the region of strict hyperbolicity. Such a point lies on the boundary of the diffusive linear instability region. Moreover, as we have shown in previous works, existence of a DRS point (satisfying certain nondegeracy conditions) implies nonuniqueness of Riemann solutions, with corresponding nontrivial asymptotic dynamics at the diffusive level and ill-posedness for the purely convective form of the equations. (2) Models employing the interpolation formula of Stone (1970) to define the relative permeabilities can be linearly unstable near a corner of the saturation triangle. We illustrate this instability with an example in which the two-phase permeabilities are quadratic.Results (1) and (2) are obtained as consequences of more general theory concerning Majda-Pego stability and existence of DRS points, developed for any two-component system and applied to three-phase flow. These results establish the need for properly modelling capillary diffusion terms, for they have a significant influence on the well-posedness of the initial-value problem. They also suggest that generic immiscible three-phase flow models, such as those employing Stone permeabilities, are inadequate for describing three-phase flow.

Introduction The standard model for three-phase ow in petroleum reservoirs is based on the two-component system of partial dierential equations @ @t @x @x @ @x U Date: November, 1999. This work was supported in part by: FAP-DF under Grant 0821 193 431/95; CAPES under Grant BEX0012/97-1; FEMAT under Grant 990003; CNPq under Grant CNPq/NSF 910029/95-4; CNPq under Grant 520725/95-6; CNPq under Grant 300204/83-3; MCT under Grant PCI 650009/97-5; FINEP under Grant 77970315-00; CNPq under Grant 301411/95-6; NSF under Grant DMS-9732876; DOE under Grant DE-FG02-90ER25084; ONR under Grant N00014-94-1-0456; and NSF under Grants DMS-9107990 and DMS-9706842. denote the saturations of two of the three-phases (viz., water, gas, and oil), f 1 and f 2 are their respective fractional ow functions, and B encodes the eects of capillary pressure. The variables u 1 and u 2 take values on the saturation triangle := f g. A physical derivation of this model is given in Sec. 4.1. It is standard practice to neglect the capillary terms and consider Eq. (1.1) from the point of view of hyperbolic conservation laws. A well-known diculty with this approach is that, generically, three-phase ow models give rise to elliptic regions, i.e., regions where the eigenvalues of the Jacobian F 0 (U) are not real [5, 37, 38, 41, 20, 21, 30]. In these regions, the initial-value problem for the equations with B(U) 0 are ill-posed, and numerical solutions of them display rapid oscillations with large-amplitudes on ner and ner scales as the mesh spacing tends to zero. However, at least at one time, there seems to have been a general feeling that the conservation laws should yield a satisfactory hyperbolic theory outside of the elliptic region. See, e.g., Refs. [18, 19, 34, 14, 13], in which the structure of Riemann solutions is studied for certain Stone models in the absence of capillarity eects. The role of elliptic regions in three-phase ow models has been investigated by several authors. Bell, Shubin, and Trangenstein studied numerically the Riemann solution of a version of Stone's model [5]. The model they considered has a long, thin elliptic region, which appears to be unrelated to the elliptic region that occurs in the present paper [25]. Keytz [24, 25, 26] studied rigorously the wave structure for a model having a strip as an elliptic region. She explained many of the wave features observed near the boundary of the elliptic region of Ref. [5]. Riemann solutions for other systems of conservation laws having innite strips as elliptic regions have been studied extensively by Slemrod [39] and Shearer [36]. In all these works, simplied or numerical diusion terms were utilized to determine admissibility conditions for shock waves. More recently, results in Refs. [27, 8, 2, 3, 17] have emphasized a more primary cause of anomalous behavior: linearized instability of the complete diusive system, Eq. (1.1), rather than ill-posedness of its purely convective form. These two concepts coincide in the case of articial diusion B I, or more generally when convective and diusive eects commute; however, as shown by Majda and Pego, the role of dissipation structure is quite important when other than such special diusion terms are considered. In Ref. [29], they developed a useful sucient condition in terms of B(U) and F 0 (U) for the linearized instability of Eq. (1.1). The corresponding \Majda-Pego instability region" contains, but is typically larger than, the elliptic region. Nonuniqueness [1, 3] or nonexistence [6, 7, 33] of solutions for Riemann problems, the latter manifested as highly oscillatory waves (which are measure-valued solutions) [17, 16, 15], can occur in the Majda-Pego instability region, even in zones of strict hyperbolicity. Accordingly, we study here a model with a physically correct diusion matrix, taking full account of capillarity. The resulting diusion matrix is not a multiple of the identity matrix and gives rise to the eects mentioned above. Notice that numerical diusion typically commutes with convective terms (indeed, for standard schemes the diusion matrix B(U) is a polynomial in F 0 (U)); hence these eects (having to do with noncommutativity) do not arise or are not captured by standard hyperbolic dierence schemes. Of course, numerical simulations which fully resolve the parabolic equation (1.1) do capture the eects discussed here. More precisely, we study a model with (a) permeabilities obtained using Stone's interpolation formula from quadratic two-phase permeabilities and (b) Leverett's capillary pressure functions. Stone's model is widely used in petroleum engineering. The purely convective form of this model generically features an open elliptic region in the interior of the saturation triangle along with three isolated points of nonstrict hyperbolicity (umbilic points) at the corners of . For articial or numerical diusion, the linear instability region consists of the elliptic region and the three corners. We nd that, when physical (viz., Leverett) capillarity is taken into account, each of these regions of linear instability expands into the region of strict hyperbolicity. In particular, we identify model parameters for which there are wedges at the corners inside (respectively, outside) of which the model is linearly unstable (resp., stable). This phenomenon is rather unexpected, since the instability occurs near the boundary, where modeling should be relatively accurate and the system might be expected to be well-behaved. On the other hand, if a particular Stone's model is well-behaved near the boundary, or the corner instability regions are suciently small, we show that a Dumortier-Roussarie- Sotomayor (DRS) bifurcation point must exist in the interior of . Such a point lies on the boundary of the Majda-Pego instability region (see the denition in Sec. 2) and generically lies in the strictly hyperbolic region (where the eigenvalues of F 0 (U) are real and distinct). As shown in Ref. [3], a nondegenerate DRS point gives rise, in its vicinity, to multiple solutions of Riemann problems for the purely convective form of the governing equations. Both linear instability and nonuniqueness are outside the range of \good" behavior for hyperbolic conservation laws. The paper divides into two parts. In the rst part, consisting of Secs. 2 and 3, we investigate existence of DRS points in general two-component models following a degree-theoretic approach. Our main result is to show that generically there exists a DRS point within any simple closed curve on which: (i) all points are strictly hyperbolic and Majda-Pego stable; and (ii) the eigendirections of F 0 (U) rotate by an odd multiple of as is traversed. This greatly generalizes an existence theorem proved in Ref. [3] for special, quadratic ux models by direct calculation. The proof relies on a detailed decomposition of the Majda-Pego instability region. In the second part, consisting of Secs. 4{6, we consider the three-phase ow model described above. We determine the Majda-Pego stability of edge and corner points and calculate the rotation of the eigendirections of F 0 (U) around a contour near @, thereby verifying the hypotheses for existence of DRS points. At the same time, we give a detailed discussion of capillarity and its relationship to well-posedness of Eq. (1.1), which we hope is of general use to the reader. 2. Linear Stability and DRS Bifurcation Consider a general two-component system of conservation laws of the form in Eq. (1.1), where F is C 3 and B is C 2 in a particular open subset U of state space; further, assume that B is strictly parabolic in the sense that, for each U 2 U , the eigenvalues of B(U) have strictly positive real parts. In this section, we explore a connection between linear stability of a constant solution and DRS bifurcation via the stability condition of Majda and Pego [29]. 2.1. Majda-Pego stability condition. We begin by recalling, and slightly extending, the results of Ref. [3] on bifurcation of admissible shock waves. An admissible shock wave is a traveling wave solution U(x; t) of system (1.1). If denotes its speed and U(x; x !1, then such a traveling wave corresponds to an orbit of the system of two ordinary dierential equations _ which we regard as a dynamical system parameterized by (U In Ref. [3], the bifurcation of nonclassical (overcompressive and transitional) shock waves from a constant solution was investigated. It was shown that this bifurcation corresponds to a codimension-three bifurcation, studied under certain nondegeneracy conditions by Du- mortier, Roussarie and Sotomayor [12, 11], that occurs at a DRS point, viz., a point (U ; U R satisfying: (D1) is an eigenvalue of F 0 (U ), with associated right and left eigenvectors r and ' ; Associated with DRS bifurcation are the phenomena of nonuniqueness of Riemann solutions and nontrivial asymptotic behavior, as described in the introduction. Remark. We have substituted the condition ( e for the original condition One consequence of our analysis below is that, given condition (D1), conditions (D3) and ( e are equivalent except if U is an umbilic point, i.e., F 0 (U will prove to be more convenient in calculations. One of the main observations of Ref. [3] is a link between DRS bifurcation and linear instability of the constant solution U(x; t) U of system (1.1). This link, however, was explored only when U is a point of strict hyperbolicity. Our rst task, therefore, is to establish a relationship between DRS bifurcation and linear instability when strict hyperbolicity fails at U . By Fourier analysis, L 2 linearized stability of the constant solution U(x; t) U of system (1.1) requires the following condition: for all real k, each eigenvalue of ikF 0 (U nonpositive real part. (2.2) Standard matrix perturbation theory shows that the eigenvalues s (k) and f (k) of the have the following expansion around where a s and a f denote the eigenvalues of F 0 (U ). Moreover, if the eigenvalues a are distinct, and ' denote left and right eigenvectors associated to a normalized so that Therefore necessary conditions for linear stability are hyperbolicity at U and, in case U is a point of strict hyperbolicity, Denition 2.1. We will refer to the inequality as the Majda-Pego stability condition for family j, and we let the Majda-Pego stable region MP be the set of strictly hyperbolic points U for which the Majda-Pego stability condition holds for both families. Evidently, condition (D3), in the context of condition (D1), represents neutral failure of the Majda-Pego stability condition (2.6). As mentioned above, condition (D3) can be replaced by condition ( e within the region of strict hyperbolicity [3]. Conditions (D1) and ( e imply that the trace and determinant of B(U Following Ref. [8], we dene the Bogdanov-Takens locus (or BT locus) as Bogdanov-Takens bifurcation in the structure of weak traveling waves occurs at points of BT satisfying certain nondegeneracy conditions [25, 8, 9]. Similarly, we dene the coincidence locus, where the characteristic speeds coincide, to be It was observed in Refs. [8, 3] that if U is a point of strict hyperbolicity and U 2 @MP , then U 2 BT . We will show in this section that the assumption of strict hyperbolicity is super uous; in other words, the boundary of the Majda-Pego stable region consists entirely of points satisfying conditions (D1) and ( e D3). To prove this statement, let us introduce the following real-valued functions dened for U The signicance of E and BT is that solves the equation tr[F 0 (U) and solves the equation tr[B(U) 1 Lemma 2.2. Let U 2 U . If a s and a f denote the eigenvalues of F 0 (U), then Moreover, is 1=4 times the discriminant of F 0 (U), and if F 0 (U) is diagonalizable (in particular, if right eigenvectors of F 0 (U) associated to a j , normalized so that 6 AZEVEDO, MARCHESIN, PLOHR, AND ZUMBRUN Proof. Equations (2.15) and (2.16) follow immediately from the denitions of g E and g BT . Substituting Eq. (2.15), we obtain Eq. (2.17). Similarly, as B(U) is a 2 2 matrix, which yields Eq. (2.18) when substituted in Eq. (2.16). Equations (2.9), (2.15), and (2.16) imply the following result. Proposition 2.3. The functions g E and g BT are related by Another consequence of Lemma 2.2 concerns E \ BT . Proposition 2.4. If U 2 E is not an umbilic point, then the following are equivalent: (a) U 2 BT ; (c) the unique eigendirection of F 0 (U) is an eigendirection of B(U). Proof. Statements (a) and (b) are equivalent by Eq. (2.21). As the matrix F 0 (U) E (U)I is nilpotent; and because U is not an umbilic point, this matrix is nonzero. Let r 6= 0 span its kernel. Since the square of this matrix is zero, r also spans its range. In other words, F 0 (U) If statement (a) holds, then with denoting the common value of E (U) and BT (U ), nilpotent and nonzero, and r belongs to its kernel. In fact, since the square of this matrix is zero, r also spans its range. Let s be a vector such that cr for some c, by nilpotency, and therefore statement (c) is true. Conversely, if statement (c) holds, trfB(U) 1 [F 0 (U) Hence which implies statement (b). Proposition 2.5. We have that In particular, @MP g. Proof. By the strict parabolicity of B, 0 < tr Therefore Eq. (2.18) shows that, given strict hyperbolicity at U , g BT (U) < 0 if and only if U 2 MP. By Prop. 2.3, strict hyperbolicity (g E (U) < 0) follows from the condition g BT (U) < 0. This result was proved for strictly hyperbolic points in Prop. 2.1 in Ref. [3]. Therefore the new content concerns points of E . In this regard, it is useful to determine the behavior of eigenvectors of F 0 (U) as U approaches a point U 2 E from within the strictly hyperbolic region. Lemma 2.6. Suppose that U 2 U has an open neighborhood V 0 containing no umbilic points. Also assume that g E (U ) 0, and that DgE (U has an open neighborhood V V 0 such that the following statements hold. (a) The sets VH := f U of hyperbolic and strictly hyperbolic points in V are connected. (b) In the hyperbolic region VH , the eigenvalues of F 0 are continuous functions a s a f . (c) Corresponding to the eigenvalues a j , for there exist: continuous right eigenvector elds r j on VH , normalized so that jr 1, such that det(r s ; r f ) has a xed sign continuous left eigenvector elds ' j in normalized so that ' j r (d) Suppose that U 2 E \V. If U ! U from within V SH , then ' s (U)=j' s Proof. By the hypothesis concerning g E and the Implicit Function Theorem, U has an open neighborhood statement (a) holds. We can also assume that if U Statement (b) holds because F is C 1 . For each U 2 VH and the matrix F 0 (U) a j (U)I has rank one (because there are no umbilic points in V 0 ); let R j (U) denote its kernel. Then by statement (b), R j is a continuous line eld on VH . Moreover, Let r vectors lying in R j (U ); if U 2 E , take r f . By choosing smaller, if necessary, we can assume that the angle between R j (U) and R j (U ) is less than =2 for all U 2 VH . Then for each U 2 VH be the unit vector that has angle less than =2 relative to r . Note that det(r s ; r f ) does not vanish in V SH , and hence has xed sign by connectivity. Therefore we can dene ' s (U) and ' f (U) to be the rows of the inverse of the matrix (r s (U); r f (U)). The vectors r j and ' j so dened satisfy statement (c). Statement (d) is a consequence of the normalization ' j r and the fact that r Remark. In the context of this lemma, consider a point U 2 E \ V, and suppose that 2 BT . By Prop. 2.4, the unique eigendirection of F 0 (U ) is not an eigendirection of . By Prop. 2.3, g BT (U ) > 0, so that g BT > 0 throughout an open neighborhood W V of U . Hence, by formulae (2.17) and (2.18), ' s Br s and ' f Br f have xed and opposite signs throughout W SH := W \V SH . In fact, by statement (d) of Lemma 2.6, these two quantities have the signs of s (r ) ? B(U )r , respectively. Thus we see that, if s (r ) ? B(U )r is positive (respectively, negative), W SH consists of points at which the Majda-Pego stability condition is violated for the slow (resp., fast) family. Moreover, with . Referring to Eq. (2.4), we see that this result re ects the transition from the second-order (in instability in the strictly hyperbolic region to the rst-order instability in the strictly elliptic region. 2.2. Decomposition of the Majda-Pego unstable region. Let us dene to be the Majda-Pego unstable region. Also let denote the elliptic region, slow-family instability region, and fast-family instability region, respectively. Here, for a strictly hyperbolic point U , ' j and r j denote any left and right eigenvectors associated to the eigenvalue a j of F 0 (U ), normalized so that ' j r Proposition 2.7. The sets s , and f form a partition of . In particular, the elliptic region is contained in the Majda-Pego unstable region. Moreover, the boundaries @ @ s , and @ f are contained in E [ BT . Proof. and by strict parabolicity, s 2.3, If, on the other hand, g E (U) < 0, then, by Eq. (2.18), g BT (U) 0 if and only if one of l s B(U)r s and ' f B(U)r f is nonpositive. Thus s f . The statement about boundaries is evident from the denitions of the sets involved. We now make the following nondegeneracy hypotheses on the model parameters of system (1.1) for the domain U of interest: (H1) at each point U 2 U such that g E (H2) at each point U 2 U such that g BT By the Implicit Function Theorem, the loci are smooth curves. Also, E is the same as the elliptic boundary @ and BT is the same as the Majda-Pego boundary Remark. A simple calculation shows that DgE is an umbilic point, i.e., F 0 (U) is a multiple of the identity matrix. Therefore hypothesis (H1) excludes umbilic points. In particular, we can restate one of the conclusions of Prop. 2.4 as: if g E only if the unique eigendirection of F 0 (U) is an eigendirection of B(U ). Let U be such that g E also shows that D(gE g BT In other words, E and BT are tangent where they meet. In view of this result, we are led to adopt a higher-order nondegeneracy assumption: (H3) at each point U 2 U such that g E for all vectors V 6= 0 such that DgE (U) Under this hypothesis, the points where E and BT intersect are isolated. Proposition 2.8. Assume hypotheses (H1){(H3). Away from the discrete set E \ BT , the boundaries @ s and @ f are smooth curves coinciding locally with either E or BT . Moreover, @ s \ @ Proof. Consider a point U 2 @ s that does not lie in E \ BT . (The proof if U 2 @ f is analogous.) Then, by Prop. 2.3 and Eq. (2.18), either (1) g E (U Case (1). By assumption (H1), there exists an open neighborhood V of U such that g. As g BT (U ) > 0, we can assume CAPILLARY INSTABILITY IN MODELS FOR THREE-PHASE FLOW 9 that (ii) g BT > 0 throughout V. By Lemma 2.6, we can further assume that properties (a){ (c) hold as well. Properties (ii) and (a){(c), together with Eqs. (2.17) and (2.18), imply that Therefore so that (@ Thus @ s coincides with E in V. Notice also that U cannot lie in the closure because V is a neighborhood of U that does not intersect denition comprises points of V SH for which ' f B r f 0, Case (2). By assumption (H2), there exists an open neighborhood V of U such that g. As we can assume that (ii) g E < 0 throughout V. By Lemma 2.6, we can further assume that properties (a){(c) hold as well. Because B is strictly parabolic and ' s Br s 0 at U , we must have that ' f Br f > 0 at U ; therefore we can assume that (iii) ' f Br f > 0 throughout V. Properties (ii), (iii), and (a){(c), together with Eqs. (2.17) and (2.18), imply that so that (@ @ s coincides with BT in V. Finally, consider a point U 2 @ s \ @ f . If g E (U ) were negative, then both ' s B(U )r s and ' f B(U )r f would vanish, which is impossible because of the strict parabolicity of B. Hence so that g BT (U ) 0 by Prop. 2.3. But it was just demonstrated that cannot belong to both s and f . Therefore Remark. As we traverse @ the family for which the Majda-Pego condition (2.6) fails can switch only at points common to s and f , which necessarily belong to the elliptic boundary @ To see how this switch can occur, let U 2 E \ BT and consider traversing instead the elliptic boundary (which is tangent to BT ). s be as in Lemma 2.6. By taking V smaller if necessary, we may assume that E \ V and BT \ V are smooth curves meeting only at U , where they have rst-order, but not second-order, contact. Let U 2 E \ V. According to the discussion in the remark following Lemma 2.6, if U belongs to @ s (respectively, @ according as s (r ) ? B(U )r is positive (resp., negative). Therefore the family for which the Majda-Pego stability condition (2.6) fails switches if and only if changes sign as U passes U . This change could be assured, for example, by another nondegeneracy hypothesis. These considerations show that, from the original perspective of traversing BT , the family for which stability fails typically switches at elliptic points of BT . An example of this phenomenon is illustrated in Fig. 3.1 below. 3. Existence of a DRS Point We are now ready to establish a general result concerning existence of DRS points. We seek a point at which conditions (D1){( e are satised. By Prop. 2.5, it is sucient to identify a point U on the boundary of the Majda-Pego stable region MP for which condition (D2) holds, i.e., where r and ' are the eigenvectors of F 0 (U ) determined by conditions (D1) and (D3). Our strategy is to show that the directions of ' and r rotate by an odd multiple of as U traverses @MP . Since the left-hand side of Eq. (3.1) is homogeneous of odd degree in ' and r , we can conclude that it vanishes at some point on @MP . 3.1. Degree. In the hyperbolic region f U g, the eigenvalues of the are continuous functions a s (U) a f (U ). Under hypothesis (H1), the matrix F 0 (U) a j (U)I has rank one throughout the hyperbolic region, as there are no umbilic points (see the remark after condition (H2)). Therefore the left and right kernels of F 0 (U) a j (U)I are continuous left and right line elds L j and R j . Notice that for U and R s Let be a continuous, oriented closed curve in R 2 , and let R be a continuous line eld on . Then the degree of R with respect to , denoted by deg(R; ), is 1= times the angle through which R(U) rotates as U traverses once around (see Ref. [22, Chapter III, Sec. 1]). For our purposes, we shall require a slightly more general notion of degree, dened for boundaries of suitably nice sets. Toward this end, dene deg(R; ) as above even if is not closed. (Of course, this number is not necessarily an integer.) If is decomposed as a succession of continuous, oriented curves 1 (see Ref. [10, Chap. XVI, Sec. 1]). Denote by C the class of bounded, nonempty sets W R 2 such that @W consists of a nite union of piecewise smooth curves, intersecting at a discrete set of points, that can be oriented unambiguously by the outward-normal convention (i.e., at each point of @W that is not one of the intersection points, W is locally dieomorphic to a half-plane). In particular, by Props. 2.7 and 2.8, MP, s and s f belong to C. If W 2 C and R is a continuous line eld on @W, we can decompose @W as a succession of continuous, oriented curves 1 deg(R; @W) := This number is independent of the decomposition. Moreover, it is an integer because any decomposition of @W into continuous, oriented curves can be grouped into a union of closed curves by forming the boundaries of the connected components of the set obtained from W by removing the nitely many intersection points. The following useful properties of the degree hold. (P1) The degree deg(R; ) is invariant under homotopies of both and R disjoint members of C, and suppose that W := belongs to C. If R is a continuous line eld on @W 1 , @W 2 and @W, then deg(R; R is a continuous line eld on W , then deg(R; For property (P1), see, e.g., , Ref. [42, Sec. 34]. Property (P2) holds because @W 1 and consist of @W together with a nite number of pairs of coinciding curves with opposite orientations. Property (P3) is proved for a simply connected region W by shrinking @W to a point and invoking homotopy invariance; it is proved in general by subdividing the compact set W into simply connected regions. 3.2. Main result. We can now state the main result of this section. Theorem 3.1. Let hypotheses (H1){(H3) hold on a bounded region U , and suppose that is a (nontrivial) piecewise smooth simple closed curve lying within MP. If deg(R , then there exists a DRS point strictly inside . We will prove this theorem using a series of lemmas. Let denote the closed bounded region with boundary . Then belongs to the class C. Without loss of generality, we may assume that the Majda-Pego unstable region contained in the interior of . Notice that is nonempty; otherwise would be contained in the strictly hyperbolic region and R k would be a continuous line eld on , so that deg(R would vanish by property (P3). Lemma 3.2. Under the hypotheses of Theorem 3.1, deg(R and the degrees deg(R s ; @ , and deg(L f ; @ all equal this common value. Proof. Since, for each U in the strictly hyperbolic region, R s (U) and R f (U) are linearly independent directions in the plane, R s and R f have equal degrees around . Likewise, by orthogonality of left and right eigenvectors, deg(L For the line eld R j is continuous throughout because this set is contained in the (non-strictly) hyperbolic region. Further, belongs to class C by the assumed smoothness of @ and its separation from = @. Therefore Similarly deg(L @On s f , dene the new line elds e R s in in and e in in The line eld e R is well-dened and continuous on because, by Prop. 2.8, R s and on s Similarly, e L is well-dened and continuous. Moreover, for @ e Lemma 3.3. Under the hypotheses of Theorem 3.1, deg( e and deg( e Proof. By properties (P2) and (P3) of the degree, @ @ @ @ R. But by Eq. (3.6), deg(R; @ independent of the choice R R f , or e R. Therefore Lemma 3.2 implies that deg( e Similarly, deg( e @ equals this value. Proof of Theorem 3.1. By hypothesis (H1), conditions (D1) and ( e are satised on @ R and ' 2 e L. It remains only to show that for some U 2 @ On the hyperbolic region, in particular on BT , the continuous line elds e R and e locally continuous, nonzero vector elds r and ' ; indeed, each line eld locally generates precisely two branches. Choose a connected component of @ around which e R and e L have odd degree, and choose a particular branch for r and ' at some initial point on this component. Following this branch while traversing once around this component, we nd by Lemma 3.3 that the quantity ' F 00 (U )(r ; r ) must change sign, since both e L and e R rotate by an odd multiple of . Thus it vanishes somewhere on @ Remark. Arguments with a similar topological avor have been used to establish the existence of points at which strict hyperbolicity fails [37, 38]. In [20, 21, 41, 30], strict hyperbolicity failure was established with dierent arguments. Remark. The DRS point generically lies in the region of strict hyperbolicity. If it were to occur on E , Prop. 2.4 indicates that r would have to be an eigendirection of B(U ). Example 3.4. For generic two component models with quadratic ux F and constant matrix B, the elliptic boundary and the BT locus intersect in precisely two points, unless B is a multiple of the identity matrix, in which case they coincide. Indeed, one can show that the eigenvector r undergoes a rotation of angle as U traverses the elliptic boundary (it suces to compute this rotation for homogeneous quadratic models). Thus the direction of r coincides precisely once with each of the eigendirections of B. If B has more than two eigendirections, then B is a multiple of the identity matrix, in which case E and BT coincide. Otherwise B has only two eigendirections, and there are precisely two points in may verify that the family in which Majda-Pego stability fails switches at these two points, illustrating the conclusions of the remark at the end of Sec. 2.2. Figure 3.1 shows MP and the decomposition of s f for a model such that is compact. The points in E \ BT are indicated by two small open circles, the three lled circles represent DRS points, and the rectangles mark line eld singularities [31, 23]. The curves through the rectangles constitute the in ection locus I, along which genuine nonlinearity fails in the family f or s indicated by whether the curve is solid or dashed. 4. Three-Phase Flow In this section, we introduce a class of systems of the type given by Eq. (1.1), which are used in Petroleum Engineering to model three-phase ow in porous media. We discuss the properties of the model that have a role in determining features such as Majda-Pego stability regions and DRS points. One of the main consequences of the analysis of important classes of three-phase ow models in Sec. 5, under the generic assumptions that such models satisfy (H1){(H3), follows from Theorem 3.1. It is the following. Theorem 4.1. Either the Majda-Pego instability region accumulates at the boundary of the saturation triangle @ or else there exists a DRS point interior to . MP s f I Figure 3.1. The Majda-Pego stable region MP , the decomposition of the Majda-Pego unstable region s f , and the coincidence, Bogdanov- Takens, and in ection loci E , BT , and I for a quadratic model. 4.1. The basic equations. We consider one-dimensional, horizontal ow of three immis- cible uid phases in a porous medium [32]. For concreteness, we consider a uid composed of gas, oil and water, mixed at macroscopic level. The dierences among these phases lie in some ow properties. We assume that the whole pore space is occupied by the uid and that there are no sources or sinks. Compressibility, thermal and gravitational eects are considered to be negligible. The equations expressing conservation of mass of water, gas, and oil are @ @t @x respectively, where denotes the porosity of the porous medium. For the phase i, s i denote the saturation, i the density and v i is the seepage velocity (the product of the saturation by the particle velocity of the phase i). Since the uid occupies the whole pore space, the saturations satisfy As a consequence, any pair of saturations in the saturation triangle may be chosen to describe the state of the uid. The theory of multiphase ow in porous media is based on the following form of Darcy's law of force [32, 35, 4]: @ @x where K denotes absolute permeability of the porous medium, i 0 is the mobility of phase i, and p i is the pressure of phase i. The mobility is usually expressed as the ratio of the relative permeability k i and the viscosity i of phase i. 14 AZEVEDO, MARCHESIN, PLOHR, AND ZUMBRUN The porosity and absolute permeability K are associated to the rock; we take them to be constant. Neglecting thermal eects and compressibility, i and i are constant too, and we can rewrite Eqs. (4.1) without i . Each relative permeability k i depends on the saturations. Experimentally, k i increases when s i increases, and the relative permeabilities never vanish simultaneously. Let us denote the dierence between the pressures in phases i and j (i 6= j) by This pressure dierence, called the capillary pressure, is measured experimentally as a function of the saturations. Dene the total mobility and the fractional ow functions f i by Of course, 1. Introducing the total seepage velocity using algebraic manipulation, we can write that @ @x Adding and subtracting v i in the last equation and noting that @ we see that @ @x Therefore, by Eqs. (4.1) and (4.7), the equations governing the ow are @ @x Kw [f g @ @ wo @ @x K g [f w @ @ @ @t @x @ ow @ Summing Eqs. (4.1), we nd @ so that v is a function of t alone. (This simplication occurs only for ow in one spatial dimension. In general, we obtain an elliptic equation for the pressure, with coecients depending on the saturations.) Assuming that v never vanishes, it is possible to change the variable t so that v is constant. We do not consider the case v(t) 0 because in this case the system (4.8) does not contain the terms related to transport. As v is nonzero, we can set removing v, , and K from system (4.8). For simplicity of notation, we drop the tildes. Of course any one of the equations in the three-component system (4.8) is redundant and the system can be reduced to a two-component system. As a result of the redundancy in the system (4.8), one of its characteristic speeds is 0. The two other characteristic speeds are the same for subsystem of two equations; see Ref. [30]. We will nd it convenient later to be exible in our choice of the two saturations entering this two-component system. Hence we will use u 1 and u 2 to denote two of the saturations in the reduced system, where the other saturation u 3 is replaced by 1 Also it is useful to replace p 12 by . Thus we obtain the two-component system for three-phase ow @ @t @ @ @t @ This system can be written in compact form as Eq. (1.1) with and and The quantities U , F and B(U) are called the state vector, ux function and diusion matrix, respectively. We refer to Q(U) and P 0 (U) as balance matrix and capillary pressure Jacobian. There are two major assumptions that we make concerning the mobilities. The rst is that (More generally, i vanishes when s i is below the irreducible saturation for phase i, which, for simplicity, we take to be zero.) Because the edge for system (4.8). Indeed, system (4.9) reduces to the scalar conservation law @ @t on the edge and a similar reduction occurs on the other edges. In other words, three-phase ow equations reduce to the Buckley-Leverett equation for two-phase ow. There are two characteristic speeds for system (4.9), one of which reduces to the Buckley- characteristic speed on the edges. Our second assumption is that the other characteristic speed is positive and strictly smaller than the Buckley-Leverett characteristic speed near the edges. For example, on the edge Thus we see that the characteristic speeds are f 1;1 and f 2;2 , that f 1;1 is the Buckley-Leverett speed, and that the second assumption can be written as 4.2. Assumptions on the fractional ow functions. In this section we present assumptions used in the rest of the paper. For brevity we use a subscript \; j" to denote the partial derivative with respect to u j for 2. The Jacobian matrix of F is therefore According to Eq. (4.5), 2: (4.17) We shall make the following assumptions concerning the fractional ow functions: f 1 , f 2 , and f 3 are continuously dierentiable functions on the closed saturation triangle such that on the open edge f 1f 2;1 has a continuous extension to the open edge where in the interior of near are bounded away from zero in the interior of near Each of the assumptions above actually represents three dierent assumptions, corresponding to a choice of two saturations u 1 and u 2 among s w , s g and s The rst part of assumption (4.18) corresponds to the requirement that the permeabilities be nonnegative. By the second part of this assumption, f on the edge so that the Jacobian F 0 is upper triangular and its eigenvalues are f 1;1 and f 2;2 . Hence assumption (4.19) states that the eigenvalues are distinct on each open edge, and that the fast eigenvector is parallel to the edge. Thus the three-phase ow reduces to two-phase ow along each edge, with the fast eigenvalue corresponding to the Buckley-Leverett speed (see Eq. (4.13)). In particular, f at the corner re ecting the immiscibility of two-phase ow. Assumption (4.21) implies that the eigenvalues are distinct in the interior of near each corner. (Necessarily, the eigenvalues coincide at each corner. Thus, the model is strictly hyperbolic near the boundary except at the corners.) This assumption also implies that the saturations u 1 and u 2 vary in opposite directions across slow rarefaction waves near each corner. Assumptions (4.20) and (4.22) simplify the conditions for stability at the boundary of . We will show that these assumptions hold for certain models employed in petroleum reservoir engineering in Sec. 6. 4.3. Assumptions on the capillary pressures. In principle, the capillary pressures p ow and p og are experimentally measured functions of all saturations. (The third capillary pressure is obtained as In general, the capillary pressure Jacobian (4.11) is We make the following assumptions: positive denite and diagonally dominant inside ; (4.24) on the closed edge Assumption (4.25), which guarantees that the two-phase ow equation (4.13) for the edge has a positive diusion coecient, actually represents three dierent assumptions, corresponding to a choice of two saturations u 1 and u 2 among s w , s g and s 4.4. Well-posedness. Next, we investigate the eects of capillary pressure in system (1.1), as represented by the diusive term (B(U)U x ) x . We rst verify that this term is strictly parabolic (i.e., the eigenvalues of B(U) have strictly positive real parts) in the interior of the saturation triangle . Lemma 4.2. Under assumption (4.18), the balance matrix Q(U) is symmetric and positive denite in the interior of the saturation triangle , and det Proof. Rewriting it is apparent that Q(U) is symmetric and diagonally dominant, with nonnegative diagonal entries that are positive in the interior of , by assumption (4.18). One can verify that det so that det Lemma 4.3. If P 0 satises assumptions (4.24) and (4.25), then the eigenvalues of the diffusion matrix B(U) have positive real part in the interior of , whereas on the boundary of , one eigenvalue of B(U) is zero and the other is positive. Proof. A real matrix has eigenvalues with positive real part provided that its trace and determinant are positive. The determinant of both Q and P 0 are non-negative, by diagonal dominance (4.24), so that det strict inequality in the interior of . Consulting Eq. (4.26), we nd that by assumptions (4.24) and (4.25). Remark. It is easily checked (see the calculations below) that the normal to is a left eigenvector of F 0 (U) and a left null vector of B(U) on @. This is a necessary condition in order that be invariant under the evolution of system (1.1), see Theorem 1.47, p. 200 of [40]. However, suciency does not follow from this standard result, due to degeneracy of B(U) on the boundary; indeed, the question of invariance in this degenerate situation seems to be an interesting one in its own right. 5. Linear Stability Conditions and Degree In this section we compute the degree for the fast-family line eld for three-phase ow models. We also nd conditions under which the Majda-Pego stability conditions (2.6) hold near @. 5.1. Eigenvalues and eigenvectors. It proves convenient to work with the scaled Jacobian matrix and to introduce the following notation: Let v s and v f denote the slow and fast eigenvalues of A, respectively. Then a d b c a d We may choose the matrix of right eigenvectors to be This a good choice for calculations near the edge where The corresponding matrix of left eigenvectors is given by d) a Using the relation d bc, we nd that the determinant of R in Eq. (5.5) 2d[a On the other hand, for calculations near the corner where choice for the right eigenvector matrix is (The reason is that the projections (1; 1) r s and (1; 1) r f are positive.) The corresponding matrix of left eigenvectors is d) Again by the relation d bc, the determinant of R in Eq. (5.7) is 4d[d 5.2. Degree. In this section we show that, for models satisfying the assumptions of Sec. 4.2, the fast-family line eld has winding number 1 around a suitable contour close to the boundary of the saturation triangle. Lemma 5.1. For a model satisfying assumptions (4.18), (4.19), and (4.21), consider the arc of a circle centered at leading from the edge to the edge If the radius of the arc is suciently small, the fast-family line eld turns by angle =2 upon traversing the arc. Proof. The projection of the eigenvector r f , as dened in Eq. (5.7), onto the vector (1; 1) is assumption (4.21), this quantity is positive along any open arc of suciently small radius. At the edge assumption (4.19) and assumption (4.18), so that d = a and r which is a positive multiple of (1; . Similarly, at the edge so that d = a and r which is a positive multiple of (0; 1) T . Therefore r f turns by angle =2. Denition 5.2. Let the contour be the boundary of the set of points in at least distance from the boundary. Proposition 5.3. Consider a model satisfying assumptions (4.18), (4.19), and (4.21). For suciently small > 0, the degree of the fast-family line eld around is 1. Proof. According to the preceding lemma, the fast-family line eld rotates by =2 at the corner Taking account the changes of coordinates, the same lemma shows that the fast-family line eld rotates by =4 at each of the two corners traversing edges, the fast-family line eld does not rotate because it is parallel to the edge. Therefore the line eld rotates by around . CAPILLARY INSTABILITY IN MODELS FOR THREE-PHASE FLOW 19 5.3. Stability analysis near edges. In this section we determine a necessary and sucient condition for a model to be linearly stable near the open edges of the saturation triangle, in that conditions (2.6) hold with strict inequality except on the edge. Without loss of generality, we focus on the edge The calculations presented in this section are not applicable at the corners, so we assume that 0 < We choose the eigenvector matrices (5.5) and (5.6). If u assumption (4.18) and a > 0 by assumption (4.19). In particular, d = a and det Therefore, when evaluated along the edge, the fast family eigenvectors r f and ' f are positive multiples of (1; 0) T and (1; b=(2a)), respectively. Assumption (4.18) also entails that, when the only nonzero entry of the balance matrix Q is Q sequently, by denition (4.10) of the diusion matrix B, ' f B r f is a positive multiple of along the edge We have proved the following result. Proposition 5.4. For a model satisfying assumptions (4.18), (4.19), and (4.25), the fast- family Majda-Pego condition holds in an open neighborhood of the open edge The right eigenvector for the slow family is r However, we shall see that ' s hence to verify that ' s B r s 0, we cannot simply set As ' s is a positive multiple of (c; (a +d)), Eq. (4.11) implies that ' s Q is a positive multiple of the vector d)). By assumption (4.20), we may extend 1 Q, to be a continuous function at positive multiple of ( 1 It proves useful to write this last vector as r T nd a simplied expression for First we note that when u Using these results in the denition (5.9) of , we nd that (The quantity 3;1 3;2 is the derivative of 3 with respect to u 1 with u 3 xed.) We can now formulate a condition equivalent to the Majda-Pego condition for the slow family. Proposition 5.5. For a model satisfying assumptions (4.18), (4.19), (4.20), and (4.25), let let be given by Eq. (5.13). Then r T is a necessary and sucient condition for the model to satisfy the slow-family Majda-Pego condition in an open neighborhood of the open edge strict inequality when u 5.4. Stability analysis at corners. In this section we determine a necessary and sucient condition for a model to be linearly stable near the corners of the saturation triangle, in that conditions (2.6) hold with strict inequality except on the edges. Without loss of generality, we focus on the corner Condition (2.5) is equivalent to the nonnegativity of the diagonal elements of LQP 0 R. For the eigenvector matrices R and L we choose those dened by Eqs. (5.7) and (5.8), respectively. Using the balance matrix dened in Eq. (4.11), we see that (det R)' s (det R)' f By assumption (4.21), det positive in the interior of in a neighborhood of the corner. Proposition 5.6. For a model satisfying assumptions (4.18), (4.22), and (4.25), the fast- family Majda-Pego condition holds in the interior of in a neighborhood of the corner only if in the interior of in a neighborhood of this corner, where a, b, c, and d are given by Eqs. (5.2) and (5.3). Proof. The stability condition for the fast family is that (det R)' f QP 0 r f > 0. By assumption (4.18) (and the corresponding assumption along the edge tend to zero at the corner. In particular, we may neglect f 1 and f 2 relative to 1 in the expression for (det R)' f Q. Moreover, by assumptions (4.18) and (4.19), a, b, c, and hence d vanish at the corner; therefore, by assumption (4.22), f 1 (a d) is negligible relative to d) is negligible relative to . Thus (det R)' f Q can be approximated by d) T , the inequality (det R)' f QP 0 r f > 0 is equivalent to f > 0 near the corner. By an analogous argument, we can prove the following result. Proposition 5.7. For a model satisfying assumptions (4.18), (4.22), and (4.25), the Majda- Pego condition for the slow family holds in the interior of in a neighborhood of the corner only if d) in the interior of in a neighborhood of this corner, where a, b, c, and d are given by Eqs. (5.2) and (5.3). 5.5. Summary . Combining the results of this section, we have the following theorem. Theorem 5.8. Consider a model satisfying assumptions (4.18){(4.22) and (4.25) along with conditions (5.14), (5.17), and (5.18). For suciently small > 0, the contour of Deni- tion 5.2 lies within the interior of MP and the degree of the fast-family line eld around is 1. 6. A Three-Phase Flow Model In this section, we present a model for the relative permeabilities and the capillary pressures used in petroleum reservoir engineering and verify the assumptions of Secs. 4.2 and 4.3. Using the results of Sec. 5, we show that the model is linearly stable near the boundary of the saturation triangle except in an open set containing two of the corners in its closure. 6.1. Stone's Permeability model. The permeability of each of the three phases is an experimentally measured function that depends on the saturations of the phases. Leverett and Lewis [28] found that, for many reservoirs, kw depends only on s w and k g depends only on s g . We make the same assumption here. For most reservoirs, the oil permeability depends on two saturations. As there is little experimental data concerning the three-phase region, this function is usually measured only along two edges of the saturation triangle and then interpolated to the interior of . Restricted to the s reduces to the relative oil-water permeability, k ow ; similarly, edge is the relative oil-gas permeability k og . To satisfy assumption (4.12), we require that k ow (s w ) and k og (s g ) are non-negative and that k ow (s w proposed an interpolation scheme for determining k throughout from k ow and k og : ow (s w )k og (s g ) The particular model we examine is the quadratic Stone's model, which uses kw (s w ow (s w so that w , and k In the notation of system (4.9), the mobilities of the quadratic Stone's model are if we associate u 1 to s w and u 2 to s g , whereas if we associate u 1 to s w or s g and u 2 to s 6.2. Leverett's Capillary pressure model. Based on the work of Leverett [28], Aziz and Settari [4] propose that the the capillary pressure functions for three-phase ow can be taken to have the form p wo ow ow and P og are certain monotone decreasing functions. Denoting , we assume that Associating u 1 to s w and u 2 to s g in system (4.9), p 13 and p 23 represent wo and p go , respectively. Therefore the Leverett capillary pressure Jacobian is On the other hand, considering the oil to be one of the phases in Eq. (4.9), the Leverett capillary pressure Jacobian takes a dierent form. Associating u 1 to s w and u 2 to s 22 AZEVEDO, MARCHESIN, PLOHR, AND ZUMBRUN loss of generality, we have that u 3 is s g , p 13 is p . In this case, It is easy to see that assumptions (4.24) and (4.25) of Sec. 4.3 hold for the Leverett capillary pressures under assumption (6.5). 6.3. Verication of assumptions. We now verify the assumptions of Sec. 4.2. Lemma 6.1. The quadratic Stone's model satises assumptions (4.18){(4.22) if we associate u 1 to s w or s g and u 2 to s that the oil viscosity is no less than half of the water and gas viscosities, i.e.,2 w Proof. The mobilities are given by Eqs. (6.4). Assumption (4.18) is evidently true. Let us verify assumption (4.19). Since we see from Eq. (4.17) that is nonnegative everywhere and reduces to u2=0 on the edge On the other hand, so that reduces to u2=0 In particular, f at the corner on the open edge only if These inequalities hold by virtue of assumption (6.8). Now we verify assumptions (4.20), (4.21) and (4.22). Since dividing by 2 gives so that f 1 f 2;1 has a continuous extension to to the open edge where In f 2;1 has a positive limit at (0; 0). Moreover, so that f 1 bounded away from zero in the interior of in a neighborhood of (0; 0). Similarly, so that f 1;2 > 0 in the interior of in a neighborhood of (0; 0). In fact, f 1 f 1;2 has a positive limit at (0; 0). Lemma 6.2. The quadratic Stone's model satises assumptions (4.18){(4.22) if we associate u 1 to s w and u 2 to s g . Proof. The mobilities are given by Eqs. (6.3). Assumption (4.18) is evidently true. Since we see by Eq. (6.10) that f 2;2 is nonnegative near is zero on the edge On the other hand, so that by Eq. (6.13), f 1;1 on u by Eq. (6.14). Therefore f 1;1 > f 2;2 on the open edge assumption (4.19) holds. shows that f 1 f 2;1 has a continuous extension to the edge and tends to 2 at the corner (0; 0). Similarly, by Eq. (6.20), f 1 tends to 2 at the corner. Thus assumptions (4.20), (4.21), and (4.22) hold. 6.4. Linear stability near edges. In this section we prove the following result. Theorem 6.3. Assuming conditions (6.5) and (6.8), the quadratic Stone's model with the capillary pressures satises the stability conditions (2.5) near each open edge of , with strict inequality away from the edges. Proof. By Props. (5.4) and (5.5) and the results of Secs. 6.2 and 6.3, the proof reduces to verifying condition (5.14) of Prop. (5.5). First, we associate u 1 to s w or s g and u 2 to s o . By Eqs. (6.14), (6.11), and (6.21) the vector r reduces to on Eqs. (6.12) and (6.12), 1 so that Using the capillarity matrix P 0 given in Eq. (6.7), we nd r T Therefore assumptions (6.5) and (6.8) imply condition (5.14). Second, we associate u 1 to s w and u 2 to s g . By the calculations in the proof of Lemma 6.2, and Using the capillary pressure Jacobian (6.6), we nd that Thus assumption (6.5) is implies condition (5.14). 6.5. Linear stability and instability at corners. In this section we investigate Majda- Pego conditions at the corners of the saturation triangle . Theorem 6.4. Consider a model satisfying assumptions (4.18), (4.22), and (4.25) and using the Leverett capillary pressures. This model satises the Majda-Pego condition (2.6) for the fast family in the interior of in neighborhoods of each corner; it also satises the Majda- Pego condition for the slow family in the interior of in neighborhood of the corner s Proof. First associate u 1 to s w and u 2 to s g . Then Leverett's capillarity matrix P 0 is given by Eq. (6.6) and inequalities (5.18) and (5.17) become d) d) > 0; (6.31) d) d) > 0: (6.32) In the interior of in a neighborhood of the corner, b and c are positive and d > jaj, so that the coecients of and in s wg and f wg are positive; thus s wg > 0. Now associate u 1 to s w or s g and u 2 to s by Eq. (6.7), and the inequality (5.17) becomes d) c) > 0: By the same argument as given above, the condition f wo > 0 is satised in the interior of in a neighborhood of the corner. However, the Majda-Pego condition for the slow family is violated for the quadratic Stone's model and Leverett capillary pressures, as we now show. Theorem 6.5. Consider the quadratic Stone's model with the Leverett capillary pressures. Assume that the oil viscosity is no less than half of the water and gas viscosities. This model in an open set containing the corners s in its closure. Proof. Without loss of generality, we may associate u 1 to s w and u 2 to s Consider the set where Eqs. (6.13), (6.10), (6.12), and (6.9), so that in a neighborhood of the corner, the set a = 0 is a curve through the corner tangent to the line When c ](( (det R)' s as seen from Eqs. (5.7) and (5.15). By Eqs. (6.21) and (6.19), so that c) as along the curve a = 0. Therefore near the corner. 7. Conclusion Linear stability is a desirable property. We have numerical evidence that certain three-phase models satisfying assumptions (4.18){(4.22) are linearly stable near the boundary of the saturation triangle and a curve satisfying the hypotheses of Theorem 4.1 can be constructed. This is not the case for quadratic Stone's model with Leverett's capillary pressures: as we can see from Theorem 6.5, there is an instability region near the two vertices. On the arc with radius 0:04, the instability region corresponds to the negative part of the lowest curve in Fig. 7.1. However this gure shows that for larger radii, the arcs avoid the instability region. We have veried numerically that the eigenvectors wind correctly along these arcs. Based on these facts, we have constructed curves which also satisfy the hypotheses of Theorem 4.1. Also for the quadratic Stone's model we found numerically regions in parameter space for which the hypotheses (H1){(H3) are satised. Theorem 4.1 establishes the existence of a DRS point inside , which generically implies other instabilities. (Indeed, the above discussion shows that a DRS point is to be expected when the instability regions near the vertices are disconnected from the elliptic region.) For some nongeneric values of parameters, the elliptic region may collapse into umbilic points that coincide with the DRS points. The same collapse occurs for certain other three- phase ow models. We have no evidence of DRS-instabilities for such degenerate DRS points. We regard the stability inequalities as providing mathematical guidelines for the construction of adequate three-phase ow models. Three-phase ow uses quantities such as 26 AZEVEDO, MARCHESIN, PLOHR, AND ZUMBRUN2610140 f Figure 7.1. Plot of ' s B r s =r 3 (multiplied by 10 3 det R) versus polar angle for 0:08. For this gure, the viscosities are the values of the capillary pressure derivatives, evaluated at the corner, are permeabilities and capillary pressures that can only be measured accurately at the boundary of the saturation triangle. We may regard the stability conditions as constraints on interpolation schemes for this data into the interior of state space. Acknowledgments We thank Beata Gundelach for careful assistance in preparing the manuscript. We also thank Somsak Orankitjaroen for thoroughly reading the manuscript and locating errors in a preliminary version. --R Multiple viscous solutions for systems of conservation laws Nonuniqueness of nonclassical solutions of Riemann problems Elsevier Applied Science Conservation laws of mixed type describing three-phase ow in porous media Cubic Li Features of three-component Existence and asymptotic behavior of measure-valued solutions for three-phase ow in porous media Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type On the Riemann problem for a prototype of a mixed type conservation law On the strict hyperbolicity of the Buckley-Leverett equations for three-phase ow in a porous medium A global formalism for nonlinear waves in conservation laws Steady ow of gas-oil-water mixtures through unconsolidated sands Stable viscosity matrices for system of conservation laws Stable hyperbolic singularities for three-phase ow models in oil reservoir simulation Fundamentals of numerical reservoir simulation The physics of ow through porous media The Riemann problem for a class of conservation laws of mixed type Loss of real characteristics for models of three-phase ow in a porous medium Admissibility criteria for propagating phase boundaries in a van der waals uid Shock waves and reaction-diusion equations Departamento de Matem --TR Conservation laws of mixed type describing three-phase flow in porous media Loss of strict hyperbolicity of the Buckley-Leverett equations for three phase flow in a porous medium On the strict hyperbolicity of the Buckley-Leverett equations for three-phase flow in a porous medium Admissibility conditions for shocks in conservation laws that change type Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type Nonuniqueness of solutions of Riemann problems On the oscillatory solutions in hyperbolic conservation laws Fundamentals of Numerical Reservoir Simulation Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering --CTR E. Abreu , J. Douglas, Jr. , F. Furtado , D. Marchesin , F. Pereira, Three-phase immiscible displacement in heterogeneous petroleum reservoirs, Mathematics and Computers in Simulation, v.73 n.1, p.2-20, 6 November 2006

capillary pressure instability;nonunique Riemann solution;flow in porous media

634972

Automated techniques for provably safe mobile code.

We present a general framework for provably safe mobile code. It relies on a formal definition of a safety policy and explicit evidence for compliance with this policy which is attached to a binary. Concrete realizations of this framework are proof-carrying code, where the evidence for safety is a formal proof generated by a certifying compiler, and typed assembly language, where the evidence for safety is given via type annotations propagated throughout the compilation process in typed intermediate languages. Validity of the evidence is established via a small trusted type checker, either directly on the binary or indirectly on proof representations in a logical framework.

Introduction Integrating software components to form a reliable system is a long-standing fundamental problem in computer science. The problem manifests itself in numerous guises: (1) How can we dynamically add services to an operating system without compromising its integrity? (2) How can we exploit existing software components when building a new application? (3) How can we support the safe exchange of programs in an untrusted environment (4) How can we replace components in a running system without disrupting its operation? These are all questions of modularity. We wish to treat software components as "black boxes" that can be safely integrated into a larger system without fear that their use will compromise, maliciously or otherwise, the integrity of the composite system. Put in other terms, we wish to ensure that the behavior of a system remains predictable even after the addition of new components. Three main techniques have been proposed to solve the problem of safe component (1) Run-time checking. Untrusted components are monitored at execution time to ensure that their interactions with other components are strictly limited. Typical techniques include isolation in separate hardware address spaces and software fault isolation [1]. These methods impose serious performance penalties in the interest of safety. Moreover, there is often a large semantic gap between the low-level properties that are guaranteed by checking (e.g., address space isolation) and the high-level properties that are required (e.g., black box abstraction). (2) Source-language enforcement. All components are required to be written in a designated language that is known, or assumed, to ensure "black box" abstraction. These techniques suffer from the requirement that all components be written in a designated, safe language, a restriction that is the more onerous for lack of widely-used safe languages. More- over, one must assume not only that the language is properly defined, but also that its implementation is correct, which, in practice, is never the case. (3) Personal authority. No attempt is made to enforce safety, rather the component is underwritten by a person or company willing to underwrite its safety. Digital signature schemes may be used to authenticate the underwritten code. In practice few, if any, entities are willing to make assurances for the correctness of their code. What has been missing until now is a careful analysis of what is meant by safe code exchange, rather than yet another proposal for how one might achieve a vaguely-defined notion of safe integration. Our contention is that safe component integration is fundamentally a matter of proof. To integrate a component into a larger system, the code recipient wishes to know that the component is suitably well-behaved - that is, compliant with a specified safety policy. In other words, it must be apparent that the component satisfies a safety specification that governs its run-time behavior. Checking compliance with such a safety specification is a form of program verification in which we seek to prove that the program complies with the given safety policy. When viewed as a matter of verification, the question arises as to who (the code producer or the code recipient) should be responsible for checking compliance with the safety policy. The problem with familiar methods is that they impose the burden on the recipient. The code producer insists that the recipient employ run-time checks, or comply with the producer's linguistic restrictions, or simply trust the producer to do the right thing. But, we argue, this is exactly the wrong way around. To maximize flexibility we wish to exploit components from many different sources; it is unreasonable to expect that a code recipient be willing to comply with the strictures of each of many disparate methods. Rather, we argue, it is the responsibility of the code producer to demonstrate safety. It is (presumably) in the producer's interest for the recipient to use its code. Moreover, it is the producer's responsibility (current practices notwith- standing) to underwrite the safety of its product. In our framework we shift the burden of proof from the recipient to the producer. Having imposed the burden of proof on the producer, how is the consumer to know that the required obligations have been fulfilled? One method is to rely on trust - the producer signs the binary, affirming the safety of the component. This suffers from the obvious weakness that the recipient must trust not only the producer's integrity, but also must trust the tools that the producer used to verify the safety of the component. Even with the best intentions, it is unlikely that the methods are foolproof. Consequently, few producers are likely to make such a warrant, and few consumers are likely to rely on the code they receive. A much better method is one that we propose here: require the producer to provide a formal representation of the proof that the code is compliant with the safety policy. After all, if the producer did carry out such a proof, it can easily supply the proof to the consumer. Moreover, the recipient can use its own tools to check the validity of the proof to ensure that it really is a genuine proof that the given code complies with the safety specification. Importantly, it is much easier to check a proof than it is to find a proof. Therefore the code recipient need only trust its own proof checker, which is, if the method is to be effective, much simpler than the tools required to find the proof in the first place. The message of this paper is that this approach can, in fact, be made to work in practice. We are exploring two related techniques for implementing our approach to safe component exchange: proof-carrying code and typed object code. In both cases mobile code is annotated with a formal warrant of its safety, which can be easily checked by the code recipient. To produce such a warrant, we are exploring the construction of certifying compilers that produce suitably-annotated object code. Such a compiler could be used by a code producer to generate certified object code. Two points should be kept in mind when reading this paper: (1) The tools and techniques of logic, type theory, and semantics are indispensable (2) These methods have been implemented and are available today. The first component in a system for safe mobile code is the safety infrastruc- ture. The safety infrastructure is the piece of the system that actually ensures the safety of mobile code before execution. It forms the trusted computing base of the system, meaning that all consumers of mobile code install it and depend on it, and therefore it must work properly. Any defect in the trusted computing base opens a possible security hole in the system. A fundamental concern in the design of the trusted computing base is that it be small and simple. Large and/or complicated code bases are very likely to contain bugs, and those bugs are likely to result in exploitable security holes. For us to have confidence in our safety infrastructure, its trusted components must be small and simple enough that they are likely to be correct. The design of the safety infrastructure consists of three parts. First, one must define a safety policy. Second, one specifies what will be acceptable as evidence of compliance with the safety policy. Suppliers of mobile code will then be required also to supply evidence of compliance in an acceptable form. Third, one must build software that is capable of automatically checking whether purported evidence of safety is actually valid. 2.1 Safety Policies The first task in the design of the safety infrastructure is to decide what properties mobile code must satisfy to be considered safe. In this paper we will consider a relatively simple safety policy, consisting of memory safety, control-flow safety, and type safety. (1) Memory safety is the property that a program never dereferences an invalid pointer, never performs an unaligned memory access, and never reads or writes any memory locations to which it has not been granted access. This property ensures the integrity of all data not available to the program, and also ensures that the program does not crash due to memory accesses. (2) Control-flow safety is the property that a program never jumps to an address not containing valid code, and never jumps to any code to which it has not been granted access. This property ensures that the program does not jump to any code to which it is not allowed (e.g., low-level system calls), and also ensures that the program does not crash due to jumps. (3) Type safety is the property that every operation the program performs is performed on values of the appropriate type. Strictly speaking, this property subsumes memory and control-flow safety (since memory accesses and jumps are program operations), but it also makes additional guarantees. For example, it ensures that all (allowable) system calls are made using appropriate values, thereby ruling out attacks such as buffer overruns on other code in the system. The additional guarantees provided by type safety are often very expensive to obtain using dynamic means, but the static means we discuss in this paper can provide them at no additional cost. Stronger safety policies are also possible, including guarantees of the integrity of data stored on the stack [2], limits on resource consumption [3,4], and policies specified by allowable traces of program operations [5]. However, for policies such as these, the evidence of compliance (which we discuss in the next section) can be more complicated, thereby requiring greater expense both to produce and to verify that evidence, and possibly reducing confidence in the system's correctness. Thus the choice of safety policy in a practical system involves important trade-offs. It is also worth observing that stronger policies are not always better if they rule out too many programs. For example, a policy that rejects all programs provides great safety (and is cheap to implement), but is entirely useless for a safety infrastructure. Therefore, it is important to design safety policies to allow as many programs as possible, while still providing sufficient safety. 2.2 Evidence of Compliance The safety policy establishes what properties mobile programs must satisfy in order to be permitted to execute on a host. However, it is impossible in general for a code consumer to determine whether an arbitrary program complies with that policy. Therefore, we require that suppliers of mobile code assist the consumer by providing evidence that their code complies with the safety policy. This evidence, which we may think of as a certificate of safety, is packaged together with the mobile program and the two together are referred to as certified code. Upon obtaining certified code, the code consumer (automatically) verifies the validity of the evidence before executing the program code. The second task in the design of the safety infrastructure is to decide what form the evidence of compliance must take. This decision is made in the light of several considerations: (1) Since evidence of safety must be transferred over the network along with the program code they certify, we wish the evidence to be as small as possible in order to minimize communication overhead. evidence must be checked before running any program code, we desire verification of evidence to be as fast as possible. Clearly, smaller evidence can lead to faster checking, but we can also speed evidence verification by careful design of the form of evidence. (3) As discussed above, the evidence verifier is an essential part of the trusted computing base; it must work properly or there will be a potential security hole in the system. For us to have confidence that the verifier works properly, it must be simple, which means that the structure of the evidence it checks must also be simple. Thus, not only is simplicity desirable from an aesthetic point of view, but it is also essential for the system to work. (4) Finally, to have complete confidence that our system provides the desired safety, we must prove with mathematical rigor that programs carrying acceptable evidence of safety really do comply with the safety policy. This proof is at the heart of the safety guarantees that the system provides. For such proofs to be feasible, the structure of the evidence must be built on mathematical foundations. In light of these considerations, we now discuss two different forms that evidence of compliance may take: explicit proofs, which are employed in the Proof-Carrying Code infrastructure [6], and type annotations, which are employed in the Typed Assembly Language infrastructure [7,2]. Explicit Proofs The most direct way to provide evidence of safety is to provide an explicit formal proof that the program in question complies with the safety policy. This is the strategy employed by Proof-Carrying Code (PCC). It requires a formal language in which safety proofs can be expressed. Any such language should be designed according to the following criteria. Effective Decidability: It should be efficiently decidable if a given object represents a valid safety proof. Compactness: Proofs should have small encodings. Generality: The representation language should permit proofs of different safety properties. Ideally, it should be open-ended so that new safety policies can be developed without a change in the trusted computing base. Simplicity: The proof representation language should be as simple as possi- ble, since we must trust its mathematical properties and the implementation of the proof checker. Our approach has been to use the LF logical framework [8] to satisfy these requirements. A logical framework is a general meta-language for the representation of logical inferences rules and deductions. Various logics or theories can be specified in LF at a very high level of abstraction, simply by stating valid axioms and rules of inference. This provides generality, since we can separate the theories required for reasoning about safety properties such as arithmetic memory update and access from the underlying mechanism of checking proofs. It is also simple, since it is based on a pure, dependently typed -calculus whose properties have been deeply investigated [9,10]. Proofs in a logic designed for reasoning about safety properties are represented as terms in LF. Checking that a proof is valid is reduced to checking that its representation in the logical framework is well-typed. This can be carried out effectively even for very large proof objects. Experiments in certifying compilation [11] and decision procedures [12] yield proofs whose representation is more than 1 MB, yet can still be checked. On the other hand, proofs in LF are not compact without additional techniques for redundancy elimination. Following some general techniques [13], Necula [14] has developed optimized representations for a fragment of LF called which is sufficient for its use in PCC applications. The experimental results obtained so far have validated the practicality of this proof compression technique [11] for the safety policies discussed here. Current research [15] is aimed at extending and improving these methods to obtain further compression without compromising the simplicity of the trusted computing base. Type Annotations A second way to provide evidence of safety is using type annotations. In this approach a typing discipline is imposed on mobile programs, and the architects of the system prove a theorem stating that any program satisfying that type discipline will necessarily satisfy the safety policy as well [7]. However, determining whether a program satisfies a type discipline involves finding a consistent type scheme for the values in the program, and such a type scheme cannot be inferred in general. Therefore, in this approach programs are required to include enough type annotations for the type checker to reconstruct a consistent type scheme. Such type annotations constitute the evidence of safety, provided they are taken in conjunction with a theorem stating that well-typed programs comply with the safety policy. A principal advantage of the type annotation approach over the explicit proof approach is that the soundness of the type system can be established once and for all. In contrast, validity of explicit proofs does not establish the soundness of the system of proof rules, and in practice the proof rules are freely customized to account for the safety requirements of each application. The main drawback of type annotations is that any program that violates a type sys- tem's invariants will not be typeable under that type system, and therefore cannot be accepted by the safety infrastructure, even if it is actually safe. With explicit proofs, such invariants are not built in, so it is possible to work around cases in which they do not hold. The idea of using types to guarantee safety is by no means new. Many modern high-level languages (e.g., ML, Modula-3, Java) rely on a type system to ensure that all legal programs are safe. Such languages have even been used for safety infrastructures; for example, the SPIN operating system [16] required that operating system extensions be written in Modula-3, thereby ensuring their safety. The drawback to using a high-level language to ensure safety is that programs are checked for safety before compilation, rather than after, thereby requiring that the entire compiler be included in the trusted computing base. As discussed above, the confidence that one can have in the safety architecture is inversely related to the size of its trusted computing base. The Typed Assembly Language (TAL) infrastructure resolves this problem by employing a type safe low-level language. In TAL, a type discipline is imposed on executable code, and therefore the program code being checked for safety is the exact code that will be executed. There is no need to trust a compiler, because if the compiler is faulty and generates unsafe executables, those executables will be rejected by the type checker. The principal exercise in developing a type system for executable code is to isolate low-level abstractions satisfying two conditions: ffl The abstractions should be independently type checkable; that is, to whatever extent type checking of the abstractions depends on surrounding code and data, it should only depend on the types of that code and data, and not on additional information not reflected in the types. ffl The atomic operations on the abstractions should be single machine instructions As an example consider function calls. High-level languages usually provide a built-in notion of functions. Functions can certainly be type checked indepen- dently, but they are not dealt with by a single machine instruction. Rather, function calls are processed using separate call and return instructions and the intervening code is by no means atomic: the return address is stored in accessible storage and can be modified or even disregarded. To satisfy the second condition, TAL's corresponding abstraction is the code block, and code blocks are invoked using a simple jump instruction. Functions are then composed from code blocks by writing code blocks with an explicit extra input containing the return address. (This decomposition corresponds to the well-known practice in high-level language of programming in continuation-passing style [17].) The first condition is satisfied by requiring code blocks to specify the types of their inputs, just as functions in high-level languages specify the types of their arguments and results. Without such specifications, it would be impossible to check the safety of a jump without inspecting the body of the jump's target. For example, consider the TAL code below for computing factorial. This code does not exhibit many of the complexities of the TAL type system, but it serves to give the flavor of TAL programs. (More exhaustive examples appear in Morrisett, et al. [7,2].) codefr1:int,r2:fr1:intgg. mov r3,1 % set up accumulator for loop jmp loop loop: codefr1:int,r2:fr1:intg,r3:intg. bz r1,done % check if done, branch if zero mul r3,r3,r1 sub r1,r1,1 jmp loop done: codefr1:int,r2:fr1:intg,r3:intg. mov r1,r3 % move accumulator to result register jmp r2 % return to caller In this code, the type for a code block is written fr1:- 1 indicating that the code may be called when registers having . The fact code block is given type fr1:int ; r2:fr1:intgg, indicating that when fact is called, it must be given in r1 an integer (the argument), and in r2 a code block (the return address) that when called must be given an integer (the return value) in r1. When called, fact sets up an accumulator register (with type int) in r3 and jumps to loop. Then loop computes the factorial and, when finished, branches to done, which moves the accumulator to the return address register (r1) and returns to the caller. In the final return to the caller, the extra registers r2 and r3 are forgotten to match the precondition on r2, which only mentions r1. As an alternative to typed assembly language, one can also strike a compromise between high- and low-level languages by exploiting typed intermediate languages for safety [18]. Using typed intermediate languages enlarges the trusted computing base, since some part of the compiler must be trusted, but it loosens the second condition on type systems for executable code. This provides a spectrum of possible designs, the closer an intermediate language is to satisfying the second condition, the lesser the amount of the compiler that needs to be trusted. Moreover, as we discuss in Section 3, typed intermediate languages are valuable for automated certification, even if the end result is a typed executable. 2.3 Automated Verification Since it plays such a central role for provably safe mobile code, we now elaborate on the mechanisms for verifying safety certificates. Explicit Proofs As discussed above, the Proof-Carrying Code infrastructure employs the LF logical framework. In the terminology of logical frameworks, a judgment is an object of knowledge which may be evident by virtue of a proof. Typical safety properties require only a few judgments, such as the truth of a proposition in predicate logic, or the equality of two integers. In LF, a judgment of an object logic is represented by a type in the logical framework, and a proof by a term. If we have a proof P for a judgment J , then the representation P has type J , where we write () for the representation function. The adequacy theorem for a representation function guarantees this property and its inverse: whenever we have a term M of type J then there is a proof P for J . Both directions are critical, because together they mean that we can reduce the problem of checking the validity of a proof P to verifying that its representation P is well-typed. So in PCC, checking compliance with a safety policy can be reduced to type checking the representation of a safety proof in the logical framework. But how does this technique allow for different safety policies? Since proofs are represented as terms in LF, an inference rule is represented as a function from the proofs of its premises to the proof of its conclusion. To represent a complete logical system we only need to introduce one type constant for each basic judgment and one term constant for each inference rule. The collection of these constant declarations is called a signature. 1 So a particular safety policy consists of a verification condition generator, which extracts a proof obligation from a binary, and signature in LF, which expresses the valid proof principles for the verification condition. This means that different policies can be expressed by different signatures, and that the basic engine that verifies evidence (the LF type checker) does not change for different policies. However, we do have to trust the correctness of the LF signature representing a policy- an inconsistent signature, for example, would allow arbitrary code to pass the safety check. Type checking in LF is syntax-directed and therefore in practice quite efficient [13], especially if we avoid checking some information which can be statically shown to be redundant [14]. Currently, the Touchstone compiler for PCC discussed in Section 3.1 uses a small, efficient type checker for LF terms written in C. Related projects on proof-carrying code [19,20] and certifying decision procedures [12] use the Twelf implementation [21-23]. For more information on logical frameworks, see [24]. Type Annotations In case the safety policy is expressed in the form of typing rules, checking compliance immediately reduces to type checking. In this case we have to carefully design the language of annotations so that type checking is practical. Generally, the more complicated the safety property the more annotations are required. Once the safety property is fixed, there is a trade-off between space and time: the more type annotations we have, the easier the type checking problem. One extreme consists of no type annotations at all, which means that type checking is undecidable. The other extreme is a full typing derivation (represented, for example, in the logical framework) which is quite similar to a proof in the PCC approach. For the safety policies we have considered so far, it has not been difficult to find appropriate compromises between these extremes that are both compact and permit fast type checking [25]. It is worth noting that in both cases (explicit proofs and type annotations) the verification method is type checking. For proof-carrying code, this is always This should not be confused with a digital signature used to certify authenticity. type checking in the LF logical framework with some optimizations to eliminate redundant work. For typed assembly language the algorithm for type checking varies with the safety property that is enforced, although the basic nature of syntax-directed code traversal remains the same. 3 Automated Certification How are certificates of safety to be obtained? In principle we may use any means at our disposal, without restriction or limitation. This freedom is assured by the checkability of safety certificates - it is always possible to determine mechanically whether or not a given certificate underwrites the safety of a given program. Since the code recipient can always check the validity of a safety certificate, there is no need to rely on the means by which the certificate was produced. Two factors determine how hard it is to construct a safety certificate for a program: (1) The strength of the assurances we wish to make about a program. The stronger the assurances, the harder it is to obtain a certificate. (2) The complexity of the programming language itself. The more low-level the language, the harder it is to certify the safety of programs. As a practical matter, the easier it is to construct safety certificates, the more likely that code certification will be widely used. The main technique we have considered for building safety certificates is to build a certifying compiler for a safe, high-level language such as ML or Java (or any other type-safe language, such as Ada or Modula). A certifying compiler generates object code that is comparable (and often superior) in quality to that of an ordinary compiler. A certifying compiler goes beyond conventional compilation methods by augmenting the object code with a checkable safety certificate warranting the compliance of the object code with the safety properties of the source language. In this way we are able to exploit the safety properties of semantically well-defined high-level languages without having to trust the compiler itself or having to ensure the integrity of code in transit from producer to consumer. The key to building a certifying compiler is to propagate safety invariants from the source language through the intermediate stages of compilation to the final object code. This means that each compilation phase is responsible for the preservation of these invariants from its input to its output. Moreover, to ensure checkability of these invariants, each phase must annotate the program with enough information for a code recipient to reconstruct the proof of these invariants. In this way the code recipient can check the safety of the code, without having to rely on the correctness of the compiler. In the (common) case that the compiler contains errors, the purported safety certificate may or may not be valid, but the recipient can detect the mistake. Since each compilation phase can be construed as a recipient of the code produced by the preceding stage, the compiler can check its own integrity by verifying the claimed invariants after each stage. This has proved to be an invaluable aid to the compiler writer [11,18]. 3.1 Constructing Evidence of Safety We have explored two main methods for propagating safety invariants during compilation: (1) Translation between typed intermediate languages [26]. Safety invariants are captured by a type system for the intermediate languages of the compiler. The type system is designed to ensure that well-typed expressions are safe, and enough type information attached to intermediate forms to ensure that we may mechanically check type correctness. The typed intermediate forms are "self-certifying" in the sense that the attached type information serves as a checkable certificate of safety. (2) Compilation to proof-carrying code [11]. Safety invariants are directly expressed as logical assertions about the execution behavior of conventional intermediate code. The soundness of the logic ensures that these assertions correctly express the required safety properties of the code. The safety of the object code is checked by a combination of verification condition generation and automatic theorem proving. By equipping the theorem prover with the means to generate a formal representation of a proof, we may generate checkable safety certificates for the object code. These two methods are not mutually exclusive. We are currently exploring their integration using dependent types which allow assertions to be blended with types in a single type-theoretic formalism. This technique is robust and can be applied to high-level languages [27,28] as well as low-level languages [29], thereby providing an ideal basis for their use in certifying compilers 3.2 Typed Intermediate Languages To give a sense of how type information might be attached to intermediate code, we give an example derived from the representation of lists. At the level of the source language, there are two methods for creating lists: (1) nil, which stands for the empty list; (2) cons(h, which constructs the non-empty list with head h and tail t. These values are assigned types according to the following rules: 2 (1) nil has type list; (2) If h has type int and t has type list, then cons(h,t) has type list. There are a variety of operations for manipulating lists, including the car and cdr operations, which have the following types: (1) If l has type list, then car(l) has type int; (2) If l has type list, then cdr(l) has type list. The behavior of these operations is governed by the following transitions in an operational semantics for the language: (1) car(cons(h,t)) reduces to h; (2) cdr(cons(h,t)) reduces to t. One task of the compiler is to decide on a representation of lists in memory, and to generate code for car and cdr consistently with this representation. A typical (if somewhat simple-minded) approach is to represent a list by (1) A pointer to . (2) . a tagged region of memory containing . (3) . a pair consisting of the head and tail of the list. The tag field distinguishes empty from non-empty lists, and the pointer identifies the address of the node in the heap. This representation can be depicted as the following compound term: ptr(tag[cons](pair(h, What is interesting is that each individual construct in this expression may be thought of as a primitive of a typed intermediate language. Specifically, 2 For simplicity we consider only lists of integers. (1) ptr(v) has type list if v has type [nil:void,cons:int*list]. The bracketed expression defines the tags (nil and cons), and the type of their associated data values (none, in the case of nil, a pair in the case of a cons). (2) tag[t](v) has type [t:-,.] if v has type - . In particular, tag[cons](pair(h,t)) has type [nil:void,cons:int*list] if h has type int and t has type list. (3) if l has type - l and r has type - r . In particular, int*list if h has type int and t has type list. Corresponding to this representation we may generate code for, say, car(l) that behaves as follows: (1) Dereference the pointer l. The value l must be a pointer because its type is list. (2) Check the tag of the object in the heap to ensure that it is cons. It must be either cons or nil because the type of the dereferenced pointer is [nil:void,cons:int*list]. (3) Extract the underlying pair and project out its first component. It must have two components because the type of the tagged value is int*list. When expressed formally in a typed intermediate language, the generated code for the car operation is defined in terms of primitive operations for performing these three steps. The safety of this code is ensured by the typing rules associated with these operations - a type correct program cannot misinterpret data by, for example, treating the head of a list as a floating point number (when it is, in fact, an integer). A type-directed compiler [26] is one that performs transformations on typed intermediate languages, making use of type information to guide the trans- lation, and ensuring that typing is preserved by each transformation stage. In a type-directed compiler each compilation phase translates not only the program code, but also its type, in such a way that the translated program has the translated type. How far this can be pushed is the subject of ongoing research. In the TILT compiler we are able to propagate type information down to the RTL (register transfer language level), at which point type propagation is abandoned. The recent development of Typed Assembly Language (TAL) [7,2] demonstrates the feasibility of propagating type information down to x86-like assembly code. The integration of TILT and TAL is the subject of ongoing research. What does the propagation of type information have to do with safety? A well-behaved type system is one for which we can prove a soundness theorem relating the execution behavior of a program to its type. One consequence of the soundness theorem for the type system is that it is impossible for well-typed programs to incur type errors, memory errors, or control errors. That is, well-typed programs are safe. Of course not every safe program is well-typed - typing is a sufficient condition for safety, but not a necessary one. However, we may readily check type correctness of a program using lightweight and well-understood methods. The technique of type-directed compilation demonstrates that a rich variety of programs can be certified using typed intermediate languages. Whether there are demands that cannot be met using this method remains to be seen. 3.3 Logical Assertions and Explicit Proofs Another approach to code certification that we are exploring [30,11] is the use of a combination of logical assertions and explicit proofs. A certifying compiler such as Touchstone works by augmenting intermediate code with logical assertions tracking the types and ranges of values. Checking the validity of these assertions is a two-step process: (1) Verification condition generation (vcgen). The program is "symbolically evaluated" to propagate the implications of the logical assertions through each of the instructions in the program. This results in a set of logical implications that must hold for the program to be considered properly annotated. (2) Theorem proving. Each of the implications generated during vcgen are verified using a combination of automatic theorem proving techniques, including constraint satisfaction procedures (such as simplex) and proof search techniques for first-order logic. In this form the trusted computing base must include both the vcgen procedure and the theorem prover(s) used to check the verification conditions. In addition, a specification of a safety policy which describes the conditions for safe execution as well as pre- and post-conditions on all procedures supplied by the host operating system and required of the certified code. To reduce the size of the trusted computing base, we may regard the combination of vcgen and theorem proving as a kind of "post-processing" phase in which the validity of the annotated program is not only checked, but a formal representation of the proof for the validity of the verification conditions is attached to the code. This is achieved by using certifying theorem provers [11,12] that not only seek to prove theorems, but also provide an explicit representation of the proof whenever one is found. Once the proofs have been obtained, it is much simpler to check them than it is to find them. Indeed, only the proof checker need be integrated into the trusted computing base; the theorem provers need not be trusted nor be protected from tampering. To gain an understanding of what is involved here, consider the array subscript operation in a safe language. Given an array A of length n and an integer i, the operation sub(A,i) checks whether or not 0 - retrieves the ith element of A. At a high level this is an atomic operation, but when compiled into intermediate code it is defined in terms of more primitive operations along the following lines: if (0 != i && return A[i+1] /* unsafe access */ else f . signal an error . Note that *A refers to the length of array A. Here we assume that an integer array is represented by a pointer to a sequence of words, the first of which contains the array's length, and the rest of which are its contents. Annotating this code with logical assertions, we obtain the following: /* int i, array A */ if (0 != i && return A[i+1] else f . signal an error . The assertion that A is an array corresponds to the invariants mentioned above; in practice, a much lower-level type system is employed [11]. It is a simple matter to check that the given assertions are correct in this case. Observe that the role of the conditional test is to enable the theorem prover to verify that the index operation A[i+1] is memory-safe - it does not stray beyond the bounds of the array. In many cases the run-time test is redundant because the compiler is able to prove that the run-time test must come out true, and therefore can be eliminated. For example, if the high-level code were a simple loop such as the following, we can expect the individual bounds checks to be elided: int for (i=0; i!length(A); i++) f sum += sub(A,i); At the call site for sub the compiler is able to prove that 0 - n is the length of A. Propagating this through the code for sub, we find that the conditional test can be eliminated because the compiler can prove that the test must always be true. This leads to the following code (after further int for (i=0; i!*A; i++) f sum += A[i+1]; Given this annotation, we can now perform verification condition generation and theorem proving to check that the required precondition on the unsafe array subscript operation is indeed true, which ensures that the program is safe to execute. However, rather than place this additional burden on the programmer, we can instead attach a formal representation - of the proof of this fact to the assertions: int for (i=0; i!*A; i++) f sum += A[i+1] The proof term - is a checkable witness to the validity of the given assertions that can be checked by the code recipient. In practice this witness is a term of the LF -calculus for which proof checking is simply another form of type checking (see Section 2.3). 4 Experimental Results As mentioned earlier, we have implemented several systems to test and demonstrate the ideas of certified code, typed intermediate languages, certifying com- pilers, and certifying theorem provers. The results of our experiments with these systems confirm several important claims about the general framework for safety certification of code that we have presented in this paper. (1) Approaches to certified code such as PCC and TAL allow highly optimized code to be verified for safety. This means that few if any compromises need to be made between high performance and safety. (2) The various approaches to certifying compilers that we have explored, such as typed intermediate languages and logical assertions, can be scaled x vs. "GNU gcc GNU gcc -O4 2.33 3.82 3.51 2.97 2.44 2.62 5.50 2.92 3.17 DEC cc -O4 2.92 3.68 3.52 2.79 2.44 2.76 6.88 11.52 3.92 Cert Comp 2.64 3.89 3.52 3.86 1.93 2.20 4.00 9.16 3.48 blur sharpen qsort simplex kmp unpack bcopy edge GMEAN1000.03000.05000.0Time (ms) Proof Checking (ms) 7.7 23.5 16.3 120.3 9.6 92.5 3.6 14.7 Proving (ms) 81.0 257.0 127.0 1272.0 108.0 1912.0 25.0 143.0 VC Generation (ms) 6.3 20.9 11.5 73.9 8.4 70.6 3.5 9.7 Code Generation (ms) 271.0 818.0 560.0 4340.0 348.0 1885.0 136.0 697.0 blur sharpen qsort simplex kmp unpack bcopy edge50001500025000 binary size (bytes) Proof 1372 4532 2748 20308 1874 17260 620 2218 Code 320 1248 576 3792 496 2528 128 640 blur sharpen qsort simplex kmp unpack bcopy edge Fig. 1. Comparison of generated object-code performance between the Touch- stone, GCC, and DEC CC optimizing compilers. The height of the bars shows the speedup of the object code relative to unoptimized code as produced by gcc. up to languages of realistic scale and complexity. Furthermore, they provide an automatic means of obtaining code that is certified to hold standard safety properties such as type safety, memory safety, and control safety. (3) The need to include annotations and/or proofs with the code is not an undue burden. Furthermore, checking these certificates can be performed quickly and reliably. In order to support these claims and give a better feel for the practical details in our systems, we now present some results of our experiments. 4.1 The Touchstone Certifying Compiler Touchstone is a certifying compiler for an imperative programming language with a C-like syntax. Although the source programs look very much like C programs, the language compiled by Touchstone is made "safe" by having a strong static type system, eliminating pointer arithmetic, and ensuring that all variables are initialized. Although this language makes restrictions on C, it is still a rich and powerful language in the sense of allowing recursive pro- cedures, aliased variables, switch statements, and dynamically allocated data structures. Indeed, it is straightforward to translate many practical C source programs into the language compiled by Touchstone [31]. Given such a source program, Touchstone generates a highly optimized native code target program for the DEC Alpha architecture with an attached proof of its type, memory, and control safety. Figure 1 shows the results of a collection of benchmark programs when compiled with Touchstone, the Gnu gcc C compiler, and the DEC cc C compiler. The benchmark programs were obtained from standard Unix utility applica- xSpeedup vs. "GNU gcc GNU gcc -O4 2.33 3.82 3.51 2.97 2.44 2.62 5.50 2.92 3.17 DEC cc -O4 2.92 3.68 3.52 2.79 2.44 2.76 6.88 11.52 3.92 Cert Comp 2.64 3.89 3.52 3.86 1.93 2.20 4.00 9.16 3.48 blur sharpen qsort simplex kmp unpack bcopy edge GMEAN1000.03000.05000.0Time (ms) Proof Checking (ms) 7.7 23.5 16.3 120.3 9.6 92.5 3.6 14.7 Proving (ms) 81.0 257.0 127.0 1272.0 108.0 1912.0 25.0 143.0 VC Generation (ms) 6.3 20.9 11.5 73.9 8.4 70.6 3.5 9.7 Code Generation (ms) 271.0 818.0 560.0 4340.0 348.0 1885.0 136.0 697.0 blur sharpen qsort simplex kmp unpack bcopy edge50001500025000 binary size (bytes) Proof 1372 4532 2748 20308 1874 17260 620 2218 Code 320 1248 576 3792 496 2528 128 640 blur sharpen qsort simplex kmp unpack bcopy edge Fig. 2. Breakdown of time required to generated the proof-carrying code binaries tions (such as the xv and gzip programs) and then edited in a completely straightforward way to replace uses of pointer arithmetic with array-indexing syntax. (Recall that the C-like language compiled by Touchstone does not support pointer arithmetic.) The bars in the figure were generated by first compiling each program with the Gnu gcc compiler with all optimizations turned off. Then, Touchstone, Gnu gcc, and DEC cc were used to compile the programs with all optimizations turned on. The bars in the figure show the relative speed improvements produced by each optimizing compiler relative to the unoptimized code. The figure shows that the Touchstone compiler generates object code which is comparable in speed to that produced by the gcc and cc compilers, and in fact is superior to gcc overall. This result is particularly surprising when one considers that Touchstone is obligated to guarantee that all array accesses and pointer dereferences are safe (that is, Touchstone must sometimes perform array-bounds and null-pointer checks), whereas the gcc and cc compilers do not do this. In fact, Touchstone is able to optimize away almost all array-bounds and null-pointer checks, and generates proofs that can convince any code recipient that all array and pointer accesses are still safe. In Figure 2 we provide a breakdown of the time required to compile each benchmark program into a PCC binary. Each bar in the figure is divided into four parts. The bottom-most part shows the "conventional" compile time. This is the time required to generate the DEC Alpha assembly code plus invariant annotations required by the underlying PCC system. Because Touchstone is a highly aggressive optimizing compiler, it is a bit slower than typical compilers. However, on average it is comparable in compiling times to the DEC cc compiler with all optimizations enabled. The second part shows the time required to generate the verification conditions. Finally, the third and fourth parts show the times required for proof generation and proof checking, respectively. One can see that very little time is required for the verification-condition generation and proof checking. This is important because it is these two steps that must also be performed by any recipient of the generated code. The fact that these two parts are so small is an indication that the code recipient in fact has very little work to do. Early measurements with the Touchstone compiler showed that the proofs were about 2 to 4 times larger than the code size [31]. Since the time that those experimental results were obtained, we have made considerable progress on reducing the size of the proofs, without increasing the time or effort required to check them. These reductions lead to proof sizes on the order of 10% to 40% of the size of the code. In addition, we have been experimenting with a new representation (which we refer to as the "oracle string" representation) which, for the types of programs described here, further reduces proof sizes to be consistently less than 5% of the code size, at the cost of making proof checking about 50% slower. We hope to be able to describe these techniques and show their effects in a future report. 4.2 The Cedilla Systems Special J Compiler The experimental results shown above are admittedly less than convincing, due to the relatively small size of the test programs. Recently, however, we have "spun off" a commercial enterprise to build an industrial-strength implementation of a proof-carrying code system. This enterprise, called Cedilla Systems Incorporated, is essentially an experiment in technology transfer, in the sense that it is attempting to take ideas and results directly out of the laboratory and into commercial practice. Cedilla Systems has shown that the ideas presented in this paper can be scaled up to full-scale languages. This is shown most clearly in an optimizing native-code compiler for the full Java programming language, called Special J [32], which successfully compiles over 300 "real-world" Java applications, including rather large ones such as Sun's StarOffice application suite and their HotJava web browser. The operation of Special J is similar to Touchstone, in that Special J produces optimized target code annotated with invariants that make it possible to construct a proof of safety. A verification-condition generator is then used to extract a verification condition, and a certifying theorem prover generates the proof which is attached to the target code. To see a simple example of this process, consider the following Java program: public class Bcopy1 f public static void bcopy(int[] src, f int int This source program is compiled by Special J into the target program for the Intel x86 architecture shown in Figure 3. Included in this target program are numerous data structures to support Java's object model and run-time system. The core of this output, however, is the native code for the bcopy method shown above. This code is largely conventional except for the insertion of several invariants, each of which is marked with a special "ANN " macro. These annotations are "hints" from the compiler that help the automatic proof generator do its job. They do not generate code, and they do not constrain the object code in any way. However, they serve an important engineering purpose, as we will now describe. The ANN LOCALS annotation simply says that the compiled method uses three locals. In this case, the register allocator did not need any spill space on the stack, so the only locals are the two formal parameters and the return address. This hint is useful for proving memory safety. The prover could, in principle, analyze the code itself to reverse-engineer this information; but it is much easier for the compiler to communicate what it already knows. Since one of our engineering goals is to simplify as much as possible the size of the trusted computing base, it is better to have the compiler generate this information, leaving only the checking problem to the PCC infrastructure. The ANN UNREACHABLE annotations come from the fact that the safety policy specifies that array accesses must always be in bounds and null pointers must never be dereferenced. In Java, such failures result in run-time exceptions, but ANN LOCALS( bcopy 6arrays6Bcopy1AIAI, .globl bcopy 6arrays6Bcopy1AIAI bcopy 6arrays6Bcopy1AIAI: cmpl $0, 4(%esp) ;src==null? je L6 movl 4(%esp), %ebx movl 4(%ebx), %ecx testl %ecx, %ecx ;l==0? jg L22 ret L22: xorl %edx, %edx ;initialize i cmpl $0, 8(%esp) ;dst==null? je L6 movl 8(%esp), %eax movl 4(%eax), %esi ;dst.length L7: ANN INV(ANN DOM LOOP, (/" (csubneq eax (/" (csubb edx ecx) (of rm mem)))) LF, cmpl %esi, %edx ;i!dst.length? movl 8(%ebx, %edx, 4), %edi ;src[i] movl %edi, 8(%eax, %edx, incl %edx ;i++ cmpl %ecx, %edx ;i!l? jl L7 ret ANN INV(ANN DOM LOOP, LF true LF, ret L13: call Jv ThrowBadArrayIndex ANN UNREACHABLE nop call Jv ThrowNullPointer ANN UNREACHABLE nop Fig. 3. Special J output code the safety policy in our example requires a proof that these exceptions will never be thrown. Therefore, the compiler points out places that must never be reached during execution so that the proof generator does not need to reverse-engineer where the source-code array accesses and pointer dereferences ended up in the binary. The first ANN INV annotation is by far the most interesting of all the annota- tions. Note that the Special J compiler has optimized the tight loop: ffl Both required null checks are hoisted. (Note that the null check on dst cannot be hoisted before the loop entry because the loop may never be entered at all; but it can be hoisted to the first iteration.) ffl The bounds check on src is hoisted. (Note that hoisting the bounds check on dst would be a more exotic optimization, because in the case that dst is not long enough, the loop must copy as far as it can and then throw an exception.) The proof generator must still prove memory safety, so it must prove that inside the loop there are no null-pointer dereferences or out-of-bounds memory accesses. Essentially, the proof generator needs to go through the same reasoning that the compiler went through when it hoisted those checks outside the loop. Therefore, to help the proof generator, the compiler outputs the relevant loop invariants that it discovered while performing the code-hoisting optimizations. In this case, it discovered that: ffl src (in register ebx) is not null: (csubneq ebx 0). ffl dst (in register eax) is not null: (csubneq eax 0). The "csub" prefix denotes the result of a Pentium comparison. Other things in the loop invariant specify: ffl which registers are modified in the loop: RB(.), ffl that memory safety is a loop invariant: (of rm mem) Pseudo-register rm denotes the computer's memory, and (of rm mem) means that no unsafe operations have been performed on the memory. After this target code is generated by Special J, the Cedilla Systems proof generator reads it and outputs a proof that the code satisfies the safety policy. The first step to doing this is to generate a logical predicate, called a verification condition (or simply VC), whose logical validity implies the safety of the code. It is important that the same VC be used by both the producer and the recipient of the code, so that the recipient can guarantee that the "right" safety proof is provided, as opposed to a proof of some unrelated or irrelevant property. As we explained earlier, both the proofs and the verification conditions are expressed in a language called the Logical Framework (LF). Space prevents us from including the entire VC for our bcopy example. However, the following excerpt illustrates the main points. (Note that X0 is the dst parameter, X1 is the src parameter, X2 is a pseudo-register representing the current state of the heap, and X3 is the variable i. (=? (csubb X3 (sel4 X2 (add X1 4))) (=? (csubneq X0 (=? (csubneq X1 (=? (csubb X3 (sel4 X2 (add X0 4))) (/" (saferd4 (add X1 (add (imul X3 (/" (safewr4 (add X0 (add (imul X3 (add X1 (add (imul X3 (/" (=? (csublt (add X3 1) (sel4 X2 (add X1 4))) (/" (csubneq X1 (/" (csubneq X0 (/" (csubb (add X3 1) (sel4 X2 (add X1 4))) This excerpt of the VC says that, given the loop-invariant assumptions ffl (csubb X3 (sel4 X2 (add X1 4))) (i.e., src[i] is in bounds), (i.e., dst is non-null), and (i.e., src is non-null), and given the bounds check that was emitted for dst: as well as some additional assumptions outside the loop (not shown in this snippet), proofs are required to establish the safety of the read of the src array and the write to the dst array. Furthermore, given the additional loop-entry condition proofs are required to reestablish the loop invariants. Here, X0 corresponds to eax (dst in the source), X1 to ebx (src in the source), X2 to rm (the memory pseudo-register), and X3 to edx (i in the source). Note that src.length is (sel4 X2 (add X1 4)), because the length is stored at byte-offset 4 in an array object. The safety policy, and hence the VC, specifies and enforces these requirements on data-structure layout. The proof generator reads the VC and outputs a proof of it. A tiny excerpt of this proof is shown below: ([ASS10: pf (csubb X3 (sel4 X2 (add X1 4)))] ([ASS12: pf (csubneq X1 0)] ([ASS13: pf (csubb X3 (add X0 4)))] (andi szint (below1 ASS10))) The proofs are shown here in a concrete syntax for LF developed for the Elf system [22,21]. In this very small snippet of the proof, one can see that assumptions (marked with the "ASS." identifiers) are labeled and then used in the body of the proof. Logical inference rules such as "impi", which in this case stands for the "implication-introduction" rule, are specified declaratively in the LF language, and included with the PCC system as part of the definition of the safety policy. Finally, a binary encoding of the proof is made and attached to the target code. The proof is included in the data segment of a standard binary in the COFF format. In this case, the proof takes up 7.1% of the total object file. We note that we currently use an unoptimized binary encoding of the proof in which all proof tokens are 16 bits long. Huffman encoding produces an average token size of 3.5 bits, and so a Huffman-encoded binary is expected to be about 22% of the size of a non-Huffman-encoded binary. In this case, that would make the size of the proof approximately 45 bytes, or less than 2% of the object file. While Huffman encoding would indeed be an effective means of reducing the size of proofs, we have found that other representations such as "oracle strings" do an even better job, without incurring the cost of decompression. In the case of the current example, the oracle string representation of the proof requires less than 6 bytes. We hope to report in detail on this representation in a future report. 5 Conclusion and Future Work We have presented a general framework for the safety certification of code. It relies on the formal definition of a safety policy and explicit evidence for compliance attached to mobile code. This evidence may take the form of formal safety proofs (in proof-carrying code) or type annotations (in typed assembly language). In both cases one can establish with mathematical rigor that certified code is tamper-proof and can be executed safely without additional run-time checks or operating system protection boundaries. Experience with the approaches has shown the overhead to be acceptable in practice, both in the time to validate the certificate and the space to represent it, using advanced techniques from logical frameworks and type theory. We also sketched how certificates can be obtained automatically through the use of certifying compilers and theorem provers. The approach of typed intermediate languages propagates safety properties which are guaranteed for the high-level source language throughout the compilation process down to low-level code. Safety remains verifiable at each layer through type-checking. A certifying compiler such as Touchstone uses logical assertions throughout compilation in a similar manner, except that the validity of the logical assertions must be assured by theorem proving. This is practical for the class of safety policies considered here, since the compiler can provide the information necessary to guarantee that a proof can always be found. Finally, a certifying theorem prover does not need to be part of the trusted computing base since it produces explicit proof terms which can be checked independently by an implementation of a logical framework. The key technology underlying our approaches to safety is type theory as used in modern programming language design and implementation. The idea that type systems guarantee program safety and modularity for high-level languages is an old one. We see our main contribution in demonstrating in practical, working systems such as Touchstone, the TILT compiler, and the Twelf logical framework, that techniques from type theory can equally be applied to intermediate and low-level languages down to machine code in order to support provably safe mobile code for which certificates can be generated automatically. --R Efficient software-based fault isolation ACM Symposium on Principles of Programming Languages A type system for expressive security policies Safe kernel extensions without run-time checking From System F to typed assembly language A framework for defining logics On equivalence and canonical forms in the LF type theory Generating proofs from a decision procedure An empirical study of the runtime behavior of higher-order logic programs Efficient representation and validation of logical proofs Algorithms for equality and unification in the presence of notational definitions Definitional interpreters for higher-order programming languages A semantic model of types and machine instructions for proof-carrying code System description: Twelf - a meta-logical framework for deductive systems Elf: A meta-language for deductive systems Logic programming in the LF logical framework A realistic typed assembly language TIL: A type-directed optimizing compiler for ML Eliminating array bound checking through dependent types Dependent types in practical programming A dependently typed assembly language The design and implementation of a certifying compiler A certifying compiler for Java --TR Logic programming in the LF logical framework A framework for defining logics Efficient software-based fault isolation Extensibility safety and performance in the SPIN operating system Safe kernel extensions without run-time checking Proof-carrying code From system F to typed assembly language Eliminating array bound checking through dependent types The design and implementation of a certifying compiler Dependent types in practical programming Proof-carrying authentication Resource bound certification A semantic model of types and machine instructions for proof-carrying code A type system for expressive security policies A certifying compiler for Java Algorithms for Equality and Unification in the Presence of Notational Definitions Stack-Based Typed Assembly Language Elf System Description Logical frameworks Efficient Representation and Validation of Proofs Definitional interpreters for higher-order programming languages A Dependently Typed Assembly Language Compiling with proofs Higher-order rewriting with dependent types (lambda calculus) --CTR Andr Pang , Don Stewart , Sean Seefried , Manuel M. T. Chakravarty, Plugging Haskell in, Proceedings of the 2004 ACM SIGPLAN workshop on Haskell, September 22-22, 2004, Snowbird, Utah, USA Mike Jochen , Anteneh Addis Anteneh , Lori L. Pollock , Lisa M. Marvel, Enabling control over adaptive program transformation for dynamically evolving mobile software validation, ACM SIGSOFT Software Engineering Notes, v.30 n.4, July 2005

certifying compilers;type systems;typed assembly language;type safety;proof-carrying code

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Abstracting soft constraints.

Soft constraints are very flexible and expressive. However, they are also very complex to handle. For this reason, it may be reasonable in several cases to pass to an abstract version of a given soft constraint problem, and then to bring some useful information from the abstract problem to the concrete one. This will hopefully make the search for a solution, or for an optimal solution, of the concrete problem, faster.In this paper we propose an abstraction scheme for soft constraint problems and we study its main properties. We show that processing the abstracted version of a soft constraint problem can help us in finding good approximations of the optimal solutions, or also in obtaining information that can make the subsequent search for the best solution easier.We also show how the abstraction scheme can be used to devise new hybrid algorithms for solving soft constraint problems, and also to import constraint propagation algorithms from the abstract scenario to the concrete one. This may be useful when we don't have any (or any efficient) propagation algorithm in the concrete setting.

Introduction Classical constraint satisfaction problems (CSPs) [18] are a very convenient and expressive formalism for many real-life problems, like scheduling, resource al- location, vehicle routing, timetabling, and many others [23]. However, many of these problems are often more faithfully represented as soft constraint satisfaction problems (SCSPs), which are just like classical CSPs except that each assignment of values to variables in the constraints is associated to an element taken from a partially ordered set. These elements can then be interpreted as levels of preference, or costs, or levels of certainty, or many other criteria. There are many formalizations of soft constraint problems. In this paper we consider the one based on semirings [4, 3], where the semiring species the partially ordered set and the appropriate operation to use to combine constraints together. Although it is obvious that SCSPs are much more expressive than classical CSPs, they are also more di-cult to process and to solve. Therefore, sometimes it may be too costly to nd all, or even only one, optimal solution. Also, although classical propagation techniques like arc-consistency [17] can be extended to SCSPs [4], even such techniques can be too costly to be used, depending on the size and structure of the partial order associated to the SCSP. For these reasons, it may be reasonable to work on a simplied version of the given problem, trying however to not loose too much information. We propose to dene this simplied version by means of the notion of abstraction, which takes an SCSP and returns a new one which is simpler to solve. Here, as in many other works on abstraction [16, 15], \simpler" may mean many things, like the fact that a certain solution algorithm nds a solution, or an optimal solution, in a fewer number of steps, or also that the abstracted problem can be processed by a machinery which is not available in the concrete context. There are many formal proposals to describe the process of abstracting a notion, be it a formula, or a problem [16], or even a classical [11] or a soft CSP [21]. Among these, we chose to use one based on Galois insertions [5], mainly to refer to a well-know theory, with many results and properties that can be useful for our purposes. This made our approach compatible with the general theory of abstraction in [16]. Then, we adapted it to work on soft constraints: given an SCSP (the concrete one), we get an abstract SCSP by just changing the associated semiring, and relating the two structures (the concrete and the abstract one) via a Galois insertion. Note that this way of abstracting constraint problems does not change the structure of the problem (the set of variables remains the same, as well as the set of constraints), but just the semiring values to be associated to the tuples of values for the variables in each constraint. Once we get the abstracted version of a given problem, we propose to 1. process the abstracted version: this may mean either solving it completely, or also applying some incomplete solver which may derive some useful information on the abstract problem; 2. bring back to the original problem some (or possibly all) of the information derived in the abstract context; 3. continue the solution process on the transformed problem, which is a concrete problem equivalent to the given one. All this process has the main aim of nding an optimal solution, or an approximation of it, for the original SCSP, within the resource bounds we have. The hope is that, by following the above three steps, we get to the nal goal faster than just solving the original problem. A deep study of the relationship between the concrete SCSP and the corresponding abstract one allows us to prove some results which can help in deriving useful information on the abstract problem and then take some of the derived information back to the concrete problem. In particular, we can prove the following { If the abstraction satises a certain property, all optimal solutions of the concrete SCSP are also optimal in the corresponding abstract SCSP (see Theorem 4). Thus, in order to nd an optimal solution of the concrete prob- lem, we could nd all the optimal solutions of the abstract problem, and then just check their optimality on the concrete SCSP. { Given any optimal solution of the abstract problem, we can nd upper and lower bounds for an optimal solution for the concrete problem (see Theorem 5). If we are satised with these bounds, we could just take the optimal solution of the abstract problem as a reasonable approximation of an optimal solution for the concrete problem. { If we apply some constraint propagation technique over the abstract problem, say P , obtaining a new abstract problem, say P some of the information in can be inserted into P , obtaining a new concrete problem which is closer to its solution and thus easier to solve (see Theorem 6 and 8). This however can be done only if the semiring operation which describes how to combine constraints on the concrete side is idempotent (see Theorem 6). { If instead this operation is not idempotent, still we can bring back some information from the abstract side. In particular, we can bring back the inconsistencies (that is, tuples with associated the worst element of the semiring), since we are sure that these same tuples are inconsistent also in the concrete SCSP (see Theorem 8). In both the last two cases, the new concrete problem is easier to solve, in the sense, for example, that a branch-and-bound algorithm would explore a smaller (or equal) search tree before nding an optimal solution. The paper is organized as follows. First, in Section 2 we give the necessary notions about soft CSPs and abstraction. Then, in Section 3 we dene our notion of abstraction, and in Section 4 we prove properties of our abstraction scheme and discuss some of their consequences. Finally, in Section 7 we summarize our work and give hints about future directions. Background In this section we recall the main notions about soft constraints [4] and abstract interpretation [5], that will be useful for the developments and results of this paper. 2.1 Soft constraints In the literature there are many formalizations of the concept of soft constraints [21, 8, 13, 10]. Here we refer to the one described in [4, 3], which however can be shown to generalize and express many of the others [4, 2]. In a few words, a soft constraint is just a classical constraint where each instantiation of its variables has an associated value from a partially ordered set. Combining constraints will then have to take into account such additional values, and thus the formalism has also to provide suitable operations for combination () and comparison (+) of tuples of values and constraints. This is why this formalization is based on the concept of semiring, which is just a set plus two operations. Denition 1 (semirings and c-semirings). A semiring is a tuple hA; such { A is a set and 0; 1 2 A; { + is commutative, associative and 0 is its unit element; { is associative, distributes over +, 1 is its unit element and 0 is its absorbing element. A c-semiring is a semiring is idempotent with 1 as its absorbing element and is commutative. Let us consider the relation S over A such that a S b i a it is possible to prove that (see [4]): { S is a partial { and are monotone on S ; { 0 is its minimum and 1 its maximum; { lattice and, for all a; b 2 A, a Moreover, if is idempotent, then hA; S i is a complete distributive lattice and is its glb. Informally, the relation S gives us a way to compare (some of the) tuples of values and constraints. In fact, when we have a S b, we will say that b is better than a. Denition 2 (constraints). Given a c-semiring set D (the domain of the variables), and an ordered set of variables V , a constraint is a pair hdef; coni where con V and A. Therefore, a constraint species a set of variables (the ones in con), and assigns to each tuple of values of D of these variables an element of the semiring set A. This element can then be interpreted in several ways: as a level of preference, or as a cost, or as a probability, etc. The correct way to interpret such elements depends on the choice of the semiring operations. Constraints can be compared by looking at the semiring values associated to the same tuples: Consider two constraints c with (t). The relation vS is a partial order. In the following we will also use the obvious extension of this relation to sets of constraints, and also to problems (seen as sets of con- straints). Therefore, given two SCSPs P 1 and P 2 with the same graph topology, we will write P 1 vS P 2 if, for each constraint c 1 in P and the corresponding constraint c 2 in P 2 , we have that c 1 vS c 2 . Denition 3 (soft constraint problem). A soft constraint satisfaction problem (SCSP) is a pair hC; coni where con V and C is a set of constraints. Note that a classical CSP is a SCSP where the chosen c-semiring is: Fuzzy CSPs [8, 19, 20] can instead be modeled in the SCSP framework by choosing the c-semiring: Figure 1 shows a fuzzy CSP. Variables are inside circles, constraints are represented by undirected arcs, and semiring values are written to the right of the corresponding tuples. Here we assume that the domain D of the variables contains only elements a and b. . 0.9 b . 0.5 a . 0.9 b . 0.1 aa . 0.8 ab . 0.2 ba . 0 bb . 0 Fig. 1. A fuzzy CSP. Denition 4 (combination). Given two constraints c combination cc 2 is the constraint hdef; coni dened by . The combina- tion operator can be straightforward extended also to sets of constraints: when applied to a set of constraints C, we will write In words, combining two constraints means building a new constraint involving all the variables of the original ones, and which associates to each tuple of domain values for such variables a semiring element which is obtained by multiplying the elements associated by the original constraints to the appropriate subtuples. Using the properties of and +, it is easy to prove that: { is associative, commutative, and monotone over vS ; { if is idempotent, is idempotent as well. Denition 5 (projection). Given a constraint c = hdef; coni and a subset I of V , the projection of c over I, written c I , is the constraint t=t# con I\con =t 0 def(t). Informally, projecting means eliminating some variables. This is done by associating to each tuple over the remaining variables a semiring element which is the sum of the elements associated by the original constraint to all the extensions of this tuple over the eliminated variables. Denition 6 (solution). The solution of a SCSP problem is the constraint That is, to obtain the solution of an SCSP, we combine all constraints, and then project over the variables in con. In this way we get the constraint over con which is \induced" by the entire SCSP. For example, each solution of the fuzzy CSP of Figure 1 consists of a pair of domain values (that is, a domain value for each of the two variables) and an associated semiring element (here we assume that con contains all variables). Y we mean the projection of tuple t, which is dened over the set of variables X, over the set of variables Y X. Such an element is obtained by looking at the smallest value for all the subtuples (as many as the constraints) forming the pair. For example, for tuple ha; ai (that we have to compute the minimum between 0:9 (which is the value (which is the value for (which is the value for Hence, the resulting value for this tuple is 0:8. Denition 7 (optimal solution). Given an SCSP problem P , consider Sol(P hdef; coni. An optimal solution of P is a pair ht; vi such that there is no t 0 such that v <S def(t 0 ). Therefore optimal solutions are solutions which have the best semiring element among those associated to solutions. The set of optimal solutions of an SCSP P will be written as Opt(P ). Denition 8 (problem ordering and equivalence). Consider two problems then they have the same solution, thus we say that they are equivalent and we The relation vP is a preorder. Moreover, is an equivalence relation. SCSP problems can be solved by extending and adapting the technique usually used for classical CSPs. For example, to nd the best solution we could employ a branch-and-bound search algorithm (instead of the classical backtrack- ing), and also the successfully used propagation techniques, like arc-consistency [17], can be generalized to be used for SCSPs. The detailed formal denition of propagation algorithms (sometimes called also local consistency algorithms) for SCSPs can be found in [4]. For the purpose of this paper, what is important to say is that the generalization from classical CSPs concerns the fact that, instead of deleting values or tuples, obtaining local consistency in SCSPs means changing the semiring values associated to some tuples or domain elements. The change always brings these values towards the worst value of the semiring, that is, the 0. Thus, it is obvious that, given an SCSP problem P and the problem P 0 obtained by applying some local consistency algorithm to P , we must have Another important property of such algorithms is that Sol(P local consistency algorithms do not change the set of solutions. 2.2 Abstraction Abstract interpretation [1, 5, 6] is a theory developed to reason about the relation between two dierent semantics (the concrete and the abstract semantics). The idea of approximating program properties by evaluating a program on a simpler domain of descriptions of \concrete" program states goes back to the early 70's. The inspiration was that of approximating properties from the exact (concrete) semantics into an approximate (abstract) semantics, that explicitly exhibits a structure (e.g., ordering) which is somehow present in the richer concrete structure associated to program execution. The guiding idea is to relate the concrete and the abstract interpretation of the calculus by a pair of functions, the abstraction function and the concretization function , which form a Galois connection. Let (C; v) (concrete domain) be the domain of the concrete semantics, while (abstract domain) be the domain of the abstract semantics. The partial order relations re ect an approximation relation. Since in approximation theory a partial order species the precision degree of any element in a poset, it is obvious to assume that if is a mapping associating an abstract object in concrete element in (C v), then the following holds: if (x) y, then y is also a correct, although less precise, abstract approximation of x. The same argument holds if x v (y). Then y is also a correct approximation of x, although x provides more accurate information than (y). This gives rise to the following formal denition. Denition 9 (Galois insertion). Let (C; v) and (A; ) be two posets (the concrete and the abstract domain). A Galois connection h; is a pair of maps C such that 1. and are monotonic, 2. for each x 2 C; x v 3. for each y 2 A; ( Moreover, a Galois insertion (of A in C) h; connection where Property 2 is called extensivity of . The map ( is called the lower (upper) adjoint or abstraction (concretization) in the context of abstract interpretation The following basic properties are satised by any Galois insertion: 1. is injective and is surjective. 2. is an upper closure operator in (C; v). 3. is additive and is co-additive. 4. Upper and lower adjoints uniquely determine each other. Namely, A fy 2 A j x 5. is an isomorphism from ( )(C) to A, having as its inverse. An example of a Galois insertion can be seen in Figure 2. Here, the concrete lattice is h[0; 1]; i, and the abstract one is hf0; 1g; i. Function maps all real numbers in [0; 0:5] into 0, and all other integers (in (0:5; 1]) into 1. Function maps 0 into 0:5 and 1 into 1. One property that will be useful later relates to a precise relationship between the ordering in the concrete lattice and that in the abstract one. }abstract lattice concrete lattice1 a a Fig. 2. A Galois insertion. Theorem 1 (total ordering). Consider a Galois insertion from (C; v) to is a total order, also is so. Proof. It easy follows from the monotonicity of (that is, x v y implies (x) (y), and from its surjectivity (that is, there is no element in A which is not the image of some element in C via ). 2 Usually, both the concrete and the abstract lattice have some operators that are used to dene the corresponding semantics. Most of the times it is useful, and required, that the abstract operators show a certain relationship with the corresponding concrete ones. This relationship is called local correctness. Denition C be an operator over the concrete lattice, and assume that ~ f is its abstract counterpart. Then ~ f is locally correct w.r.t. f if 8x Abstracting soft CSPs Given the notions of soft constraints and abstraction, recalled in the previous sections, we now want to show how to abstract soft constraint problems. The main idea is very simple: we just want to pass, via the abstraction, from an SCSP P over a certain semiring S to another SCSP ~ P over the semiring ~ S, where the lattices associated to ~ S and S are related by a Galois insertion as shown above. Denition 11 (abstracting SCSPs). Consider the concrete SCSP problem semiring S, where { { we dene the abstract SCSP problem ~ C; coni over the semiring ~ where ~ { ~ c n g with ~ { if is the lattice associated to S and ~ i the lattice associated to ~ S, then there is a Galois insertion h; i such that : { ~ is locally correct with respect to . Notice that the kind of abstraction we consider in this paper does not change the structure of the SCSP problem. That is, C and ~ C have the same number of constraints, and c i and ~ involve the same variables. The only thing that is changed by abstracting an SCSP is the semiring. Thus P and ~ P have the same graph topology (variables and constraints), but dierent constraint denitions, since if a certain tuple of domain values in a constraint of P has semiring value a, then the same tuple in the same constraint of ~ P has semiring value (a). Notice also that and can be dened in several dierent ways, but all of them have to satisfy the properties of the Galois insertion, from which it derives, among others, that Example 1. As an example, consider any SCSP over the semiring for optimiza- tion (where costs are represented by negative reals) and suppose we want to abstract it onto the semiring for fuzzy reasoning In other words, instead of computing the maximum of the sum of all costs, we just want to compute the maximum of the minimum of all costs, and we want to normalize the costs over [0::1]. Notice that the abstract problem is in the FCSP class and it has an idempotent operator (which is the min). This means that in the abstract framework we can perform local consistency over the problem in order to nd inconsistencies. As noted above, the mapping can be dened in dierent ways. For example one can decide to map all the reals below some xed real x onto 0 and then to map the reals in [x; 0] into the reals in [0; 1] by using a normalization function, for example x . Example 2. Another example is the abstraction from the fuzzy semiring to the classical one: Here function maps each element of [0; 1] into either 0 or 1. For example, one could map all the elements in [0; x] onto 0, and all those in (x; 1] onto 1, for some xed x. Figure 2 represents this example with We have dened Galois insertions between two lattices hL; S i and h ~ ~ of semiring values. However, for convenience, in the following we will often use Galois insertions between lattices of problems hPL; vS i and h ~ contains problems over the concrete semiring and ~ PL over the abstract semiring. This does not change the meaning of our abstraction, we are just upgrading the Galois insertion from semiring values to problems. Thus, when we will say that ~ mean that ~ P is obtained by P via the application of to all the semiring values appearing in P . An important property of our notion of abstraction is that the composition of two abstractions is still an abstraction. This allows to build a complex abstraction by dening several simpler abstractions to be composed. Theorem 2 (abstraction composition). Consider an abstraction from the lattice corresponding to a semiring S 1 to that corresponding to a semiring S 2 , denoted by the pair h 1 ; 1 i. Consider now another abstraction from the lattice corresponding to the semiring S 2 to that corresponding to a semiring S 3 , denoted by the pair h 2 ; i. Then the is an abstraction as well. Proof. We rst have to prove that h; the four properties of a Galois insertion: { since the composition of monotone functions is again a monotone function, we have that both and are monotone functions; { given a value x from the rst abstraction we have that x v 1 Moreover, for any element y we have that y v 2 This holds also for thus by monotonicity of 1 we have x v 1 { a similar proof can be used for the third property; { the composition of two identities is still an identity . To prove that 3 is locally correct w.r.t. 1 , it is enough to consider the local correctness of 2 w.r.t. 1 and of 3 w.r.t. 2 , and the monotonicity of 4 Properties and advantages of the abstraction In this section we will dene and prove several properties that hold on abstractions of soft constraint problems. The main goal here is to point out some of the advantages that one can have in passing through the abstracted version of a soft constraint problem instead of solving directly the concrete version. 4.1 Relating a soft constraint problem and its abstract version Let us consider the scheme depicted in Figure 3. Here and in the following pictures, the left box contains the lattice of concrete problems, and the right one the lattice of abstract problems. The partial order in each of these lattices is shown via dashed lines. Connections between the two lattices, via the abstraction and concretization functions, is shown via directed arrows. In the following, we will call the concrete semiring and ~ 1i the abstract one. Thus we will always consider a Galois insertion h; abstract problems concrete problems a a a a Fig. 3. The concrete and the abstract problem. In Figure 3, P is the starting SCSP problem. Then with the mapping we get ~ which is an abstraction of P . By applying the mapping to ~ we get the problem us rst notice that these two problems (P and ((P ))) are related by a precise property, as stated by the following theorem. Theorem 3. Given an SCSP problem P over S, we have that P vS Proof. Immediately follows from the properties of a Galois insertion, in particular from the fact that x S for any x in the concrete lattice. In fact, ((P means that, for each tuple in each constraint of P , the semiring value associated to such a tuple in P is smaller (w.r.t. S ) than the corresponding value associated to the same tuple in Notice that this implies that, if a tuple in ((P semiring value 0, then it must have value 0 also in P . This holds also for the solutions, whose semiring value is obtained by combining the semiring values of several tuples. Corollary 1. Given an SCSP problem P over S, we have that Sol(P ) vS Proof. We recall that Sol(P ) is just a constraint, which is obtained as Thus the statement of this corollary comes from the monotonicity of and +.Therefore, by passing from P to ((P )), no new inconsistencies are intro- duced: if a solution of ((P )) has value 0, then this was true also in P . However, it is possible that some inconsistencies are forgotten (that is, they appear to be consistent after the abstraction process). If the abstraction preserves the semiring ordering (that is, applying the abstraction function and then combining gives elements which are in the same ordering as the elements obtained by combining only), then there is also an interesting relationship between the set of optimal solutions of P and that of (P ). In fact, if a certain tuple is optimal in P , then this same tuple is also optimal in us rst investigate the meaning of the order-preserving property. Denition 12 (order-preserving abstraction). Consider two sets I 1 and I 2 of concrete elements. Then an abstraction is said to be order-preserving if ~ Y ~ Y Y Y x where the products refer to the multiplicative operations of the concrete ( the abstract ( ~ semirings. In words, this notion of order-preservation means that if we rst abstract and then combine, or we combine only, we get the same ordering (but on dierent semirings) among the resulting elements. Example 3. An abstraction which is not order-preserving can be seen in Figure 4. Here, the concrete and the abstract sets, as well as the additive operations of the two semirings, can be seen from the picture. For the multiplicative operations, we assume they coincide with the glb of the two semirings. In this case, the concrete ordering is partial, while the abstract ordering is total. Functions and are depicted in the gure by arrows going from the concrete semiring to the abstract one and vice versa. Assume that the concrete problem has no solution with value 1. Then all solutions with value a or b are optimal. Suppose a solution with value b is obtained by computing we have a = 1a. Then the abstract counterparts will have values (1) 0 1. Therefore the solution with value a, which is optimal in the concrete problem, is not optimal anymore in the abstract problem.a b1g a a a a Fig. 4. An abstraction which is not order-preserving. Example 4. The abstraction in Figure 2 is order-preserving. In fact, consider two abstract values which are ordered, that is 0 0 1. Then where both x and y must be greater than 0:5. Thus their concrete combination (which is the min), say v, is always greater than 0:5. On the other hand, 0 can be obtained by combining either two 0's (therefore the images of two elements smaller than or equal to 0:5, whose minimum is smaller than 0:5 and thus smaller than v), or by combining a 0 and a 1, which are images of a value greater than 0:5 and one smaller than 0:5. Also in this case, their combination (the min) is smaller than 0:5 and thus smaller than v. Thus the order-preserving property holds. Example 5. Another abstraction which is not order-preserving is the one that passes from the semiring hN [ f+1g; min; sum; 0; +1i, where we minimize the sum of values over the naturals, to the semiring hN [ f+1g; min; max; 0; +1i, where we minimize the maximum of values over the naturals. In words, this abstraction maintains the same domain of the semiring, and the same additive operation (min), but it changes the multiplicative operation (which passes from sum to max). Notice that the semiring orderings are the opposite as those usually used over the naturals: if i is smaller than j then j S i, thus the best element is 0 and the worst is +1. The abstraction function is just the identity, and also the concretization function In this case, consider in the abstract semiring two values and the way they are obtained by combining other two values of the abstract semiring: for example, 9). In the abstract ordering, 8 is higher than 9. Then, let us see how the images of the combined values (the same values, since is the identity) relate to each other: we have sum(7; and 15 is lower than 10 in the concrete ordering. Thus the order-preserving property does not hold. Notice that, if we reduce the sets I 1 and I 2 to singletons, say x and y, then the order-preserving property says that (x) ~ S (y) implies that x S y. This means that if two abstract objects are ordered, then their concrete counterparts have to be ordered as well, and in the same way. Of course they could never be ordered in the opposite sense, otherwise would not be monotonic; but they could be incomparable. Therefore, if we choose an abstraction where incomparable objects are mapped by onto ordered objects, then we don't have the order-preserving property. A consequence of this is that if the abstract semiring is totally ordered, and we want an order-preserving abstraction, then the concrete semiring must be totally ordered as well. On the other hand, if two abstract objects are not ordered, then the corresponding concrete objects can be ordered in any sense, or they can also be not comparable. Notice that this restriction of the order-preserving property to singleton sets always holds when the concrete ordering is total. In fact, in this case, if two abstract elements are ordered in a certain way, then it is impossible that the corresponding concrete elements are ordered in the opposite way, because, as we said above, of the monotonicity of the function. Theorem 4. Consider an abstraction which is order-preserving. Given an SCSP problem P over S, we have that Opt(P ) Opt((P )). Proof. Let us take a tuple t which is optimal in the concrete semiring S, with value v. Then v has been obtained by multiplying the values of some subtuples. Suppose, without loss of generality, that the number of such subtuples is two (that is, we have two us then take the value of this tuple in the abstract problem, that is, the abstract combination of the abstractions of v 1 and We have to show that if v is optimal, then also v 0 is optimal. Suppose then that v 0 is not optimal, that is, there exists another tuple t 00 with value v 00 such that v 0 S 0 v 00 . Assume v 2 . Now let us see the value of tuple t 00 in P . If we set v 00 have that this value is us now compare v with v. Since v 0 ~ we get that v S v. But this means that v is not optimal, which was our initial assumption. Therefore v 0 has to be optimal. 2 Therefore, in case of order-preservation, the set of optimal solutions of the abstract problem contains all the optimal solutions of the concrete problem. In other words, it is not allowed that an optimal solution in the concrete domain becomes non-optimal in the abstract domain. However, some non-optimal solutions could become optimal by becoming incomparable with the optimal solutions. Thus, if we want to nd an optimal solution of the concrete problem, we could nd all the optimal solutions of the abstract problem, and then use them on the concrete side to nd an optimal solution for the concrete problem. Assuming that working on the abstract side is easier than on the concrete side, this method could help us nd an optimal solution of the concrete problem by looking at just a subset of tuples in the concrete problem. Another important property, which holds for any abstraction, concerns computing bounds that approximate an optimal solution of a concrete problem. In any optimal solution, say t, of the abstract problem, say with value ~ v, can be used to obtain both an upper and a lower bound of an optimum in P . In fact, we can prove that there is an optimal solution in P with value between (~v) and the value of t in P . Thus, if we think that approximating the optimal value with a value within these two bounds is satisfactory, we can take t as an approximation of an optimal solution of P . Theorem 5. Given an SCSP problem P over S, consider an optimal solution of (P ), say t, with semiring value ~ v in (P ) and v in P . Then there exists an optimal solution t of P , say with value v, such that v (~v). Proof. By local correctness of the multiplicative operation of the abstract semir- ing, we have that v S (~v). Since v is the value of t in P , either t itself is optimal in P , or there is another tuple which has value better than v, say v. We will now show that v cannot be greater than (~v). In fact, assume by absurd that local correctness of the multiplicative operation of the abstract semiring, we have that (v) is smaller than the value of t in (P ). Also, by monotonicity of , by that ~ (v). Therefore by transitivity we obtain that ~ v is smaller than the value of t in (P ), which is not possible because we assumed that ~ v was optimal. Therefore there must be an optimal value between v and (~v). 2 Thus, given a tuple t with optimal value ~ v in the abstract problem, instead of spending time to compute an exact optimum of P , we can do the following: { compute (~v), thus obtaining an upper bound of an optimum of P ; { compute the value of t in P , which is a lower bound of the same optimum of { If we think that such bounds are close enough, we can take t as a reasonable approximation (to be precise, a lower bound) of an optimum of P . Notice that this theorem does not need the order-preserving property in the abstraction, thus any abstraction can exploit its result. 4.2 Working on the abstract problem Consider now what we can do on the abstract problem, (P ). One possibility is to apply an abstract function ~ f , which can be, for example, a local consistency algorithm or also a solution algorithm. In the following, we will consider functions ~ f which are always intensive, that is, which bring the given problem closer to the bottom of the lattice. In fact, our goal is to solve an SCSP, thus going higher in the lattice does not help in this task, since solving means combining constraints and thus getting lower in the lattice. Also, functions ~ f will always be locally correct with respect to any function f sol which solves the concrete problem. We will call such a property solution-correctness. More precisely, given a problem P with constraint set C, f sol (P ) is a new problem P 0 with the same topology as P whose tuples have semiring values possibly lower. Let us call C 0 the set of constraints of P 0 . Then, for any constraint c In other words, f sol combines all constraints of P and then projects the resulting global constraint over each of the original constraints. Denition 13. Given an SCSP problem P over S, consider a function ~ f on f is solution-correct if, given any f sol which solves P , ~ f is locally correct w.r.t. f sol . We will also need the notion of safeness of a function, which just means that it maintains all the solutions. Denition 14. Given an SCSP problem P and a function f : PL ! PL, f is safe if Sol(P It is easy to see that any local consistency algorithm, as dened in [4], can be seen as a safe, intensive, and solution-correct function. From ~ f((P )), applying the concretization function , we get which therefore is again over the concrete semiring (the same as P ). The following property says that, under certain conditions, P and f((P ))) are equivalent. Figure 5 describes such a situation. In this gure, we can see that several partial order lines have been drawn: { on the abstract side, function ~ f takes any element closer to the bottom, because of its intensiveness; { on the concrete side, we have that f((P ))) is smaller than both P and because of the properties of f((P ))) is smaller than because of the monotonicity of f((P ))) is higher than f sol (P ) because of the solution-correctness of ~ f sol (P ) is smaller than P because of the way f sol (P ) is constructed; if is idempotent, then it coincides with the glb, thus we have that f((P ))) is higher than f sol (P ), because by denition the glb is the higher among all the lower bounds of P and a a x O a a a a a a x idempotent concrete problems abstract problems Fig. 5. The general abstraction scheme, with idempotent. Theorem 6. Given an SCSP problem P over S, consider a function ~ f on (P ) which is safe, solution-correct, and intensive. Then, if is idempotent, Sol(P Proof. Take any tuple t with value v in P , which is obtained by combining the values of some subtuples, say two: us now consider the abstract versions of v 1 and f changes these values by lowering them, thus we get ~ 1 and ~ . Since ~ f is safe, we have that v 0 ~ f is solution-correct, thus v S (v 0 ). By monotonicity of , we have that (v 0 2. This implies that (v 0 (v 0 2 )), since is idempotent by assumption and thus it coincides with the glb. Thus we have that v S ( (v 0 (v 0 To prove that P and f((P ))) give the same value to each tuple, we now have to prove that (v 0 (v 0 )). By commutativity of , we can write this as (v 1 (v 0 (v 0 and we have shown that v S (v 0 (v 0 (v 0 (v 0 theorem does not say anything about the power of ~ f , which could make many modications to (P ), or it could also not modify anything. In this last case, Figure 6), so means that we have not gained anything in abstracting P . However, we can always use the relationship between P and (P ) (see Theorem 4 and 5) to nd an approximation of the optimal solutions and of the inconsistencies of P . a f( ~ (P)) a a abstract problems concrete problems a a a Fig. 6. The scheme when ~ f does not modify anything. If instead ~ f modies all semiring elements in (P ), then if the order of the concrete semiring is total, we have that Figure 7), and thus we can work on f((P ))) to nd the solutions of P . In f((P ))) is lower than P and thus closer to the solution. Theorem 7. Given an SCSP problem P over S, consider a function ~ f on (P ) which is safe, solution-correct, and intensive. Then, if is idempotent, ~ f mod- ies every semiring element in (P ), and the order of the concrete semiring is total, we have that P wS Proof. Consider any tuple t in any constraint of P , and let us call v its semiring value in P and v sol its value in f sol (P ). Obviously, we have that v sol S v. Now take v ((v)). By monotonicity of , we cannot have v <S v 0 . Also, by solution-correctness of ~ f , we cannot have v 0 <S v sol . Thus we must have which proves the statement of the theorem. 2 Notice that we need the idempotence of the operator for Theorem 6 and 7. If instead is not idempotent, then we can prove something weaker. Figure 8 shows this situation. With respect to Figure 5, we can see that the possible non-idempotence of changes the partial order relationship on the concrete side. In particular, we don't have the problem f((P ))) any more, nor the problem f sol (P ), since these problems would not have the same solutions as P a a a l r d e r abstract problems concrete problems a a a a a Fig. 7. The scheme when the concrete semiring has a total order. and thus are not interesting to us. We have instead a new problem P 0 , which is constructed in such a way to \insert" the inconsistencies of obviously lower than P in the concrete partial order, since it is the same as P with the exception of some more 0's, but the most important point is that it has the same solutions as P . a a a a a a a x not idempotent concrete problems abstract problems Fig. 8. The scheme when is not idempotent. Theorem 8. Given an SCSP problem P over S, consider a function ~ f on (P ) which is safe, solution-correct and intensive. Then, if is not idempotent, consider P 0 to be the SCSP which is the same as P except for those tuples which have semiring value 0 in these tuples are given value 0 also in P 0 . Then we have that Sol(P Proof. Take any tuple t with value v in P , which is obtained by combining the values of some subtuples, say two: us now consider the abstract versions of v 1 and f changes these values by lowering them, thus we get ~ 1 and ~ . Since ~ f is safe, we have that v 0 f is solution-correct, thus v S (v 0 ). By monotonicity of , we have that (v 0 2. Thus we have that v S (v 0 2. Now suppose that (v 0 This implies that also Therefore, if we set again the combination of v 1 and v 2 will result in v, which is 0. 2 Summarizing, the above theorems can give us several hints on how to use the abstraction scheme to make the solution of P easier: If is idempotent, then we can replace P with f(P ))), and get the same solutions (by Theorem 6). If instead is not idempotent, we can replace P with P 0 (by Theorem 8). In any case, the point in passing from P to is that the new problem should be easier to solve than P , since the semiring values of its tuples are more explicit, that is, closer to the values of these tuples in a completely solved problem. More precisely, consider a branch-and-bound algorithm to nd an optimal solution of P . Then, once a solution is found, its value will be used to cut away some branches, where the semiring value is worse than the value of the solution already found. Now, if the values of the tuples are worse in the new problem than in P , each branch will have a worse value and thus we might cut away more branches. For example, consider the fuzzy semiring (that is, we want to maximize the minimum of the values of the subtuples): if the solution already found has value 0:6, then each partial solution of P with value smaller than or equal to 0:6 can be discarded (together with all its corresponding subtree in the search tree), but all partial solutions with value greater than 0:6 must be considered; if instead we work in the new problem, the same partial solution with value greater than 0:6 may now have a smaller value, possibly also smaller than 0:6, and thus can be disregarded. Therefore, the search tree of the new problem is smaller than that of P . Another point to notice is that, if using a greedy algorithm to nd the initial solution (to use later as a lower bound), this initial phase in the new problem will lead to a better estimate, since the values of the tuples are worse in the new problem and thus close to the optimum. In the extreme case in which the change from P to the new problem brings the semiring values of the tuples to coincide with the value of their combination, it is possible to see that the initial solution is already the optimal one. Notice also that, if is not idempotent, a tuple of P 0 has either the same value as in P , or 0. Thus the initial estimate in P 0 is the same as that of P (since the initial solution must be a solution), but the search tree of P 0 is again smaller than that of P , since there may be partial solutions which in P have value dierent from 0 and in P 0 have value 0, and thus the global inconsistency may be recognized earlier. The same reasoning used for Theorem 4 on (P ) can also be applied to ~ f((P )). In fact, since ~ f is safe, the solutions of ~ have the same values as those of (P ). Thus also the optimal solution sets coincide. Therefore we have that Opt( ~ contains all the optimal solutions of P if the abstraction is order-preserving. This means that, in order to nd an optimal solution of P , we can nd all optimal solutions of ~ use such a set to prune the search for an optimal solution of P . Theorem 9. Given an SCSP problem P over S, consider a function ~ f on P which is safe, solution-correct and intensive, and let us assume the abstraction is order-preserving. Then we have that Opt(P ) Opt( ~ Proof. Easy follows from Theorem 4 and from the safeness of ~ f . 2 Theorem 5 can be adapted to ~ thus allowing us to use an optimal solution of ~ f((P )) to nd both a lower and an upper bound of an optimal solution of P . 5 Some abstraction mappings In this section we will list some semirings and several abstractions between them, in order to provide the reader with a scenario of possible abstractions that he/she can use, starting from one of the semirings considered here. Some of these semirings and/or abstractions have been already described in the previous sections of the paper, however here we will re-dene them to make this section self- contained. Of course many other semirings could be dened, but here we focus on the ones for which either it has been dened, or it is easy to imagine, a system of constraint solving. The semirings we will consider are the following ones: { the classical one, which describes classical CSPs via the use of logical and and logical or: { the fuzzy semiring, where the goal is to maximize the minimum of some values over [0; 1]: { the extension of the fuzzy semiring over the naturals, where the goal is to maximize the minimum of some values of the naturals: { the extension of the fuzzy semiring over the positive reals: { the optimization semiring over the naturals, where we want to maximize the sum of costs (which are negative integers): { the optimization semiring over the negative reals: { the probabilistic semiring, where we want to maximize a certain probability which is obtained by multiplying several individual probabilities. The idea here is that each tuple in each constraint has associated the probability of being allowed in the real problems we are modeling, and dierent tuples in dierent constraints have independent probabilities (so that their combined probability is just the multiplication of their individual probabilities) [10]. The semiring is: { the subset semiring, where the elements are all the subsets of a certain set, the operations are set intersection and set union, the smallest element is the empty set, and the largest element is the whole given set: We will now dene several abstractions between pairs of these semirings. The result is drawn in Figure 9, where the dashed lines denote the abstractions that have not been dened but can be obtained by abstraction composition. In reality, each line in the gure represents a whole family of abstractions, since each h; pair makes a specic choice which identies a member of the family. Moreover, by dening one of this families of abstractions we do not want to say that there do not exist other abstractions between the two semirings. It is easy to see that some abstractions focus on the domain, by passing to a given domain to a smaller one, others change the semiring operations, and others change 1. from fuzzy to classical CSPs: this abstraction changes both the domain and the operations. The abstraction function is dened by choosing a threshold within the interval [0; 1], say x, and mapping all elements in [0; x] to F and all elements in (x; 1] to T. Consequently, the concretization function maps T to 1 and F to x. See Figure 2 as an example of such an abstraction. We recall that all the abstractions in this family are order-preserving, so Theorem 4 can be used. 2. from fuzzy over the positive reals to fuzzy CSPs: this abstraction changes only the domain, by mapping the whole set of positive reals into the [0; 1] interval. This means that the abstraction function has to set a threshold, say x, and map all reals above x into 1, and any other real, say r, into r x . Then, the concretization function will map 1 into +1, and each element of [0; 1), say y, into y x. It is easy to prove that all the members of this family of abstractions are order-preserving. 3. from probabilistic to fuzzy CSPs: this abstraction changes only the multiplicative operation of the semiring, that is, the way constraints are combined. In fact, instead of multiplying a set of semiring elements, in the abstracted version we choose the minimum value among them. Since the domain remains the same, both the abstraction and the concretization functions are the identity (otherwise they would not have the properties required by a Galois insertion, like monotonicity). Thus this family of abstractions contains just one member. It is easy to see that this abstraction is not order-preserving. In fact, consider for example the elements 0:6 and 0:5, obtained in the abstract domain by These same combinations in the concrete domain would be 0:7 resulting in two elements which are in the opposite order with respect to 0:5 and 0:6. 4. from optimization-N to fuzzy-N CSPs: here the domain remains the same (the negative integers) and only the multiplicative operation is modied. Instead of summing the values, we want to take their minimum. As noted in a previous example, these abstractions are not order-preserving. 5. from optimization-R to fuzzy-R CSPs: similar to the previous one but on the negative reals. 6. from optimization-R to optimization-N CSPs: here we have to map the negative reals into the negative integers. The operations remain the same. A possible example of abstraction is the one where It is not order-preserving. 7. from fuzzy-R to fuzzy-N CSPs: again, we have to map the positive reals into the naturals, while maintaining the same operations. The abstraction could be the same as before, but in this case it is order-preserving (because of the use of min instead of sum). 8. from fuzzy-N to classical CSPs: this is similar to the abstraction from fuzzy CSPs to classical ones. The abstraction function has to set a threshold, say x, and map each natural in [0; x] into F , and each natural above x into T . The concretization function maps T into +1 and F into x. All such abstractions are order-preserving. 9. from subset CSPs to any of the other semirings: if we want to abstract to a semiring with domain A, we start from the semiring with domain P(A). The abstraction mapping takes a set of elements of A and has to choose one of them by using a given function, for example min or max. The concretization function will then map an element of A into the union of all the corresponding sets in P(A). For reason similar to those used in Example 3, some abstractions of this family may be not order-preserving. S_prob S_fuzzy S_CSP Fig. 9. Several semiring and abstractions between them. 6 Related work We will compare here our work to other abstraction proposals, more or less related to the concepts of constraints. Abstracting valued CSPs. The only other abstraction scheme for soft constraint problems we are aware of is the one in [7], where valued CSPs [21] are abstracted in order to produce good lower bounds for the optimal solutions. The concept of valued CSPs is similar to our notion of SCSPs. In fact, in valued CSPs, the goal is to minimize the value associated to a complete assignment. In valued CSPs, each constraint has one associated element, not one for each tuple of domain values of its variables. However, our notion of soft CSPs and that in valued CSPs are just dierent formalizations of the same idea, since one can pass from one formalization to the other one without changing the solutions, provided that the partial order is total [2]. However, our abstraction scheme is dierent from the one in [7]. In fact, we are not only interested in nding good lower bounds for the optimum, but also in nding the exact optimal solutions in a shorter time. Moreover, we don't dene ad hoc abstraction functions but we follow the classical abstraction scheme devised in [5], with Galois insertions to relate the concrete and the abstract domain, and locally correct functions on the abstract side. We think that this is important in that it allows to inherit many properties which have already been proven for the classical case. It is also worth noticing that our notion of an order-preserving abstraction is related to their concept of aggregation compatibility, although generalized to deal with partial orders. Abstracting classical CSPs. Other work related to abstracting constraint problems proposed the abstraction of the domains [11, 12, 22], or of the graph topology (for example to model a subgraph as a single variable or constraint) [9]. We did not focus on these kinds of abstractions for SCSPs in this paper, but we believe that they could be embedded into our abstraction framework: we just need to dene the abstraction function in such a way that not only we can change the semiring but also any other feature of the concrete problem. The only dierence will be that we cannot dene the concrete and abstract lattices of problems by simply extending the lattices of the two semirings. A general theory of abstraction. A general theory of abstraction has been proposed in [16]. The purpose of this work is to dene a notion of abstraction that can be applied to many domains: from planning to problem solving, from theorem proving to decision procedures. Then, several properties of this notion are considered and studied. The abstraction notion proposed consists of just two formal systems 1 and 2 with languages L 1 and L 2 and an eective total function Much emphasis is posed in [16] onto the study of the properties that are preserved by passing from the concrete to the abstract system. In particular, one property that appears to be very desirable, and present in most abstraction frameworks, is that what is a theorem in the concrete domain, remains a theorem in the abstract domain (called the TI property, for Theorem Increasing). It is easy to see that our denition of abstraction is an instance of this general notion. Then, to see whether our concept of abstraction has this property, we rst must say what is a theorem in our context. A natural and simple notion of a theorem could be an SCSP which has at least one solution with a semiring value dierent from the 0 of the semiring. However, we can be more general that this, and say that a theorem for us is an SCSP which has a solution with value greater than or equal to k, where k 0. Then we can prove our version of the TI property: Theorem 10 (our TI property). Given an SCSP P which has a solution with value v k, then the SCSP (P ) has a solution with value v 0 (k). Proof. Take any tuple t in P with value v > k. Assume that abstracting, we have solution correctness of 0 , we have that v S (v 0 ). By monotonicity of , we have that (v) S 0 ( (v again by monotonicity of , we have thus by transitivity Notice that, if we consider the boolean semiring (where a solution has either value true or false), this statement reduces to saying that if we have a solution in the concrete problem, then we also have a solution in the abstract problem, which is exactly what the TI property says in [16]. Thus our notion of abstraction, as dened in the previous sections, on one side can be cast within the general theory proposed in [16], while on the other side it generalizes it to concrete and abstract domains which are more complex than just the boolean semiring. This is predictable, because, while in [16] formulas can be either true (thus theorems) or false, here they may have any level of satisfaction which can be described by the given semiring. Notice also that, in our denition of abstraction of an SCSP, we have chosen to have a Galois insertion between the two lattices (which corresponds to the concrete semiring S) and h ~ (which corresponds to the abstract semiring ~ S). This means that the ordering in the two lattices coincide with those of the semirings. We could have chosen dierently: for example, that the ordeing of the lattices in the abstraction be the opposite of those of those in the semirings. In that case, we would not have had property TI. However, we would have the dual property (called TD in [16]), which states that abstract theorems remain theorems in the concrete domain. It has been shown that such a property can be useful in some application domains, such as databases. 7 Conclusions and future work We have proposed an abstraction scheme for abstracting soft constraint prob- lems, with the goal of nding an optimal solution, or a good approximation of it, in shorter time. The main idea is to work on the abstract version of the problem and then bring back some useful information to the concrete problem, to make it easier to solve. This paper is just a rst step towards the use of abstraction for helping to nd the solution of a soft constraint problem in a shorter time. More properties can probably be investigated and proved, and also an experimental phase is necessary to check the real practical value of our proposal. We plan to perform such a phase within the clp(fd,S) system developed at INRIA [14], which can already solve soft constraints in the classical way (branch-and-bound plus propagation via partial arc-consistency). Another line for future research concerns the generalization of our approach to include also domain and topological abstractions, as already considered for classical CSPs. Acknowledgments This work has been partially supported by Italian MURST project TOSCA. --R Constraint Solving over Semirings. Abstract interpretation: A uni Systematic design of program analyis. The calculus of fuzzy restrictions as a basis for exible constraint satisfaction. Synthesis of abstraction hierarchies for constraint satisfaction by clustering approximately equivalent objects. Uncertainty in constraint satisfaction problems: a probabilistic approach. Eliminating interchangeable values in constraint satisfaction sub- problems Interchangeability supports abstraction and reformulation for constraint satisfaction. Partial constraint satisfaction. Compiling semiring-based constraints with clp(fd A theory of abstraction. Consistency in networks of relations. Constraint satisfaction. Fuzzy constraint satisfaction. Possibilistic constraint satisfaction problems Valued Constraint Satisfaction Problems: Hard and Easy Problems. An evaluation of domain reduction: Abstraction for unstructured csps. Practical applications of constraint programming. --TR Partial constraint satisfaction Possibilistic constraint satisfaction problems or MYAMPERSANDldquo;how to handle soft constraints?MYAMPERSANDrdquo; A theory of abstraction Semiring-based constraint satisfaction and optimization Theories of abstraction Abstract interpretation Systematic design of program analysis frameworks A CSP Abstraction Framework An Abstraction Framework for Soft Constraints and Its Relationship with Constraint Propagation Uncertainty in Constraint Satisfaction Problems Abstracting Soft Constraints Compiling Semiring-Based Constraints with clp (FD, S) AbsCon Semiring-Based CSPs and Valued CSPs --CTR Thomas Ellman , Fausto Giunchiglia, Introduction to the special volume on reformulation, Artificial Intelligence, v.162 n.1-2, p.3-5, February 2005 Didier Dubois , Henri Prade, Editorial: fuzzy set and possibility theory-based methods in artificial intelligence, Artificial Intelligence, v.148 n.1-2, p.1-9, August Salem Benferhat , Didier Dubois , Souhila Kaci , Henri Prade, Bipolar possibility theory in preference modeling: Representation, fusion and optimal solutions, Information Fusion, v.7 n.1, p.135-150, March, 2006 Stefano Bistarelli , Francesco Bonchi, Soft constraint based pattern mining, Data & Knowledge Engineering, v.62 n.1, p.118-137, July, 2007 Giampaolo Bella , Stefano Bistarelli, Soft constraint programming to analysing security protocols, Theory and Practice of Logic Programming, v.4 n.5-6, p.545-572, September 2004

fuzzy reasoning;abstraction;constraint propagation;constraint solving;soft constraints

635238

Families of non-IRUP instances of the one-dimensional cutting stock problem.

In case of the one-dimensional cutting stock problem (CSP) one can observe for any instance a very small gap between the integer optimal value and the continuous relaxation bound. These observations have initiated a series of investigations. An instance possesses the integer roundup property (IRUP) if its gap is smaller than 1. In the last 15 years, some few instances of the CSP were published possessing a gap greater than 1.In this paper, various families of non-IRUP instances are presented and methods to construct such instances are given, showing in this way, there exist much more non-equivalent non-IRUP instances as computational experiments with randomly generated instances suggest. Especially, an instance with gap equal to 10/9; is obtained. Furthermore, an equivalence relation for instances of the CSP is considered to become independent from the real size parameters.

Introduction The one-dimensional cutting stock problem (CSP) is as follows: Given an unlimited number of pieces of identical stock material of length L (e.g. wooden length, paper reels, etc.) the task is to cut b i pieces of length ' i for while minimizing the number of stock material pieces needed. Throughout this paper we will use the abbreviation E := (m; L; '; b) for an instance of the CSP with loss of generality we will assume L The classical solution approach is due to Gilmore and Gomory [5]. A non-negative integer vector a is called a (feasible) cutting pattern of cutting pattern (shortly: pattern) a is said to be maximal if ' T a+'m ? L; a is proper if a b, i.e. a i b i , ng denote the index set of all maximal patterns a If the integer variable x j denotes the times pattern a j is used, the CSP can be modelled as follows: Now the common solution approach consists of solving the corresponding continuous (LP) relaxation where IR is the set of real numbers. Then, based on an optimal solution of (2), integer solutions for (1) are constructed by means of suitable heuristics (cf. [5], [13]). Let z denote the optimal value of (1) and (2) for an instance E, respectively. Practical experience and many computational tests have shown (cf. e.g. [13], [16]) there is only a small gap \Delta(E) := z (E) \Gamma z c (E) for any instance E. These observations have initiated a number of investigations. A set P of instances E has the integer round-up property (IRUP) if \Delta(E) ! 1 for all \Delta(E) 1 is called non-IRUP instance in the following. It is well-known that the CSP does not possess the IRUP. In [6] a first counter-example was given with gap equal to 1. In the last decade instances were found having a gap larger then 1 ([3], [12]). Since the gaps of these instances are less than 2, the modified integer round-up property (MIRUP) was defined in [14]: a set P of instances E possesses the MIRUP if \Delta(E) ! 2 for all It is conjectured in [12] that the one-dimensional CSP possesses the MIRUP. Since in the numerical tests non-IRUP instances occur relatively rarely, the impression could arise that there exist only a limited number of non-IRUP instances. In this paper we will present families with an infinite number of non-equivalent non-IRUP instances. These investigations could be helpful for developing and testing of exact solution approaches for the one-dimensional CSP. Especially if existing exact solution algorithms (branch-and-bound algorithms are presented e.g. in [13] for the CSP and in [7] for the bin packing problem) are applied to non-IRUP instances then computational difficulties can arise. Moreover, they could have importance also for higher dimensional CSPs. For a comprehensive overview of recent work in connection with cutting and packing problems we refer to [2]. Throughout this paper we will use some common notations. Let M denote the set of instances possessing IRUP. Especially for theoretical investigations, a special case of the CSP is of interest which is named divisible case. An instance E belongs to the set D of 'divisible case'-instances if L is an integer multiple of any piece length. For example, the instance (presented in [11]) belongs to D and has gap \Delta(E D which is the largest for D ever found. In our investigations we allow also rational sizes, i.e. we assume ' 2 IQ m and IQ is the set of rational numbers. In the most cases we will consider equivalent instances with integer sizes. In the next section we will introduce an equivalence relation for instances of the CSP in order to characterize different instances in a better way and to become independent from the real sizes. Then in section 3, we will consider families of non-IRUP instances in the divisible case. Families of non-IRUP instances not restricted to the divisible case will be presented in section 4. Constructive methods to obtain non-IRUP instances will be discussed in the fifth section followed by some concluding remarks. 2 Equivalence of Cutting Stock Problems Among some possibilities of defining equivalence relations for instances of CSPs we will consider here only the kind of equivalence which is based on the cutting patterns. Through-out this section we assume all input-data to be integral. 2.1 Pattern equivalence Let the instances cutting patterns, respectively. Without loss of generality the cutting patterns are assumed to be sorted lexicographically decreasing. are called equivalent (pattern-equivalent) if Hence, any feasible pattern a j of E is also feasible for E and vice versa. For example, the following instances E are equivalent to the instance 26 ' 3 30, for all d; d Let A(E) denote the m \Theta n-matrix containing all maximal patterns of E. Hence, E and are equivalent if and only if Then the class K(E) of instances equivalent to E can be characterized as follows: Obviously, K(E). For that reason, K(E) can be viewed as the intersection of ZZ m+1 and a cone which is induced by A(E). 2.2 Characterization by system of inequalities Let e i denote the i-th unit vector of ZZ m (i 2 I). Assertion 1 Let a denote the maximal cutting patterns of instance b). Then the instance is equivalent to E if and only if ('; L) 2 ZZ m+1 is feasible for Proof: ")" E is equivalent to E, that means A(E) is pattern matrix of E. Hence, (4), (5) and (6) are fulfilled. "(" Let ('; L) 2 ZZ m+1 fulfill (4), (5) and (6). Because of (6) it follows 1. Because of (4) any maximal cutting pattern of E is also feasible with respect to E. Because of (5) any maximal cutting pattern of E is also maximal with respect to E. Assume there exists a maximal cutting pattern a of E which is not feasible or is not maximal with respect to E. If a is not maximal but feasible with respect to E then a with respect to E and there exists a 2 ZZ such that a with respect to E . occurs in (4) a contradiction arises to a is maximal with respect to E. If a is not feasible with respect to E , i.e. ' a there exists a pattern ea with which is maximal with respect to E . Hence, ' T ea L (because of (because of (5)). This leads to a contradiction to a is feasible with respect to E. 2.3 Dominance Let a pattern matrix In order to get an instance of the cutting stock problem with pattern matrix A, one has to determine a solution of (4) - (6). Because of the condition numerous constraints in (4) and (5), respectively, will automatically be fulfilled if other constraints are fulfilled as the following example shows. Let From ' T a 1 L and 9. That means, at most the constraint ' T a 1 L is active in (4). Analogously, if ' T (a 9 8. Hence, again only one of the inequalities in (5) is needed. Definition 2 The cutting pattern a dominates the cutting pattern a if ' T a ' T a for all fulfilling For a Assertion 2 Let the patterns a and a be given. Then: a ? d a Proof: ")" If s 1 (a) ! s 1 (a) then a 1 ! a 1 and hence, a 6? d a. f1g such that s j (a) s j (a) for We define the vector ' 2 ZZ m such that ' T a Then -z =: ff -z -z -z It follows sufficiently large since ff, fi and fl are independent of ' 0 . "(" Now we assume s i (a) s i (a) for all i and -z -z -z 0: A pattern a is said to be dominant if there does not exist any (feasible) pattern a, a 6= a, with a ! d a. If ' and L are fixed then a pattern a can easily be identified to be dominant. Let Assertion 3 The pattern a is dominant if and only if ffi(a). Then there exists i 2 I n fmg with a the pattern a dominates a, a cannot be dominant. If a is not dominant then there exists a pattern a, a 6= a, with a ! d a. The following procedure can be applied at least once: Let j be the index determined by a -z Otherwise the pattern ea is defined by a i otherwise. Then Furthermore, a ! d ea since s i because of (7), we have ' T Otherwise the procedure has to be repeated with ea instead of a. Let J d ae ng denote the index-set of dominant cutting patterns in A. Then: Assertion 4 ('; L) 2 ZZ m+1 fulfills (4) if and only if ('; L) is feasible for Proof: Since for any cutting pattern a j there exists a column a k with k 2 J d and a j ! d a k , Furthermore, a maximal pattern a is said to be non-dominant if there does not exist any maximal pattern a, a 6= a, with a ! d a. Non-dominant patterns can be identified similar to Assertion 3. Let J nd ae ng be the index-set of non-dominant cutting patterns in A. Then: Assertion 5 ('; L) 2 ZZ m+1 fulfills (5) if and only if ('; L) is feasible for Proof: Since for any cutting pattern a j there exists a column a k with k 2 J nd and a j ? d a k , In order to illustrate the usage of dominance let us consider the instance presented in [4] with \Delta(E G 1:0667. The instance EG has 41 feasible patterns. Among them are 20 maximal, 7 dominant and 5 non-dominant patterns. Hence, the class K(EG ) of equivalent instances can be described as solution set of the system of inequalities: 5: is such a solution, EG is equivalent to 3 Divisible Case In 1994 Nica ([9]) proposed a first infinite family of non-equivalent non-IRUP instances of the CSP which belongs to D. For the sake of completeness we repeat here his result. be given such that 2 . The length L of the stock material is defined to be \Pi m . Accordingly to the divisible case, the length ' i of the ith piece equals L=k i (i 2 I). Using proposed in [9], are as follows: km Note, the instance ED defined in (3) belongs to this family with Theorem 1 (Nica 1994) Let k 2 ZZ m be given such that 2 are pairwise relatively prime. If 1 b m ! km and if the order demands cannot be cut using less than cutting patterns, then Proof: Because of construction we have z c km km km and z c (E(k))+1=km ? m \Gamma 1. Since the parameters k i are assumed to be pairwise relatively prime there does not exist any non-negative integer vector a a i This means, there does not exist any proper and trim-less pattern. If represents a maximal pattern then a i )c a i a i a i since km a i If we consider any set of cutting patterns a fulfilling a mj (m Because of e we obtain a since m 3 and k 1 2 are assumed. Hence, at least one more piece of length ' m is ordered as can be cut with cutting patterns. Note, the gap km d is asymptotically bounded by 1 1=km and tends to 1 if km is increased. Next we show by means of an example that non-IRUP instances can also be obtained in the divisible case if some of the assumptions in Theorem 1 are not fulfilled. For this let us consider the instance with and the k i are not pairwise relatively prime. Using the proper relaxation bound (cf. [10]) can be verified. Moreover, Hence, separating this pattern a new instance can be derived from E 1 with Now we will consider another family of instances of D with be any positive integer and let . L is defined to be the smallest common multiple of the k i (or a multiple of it). For abbreviation, let the family of instances 9p+2 will be investigated. Notice, ED defined in (3) belongs to this family with Theorem 2 For any integer p 1, the instance E(p) does not belong to M . can be assumed. Because of E(p) 2 D we have z Because of z c 2, the instance E(p) would belong to M if and only if there exist any proper pattern a a 1 We will show that such a pattern cannot occur. Since in the divisible case feasible pattern a it follows Since the following inequality must be fulfilled: Because of (9) we have Furthermore, with it follows that the pattern a has to fulfil the inequality Summarizing (11) and (12), if E(p) 2 M then there must exist a proper pattern a with Note, the left-hand side is integral. In case a the interval has the form which contains the unique integer \Gamma2q \Gamma 1. The corresponding equation can be transformed in and we find for a 1 q \Gamma 1, a 1 integer, there exists no non-negative integer value a 2 . In the other case, a 3 5, because of the interval in formula (13) can be written in form For the interval size is less than 5 1. Hence, there is no integer contained in the interval. Therefore, no such pattern can exist. 4 Non-Divisible Case The family E(t) with is considered next. Note, the family does not contain only non-equivalent instances. For example, instances E(t) are equivalent and Theorem 3 The instances E(t), t 2 T , do not belong to M . Proof: We show z c Because of ' T 3. On the other hand, z c (E(t)) 3 since2B Suppose z the optimal solution must contain only proper patterns a with ' T a = L. We consider the pattern with a because of is the only trim-less proper pattern. But now there does not exist any proper trim-less pattern with a 2 1 since ' 5 is already used. Using the substitution p=q obtain from E(t) a further family of instances by removing the smallest piece: Theorem 4 Let p=q 2 (0; 1), then the instance E(p; q) does not belong to M . The proof is similar to that of Theorem 3. In case of p ? 1 the length of the smallest piece can be reduced to q violating the non-IRUP property. Let we have to show z 4. Because of an integer solution with 3 patterns has to have a total trim-loss of units. If such a solution exists then it must contain the pattern with the trim-loss of p units. Since any proper pattern containing at least one piece of length ' 2 but non of ' 5 has a trim-loss of at least p units, the total trim-loss is at least 2p which is a contradiction. Next we will consider the family of non-equivalent CSP instances. Theorem 5 The instances E(p) defined in (16) do not belong to M . Proof: Because 1), we have z c (E(p)) Because of ' T the total trim-loss has to be equal to assumed. But the best proper pattern a with a which has a trim-loss of must hold. Notice, lim p!1 z c which corresponds to lim p!1 k Moreover, since the gaps are asymptotically bounded by 1 the gaps tend to 1 for p !1. The maximal gap for this family is 29/28 obtained for The following procedure leads to further non-equivalent non-IRUP instances if p 5: If pieces of length ' are composed to larger ones then no better integer solution can arise. If additionally the value of the continuous relaxation does not increase then non-IRUP instances will be constructed. Let 2), such that p loss of generality, let . Then the non-equivalent instance E 0 (p) derived from E(p) is as follows in case of Using times the pattern ' 0+ p+1 instead of -times the pattern (1; 0; 0; 0; in (17), a feasible solution of the LP relaxation with the same objective function value can be obtained. For example, let 5. Then from using Furthermore, if p 5, odd, ld(p+1) 62 ZZ and because of construction, the length ' 3 now can be reduced to 2p 2 since no new proper pattern gets feasible. On the other hand, new non-proper patterns can lead to a smaller optimal value of the continuous relaxation. In the example we obtain with \DeltaE 00 Next we will consider a modification of E 00 (5). Let Here we have z c 7. Thus E(b) 62 M . (The verification of z done using the cutting plane algorithm proposed in [8].) proper pattern a of 5. If a 2 the same gap arises; for a 2 smaller gaps occur. The corresponding residual instance of E(b) is Residual (cf. [12]) means there exists an optimal solution x c of the LP relaxation with components less than 1 so that x c cannot be used for a further reduction of the order demands. Non-IRUP instances can also be found for special classes of CSPs. Let us consider instances with the property k and For short we refer such instances as (1; k)-instances. It is known, all k)-instances belong to M for k 2 f2; 3g. But for any k 4 there exist non-IRUP k)-instances. Such instances for are and A family of non-IRUP (1; k)-instances with k 6 is the following: Theorem 6 The instances E k , k 4, do not belong to M . Proof for k 6: It is easy to verify that the instance E k is a (1; k)-instance. Since at least 3 smaller pieces can be contained in a pattern a with a Because of follows. The reduced instance E 0 k obtained form E k by omitting the 3 pieces of length ' 1 and by setting the stock length to correspond to the instance E(t) with considered in Theorem 3. Therefore E 0 and hence, since z c Remember, the instance E 0 namely we have any i. An instance is said to be a 1)-instance if for any i k The proof of Theorem 6 suggests the following procedure to construct non-IRUP (1; k)- instances. Let a non-IRUP In order to obtain a non-IRUP (1; k)-instances at first an instance constructed. The piece lengths are increased by t (t sufficiently large), i.e. ' 0 1)t. Then, any feasible (with respect to E) pattern a with e T a non-feasible (with respect to E) pattern a with e T a tends to infinity then i tends to 1=(p + 1) for i 2 I. If we add dz c (E)e pieces of length ' 0(' 0sufficiently large) and increase the stock length by ' 0 can be reached for k 2p 2. Hence, a (1; k)-instance with m+1 piece types is obtained. Let us now consider the set of (k; k+1)-instances with k 2 ZZ (1,2)-instances belong to M , but for any k 2 there exist non-IRUP (k; k+1)-instances as will be shown next. For k 2 let 1)-instance for k 2. Note, in order to make more clear the inherent structur we do not sort the piece lengths in decreasing order. Theorem 7 The instance does not belong to M . z c If z supposed then no trim-loss must occur. We consider patterns a with must contain k +1 pieces. Because of construction, pattern is not proper. Furthermore, but again this pattern is not proper since b 7 = 1. there is no trim-less pattern a with a z 5 Further Constructions be an instance not belonging to M with 1. Moreover, let q 2 IQ with q 2=' m 0 . Then the instance E is defined by Theorem 8 If for any a 0 2 ZZ m then E does not belong to M too. Proof: At first we prove dz c (E)e dz c We consider an optimal solution of the LP relaxation of E 0 characterized by the set of feasible patterns and corresponding coefficients x j , Now we assign to any a 0 j (j 2 J ) an m-dimensional pattern a j as follows: a The patterns a j are feasible with respect to E. Then, Using (dz c ))-times the feasible pattern a 1 := e 1 a 1j x Supplementing (p \Gamma 1)=p-times the pattern a 2 := pe 2 and the pattern a 3 := pe 3 , a feasible solution of the continuous relaxation of E is found (which is not-necessarily optimal). Hence, z c (E) z c Next we show z (E) ? dz c 1. Because of patterns a j with a are needed to cut the b 1 pieces of length ' 1 . If we suppose an integer solution for patterns then all pieces of length ' 2 must be in one pattern a with a the total length in a not covered with ' 2 -pieces equals qL 0 +1. all the pieces with lengths ' 4 cannot be cut with these dz c (E 0 )e patterns a j . At least one piece (with length q' 0 remains unpacked. Since this piece and the piece of length ' 3 cannot be cut in pattern a. Hence, one more pattern is needed, that means z (E) ? dz c Let us consider the instance which is equivalent to that proposed by Gau ([4], cf. (8)). Here we have If we apply the procedure corresponding to Theorem 8 with 2, the instance results. For this instance we get z c obtained. This is the largest gap found so far ([15]). It is remarkable, although ' there is no way to cut the order demands b i with 3 patterns. If this procedure is applied to E with then the gaps 1.0741, 1.0556, 1.0444 and 1.0394 result. Note, the repeated application of Theorem 8 is possible but does not succeed with larger gaps. There are some other possibilities to obtain non-IRUP instances. Let an instance given. If the order demand of the m-th piece is increased by some units, say three cases can occur for If z c If z c (E ) dz c (E)e and z Otherwise, E and E have the same behavior. The instances considered in this paper are mostly residual instances. But, if the order demand b of a non-IRUP instance E is increased by an non-negative integer combination of patterns occurring in an LP solution of E then a non-IRUP instance is obtained very often. For example, if we use ED defined in (3) then we get To verify such statements stronger relaxations of the CSP are needed as proposed in [10] and [8]. 6 Concluding Remarks In this paper families of non-IRUP instances of the CSP are presented. Since the gaps tend to 1 for increasing parameters counter-examples for the MIRUP-conjecture, if such exist, have probably small ratios between stock material length and piece lengths. There remain some open questions. Especially the efficient identification of instances to be a non-IRUP instance is of importance for exact solution approaches. --R Cutting and packing. The duality gap in trim problems. A linear programming approach to the cutting-stock problem (Part I) An instance of the cutting stock problem for which the rounding property does not hold. Knapsack Problems - Algorithms and Computer Implemen- tations Solving cutting stock problems with a cutting plane method. General counterexample to the integer round-up property Tighter relaxations for the cutting stock problem About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem Theoretical investigations on the Modified integer Round-Up Property for the one-dimensional cutting stock problem A branch-and-bound algorithm for solving one-dimensional cutting stock problems exactly The modified integer round-up property of the one-dimensional cutting stock problem A new 1csp heuristic. Heuristics for the one-dimensional cutting stock problem: A computational study --TR Knapsack problems: algorithms and computer implementations Computers and Intractability

integer round-up property;continuous relaxation;one-dimensional cutting stock problem

635239

A framework for the greedy algorithm.

Perhaps the best known algorithm in combinatorial optimization is the greedy algorithm. A natural question is for which optimization problems does the greedy algorithm produce an optimal solution? In a sense this question is answered by a classical theorem in matroid theory due to Rado and Edmonds. In the matroid case, the greedy algorithm solves the optimization problem for every linear objective function. There are, however, optimization problems for which the greedy algorithm correctly solves the optimization problem for many--but not all--linear weight functions. Our intention is to put the greedy algorithm into a simple framework that includes such situations. For any pair (S,P) consisting of a finite set S together with a set P of partial orderings of S, we define the concepts of greedy set and admissible function. On a greedy set L S, the greedy algorithm correctly solves the naturally associated optimization problem for all admissible functions f consists of linear orders, the greedy sets are characterized by this property. A geometric condition sufficient for a set to be greedy is given in terms of a polytope and roots that generalize Lie algebra root systems.

Introduction This paper concerns a classical algorithm in combinatorial optimization, the greedy algorithm. The MINIMAL SPANNING TREE problem, for example, is solved by the greedy algorithm: Given a nite graph G with weights on the edges, nd a spanning tree of G with minimum total weight. At each step in the greedy algorithm that solves this problem, there is chosen a set of edges T comprising the partial tree; an edge e of minimum weight among the edges not in T (the greedy choice) is added to T so long as contains no cycle. A greedy algorithm makes a locally optimal choice in the hope that this will lead to a globally optimal solution. Clearly, greedy algorithms do not always yield the optimal solution. But for a wide range of important problems the greedy algorithm is quite powerful; a variety of such applications can be found in standard texts such as those by Lawler [8] or Papadimitriou and Steiglitz [9]. A natural question, precisely posed below, is the following. For which optimization problems does the greedy algorithm give the correct solution. In a sense this question is answered by a classical theorem in matroid theory due to Rado and Edmonds [4]. In the matroid case, the greedy algorithm always solves the optimization problem. That is, the greedy algorithm solves the optimization problem for every linear objective function. There are situations, however, for which the greedy algorithm works for many - but not all - linear objective functions. A simple framework for such problems is suggested below. To make the question precise, consider a set system (S; I) consisting of a nite set S together with a nonempty collection I of subsets of S, called independent sets, closed under inclusion. Given a weight function f R, extend this function linearly to the collection of subsets A S by dening a2A There is a natural combinatorial optimization problem associated with the set system I). OPTIMIZATION PROBLEM. Given a weight function f nd an independent set with the greatest weight. In the spanning tree problem, S is the set of edges of the graph G and the independent sets are the acyclic subsets of edges. Minimum and maximum in this problem are interchanged by negating the weights. The greedy algorithm for this optimization problem is simply: GREEDY ALGORITHM while S 6= ; do remove from S an element a of largest weight. I [ fag is independent then By the theorem of Edmonds and Rado referred to earlier, the following statements are equivalent for the set system (S; I). Here B denotes the set of bases, a basis being a maximum independent set. 1. (S; I) is a matroid. 2. The greedy algorithm correctly solves the combinatorial optimization problem associated with (S; I) for any weight function f 3. Every basis has the same cardinality and, for every linear ordering on S, there exists a such that for any A 2 B, if we with the elements of B and A both in increasing order, then a This ordering on k-element subsets of X is called the Gale order. In the spanning tree problem, the acyclic subsets of edges comprise the independent sets of a matroid. Many well known optimization problems, besides the spanning tree problem, can be put into the framework of matroids. Texts by Lawler [8] and by Papadimitriou and Steiglitz [9] contain numerous examples. Matroids are characterized by the property that the greedy algorithm correctly solves the optimization problem for any weight function. There are, however, non-matroids for which the greedy algorithm correctly solves the appropriate optimization problem for many - but not for every weight function. This is the case for the following, in order of increased generality: symmetric matroids [2], sympletic matroids [1] and the Coxeter matroids of Gelfand and Serganova [5, 6, 11]. It is our intention in this paper to put the greedy algorithm into a simple framework that includes such examples. In particular, Theorems 4.1 and 4.2 in this paper contain, as a special case, the classical matroid theorem of Rado and Edmonds stated earlier. In [1] (Theorem 16), Borovik, Gelfand and White prove that for symplectic matroids the greedy algorithm correctly solves the optimization problem for all \admissible" functions. This is also a special case of Theorem 4.1; symplectic matroids are used as an example in Sections 2, 3 and 4 of this paper. Given a Coxeter system (W; P ) consisting of a nite irreducible Coxeter group W and maximal parabolic subgroup P , in [11] (Theorem 1) a concrete realization of the set W=P of left cosets is given as a collection of subsets B of an appropriate partially ordered set S. Theorem 3 of [11], in part, states that the natural optimization problem for Coxeter matroids is solved, for all appropriate \admissible" objective functions, by the greedy algorithm applied to (S; B). This result is again a special case of Theorem 4.1. The framework in this paper, however, is conceptually simpler than the usual approaches to Coxeter matroids. The theory of greedoids, developed by Korte and Lovasz [7], concerns a framework for optimization problems for which the greedy algorithm nds the optimal for all generalized bottleneck functions. Since our results concern linear objective functions, they do not subsume results on greedoids. Our results on the greedy algorithm apply to situations not previously appearing in the literature, for example the cyclic and bipartite cases mentioned in Sections 3 and 4. Concerning open avenues of research, it would be interesting to formulate additional optimization problems, analogous to the minimum spanning tree problem for matroids, to which the theory is applicable. The basic notion in this paper is that of a pair (S; P) consisting of a nite set S together with a set P of partial orderings of S. The notions of greedy set and admissible function are dened in Section 2 and examples are given in Sections 2 and 3. It is shown in Section 4 that, for greedy sets, the greedy algorithm correctly solves the naturally associated optimization problem for all admissible functions. Indeed, when P contains only linear orders, the greedy sets are characterized by this property. It is also proved in Section 4 that there is essentially no loss of generality in assuming that P contains only linear orders. Our results naturally lead to the problem of eectively characterizing greedy sets. A geometric approach via polytopes and a generalization of Lie algebra root systems is taken in Section 5. It is proved that if every edge of the polytope of a collection L is parallel to a root, then L is a greedy set. Greedy sets and Admissible Functions A partial ordering of a set S is a binary relation on S that is re exive, transitive and antisymmetric. If, for any a and b in S, either a b or b a then the partial ordering is called a linear ordering of S. The notation a b will mean that a b but a 6= b. Consider a pair (S; P) consisting of a nite set S and a collection P of partial orderings of S. A subset L S will be called a greedy set for the pair (S; P) if L has a maximum for every ordering in P. That is, for every ordering in P, there is an a 2 L such that b a for all b 2 L. denote the collection of all k-element subsets of S. Each partial order on S induces a partial order on S k , namely the Gale order. If A; B 2 S k , then AB in the Gale order if there are arrangements of the elements of the two sets such that a i b i for all i. The set of Gale orderings induced on S k by the orderings in P will be denoted P k . A greedy set for the pair will be referred to as a rank k greedy set for (S; P). A weight function f called compatible with a partial order on S if f(a) < f(b) whenever a b. A weight function f is said to be admissible for (S; P) if f is compatible with some partial order in P. An admissible weight function f for (S; P) can be extended to an admissible weight function f dening a2A That this function is indeed admissible is the statement of the corollary below. The rst proposition is obvious from the denitions of greedy set and admissible weight function. Proposition 2.1 If L S is a greedy set for (S; P) and f is an admissible weight function, then f attains a unique maximum on L. Proposition 2.2 Let S be a partially ordered set and S k the set of all k-element subsets of S with the Gale order. For any A; B 2 S k we have BA if and only if f(B) f(A) for every weight function f compatible with the order on S. Proof: Assume that BA. Then there are orderings (b Conversely assume that it is not the case that BA. We will construct a function f that is compatible with the order on S but for which f(B) > f(A). For each element a xag. Now BA in the Gale order if and only if there is a set of distinct representatives of the sets B a ; a 2 A. Hence there is not such set of representatives, and, by Philip Hall's theorem on distinct representatives, there must be a set A 0 A such that g. Now dene this function we have It is easy to see that this function f can be perturbed slightly to be compatible with the order on S and to retain the property that f(B) > f(A). Remark. By the same reasoning as in the proof above, it is also true that, for any if and only if f(B) < f(A) for every weight function f compatible with the order on S. Corollary 2.3 If f R is admissible for This paper uses notation that is common in the literature. In particular, we use [n] for the set f1; ng of the rst n positive integers. For readability, brackets may be deleted in denoting a set; for example f2; 4; 6g may be denoted simply 246. Example 2.4 Matroids are a special case. let P be the set of all linear orderings of S. Then clearly every injective weight function f : S ! R is admissible. By denition, a rank k greedy set L is a collection of k-element subsets of S such that, for every linear ordering of [n], there is a unique maximum in L in the induced Gale ordering on S k . But this is exactly the third characterization of matroid given in the introduction. In other words, L is a rank k greedy set for (S; P) if and only if L is the set of bases of a rank k matroid. Example 2.5 Symplectic matroids. Let [n] and let By convention i be the set of all linear orderings of S with the property that i j if and only if j i for any i; j 2 S. Equivalently, the pair i and i appear symmetrically in the order. For example, with one such order is 2 1 3 . Consequently the admissible weight functions include all injective weights R such that f(i for each i 2 [n]. A symmetric matroid of Bouchet [2] is exactly a rank n greedy set L for the pair (S; P) with the additional property that for each A 2 L. More generally, the rank k greedy sets, 0 k n, satisfying this same property are exactly the symplectic matroids of Borovik, Gelfand and White [1]. The signicance of the added assumption will be discussed in the next section. 3 The Group Case Let S be a partially ordered set and G a transitive group of permutations of S. If denotes the order on S and 2 G, let denote the order dened by a b if 1 a 1 b: For rank k subsets, the corresponding Gale order will likewise be denoted A B. This shifted order will be called the -order. Let be the set of all -orders on S for 2 G. For example, if with the order 1 2 n, then P is the set of all orders (1) (2) (n) where 2 G. The pair (S; P(G)) is referred to as the group case. Although G acts transitively on S, the induced action of G on S k may not be tran- sitive. There will therefore be situations where attention will be restricted to a single orbit O k of G acting on S k . The rank k greedy sets of (S; P(G)) will then be restricted to being contained in a set O k on which G acts transitively. Example 3.1 Matroids. If with the linear order and n is the symmetric group consisting of all permutations of [n], then the greedy sets for (S; P( n are exactly the ordinary matroids of Example 2.4. Example 3.2 Symplectic matroids. Let with the linear order and let G be the hyperoctahedral group of permutations of S generated by all transpositions of the form (i i ) and all involutions of the form (i j)(i j ). Note that the set of all k-element sets A with the property that A \ A single orbit of G acting on S k ; call this orbit O k . Then the greedy sets L O k for (S; P(G)) are exactly the symplectic matroids of Example 2.5 above. Example 3.3 Cyclic case. Let be a poset with the order and let G be the cyclic group acting on [n] and generated by the cycle (1 2 n). For example, 3 4 1 2 is a cyclic ordering for 4. It is interesting that, even for this elementary example, characterization of the collection of greedy sets is evasive. For example, it is easy to check that the orbit of 652 under the action of the cyclic group C 7 acting on is a greedy set, but the orbit of 652 under the action of C 6 acting on [6] is not a greedy set. (As noted earlier, the set f6; 5; 2g is denoted simply 652.) Example 3.4 Bipartite case. Let consist of any linear order such that either all the unstarred elements precede the starred elements or all the starred elements precede the unstarred elements. This is the group case where, if [n] and [n] are considered as vertex sets of the two parts of a complete bipartite graph K n;n , then the group is the automorphism group of K n;n . 4 The Greedy Algorithm As previously, S is a nite set and P a collection of partial orderings of S. The optimization problem applied to the pair (S; P) is the following. OPTIMIZATION PROBLEM Given an admissible weight function f and a set L S k , nd an element of L that maximizes the induced weight function f Given L S k , call a subset I S independent with respect to L if I is a subset of some element of L. The greedy algorithm, precisely as stated in the introduction, applied to this optimization problem, merely chooses the largest weight element at each stage subject to the condition that the resulting set is independent with respect to L. The following result is basic. Theorem 4.1 Let (S; P) be a pair consisting of a collection P of orderings of the nite set S. If L S k is a rank k greedy set, then the greedy algorithm correctly solves the optimization problem for every admissible weight function f Proof: Assume that L is a greedy set and that f is compatible with some order, say , in P. Since L is a greedy set, it contains a unique set A that is maximumwith respect to the Gale order. We claim that the greedy algorithm selects this set A. Suppose, instead that B is chosen where BA. Order the sets that the elements of B appear in the order selected by the greedy algorithm and b i a i for 1 i k. Assume that a by the compatibility of f , that f(b j ) < f(a j ). But this contradicts the fact that, at this stage, the greedy algorithm chooses b j . In the case that P contains only linear orderings of S, the greedy sets are actually characterized by the property that the greedy algorithm correctly solves the optimization problem for every admissible weight function. The assumption that P contains only linear orderings of S is a reasonable one in light of Theorem 4.3 below. Theorem 4.2 Let P be a set of linear orderings of S and L S k . Then the greedy algorithm correctly solves the optimization problem for every admissible weight function if and only if L is a greedy set. Proof: In one direction, this result is a corollary of Theorem 4.1. To prove it in the other direction, assume that L is not a greedy set. Then there exists an order on S, say , for which L does not attain a unique maximum. Since is a linear ordering of S, order the elements in each set in L in decreasing order. Let A denote the lexicographically maximum element of L, and let B 6= A be a set in L that is a maximum with respect to Gale order. According to Proposition 2.2 there exists a weight function compatible with such that f(B) > f(A). On the other hand, the greedy algorithm chooses A. It is desirable to choose P so that there are many admissible weight functions and many rank k greedy sets. Then the greedy algorithm will correctly solve a large collection of optimization problems. Unfortunately, these two objectives - many admissible functions and many greedy sets - are often con icting. If (S; P) and (S; Q) have the same collection of admissible functions, but, for each k, each rank k greedy set for (S; P) is a rank k greedy set for (S; Q), then clearly it is preferable, for algorithmic purposes, to use (S; Q) rather than (S; P). Theorem 4.3 For any pair (S; P) there is a pair (P; Q) such that 1. Q contains only linear orders; 2. (S; P) and (S; Q) have the same admissible injective functions; and 3. for each k, each rank k greedy set for (S; P) is also a rank k greedy set for (S; Q). Proof: Let Q be the collection of all linear extensions of the orders in P. Clearly condition (1) is satised. Concerning condition (2) assume that weight function f is admissible for (S; Q). Then f is compatible with some linear ordering in Q and hence is also compatible with any ordering in P having as a linear extension. Therefore f is admissible for (S; P). Conversely assume that f is admissible for (S; P) and is injective. Then f is compatible with some ordering in P. Assume that the elements of of S are indexed such that f(x 1 by denition, the linear order x 1 x 2 x n is a linear extension of and f is compatible with this linear order. Therefore f is admissible for (S; Q). Concerning condition (3) assume that L is a rank k greedy set for (S; P). To show that L must also be a greedy set for (S; Q), let be any linear order in Q. Then is a linear extension of some order in P. Since L is greedy for (S; P), there is a unique maximum set in L such that, for any for all i for some ordering of the elements of A and B. Because is a linear extension of , it is also true that b i a i for all i. Therefore A is also the unique maximum with respect to the Gale order relative to . So L is a greedy set for (S; Q). Remark. The assumption in condition (2), that the admissible functions be injective, is not a serious restriction because, for any admissible function f , there is an injective admissible function that is small perturbation of f . The following examples are intentionally kept simple in order to illustrate the main ideas. Example 4.4 Symplectic matroids. This is a continuation of Example 2.5 of Section 2. Consider the case 3. It is not hard to check that the set is a rank 2 greedy set. Consider, as an example, the particular function f : [3][[3] ! R dened by This is an admissible function because it is compatible with the ordering 1 2 3 . The greedy algorithm applied to L chooses the set 1 3 whose total weight is 10, greater than that of any other set in L. On the other hand, for the function which is not admissible, the greedy algorithm again chooses the set 1 3 whose total weight is 7, but the total weight of 12 is 9. The greedy algorithm fails in this case. (Note that the collection L is not an ordinary matroid on the set [3] [ [3] .) Example 4.5 Cyclic case. This is a continuation of Example 3.3 of Section 3. Consider the case 4, and take the admissible weight function 2: The set is a rank 2 greedy set for (S; P). The greedy algorithm chooses 24 whose weight is 6, greater than the other element of L. On the other hand, for the weight function which is not admissible, the greedy algorithm fails for L. Example 4.6 Bipartite case. This is a continuation of Example 3.4 of Section 3. Consider the case is a rank 2 greedy set. An admissible function is one for which either f(i) < f(j) for every unstarred i and starred j or f(i) < f(j) for every starred i and unstarred j. As a simple example, let Then f is admissible, and the greedy algorithm chooses 1 2 which has total weight 7 compared to the total weight 3 of 12. On the other hand the function is not admissible. The greedy algorithm chooses 1 2 which has total weight 6, although 12 has greater total weight 7. 5 Roots and Polytopes In light of Theorems 4.1 and 4.2, it is important to have an e-cient method to determine whether a collection L S k is a greedy set. If (S; P) is such that of N linear orderings of S, then N may well be exponential as a function of n. If L is a collection of k-element subsets of S, then it will take no less than exponential time to check, using the denition, whether L is a greedy set. An alternative procedure is sought that is polynomial in n and the cardinality of L. In the matroid case, any one of many cryptomorphic denitions of matroid can be used to do this; that is one of the nice properties of matroids. For ordinary and symplectic matroids, the associated matroid polytope [10] furnishes an e-cient procedure to determine whether a set is greedy. For the general case, we give a geometric approach (Theorem 5.7 and the remarks that follow using polytopes and roots. Because of Theorem 4.3, it will be assumed throughout this section that and P contains only linear orders. Each linear order in P can be denoted by the permutation for which 1 Given the pair (S; P), we will associate a polytope (L) to each subset L S k as follows. Let be the canonical orthonormal basis for n-dimensional Euclidean space R n . For any k-element subset A of S, set Let (L) be the convex hull of the points f-A j A 2 Lg. Notice that (L) lies in the (n 1)-dimensional hyperplane in R n with equation Dene a root of (S; P) as a non-zero vector 1. 2. 3. for every 2 P there exists an 2 f1; 1g such that for each In particular, note that i j is a root of any pair (S; P) for all i 6= j. Our denition of root of (S; P) is meant as a generalization of a Lie algebra root system. The two notions coincide for root systems of Coxeter groups. In the group case, we refer to a root of (S; P(G)) as a root of G. The group acts on the set of roots. The roots in the following examples are easy to compute. Example 5.1 For either the symmetric group n or the alternating group A n acting on [n], the roots are Example 5.2 For the cyclic group Z n of Example 3.3 acting on [n], the roots are f In other words, +1 and 1 alternate in the vector r. For example, with the vector (1; 0; is a root while (1; 1; 0; 1; 1) is not. Example 5.3 For the hyperoctahedral group of Example 3.2 acting on [n] [ [n] the roots are For example, for the vector (1; 1; 0; 0; 1; 1) is a root. Example 5.4 For the bipartite group of Example 3.4 acting on [n] [ [n] , let . The roots are The idea now is, given a set L, to nd a computationally e-cient algorithm, in terms of its polytope (L), for deciding whether L is a greedy set. A vector will be called -admissible for (S; P) if 0 < x 1 < x x n . A vector that is -admissible for some 2 P will simply be called admissible. Theorem 5.5 A subset L S k is a rank k greedy set for (S; P) if and only if, for each admissible vector v, the linear function f v attains a maximum on (L) at a unique vertex. Proof: If Notice that, for we have, by Proposition 2.2 and the remark following it, that A B in the -Gale order if and only if f(A) < f(B) for all positive weight functions f compatible with . This is equivalent to i2B f(i) for all positive weight functions such that 0 < f(1) < f(2) < < f(n). But this, in turn, is the same as f v (- A vectors v. Thus the linear function f v attains a maximum on (L) at a unique vertex for each admissible vector v if and only if there is a unique -Gale maximum in L for each 2 P. But the latter condition is the denition of greedy set. Lemma 5.6 Let r be a vector satisfying conditions (1) and (2) in the denition of root. Then r is a root if and only if r ? contains no admissible vector. Proof: Assume that orthogonal to some admissible vector is admissible, there is a 2 P such that 0 < x 1 < x 2 < We have 1g. Then Assume, by way of contradiction, that r is a root. Condition (3) in the denition of root implies (without loss of generality) that there is a bijection from A onto B so that (i) > i for all i 2 A. But this implies that Conversely, assume that not a root. We use the same notation A and B as above for some xed 2 P violating condition (3). Without loss of generality we can ignore the 0 entries in r and assume that A [ [n] and that A. Dene a bijection from A to B recursively as follows. Let (1) be the least element of B. Having dened (i) for i < j, dene (j) as the least element of B not already in the image of . Let and r is not a root, C and are nonempty. A -admissible vector now be chosen so that arbitrary positive and negative values, respectively. In particular, a -admissible vector v can be chosen so that hr; Theorem 5.7 Let P be a collection of linear orderings of a nite set S and let L S k . If every edge of (L) is parallel to a root, then L is a greedy set for (S; P). Proof: Assume that L is not a greedy set. By Theorem 5.5 there exists an admissible vector such that the linear function f v achieves its maximum on at least two vertices of (L). Since (L) is convex, f v achieves its maximum on some edge e of (L). Therefore he; e is not parallel to a root. Unfortunately the converse of Theorem 5.7 is, in general, false. There exist greedy sets L for which the polytope (L) has non-root edges. As an example, consider the cyclic group Z 5 acting on the poset 5g. This is the group case Example 3.3. If then L is a greedy set: The (Gale) maximum is 35 for the order 12345; 15 for the order for the order 34512; 23 for the order 45123; and 34 for the order 51234. It is easy to check that the segment joining - 12 and - 34 is an edge e of (L) but that is not a root because condition (3) in the denition fails. Recall from Section 3 that in the group case it is natural to require that L be contained in a single orbit O k S k under the action of G. This is, however, not the case in the above example; L is not contained in a single orbit of S 2 under the action of Z 5 . This motivates the following question. Question 5.8 Let G denote a permutation group acting on S, and let O k be an orbit of G acting on S k . Is it true that L O k is a rank k greedy set for (S; P(G)) if and only if every edge of (L) is parallel to a root of G? In certain cases the question can be answered in the a-rmative. An edge joining vertices x and y of a polytope in R n will be called supporting if it is contained in a supporting hyperplane of that is orthogonal to an admissible vector v with According to Lemma 5.6, an edge of a polytope (L) not parallel to a root must be orthogonal to some admissible vector; so it is possible for such an edge to be supporting. Also according to Lemma 5.6, an edge that is parallel to a root cannot be supporting. A set L S k is called supporting if each edge of (L) is either supporting or parallel to a root. Theorem 5.9 Let P be a collection of linear orderings of a nite set S such that P is closed under the operation of taking the inverse (reversing order). Assume that L S k is supporting. Then L is a greedy set for (S; P) if and only if every edge of (L) is parallel to a root. Proof: In one direction the statement follows directly from Theorem 5.7. Conversely, assume that some edge e of (L) is not parallel to a root. Because L is supporting, there is an admissible vector v such that e is contained in a supporting hyperplane of (L) that is orthogonal to v, and the linear function f v values at the two endpoints of e. This implies that f v (x) attains a maximum on (L) at both endpoints of the edge e or a minimum at both endpoints of the edge e. By Theorem 5.5, if f v (x) attains a maximum at both endpoints, then L is not a greedy set. If f v (x) attains a minimum at both endpoints, say - A and - B , then, as in the proof of Theorem 5.5, there is a 2 P such that A and B are both minimum in the -Gale order. But a Gale minimum for is a Gale maximum for the inverse of , which is also in P. Hence L is not a greedy set. Let G be a permutation group acting on S. Then P(G) is closed under the operation of taking the inverse. Let O k be an orbit under the induced action of G on S k . Sometimes it is the case that any L O k is supporting. This is so, for example, if each vector determined by a point on the boundary of (O k ) is either admissible or orthogonal to a root. It can be shown that this is the case for ordinary and symplectic matroids. In general, it is not the case that any L O k is supporting. Again consider the cyclic group Z 5 acting on the poset f1; 2; 3; 4; 5g. If then L lies on one orbit under the action of the cyclic group acting on the set of pairs but the edge joining - 12 and - 34 in (L) is not a root and is not supporting. For e to be supporting, there would have to exist an admissible vector This would imply that either In both cases, the vertices lie on dierent sides of the hyperplane containing e and orthogonal to v. So there does not exist a supporting hyperplane of (L) containing e which is orthogonal to some admissible vector. --R Combinatorics 8 Mathematical Programming 38 Optimal assignments in an ordered set: an application of matroid theory Matroids and the greedy algorithm On a general de Combinatorial geometries and torus strata on homogeneous compact manifolds Networks and Matroids A geometric characterization of Coxeter matroids The greedy algorithm and Coxeter matroids --TR Combinatorial optimization: algorithms and complexity Greedy algorithm and symmetric matroids Symplectic Matroids The Greedy Algorithm and Coxeter Matroids

coxeter matroid;greedy algorithm;matroid

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Algorithms for the fixed linear crossing number problem.

several heuristics and an exact branch-and-bound algorithm are described for the fixed linear crossing number problem (FLCNP). An experimental study comparing the heuristics on a large set of test graphs is given. FLCNP is similar to the 2-page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal "node line" in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In this restricted version of the problem, the order of the vertices along the node line is predetermined and fixed. FLCNP belongs to the class of NP-hard optimization problems (IEEE Trans. Comput. 39 (1) (1990) 124). The heuristics are tested and compared on a variety of graphs including some "real world" instances of interconnection networks proposed as models for parallel computing. The experimental results indicate that a heuristic based on the neural network model yields near-optimal solutions and outperforms the other heuristics. Also, experiments show the exact algorithm to be feasible for graphs with up to 50 edges, in general, although the quality of the initial upper bound is more critical to runing time than graph size.

Introduction Recently, several linear graph layout problems have been the subject of study. Given a set of vertices, the problem involves placing the vertices along a horizontal \node line" in the plane and then adding edges as specied by the interconnection pattern. The node line, or \spine", divides the plane into two half-planes, also called \pages", corresponding to the two pages of an open book. Some examples of linear layout problems are the bandwith problem [8], the book thickness problem [2,29], the pagenumber problem [9,32], the boundary VLSI layout problem [50], and the single-row routing problem [37]. For Preprint submitted to Elsevier Science 22 August 2001 (a) (b) Figure 1. Fixed linear embedding of the complete graph K 6 crossings; (b) L (K 6 surveys of linear layout problems, see [4,51]. Linear layout is important in several applications, e.g., sorting with parallel stacks [49], fault-tolerant processor array design [39], and VLSI design [9]. In this paper we study a restricted version of linear graph layout in which the vertex order is predetermined and xed along the node line and each edge is drawn as an arc in one of the two pages. The objective is to embed the edges so that the total number of crossings is minimized (see Figure 1). We refer to this as the xed linear crossing number problem (FLCNP) and denote the minimum number of crossings by L (G) for a graph G. FLCNP was shown to be NP-hard in [33]. Single-row routing with the restriction that wires do not cross the node line is similar to FLCNP, although its objective is to nd a layout (if any) with no crossings. FLCNP also appears as a subproblem in communications network management graphics facilities such as CNMgraf [21]. The problem is also of general interest in graph drawing and graphical visualization systems where crossing minimization is an aesthetic criterion used to measure the quality of a graph drawing [14,48]. A variant of the problem in which the vertex positions are not xed is studied in [35] and a heuristic is given for its solution. A related parameter is the book crossing number of a graph G, k (G), which is the minimum number of crossings in a k-page embedding of G [2,29,42]. Note that vertex positions are not xed and hence it is rst necessary to nd an optimal ordering of vertices in order to determine k (G). The book crossing number problem is closely related to the pagenumber problem. The pagenumber of a graph is the minimum number of pages necessary to embed the edges of a graph (each edge on one page) without crossings. It is known that outerplanar graphs comprise the 1-page embeddable graphs [2], that subhamiltonian graphs, i.e., subgraphs of planar hamiltonian graphs, are precisely the 2-page embeddable graphs [2], and that planar graphs are 4-page embeddable [52]. Nonplanar graphs, however, require at least three pages [2]. A recent survey of the k- pagenumber and general crossing number problem on various surfaces can be found in [40]. Crossing minimization has also been studied for the case of two l Figure 2. Edge crossing condition i < j < k < l. levels of vertices in [17] and [28], and for the general case in [15]. Let (G) denote the general planar crossing number of a graph G. In [43] it is shown that 2 (G) (G). Observe that L (G) 2 (G), since the achievement of minimum crossings is dependent on an optimal ordering of vertices on the node line. ng. A 2-page drawing of a graph is represented by a pair of binary adjacency matrices A[ ] and B[ ]. For each edge ij, A[i; j] (B[i; j]) is embedded in the upper (lower) page and 0, otherwise. Then any pair of edges ik and jl cross in a drawing and both lie in the same page (see Figure 2). Hence, the following formula counts the number of crossings in a 2-page drawing D: In this paper, we present eight dierent heuristics for FLCNP as well as a branch-and-bound algorithm for nding exact solutions. We test the methods on random graphs in addition to \real world" instances of graphs which model some interconnection topologies proposed as architectures for parallel comput- ing. Our results show that a heuristic based on the neural network model of computation, which we simulate with a sequential algorithm, is a highly effective method for computing L (G). The heuristic consistently outperforms the other heuristics both in solution quality and running time. Furthermore, for graphs with up to approximately 50 edges, the exact algorithm is a practical choice, although its performance is highly dependent on the quality of the initial upper bound value. Hence, the algorithms we present serve as useful methods for computing L (G) and also for obtaining good linear 2-page layouts of various networks. Also, since L (G) (G), they provide upper bounds for the general planar crossing number of a graph G. We begin by presenting some theoretical bounds, followed by a description of the algorithms, and conclude with an experimental analysis. Theoretical Bounds We present some theoretical upper and lower bounds for L (G) which are used in assessing the performance of the algorithms. Throughout our discussion we assume good drawings of graphs, in which the following conditions hold: (i) an edge does not cross itself, (ii) edges with common endpoints do not cross, (iii) any intersection of two edges is a crossing rather than tangential, (iv) no three edges have a common crossing, and (v) any pair of edges cross at most once. It is a routine exercise to show, for any graph G, there is a good drawing of G having the minimum number of crossings. In [29], 1 (G) was dened as the outerplanar crossing number but no results were given. However, in [42] the following results are shown: Theorem 1. 1 2. Theorem 2. 1 for n 4; m 3n. Theorem 3. k (G) m 3 27knAlso, the following result can be deduced: Theorem 4. 1 (K n Proof. This is equivalent to the problem of arranging the vertices of the graph on the boundary of a circle and drawing the edges as chords. Then for each 4-tuple of vertices (i; j; k; l) along the boundary, with labels satisfying there is precisely one crossing caused by edges ik and jl. Hence,B @ n1 C A gives the correct number of crossings. 2 The following result for K n was previously shown in [24] (see also [22,23]): Theorem 5. 2 (K n Actually, equality has been shown for in the above formula. In [13], an alternate upper bound based on the adjacency matrix is given for when drawn on k pages, and tables of results for dierent n and k values are given. The results for with those of Theorem 5. Theorem 6. Proof. We construct a 2-page drawing of G with dm=2e edges in one page and bm=2c in the other. Now, assuming in the worst case that each edge crosses every other edge exactly once, we have at mostB @ dm=2e1 C A cross- ings. Expanding this sum and using the identity dn=2e at the given inequality. 2 Also, in [42] the following result and a greedy algorithm are given for constructing a k-page drawing of G from a 1-page drawing with the indicated number of crossings: Theorem 7. k (G) 1 (G)=k. 3 The Heuristics We developed and tested eight dierent heuristics. They are grouped into two general categories | greedy and non-greedy. The two greedy heuristics dier only in the order in which edges are added to the layout. Descriptions of the heuristics are given in the following sections. We assume that vertices are xed in the order 1; 2; :::; n along the node line. As a pre-processing step to each algorithm, we remove all insignicant edges. Observe that edges between consecutive vertices on the node line and the edge 1n cannot be involved in crossings according to the constraints of the problem. Also, if there is a vertex k such that no edge ij, i < k < j, exists, then edges 1k and kn cannot cause crossings. Hence, these edges are insignicant and may be ignored without aecting the nal solution. At the same time, the problem size is reduced so that larger instances can be solved. The output of each heuristic is the minimum number of crossings obtained and the corresponding embedding. 3.1 Greedy Heuristics The Greedy heuristic adds edges to the layout in row-major order of the adjacency matrix of the graph, that is, rst all edges 1i are added in increasing order of i-value, then all edges 2i in increasing i-value order, etc. At each step, an edge is embedded in the page (upper or lower) which results in the smallest increase in the number of crossings. Ties are broken by placing the edge in the upper page. Heuristic Gr-ran uses the same approach but adds edges in random order. 3.2 Maximal Planar Heuristic Heuristic Mplan nds a maximal planar subgraph in each page. In the rst phase, edges are added in row-major order of the adjacency matrix to the upper page. If an edge causes a crossing, it is put aside until the second phase. In the second phase, all edges put aside in the rst phase are added to the lower page. If an edge causes a crossing, it is once again put aside. In the third phase, any edges put aside in the second phase are added to the page with the smallest increase in crossings. 3.3 Edge-Length Heuristic Heuristic E-len initially orders all edges non-increasingly by their \length", i.e., ju vj for edge uv. The intuition here is that longer edges have a greater potential for crossings than shorter edges and hence should be embedded rst. Each edge is added one at a time to the page of smallest increase in crossings. 3.4 One-Page Heuristic This is essentially the same method described in [42] and implied by Theorem 7 with 2. Heuristic 1-page initially embeds all edges in the upper page. Figure 3. Fixed embeddings for base cases of dynamic programming heuristic. This is followed by a \local improvement" phase in which each edge is moved to the lower page if it results in fewer crossings. Edges are considered for movement in order of non-increasing local crossing number, i.e., the number of crossings involving an edge. 3.5 Dynamic Programming Heuristic Unfortunately, FLCNP does not satisfy the principle of optimality which says that in an optimal sequence of decisions each subsequence must also be opti- mal. Subgraphs embedded optimally earlier in the process do not necessarily lead to optimal embeddings of larger subgraphs when edges are added between the smaller subgraphs later on. However, this does not preclude the potential benet of a dynamic programming approach to the problem as a heuristic so- lution. If most crossings are localized within relatively small subgraphs along the node line for a given graph, a dynamic programming method may produce a good solution. Let G i::j denote the subgraph induced by consecutive vertices i::j along the node line, and let cr[i; k; j] be the number of crossings in the subgraph G i::k is the set of \link edges" between G i::k and G k+1::j . We compute cr[i; k; j] greedily by adding each link edge to the page with the smallest increase in crossings. This leads to a recurrence for the number of crossings, nc[1; n], computed by a dynamic programming solution: The base cases for the algorithm are the subgraphs of order 2 4. Optimal embeddings for these are predetermined and shown in Figure 3. 3.6 Bisection Heuristic This heuristic uses a straightforward divide-and-conquer approach. The original graph G 1::n is initially bisected into two smaller subgraphs G 1::b nc and G b nc+1::n by temporarily removing the link edges between them. Each sub-graph is then bisected recursively in the same manner until subgraphs of order 4 or less are obtained. Embeddings for these base cases are the same as those shown in Figure 3. When combining smaller subgraphs, link edges between the subgraphs are embedded in greedy fashion as before. A similar method is described in [3], although the way in which edges are re-inserted into the embedding after the bisection phase is not clearly specied. 3.7 Neural Network Heuristic This heuristic is based on the neural network model of parallel computation [27]. In this model there are a large number of simple processing elements called We assume the McCulloch-Pitts binary neuron in which each element has a binary state [34]. This model has also been used for the graph planarization problem [47]. For testing purposes, a sequential simulator of the actual parallel algorithm was used. The model uses 2m neurons for a graph with m edges. With each edge is associated an \up" and a \down" neuron, representing the two pages of the plane. Brie y, two kinds of forces, 'excitatory' and `inhibitory', are present in the neural network. The presence of an edge uv in a graph encourages the two neurons for the edge to re as the excitatory force, while neurons of crossing edges are discouraged from ring as the inhibitory force. At each iteration of the main processing loop, neuron values are recalculated according to specied motion equations. Eventually, after several iterations, either an 'up' or `down' neuron for each edge is in an excitatory state and a nal embedding is obtained. It is straightforward to simulate the parallel algorithm with a sequential algo- rithm. Whereas, in the parallel algorithm, the output values of the neurons are simultaneously updated outside of the motion equation loop, in the sequential simulator, the output value of each neuron is individually computed in sequence as soon as the input of the neuron is evaluated inside of the motion equation loop. A drawback to neural network algorithms is the possibility of non-convergence. Typically, a constant limit is imposed upon the number of iterations of the motion equation computation loop, and the process is terminated if convergence to the equilibrium state has not occurred by the limit. Full details of the heuristic are given in [11]. Table Time Complexities of the Heuristics. Heuristic Time Complexity Greedy O(m 2 ) Gr-ran O(m 2 ) Mplan O(m 2 ) 1-page O(m 2 ) dynamic O(m 2 bisection O(n 4 ) neural O(m) 4 Time Complexities of the Heuristics The time complexities of the heuristics and exact algorithm are given in Table 1. For the greedy, maximal planar, edge-length, and one-page heuristics, the total time is dominated by the time to calculate the number of crossings after each edge is added to the layout, which is O(n 4 ) if eq. (1) is directly applied. Instead, however, we use a \dynamic" crossing recalculation method which only checks for crossings involving the edge just added. This lowers the recalculation time to O(m) for each edge added. For the dynamic programming heuristic, there are a total of O(n 2 ) subgraphs to process, and each subgraph requires O(m 2 ) time to add link edges and recalculate crossings. The time for the bisection heuristic is given by the recurrence T where O(n 4 ) is the time to merge each pair of subgraphs, and this recurrence has the solution O(n 4 ). The sequential simulator of the neural network heuristic has a main loop with a number of iterations dependent on the rate of convergence of the system to a stable state, which is not bounded by any function of the input size. However, in experimental testing, the maximum number of loop iterations observed for any test graph was 84. There are O(1) operations performed on each of the m edge neurons per iteration. Hence, the time complexity is O(m). 5 An Exact Algorithm The number of xed linear layouts of a graph is 2 m . Ignoring up to n insignificant edges, this yields at most 2 n(n 3)=2 dierent layouts. Since any layout has a \mirror image" (symmetric) drawing with the same number of crossings obtained by switching the embeddings between the two pages, only one half of this number need be checked, or 2 n(n 3)=2 1 . A branch-and-bound algorithm was developed to nd optimal solutions by enumerating all possible embeddings of edges subject to optimization bound conditions. Two bounding conditions were applied to prune partial solution paths in the search tree. A path (branch) of the tree is pruned if: (1) the number of crossings in the partial solution exceeds the current global upper bound, or (2) the number of crossings in the partial solution plus the number of extra crossings resulting from adding each remaining edge to the partial embeddding greedily and independently of other remaining edges exceeds the current global upper bound. A backtracking algorithm was developed to enumerate the embeddings and apply the bounding conditions. An initial global upper bound was obtained from the best solution generated by the theoretical bounds and the heuristics. As with the heuristics, the output of the algorithm is the number of crossings obtained and the corresponding embedding. 6 Test Graphs Several classes of test graphs were generated, and they are brie y described in the following sections. Since we were interested in obtaining good upper bounds for the planar crossing number of families of graphs, we generated several types of hamiltonian graphs. Many networks proposed as models of parallel computer architectures are hamiltonian. For each graph G, the strategy was to x the vertices along the node line in the order of a hamiltonian cycle, if possible, since G may have a crossing-minimal drawing in which there is a hamiltonian cycle which is not crossed by an edge. If the vertices are then positioned along the node line in the given hamiltonian order, and the edges are optimally drawn, then such a drawing would have (G) crossings. However, as is discussed in [9], not all hamiltonian cycle orderings of vertices correspond to an optimal vertex ordering, that is, one that leads to a linear layout with a number of crossings equivalent to the planar crossing number of the graph. To nd an optimal cycle, for example, for the d-dimensional hypercube , as many as 2 d 3 d! cycles [41], in the worst case, would have to be generated and tested, and this would be impractical for large d. Nevertheless, by using a hamiltonian ordering of vertices on the node line, the likelihood of computing the planar crossing number of the graph was increased. 6.1 Random Graphs We used the traditional model G n;p of random graphs [6] formed by independently including each edge of K n with probability random graphs of order was made to nd a hamiltonian cycle in any graph, due to the computational di-culty of this problem. Hence, the vertices were simply positioned along the node line in the order 6.2 Interconnection Network Graphs Many interconnection topologies have been proposed for parallel computing architectures. While optimal book embeddings have been investigated for several of the networks (e.g., [9,26]), the xed linear crossing number has not to our knowledge been investigated. Detailed descriptions of many of these networks can be found in [30,53]. Here we provide only brief descriptions. All of the network graphs generated are hamiltonian. Since hamiltonian cycles may be easily found in these graphs, vertices were positioned along the node line in hamiltonian order for the testing, with the hope of obtaining better approximations to the planar crossing number of the graph, as discussed earlier. 6.2.1 Hypercubic Networks The hypercube, Q d , of dimension d, is a d-regular graph with 2 d vertices and Each vertex is labelled by a distinct d-bit binary string, and two vertices are adjacent if they dier in exactly one bit. Q 3 is shown in Figure 4(a). Hypercubes of dimension Several derivatives of the hypercube have also been proposed. These are generally referred to as hypercubic networks. While the hypercube has unbounded vertex degree, according to its dimension, most of the hypercubic networks have constant degree bounds, usually 3 or 4, making them less dense with increasing order. The cube-connected-cycles [36], CCC d , of dimension d is formed from Q d by replacing each vertex u with a d-cycle of vertices in CCC d and then joining each cycle vertex to a cycle vertex of the corresponding neighbor of u in Figure 4(b) shows CCC 3 . CCC d has d2 d vertices, 3d2 d 1 edges, and is 3-regular. The instances The twisted cube [7], TQ d , has the same order, size, and regularity as Q d . TQ d is formed by twisting one pair of edges in a shortest cycle (4-cycle) of Q d . Figure 4(c) displays TQ 3 . TQ d of dimension The crossed cube, CQ d , is dened in [18]. Like Q d , CQ d has 2 d vertices, d2 d 1 edges, and is d-regular. Figure 4(d) displays CQ 3 . CQ d of dimension were generated. The folded cube [19], FLQ d , is formed from Q d by adding the 2 d 1 extra complementary edges fu; ug for each vertex u where u is a d-bit binary string. Figure 4(e) displays FLQ 3 . FLQ d of dimension The hamming cube [16], HQ d , of dimension d, has 2 d vertices and (d+2)2 d 1 2 edges. HQ d has minimum degree d + 1, maximum degree 2d 1, and is a supergraph of Q d . Figure 4(f) displays HQ 3 . HQ d of dimension generated. The binary de Bruijn graph [12], DB d , is a directed graph of 2 d vertices and arcs. The vertices are labelled by the 2 d binary d-tuples. There is an arc from vertex x 1 :::x d to vertex y 1 :::y d i x 2 :::x As a result, vertices 00:::0 and 11:::1 have self-loops. Undirected de Bruijn graphs [1], UDB d , are formed from directed de Bruijn graphs by ignoring the orientations of the edges and deleting the two self-loops, which are irrelevant in determining the crossing number. UDB d has 2 d vertices, 2 d+1 2 edges, and maximum degree 4. Figure 4(g) displays UDB 3 which is planar. UBD d of dimension were generated. The wrapped butter y graph [30], WBF d , of dimension d has d2 d vertices and d2 d+1 edges. The graph is 4-regular. Figure 5(f) displays WBF 3 . WBF d of dimension The shue-exchange graph [30] SX d , has 2 d vertices and 3 2 d 1 edges. uv is an edge of SX d if either u and v, which are d-bit binary strings, dier in precisely the last bit, or u is a left or right cyclic shift of v. For embedding purposes, we ignore the two self-loop edges at vertices 00:::0 and 11:::1, which results in a graph with 3 2 d 1 2 edges. Pendant vertices 00:::01 and 11:::10 may also be ignored to facilitate a hamiltonian vertex ordering along the node line. Figure 5(d) displays SX 3 . SX d of dimension (a) (b) (c) (d) (e) Figure 4. Some interconnection networks: (a) hypercube conected cycles CCC 3 ; (c) twisted cube TQ 3 ; (d) crossed cube CQ 3 ; (e) folded cube hamming cube HQ 3 ; (g) undirected de Bruijn graph UDB 3 . 6.2.2 Other Networks The d d torus, T d;d , is the graphical cross product of the cycles C d and C d . Figure 5(a) displays T 4;4 . Torii T d;d , for The star graph [45], ST d , has d! vertices labelled by all permutations of f1; 2; 3; :::; dg. Two vertices are adjacent i the corresponding permutations dier only in the rst and one other position. Hence, ST d has (d 1)d!=2 edges and is (d 1)- regular. Figure 5(b) displays ST 4 . ST d for The pancake graph [45], PK d , has d! vertices labelled by all permutations of the elements dg. Two vertices are adjacent i one can be obtained by ipping the rst i elements of the other for some i 2. PK d has the same order and size as ST d and is also (d 1)-regular. Figure 5(c) displays PK 4 . PK d for The pyramid graph [30], PM d , has d levels of vertices, with each level k, 0 k < d, having 4 d vertices, for a total of (4 d 1)=3 vertices. The interconnection structure of PM 3 is shown in Figure 5(e). PM d for (a) (c) (d) (b) Figure 5. Additional interconnection networks: (a) torus T 4;4 ; (b) star graph wrapped butter 6.3 Other Graph Families Also included in the testing were complete graphs, K n , for circulant graphs. The circulant graph [5], C n (a a k < (n + 1)=2, is a regular hamiltonian graph with n vertices, with vertices n) adjacent to each vertex i. C n (a 20::46 and various a i values were generated. C 8 (1; 2; 4) is shown in Figure 5(g). Vertices were placed along the node line in hamiltonian order for the testing. 7 Experimental Results All algorithms were implemented in the C language on a DEC AlphaServer 2100A 5/300 workstation with 300 MHz cpu speed and 512 megabytes of RAM. Test graph size was limited by the excessive time requirements of the bisection and dynamic heuristics and the branch-and-bound algorithm. The largest test graph contained 1016 edges. Due to the memory needed by the program to store all of the subgraphs generated by bisection and dynamic, it was impossible to test graphs larger than this with the current implementation and hardware. The two greedy heuristics and heuristics Mplan, 1-page, Neural, and E-len, on the other hand, can accommodate much larger graphs. Results for the dierent classes of graphs are shown in Tables 2-5 and Figures 6-10. Figure 11 shows a plot of heuristic performance on all 196 test graphs. For complete graphs (Table 2), both Neural and E-len found optimal solutions in all cases. The Greedy heuristic had the worst performance on these graphs. For the hypercubic networks, as indicated by Table 3 and Figure 8, bisect and dynamic performed poorly, while the remaining heuristics had much better performance. In particular, neural found the optimal solution for 15 of the 39 test cases. For the random graphs tested (see Figure 7), the edge density was approximately the same ( 0:5) for all instances, with some minor variation due to normal inconsistencies in the pseudorandom number generator. Here, we dene the edge density of a graph to be m=jE(K n )j for an n-vertex, m-edge graph. In Figure 11, we plotted the number of crossings versus the edge density for all test graphs. It is interesting to observe that the number of crossings found by all heuristics increases dramatically after 80 vertices, even though the edge density is constant. It is likely that the xed vertex orderings along the node line become more of a factor in heuristic performance as the number of vertices increases. Due to the rather large scale of the gure, the plot is undetectable for sparse graphs, until the density exceeds about 0:5. Any dierences in performance between six of the heuristics (excluding bisect and dynamic) are still hard to observe after that. Hence, to gain a better perspective of relative performance, we also compared the heuristics according to a ranking scheme, where the rank of a heuristic A, was dened to be k, 1 k 8, if A obtained the k th best solution among the eight heuristics for a given instance. Table 6 shows the overall rankings on all test graphs. Heuristic Neural either had or tied for the best average rank in 15 of the 17 classes, and also had the best composite average rank (1:35) of the eight heuristics for all test graphs. At the other extreme, bisect exhibited the poorest overall performance, nish- ing last or tied for last in average rank on all graph classes. Its performance can probably be improved by a more clever method for adding link edges between subgraphs after the bisection process. The same applies to the dynamic programming heuristic, although its performance was signicantly better than bisect. It is also interesting to examine the degree of optimality of the heuristics in cases where the optimal solution is known. This is summarized in Table 7. Optimal solutions were obtained for 95 of the 196 test graphs. Neural found the optimal solution in 75 (79%) of these cases. Its solution deviated from the optimal by a total of 38 crossings in the remaining 20 cases, for an average deviation of 1:9 crossings. The maximum deviation of 6 crossings by Neural occurred on the circulant graph C 24 (1; 3; 5). For the classes of complete graphs, torii, de Bruijn, and hypercubic graphs (except hypercubes), Neural found the optimal or conjectured optimal solution for all test cases with known solutions. The random greedy heuristic nished second in overall performance with an average deviation of 1:9 crossings from the optimal solution on 95 graphs. The standard greedy heuristic, on the other hand, was second to last in perfor- mance. Hence, the advantage of random edge selection is apparent. In the tables, the column headings are abbreviated as follows: Opt.: optimal solution from branch-and-bound algorithm Greedy: greedy heuristic Gr-ran: random greedy heuristic Mplan: maximal planar heuristic E-len: edge length heuristic 1-page: one-page heuristic Dyn: dynamic programming heuristic Bisect: bisection heuristic Neural: neural network heuristic Within the Opt. column, in cases where an optimal solution was not obtainable due to problem size, a pair of values LB:UB indicates the best known theoretical lower and upper bounds. Except where noted, bounds were obtained from Theorems 3-7. We note that for the hypercube, Q 5 , the optimal solution obtained by the exact algorithm contained 60 crossings. However, a drawing is given in [31] with only 56 crossings. Hence, the particular hamiltonian cycle used here was Results for Complete Graphs. Number of Crossings Found by Heuristic Graph Opt: Greedy Gr-ran Mplan E-len 1-page Dyn Bisect Neural K8 optimal value conjectured optimal value not an optimal vertex ordering necessary to achieve the minimum. The same can be noted for the solutions obtained for the torii T 5;5 and T 7;7 (20 and 48, resp.), which are higher than the known optimal values of 15 and 35, respectively. For heuristics Neural and Gr-ran, the best solution obtained from 10 trials per instance is indicated. As a practical limit, Neural was allowed to iterate up to 5000 times per instance before non-convergence was assumed. However, the maximum number of iterations observed for any test graph was 84. Moreover, graph size had no observable eect on the number of iterations. A comparison of running times for the heuristics on a sampling of 48 graphs is shown in Figure 12. The times for E-len, Mplan, and 1-page are dwarfed by those of the dynamic programming heuristic and thus are barely discernible along the x-axis in the plot. Running times for the bisection heuristic were somewhat longer than these but still much shorter than the dynamic programming heuristic. Figure 12 shows the running times of the heuristics for a representative sampling of the test cases selected from each of the dierent classes of graphs. The sharp spikes in the plots for the dynamic and bisection heuristics may be attributed to instances with a large number of vertices, since this has a dramatic eect on the depth of recursion and resulting cpu overhead for both heuristics. The cpu times for the remaining heuristics were all comparatively fast. In particular, the neural network heuristic required at most 84 iterations on any instance and ran noticeably faster than the other heuristics in most cases. The cpu times for the exact algorithm are shown in Table 8 on a sampling of 17 graphs. The table includes the percentage of the total search space explored. In Results for Hypercubic Networks. Number of Crossings Found by Heuristic Graph n m Opt: Greedy Gr-ran Mplan E-len 1-page Dyn Bisect Neural 28 34 28 29 29 29 29 34 28 single value indicates optimal solution; pair of values indicates theoretical lower and upper bounds 1 lower bound d(d 1)2 d[31] 2 upper bound 1654 d(2d 4 upper bound 34 d3 2 d d2 d [10] number of crossings number of vertices Greedy Gr-ran Mplan E-len 1-page Dyn Bisect Neural Figure 6. Heuristic results for complete graphs. general, exact solutions were feasible only for graphs with up to approximately 50 signicant edges. However, the quality of the initial upper bound is also very critical to the running time. For example, we were able to process the cube-connected cycles CCC 4 , with 96 edges, in only 3 cpu seconds due to the optimality of the heuristic solution (and initial upper bound). 8 Conclusion and Remarks We have presented several heuristics and an exact algorithm for computing the xed linear crossing number of a graph. An experimental analysis of their performance on a variety of test graphs has been given. Our main conclusion is that a heuristic based on the neural network model of computation is a highly eective method for solving the problem, giving near-optimal solutions in most cases and consistently better solutions than other popular heuristics in other cases. An exact algorithm has been shown eective for graphs with up to 50 signicant edges, in general, although it can handle much larger graphs number of crossings number of vertices Greedy Gr-ran Mplan E-len 1-page Dyn Bisect Neural Figure 7. Heuristic results for 100 random graphs. if the initial upper bound is fairly tight. The algorithms are useful in providing upper bounds to the book crossing number and planar crossing number of a graph as well as for nding crossing- minimal 2-page layouts of parallel interconnection networks. In future work, we plan to study the worst-case performance of the heuristics and to investigate their adaptation to the unxed linear crossing number problem. Since this requires nding an optimal vertex ordering on the node line, the problem complexity is greater than that of FLCNP. Also, recently some other algorithms have been brought to the attention of the author, i.e., [46,38], and these may be included along with the present set of algorithms in future experiments. number of crossings number of edges Greedy Gr-ran Mplan E-len 1-page Dyn Bisect Neural Figure 8. Heuristic results for hypercubic networks. --R a competitor for the hypercube? The book thickness of a graph A framework for solving VLSI graph layout problems Embedding graphs in books: a survey Circulants and their connectivities A new variation on hypercubes with smaller diameter Embedding graphs in books: a layout problem with applications to VLSI design Topological properties of some interconnection network graphs A neural network algorithm for a graph layout problem An upper bound to the crossing number of the complete graph drawn on the pages of a book Algorithms for drawing graphs: an annotated bibliography An experimental comparison of four graph drawing algorithms A theoretical network model and the Hamming cube networks Heuristics for reducing crossings in 2-layered networks The crossed cube architecture for parallel computation On the Eggleton and Guy conjectured upper bound for the crossing number of the n-cube Crossing number of graphs Latest results on crossing numbers The toroidal crossing number of the complete graph Toroidal graphs with arbitrarily high crossing numbers Kautz and shu The book thickness of a graph Introduction to Parallel Algorithms and Architectures: Arrays Bounds for the crossing number of the n-cube On the page number of graphs Crossing minimization in linear embeddings of graphs A logical calculus of ideas imminent in nervous activity Permutation procedure for minimising the number of crossings in a network The cube-connected cycles: a versatile network for parallel computation IEEE Trans. Using simulated annealing to The DIOGENES approach to testable fault-tolerant arrays of processors On some topological properties of hypercubes A parallel stochastic optimization algorithm for Automatic graph drawing and readability of diagrams Sorting using networks of queues and stacks Computational aspects of VLSI Linear and book embeddings of graphs Four pages are necessary and su-cient for planar graphs Parallel and Distributed Computing Handbook --TR Four pages are necessary and sufficient for planar graphs Linear and book embeddings of graphs Embedding graphs in books: a layout problem with applications to VLSI design Automatic graph drawing and readability of diagrams Crossing Minimization in Linear Embeddings of Graphs Bounds for the crossing number of the <italic>N</italic>-cube Introduction to parallel algorithms and architectures A new variation on hypercubes with smaller diameter Graphs with <?Pub Fmt italic>E<?Pub Fmt /italic> edges have pagenumber<inline-equation> <f> <fen lp="par"><rad><rcd><i>E</i></rcd></rad><rp post="par"></fen> </f> </inline-equation> <?Pub Fmt italic>O<?Pub Fmt /italic> On VLSI layouts of the star graph and related networks Algorithms for drawing graphs Parallel and distributed computing handbook The book crossing number of a graph Embedding de Bruijn, Kautz and shuffle-exchange networks in books An experimental comparison of four graph drawing algorithms CNMGRAFMYAMPERSANDmdash;graphic presentation services for network management Sorting Using Networks of Queues and Stacks The cube-connected cycles: a versatile network for parallel computation Properties and Performance of Folded Hypercubes The Crossed Cube Architecture for Parallel Computation A Theoretical Network Model and the Hamming Cube Networks Book Embeddings and Crossing Numbers

linear layout;neural network;crossing number;branch-and-bound;heuristic

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Some recent advances in validated methods for IVPs for ODEs.

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution.

Preprint submitted to Elsevier Preprint 20 November 2000 is D. The condition (2) permits the initial value to be in an interval, rather than specifying a particular value. We assume that the representation of f contains a nite number of constants, variables, elementary operations, and standard functions. Since we assume f 2 C k 1 (D), we exclude functions that contain, for example, branches, abs, or min. For expositional convenience, we consider only autonomous systems. This is not a restriction of consequence, since a nonautonomous system of ODEs can be converted into an autonomous one. Moreover, the methods discussed here can be extended easily to nonautonomous systems. We consider a grid t 0 < which is not necessarily equally spaced, and denote the stepsize from t j 1 to t j by h . The step from is referred to as the jth step. We denote the solution of (1) with an initial condition For an interval, or an interval vector [y j 1 ], we denote by y(t; t the set of solutions Our goal is to compute interval vectors [y j m, that are guaranteed to contain the solution of (1{2) at Standard numerical methods for IVPs for ODEs attempt to compute an approximate solution that satises a user-specied tolerance. These methods are usually robust and reliable for most applications, but it is possible to nd examples for which they return inaccurate results. On the other hand, if a validated method for IVPs for ODEs returns successfully, it not only produces a guaranteed bound for the true solution, but also veries that a unique solution to the problem y exists for all y There are situations when guaranteed bounds are desired or needed. For ex- ample, a guaranteed bound on the solution could be used to prove a theorem [28]. Also, some calculations may be critical to the safety or reliability of a system. Therefore, it may be necessary or desirable to ensure that the true solution is within the computed bounds. One reason validated solutions to IVPs for ODEs have not been popular in the past is that their computation typically requires considerably more time and memory than does that of standard methods. However, now that \chips are cheap", it seems natural to shift the burden of determining the reliability of a numerical solution from the user to the computer | at least for standard problems that do not require a very large amount of computer resources. In addition, there are situations where interval methods for IVPs for ODEs may not be computationally more expensive than standard methods. For ex- ample, many ODEs arising in practical applications contain parameters. Often these parameters cannot be measured exactly, but are known to lie in certain intervals, as, for example, in economic models or in control problems. In these situations, a user might want to compute solutions for ranges of parameters. If a standard numerical method is used, it has to be executed many times with dierent parameters, while an interval method can \capture" all the solutions at essentially no extra cost. Important developments in the area of validated solutions of IVPs for ODEs are the interval methods of Moore [19], Kruckeberg [13], Eijgenraam [8], and Lohner [17]. All these methods are based on Taylor series. One reason for the popularity of this approach is the simple form of the error term. In addition, the Taylor series coe-cients can be readily generated by automatic dierenti- ation, and both the order of the method and its stepsize can be changed easily from step to step. Usually, validated methods for IVPs for ODEs are one-step methods, where each step consists of two phases: Algorithm I: validate existence and uniqueness of the solution with some stepsize, and Algorithm II: compute a tight enclosure for the solution. The main di-culty in Algorithm I is how to validate existence and uniqueness of the solution with a stepsize that corresponds to the order of Algorithm II. The main obstacle in Algorithm II is how to reduce the so-called wrapping eect, which arises when the solution set, which is not generally a box, is enclosed in | or wrapped by | a box on each integration step, thus introducing overestimations in the computed bounds. Currently, Lohner's QR- factorization method is the most-eective standard scheme for reducing the wrapping eect. The methods considered in this paper are based on high-order Taylor series expansions with respect to time. Recently, Berz and Makino [4] proposed a method for reducing the wrapping eect that employs high-order Taylor series expansions in both time and the initial conditions. Berz and Makino's scheme validates existence and uniqueness and also computes tight bounds on the solution at each grid point t j in one phase, while the methods considered here separate these tasks into two phases, as noted above. The purpose of this paper is to present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and to discuss some recent advances in the area of validated ODE solving. In particular, we discuss an interval Hermite-Obreschko (IHO) scheme for computing tight enclosures on the solution [20, 21]; instability in interval methods for IVPs for ODEs due to the associated formula for the truncation error [20, 21], which appears to make it di-cult to derive eective validated methods for sti problems; and a new perspective on the wrapping eect [22], where we view the problem of reducing the wrapping eect as one of nding a more stable scheme for advancing the solution. Section 2 introduces interval-arithmetic operations, enclosing ranges of func- tions, and automatic generation of Taylor series coe-cients. In Section 3, we brie y discuss methods for validating existence and uniqueness of the solution, explain the wrapping eect, and describe Lohner's QR-factorization method for reducing the wrapping eect. In Section 4, we outline the IHO method. Section 5 shows that the stability of the ITS and the IHO methods depends not only on the formula for advancing the solution, as in point methods for IVPs for ODEs, but also on the associated formula for the truncation error. Section 6 explains how the wrapping eect can be viewed as a source of instability in interval methods for IVPs for ODEs and how Lohner's QR-factorization scheme improves the stability of an interval method. The set of intervals on the real line R is dened by a If a a then [a] is a point interval; if a 0 then [a] is nonnegative ([a] 0); and if a then [a] is symmetric. Two intervals [a] and [b] are equal if a and a = b. Let [a] and [b] 2 IR, and - 2 =g. The interval-arithmetic operations are dened [19] by which can be written in the following equivalent form (we omit in the nota- a a b; a b min a b; ab a b; ab 1= b; 1=b The strength of interval arithmetic when implemented on a computer is in computing rigorous enclosures of real operations by including rounding errors in the computed bounds. To include such errors, we round the real intervals in (4{7) outwards, thus obtaining machine interval arithmetic. For example, when adding intervals, we round a +b down and round a+ b up. For simplicity of the discussion, we assume real interval arithmetic in this paper. Because of the outward roundings, intervals computed in machine interval arithmetic always contain the corresponding real intervals. Denition (3) and formulas (4{6) can be extended to vectors and matrices. If the components of a vector or matrix are intervals, we have interval vectors and matrices, respectively. The arithmetic operations involving interval vectors and matrices are dened by the standard formulas, except that reals are replaced by intervals and real arithmetic is replaced by interval arithmetic in the associated computations. We have an inclusion of intervals [a] [b] () a and a b: The interval-arithmetic operations are inclusion monotone. That is, for real intervals [a], [a 1 ], [b], and [b 1 ] such that [a] [a 1 ] and [b] [b 1 ], Although interval addition and multiplication are associative, the distributive law does not hold in general [1]. That is, we can easily nd three intervals [a], [b], and [c], for which However, for any three intervals [a], [b], and [c], the subdistributive law does hold. Moreover, there are important cases in which the distributive law does hold. For example, it holds if [b] [c] 0, if [a] is a point interval, or if [b] and [c] are symmetric. In particular, for 2 R, which can be interpreted as the point interval [; ], and intervals [b] and [c], we have For an interval [a], we dene width and midpoint of [a] as a a and respectively [19]. Width and midpoint are dened component-wise for interval vectors and matrices. Using (4{6) and (9), one can easily show that for any [a], [b] 2 IR and 2 R. The equalities (11) and (12) also hold when [a] and [b] are interval vectors. If A is an n n real matrix and [a] is an n dimensional interval vector, then from (11{12), where jAj is obtained by taking absolute values on each component of A. 2.1 Ranges of Functions R be a function on D R n . The interval-arithmetic evaluation of f on [a] D, which we denote by f ([a]), is obtained by replacing each occurrence of a real variable with a corresponding interval, by replacing the standard functions with enclosures of their ranges, and by performing interval- arithmetic operations instead of the real operations. From (8), the range of f , g, is always contained in f ([a]). Although f ([a]) is not unique, since expressions that are mathematically equivalent for scalars, such as x(y z) and xy may have dierent values if x, y, and z are intervals, a value for f ([a]) can be determined from the code list, or computational graph, for f . This implies that, if we rearrange an interval expression, we may obtain tighter bounds. continuously dierentiable on D R n and [a] D, then, for any y and b 2 [a], some 2 [a] by the mean-value theorem. Thus, [19]. The mean-value form, f M ([a]; b), is popular in interval methods since it often gives tighter enclosures for the range of f than the straightforward interval-arithmetic evaluation of f itself. Like f , f M is not uniquely dened, but a value for it can be determined from the code lists for f and f 0 . 2.2 Automatic Generation of Taylor Coe-cients Since the interval methods considered here use Taylor series coe-cients, we outline the scheme for their generation. More details can be found in [3] or [19], for example. If we know the Taylor coe-cients (u) i u (i) =i! and (v) i v (i) =i! for for two functions u and v, then we can compute the pth Taylor coe-cient of u v, uv, and u=v by standard calculus formulas [19]. We introduce the sequence of functions @y f For the IVP y the ith Taylor coe-cient of y(t) at t j , where is the (i 1)st Taylor coe-cient of f evaluated at y j . Using and formulas for the Taylor coe-cients of sums, products, quotients, and the standard functions, we can recursively evaluate (y j 1. It can be shown that, to generate k Taylor coe-cients, we need at most O(k 2 ) times as much computational work as we require to evaluate f(y) [19]. Note that this is far more e-cient than the standard symbolic generation of Taylor coe-cients. If we have a procedure to compute the point Taylor coe-cients of y(t) and perform the computations in interval arithmetic with [y j ] instead of y j , we obtain a procedure to compute the interval Taylor coe-cients of y(t) at t j . 3 Overview of Interval Methods for IVPs for ODEs 3.1 Algorithm I: Validating Existence and Uniqueness of the Solution Using the Picard-Lindelof operator and the Banach xed-point theorem, one can show that if h j 1 and [~y 0 then (1) with has a unique solution y(t; We refer to a method based on (16) as a rst-order enclosure method. Such a method can be easily implemented, but a serious disadvantage of this approach is that it often restricts the stepsize Algorithm II could take. One can obtain methods that enable larger stepsizes by using polynomial enclosures [18] or more Taylor series terms in the sum in (16), thus obtaining a high-order Taylor series enclosure method [6, 24]. In the latter, we determine holds for all has a unique solution y(t; for all In [24], we show that, for many problems, an interval method based on (17) is more e-cient than one based on (16). 3.2 Algorithm II: Computing a Tight Enclosure of the Solution Consider the Taylor series expansion its lth component n) evaluated at Using the a priori bounds [~y we can enclose the local truncation error of the Taylor series (18) on We can also bound the ith Taylor coe-cient f [i] (y Therefore, contains However, as explained below, (19) illustrates how replacing real numbers in an algorithm by intervals often leads to large overestimations Taking widths on both sides of (19) and using (11), we obtain w([y which implies that the width of [y j ] almost always increases 2 with j, even if the true solution contracts. A better approach is to apply the mean-value theorem to f [i] in (18), obtaining for any ^ I An equality is possible only in the trivial cases h for all where J is the Jacobian of f [i] with its lth row evaluated at This formula is the basis for the ITS methods of Moore [19], Eigenraam [8], Lohner [17], and Rihm [27]; see also [23]. Let and (The ith Jacobian can be computed by generating the Taylor coe-cient dierentiating it [2, 3]. Alternatively, these Jacobians can be computed by generating the Taylor coe-cients for the associated variational equation [17].) Using (21), we can rewrite (20) as We refer to a method implementing (22) as the direct method. If we compute enclosures with (22), the widths of the computed intervals may decrease. However, this approach frequently works poorly, because the interval vector signicantly overestimate the set Such overestimations accumulate as the integration proceeds. This is often called the wrapping eect. In Figure 1, we illustrate the wrapping of the parallelepiped f Ax j x 2 [x] g by the box A[x], where A =B @ 1 2 Fig. 1. The wrapping of the parallelepiped f Ax j x 2 [x] g by the box A[x]. 3.2.1 The Wrapping Eect The wrapping eect is clearly illustrated by Moore's example [19], The interval vector [y 0 ] can be viewed as a box in the (y 1 the true solution of (23) is the rotated box shown in Figure 2. If we want to Fig. 2. The rotated box is wrapped at enclose this box in an interval vector, we have to wrap it by another box with sides parallel to the y 1 and y 2 axes. On the next step, this larger box is rotated and so must be enclosed in a still larger one. Thus, at each step, the enclosing boxes become larger and larger, but the true solution set is a rotated box of the same size as [y 0 ]. Moore showed that at 2, the interval inclusion is in ated by a factor of e 2 535 as the stepsize approaches zero [19]. 3.2.2 Lohner's Method Here, we describe Lohner's QR-factorization method [16, 17], which is one of the most successful, general-purpose methods for reducing the wrapping eect. Let (j 1), where I is the identity matrix. From (20{21) and (24{25), we compute and propagate for the next step the interval vector where A j 2 R nn is nonsingular for In Lohner's QR-factorization method, where Q j is orthogonal and R j is upper triangular. Other choices for A j are discussed in [16, 17, 20, 22]. One explanation why this method is successful at reducing the wrapping eect is that we enclose the solution on each step in a moving orthogonal coordinate system that \matches" the solution set; for more details, see [16, 17, 20, 23]. In section 6, we show that this choice for A j ensures better stability of the QR-factorization method, compared to the direct method. 4 An Interval-Hermite Obreschko Method For more than Taylor series has been the only eective approach for computing rigorous bounds on the solution of an IVP for an ODE. Recently, we developed a new scheme, an interval Hermite-Obreschko (IHO) method [20, 21]. Here, we outline the method and its potential. Let c q;p (q; p, and i 0), y It can be shown that c p;q [20, 31]. The Hermite-Obreschko formula (29) is the basis of our IHO meth- od. As in an ITS method, we can easily bound the local truncation error in the IHO method, since we can readily generate the interval Taylor coe-cient The method we propose in [20] consists of two phases, which can be considered as a predictor and a corrector. The predictor computes an enclosure [y (0) of the solution at t j , and using this enclosure, the corrector computes a tighter enclosure j ] at t j . If q > 0, (29) is an implicit scheme. The corrector applies a Newton-like step to tighten [y (0) We have shown in [20, 21] that for the same order and stepsize, our IHO method has smaller local error, better stability, and requires fewer Jacobian evaluations than an ITS method. The extra cost of the Newton step is one matrix inversion and a few matrix multiplications. 5 Instability from the Formula for the Truncation Error In this section, we investigate the stability of the ITS and IHO methods, when applied with a constant stepsize and order to the test problem where and y 0 2 R, and < 0. Since we have not dened complex interval arithmetic, we do not consider problems with complex. Note also that the wrapping eect does not occur in one-dimensional problems. For the remainder of this paper, we consider ITS methods with a constant stepsize and order k truncation error and in this section, we consider the IHO scheme with a constant stepsize and order 5.1 The Interval Taylor Series Method Suppose that at t j 1 > 0, we have computed a tight enclosure [y ITS of the solution of (30) with an ITS method, and [~y ITS is an a priori enclosure of the solution on Let r Using (31), an ITS method for computing tight enclosures of the solution to (30) can be written as [y ITS [~y ITS cf. (20). Since [~y ITS we obtain from (31{32) that w([y ITS w([y ITS (Note that w([y ITS Therefore, if the ITS method given by (32) is asymptotically unstable, in the sense that lim j!1 w([y ITS This result implies that we have restrictions on the stepsize not only from the function T k 1 (h), as in point methods for IVPs for ODEs, but also from the factor jhj k =k! in the remainder term. 5.2 The Interval Hermite-Obreschko Method we assume that [y IHO computed with an IHO method, and [y IHO . The formula (29) reduces to c p;q are dened in (28). Dene R p;q and Q p;q c q;p Also let [~y IHO be an a priori enclosure of the solution on From (34{35), we compute an enclosure [y IHO [y IHO [~y IHO where w([y IHO w([y IHO Therefore, the IHO method is asymptotically unstable in the sense that lim j!1 w([y IHO In (33) and (36), are approximations to e z of the same order. In particular, R p;q (z) is the Pade rational approximation to e z (see for example [26]). For the ITS method, only if h is in the nite stability region of T k 1 (z). However, for the IHO method with 0 > 2 R, jR p;q (h)j < 1 for any h > 0 if Roughly speaking, the stepsize in the ITS method is restricted by both while in the IHO method, the stepsize is limited mainly by In the latter case, p;q =Q p;q (h) is usually much smaller than one; thus, the stepsize limit for the IHO method is usually much larger than for the ITS method. An important point to note here is that an interval version of a standard numerical method, such as the Hermite-Obreschko formula (29), that is suitable for sti problems may still have a restriction on the stepsize. To obtain an interval method without a stepsize restriction, we must nd a stable formula not only for the propagated error, but also for the associated truncation error. 6 A New Perspective on the Wrapping Eect The problem of reducing the wrapping eect has usually been studied from a geometric perspective as nding an enclosing set that introduces as little overestimation of the enclosed set as possible. For example, parallelepipeds [8, 13, 17, 19], ellipsoids [11, 12, 25], convex polygons [29], and zonotopes [14] have been employed to reduce the wrapping eect. In [22], we linked the wrapping eect to the stability of an ITS method for IVPs for ODEs and interpreted the problem of reducing the wrapping eect as one of nding a more stable scheme for advancing the solution. This allowed us to study the stability of several ITS methods (and thereby the wrapping eect) by employing eigenvalue techniques, which have proven so useful in studying the stability of point methods. In this section, we explain rst how the wrapping eect can cause instability in the direct method and then show that Lohner's QR-factorization method provides a more stable scheme for propagating the error. 6.1 The Wrapping Eect as a Source of Instability Consider the IVP 2. applied with a constant stepsize h and order k to (37), the direct method (22) reduces to Taking widths on both sides of (38) and using (11) and (13), we obtain w([y We can interpret w([y j ]) as the size of the bound on the global error and w([z j ]) as the size of the bound on the local error. We can keep w([z j ]) small by decreasing the stepsize or increasing the order. However, T [y can be a large overestimation of the set f Ty g. The reason for this is that f Ty is not generally a box, but it is enclosed, or wrapped, by the box T [y j 1 ]; cf. Figure 1. Note that we wish to compute g, but this is generally infeasible. So we compute T [y instead. That is, we normally compute on each step products of the form consequently, we may incur a wrapping on each step, resulting in unacceptably wide bounds for the solution of (37). Consider now the point Taylor series (PTS) method given by If we denote by - the global error of this method at t j , then where z j is the local truncation error of this method. An important observation is that the global error in the PTS method propagates with T , while the global error in the direct method propagates with jT j. Denote by (A) the spectral radius of A 2 R nn . It is well-known that Assuming that T and jT j can be diagonalized, we showed in [22] that then the bounds on the global error in these two methods are almost the same, and the wrapping eect is not a serious di-culty for the direct method; and then the global error in the direct method can be much larger than the global error in the PTS method, and the direct method may suer signicantly from wrapping eect. Thus, we can associate the wrapping eect in an ITS method with its global error being signicantly larger than the global error of the corresponding PTS method. In particular, if (T ) < 1 and (jT the PTS method is stable, but the associated ITS method is likely asymptotically unstable in the sense that lim j!1 kw([y j To reduce the wrapping eect, or improve the stability of an ITS method, we must nd a more stable scheme for advancing the solution. 6.2 How Lohner's QR-factorization Method Improves Stability When applied with a constant stepsize and order to (37), an ITS method incorporating Lohner's QR-factorization scheme, which together we refer to as Lohner's method, can be written as The transformation matrices A j 2 R nn satisfy where A is orthogonal, and R j is upper triangular. Using (42{43), we write (40{41) in the following equivalent form: The interval vector [r j ] in (45) can be interpreted as an enclosure of the global error (at that is propagated to the next step. Obviously, we are interested in keeping the overestimation in [r j ] as small as possible. is not much bigger than w([z j ]), which we keep su-ciently small (by reducing the stepsize or increasing the order). Since we can consider jR as the matrix for propagating the global error in the QR method. As we discuss below, the nature of the R j matrices ensures that this method is normally more stable than the direct method. A key observation here is that (42{43) is the simultaneous iteration (see for example [32]) for computing the eigenvalues of T . This iteration is closely related to Francis' QR algorithm [9] for nding the eigenvalues of T . To see this, let (j 1): (47) Note that S Now observe that the iteration (48{49) is just the unshifted QR algorithm for nding the eigenvalues of T . 6.2.1 Eigenvalues of Dierent Magnitudes Let T be nonsingular with eigenvalues f of distinct mag- nitudes. Then, from Theorem 3 in [9], under the QR iteration (48{49), the elements below the principal diagonal of S j tend to zero, the moduli of those above the diagonal tend to xed values, and the elements on the principal diagonal tend to the eigenvalues of T . Using this theorem, we showed in [22] that lim j!1 jR matrix R with eigenvalues f j g. Thus, lim Using (50), we derived in [22] an upper bound for the global error of Lohner's method and showed that this bound is not much bigger than the bound for the global error of the corresponding PTS method (39), for the same stepsize and order. We also showed on several examples, that, at each t j , the global error of Lohner's method is not much bigger than the global error of the PTS method. 6.2.2 Eigenvalues of Equal Magnitude If T is nonsingular with p eigenvalues of equal magnitude, then the matrices S j , dened in (47), tend to a block upper-triangular form with a pp block on its main diagonal [10]. The eigenvalues of this block tend to these p eigenvalues. If T is nonsingular with at least one complex conjugate pair of eigenvalues and at most two eigenvalues of the same magnitude, then S j tends to a block upper-triangular form with 1 1 and 2 2 blocks on its diagonal. The eigenvalues of these blocks tend to the eigenvalues of T . Note that S j does not converge to a xed matrix, but the eigenvalues of its diagonal blocks converge to the eigenvalues of T . For this case, we showed in [22] that, as j !1, if T has a dominant complex conjugate pair of eigenvalues, then and (R j ) oscillates; and if T has a unique real eigenvalue of maximal modulus, then we should generally expect that Since the matrices jR j j do not approach a xed matrix, as in the case of eigenvalues of distinct magnitudes, deriving a bound for the global error in this case is more di-cult. However, we illustrated with four examples in [22] that the global error in Lohner's method is not normally much bigger than the corresponding global error of the PTS method. --R Introduction to Interval Computations. FADBAD, a exible C TADIFF, a exible C Validating an a priori enclosure using high-order Taylor series The Solution of Initial Value Problems Using Interval Arithmetic. The QR transformation: A unitary analogue to the LR transformation The Algebraic Eigenvalue Problem. Automatic error analysis for the solution of ordinary di A computable error bound for systems of ordinary di Computational Methods in Ordinary Di Enclosing the solutions of ordinary initial and boundary value problems. Step size and order control in the veri Interval Analysis. Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Di An interval Hermite-Obreschko method for computing rigorous bounds on the solution of an initial value problem for an ordinary dierential equation A new perspective on the wrapping e Validated solutions of initial value problems for ordinary di Global, rigorous and realistic bounds for the solution of dissipative di A First Course in Numerical Analysis. On a class of enclosure methods for initial value problems. On the existence and the veri A heuristic to reduce the wrapping e On higher order stable implicit methods for solving parabolic di On the integration of sti Understanding the QR algorithm. --TR The algebraic eigenvalue problem Rigorously computed orbits of dynamical systems without the wrapping effect Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation --CTR Hidde De Jong, Qualitative simulation and related approaches for the analysis of dynamic systems, The Knowledge Engineering Review, v.19 n.2, p.93-132, June 2004

wrapping effect;interval methods;validated methods;QR algorithm;ordinary differential equations;simultaneous iteration;taylor series;initial value problems;stability

635471

Differential algebraic systems anew.

It is proposed to figure out the leading term in differential algebraic systems more precisely. Low index linear systems with those properly stated leading terms are considered in detail. In particular, it is asked whether a numerical integration method applied to the original system reaches the inherent regular ODE without conservation, i.e., whether the discretization and the decoupling commute in some sense. In general one cannot expect this commutativity so that additional difficulties like strong stepsize restrictions may arise. Moreover, abstract differential algebraic equations in infinite-dimensional Hilbert spaces are introduced, and the index notion is generalized to those equations. In particular, partial differential algebraic equations are considered in this abstract formulation.

Introduction When dealing with standard linear time-varying coe-cient DAEs and their adjoint equations one is confronted with a kind of unsightly dissymmetry. These equations are of dierent type. In [3], more precise formulations E(PE Humboldt University, Institute of Mathematics, D-10099 Berlin, Germany ([email protected]) are used, where PE denotes a projector function along ker E, say These formulations show a little bit more symmetry. However, (1.4) is treatet in [3] via an auxiliary enlarged system that has a correspondig projector inside the derivative like (1.3). This formal uncompleteness makes us to treat general equations of the form ([2]) where A; D and B are continuous matrix functions, and A and D are well matched so that D precisely gures out which derivatives are actually in- volved. Neither D nor A have to be projectors. The adjoint equation now ts nicely into this form. There are remarkable benets of this new The inherent regular dierential equation of a DAE (1.5) is now uniquely determined (cf. [2] and Section 2). Nice generalizations of fundamental classical results (like the relation Y fundamental matrices) are obtained, but also further symmetries concerning the characteristic subspaces and the index of (1.5) and (1.6), respectively ([2]). Further, the boundary value problem resulting from linear-quadratic control problems looks now nice and transparent (cf. Section 5). The nonlinear counterparts of (1.5) and should be very welcome in applications, e. g. in circuit simulation. In [13], [14], those nonlinear equations are studied as well as numerical integration methods. To be more transparent we do not touch at all the nonlinear cases in the present paper. On the one hand, the new approach (1.5) allows to maintain all theoretical and numerical results concerning (1.1) via (1.3), but also those via the formulation where RE is a projector function along im E. On the other hand, a generalization to abstract dierential algebraic equations in innite dimensional spaces becomes possible (cf. Sections 4, 5). In particular, this seems to be a useful tool for treating so-called partial dier- ential algebraic equations. A really surprising benet of the new approach concerns numerical integration methods. At this point it should be mentioned that, e. g. in the early paper [18], the formulation (1.5) was used with the seemingly technical condition on im D(t) to be time-invariant. Now we understand that this condition is exactly the one allowing the DAE-decoupling into its essential parts and the discretization of the DAE to commute in the index-1 case (cf. [13]). In the index-2 case, if im D(t) is decomposed into certain two further constant subspaces, the decoupling and the discretization also commute in some sense. Hence, neither phenomena converting e. g. the implicit Euler method within the inherent regular ODE into the explicit one ([11], [10]) nor additional stepsize restrictions may arise. Since this is primarily a property of the DAE itself, we call those equations numerically well formulated (cf. Section 3, [13], [14]). Obviously, now one should try to have numerically well formulated equations at the very beginning. If this is not the case, refactorizations A ~ D (or factorizations leading to numerically well formulated versions might be useful (cf. Examples 3.1, 3.2, 3.3 below). In particular, the Euler backward method applied to the famous index-2 example given in [20] is well-known to get into great di-culties and to fail. However, in a slight modication (Example 3.1) it works best, and we understand why this happens. By the way, the results on the contractivity of standard methods applied to given (e. g. in [8], [10]) for the case of a constant nullspace ker E, and those of modied methods (e. g. [12]) for the case of a constant im E are now completely clear. This paper ist organized as follows. In Section 2 the basic decoupling is described in the nessesary details. With this, we follow the lines of [2], but now the index-1 and index-2 cases are put together such that index-1 appers to be a special case. Furthermore, the matrix functions A and D may be rectangular ones now, i. e., not necessarily quadratic. In Section 3 we deal with the BDF as a prototype of a discretization (the same can be done, with some more eort, with Runge-Kutta methods, cf. [13], [14]), and show the advatages of numerically well formulated DAEs. A rst attempt to formulate and to apply abstract differential algebraic equations is presented in Section 4. A linear-quadratic control problem for a DAE (in abstract formulation) is proved to be solvable in Section 5. Thereby, a Hamiltonian property turns out to be a further benet of the numerically well formulated case. Linear DAEs with properly stated leading Consider equations with matrix coe-cients The coe-cients A and D that determine the leading term of equation (2.1) are assumed to be well matched in the following sense (cf.[2]). Condition C1 for the case that Denition 2.1 The ordered pair of continuous matrix functions A and D is said to be well matched if im D(t)ker and these subspaces are spanned by continuously dierentiable bases. The leading term is properly stated if A and D are well matched. Obviously, if A and D are well matched, there is a uniquely determined projector function R(t) 2 L(R n ); Remark 2.1 The decomposition (2.2) implies that, for all t 2 I, and vice versa. Roughly speaking, D plays the role of an incidence matrix. It gures out the really involved derivatives of the unknown function. Remark 2.2 Starting with a standard form equation E(t)x 0 (t)+F we may use any factorization matched A and D, to obtain In particular, the factorizations can be used if PE (t) is a projector along the nullspace of E(t), but RE (t) denotes a projector onto the image of E(t). Naturally, a solution of equation (2.1) should be a function x 2 that has a continuously dierentiable product Dx, and satises the equation at all t 2 I. Let D denote the respective function space. Next we introduce certain subspaces and matrices to be used throughout this paper: Further, D(t) denotes the re exive generalized inverse of D(t) with Note that D(t) is uniquely determined if P 0 (t) is given and vice versa. Denition 2.2 (cf. [2]): The DAE (2.1) with properly formulated leading term and nontrivial N 0 (t) has index 2 f1; 2g if N j (t) \S j (t) has constant dimension j+1 on I and j > 0 for j < , Remark 2.3 In terms of the matrices G j (t), the index case is characterized by rank G j+1 (t) =: r Namely, using the auxiliary matrix and the re exive generalized inverse G j determined by but I G remains nonsingular. Remark 2.4 In an analogous way, considering a somehow more complex contruction of the matrices B j we could continue the chain (2.6) for i > 1 dene also an index higher than two. However, we dispense with that here in favour of more transparency, and since in the sections below we fully focus on the lower index case. Turn, for a moment, to the well understood case of Hessenberg size-two DAEs Letting I I we may rewrite (2.9) as a DAE (2.1) with properly stated leading term. Then we have I Furthermore is the projection onto im B 12 (t) along ker B 21 (t). In our context, are projectors that realize the decomposition (we have ker here). The second equation in (2.9) gives simply while the rst one leads to and In order to extract a regular explicit ODE with respect to the unknown component one has to dierentiate I H and to replace (I H)x 0 Then as an inherent regular ODE, the equation results. The solutions of (2.9) are given by the formula I Note that there is no need to dierentiate the second equation of (2.9) or any part of the coe-cients { except for the projection H and the term q. Thus we obtain solvability for all supposed H belongs to the class C 1 . In the following, we will derive an anlogous result for general equations (2.1), where H and I H correspond to DQ 1 D and DP 1 D . Lemma 2.1 Given an index-2 DAE (2.1). denote the projector onto N 1 (t) along S 1 Then, for all t 2 I, the decomposition holds true, and are the uniquely determined projectors that realize this decomposition. If, additionally, D(t)S 1 (t) and D(t)N 1 (t) are spanned by continuously dier- entiable on I functions, then Proof. This assertion can be veried in the same way as [2], Lemma 2.5 and Lemma 2.6 were proved for the case of In contrast to the index-1 case, im D is once more smoothly decomposed into the further subspaces DS 1 and DN 1 for index-2 DAEs. Let us mention, that if we think of index-3, we have to decompose DS 1 again and so on. Formally, we put the index-1 and index-2 cases together for brevity, as we have trivially for index-1 DAEs. Hence, G 2 (t) is nonsingular on I for index-1 and index-2 DAEs, i. e., In the following, Q 1 (t) always denotes the projector onto N 1 (t) along S 1 (t). Recall from [8], Appendix A the respresentation Relation (2.17) leads to the properties Further, due to the chain construction we compute Consequently, scaling (2.1) by G 1 2 leads to Multiplying we decouple this equation into the following three ones: Now, given any solution x D By this, we nd the solution representation where the inherent regular ODE and K := I Q is a nonsingular matrix function. Clearly, this generalizes the well-known representation described above for Hessenberg size-two forms (2.9). In the index-1 case (2.23), (2.24) simplify to Of course, the regular explicit ODE (2.24) can be formed without assuming the existence of a solution of the DAE just by giving the coe-cients A; D; B and the right-hand side q. Denition 2.3 Given a DAE (2.1) of index 2 f1; 2g. The explicit ODE (2.24) is said to be the inherent regular ODE of the DAE. Lemma 2.2 Given an index- DAE (2.1), 2 f1; 2g. Let DP 1 be continuously dierentiable. (i) Then, the subspaces D(t)S 1 (t) and D(t)N 1 (t) as well as the inherent regular ODE are uniquely determined by the problem data. is a time-varying invariant subspace of the inherent regular ODE, i. e., if a solution belongs to this subspace at a certain point, it runs within this subspace all the time. (iii) If D(t)S 1 (t) and D(t)N 1 (t) do not vary with time t, then, solving the IVP for (2.24) with the initial condition u(t yields the same solution as solving this IVP for Proof. (i) (cf. [2]) When constructing the ODE (2.24) the only arbitrariness is the choice of the projector P 0 and the corresponding D . So we start with two dierent P 0 and ~ and the corresponding D and ~ D . Recall that then DD ~ ~ ~ ~ ~ ~ ~ ~ Consequently, since the projectors DP 1 D ; DQ 1 D do not depend on the choice of P 0 , their images do not so, either. Further, due to ~ ~ ~ ~ I; we have ~ ~ G 1= D ~ ~ ~ ~ ~ ~ (ii) Given any solution ~ some into (2.24) and multiply the resulting identity by hence, for ~ vanishes identically as the solution of a homogeneous regular initial value problem. (iii) (cf. [13]) If D(t)S 1 (t) and D(t)N 1 (t) are constant, we may use constant projectors subspaces. Then it holds that In the consequence the term (DP 1 D disappears. In comparison with dierent notions used in the standard DAE theory (un- derlying ODE, essentially underlying ODE, inherent regular ODE), where certain transforms resp. projectors are not completely determined by the problem data, the decoupling described above is fully given by the problem. This is a real advantage. If the nullspace of A(t) is trivial, i. e., im D(t) has dimension n as it is the case for the Hessenberg size-two DAE (2.9), we can also speak of an inherent state space system (in minimal coordinates), supposed D(t)S 1 (t) does not vary with t. Then, the inherent regular ODE (2.24) is essentially nothing else but an explicit ODE located in the constant state space DS 1 . However, in the index-2 case D(t)S 1 (t) and D(t)N 1 (t) (cf. H(t) in (2.9)) vary with time in general. Theorem 2.3 Given a DAE (2.1) with index ; 2 f1; 2g, (i) For any equation (2.1) is solvable on C 1 D (ii) The initial value problem for (2.1) with the initial condition D (iii) Supposed I is compact, the inequality becomes true, i. e., the DAE has perturbation index . (iv) The subspace S ind (t) := K(t)D(t) D(t)S 1 the geometric solution space of the homogeneous equation (2.1) with ind (t) is lled by solutions and all solutions run within this subspace S ind Proof. (cf. (i) is a simple consequence of (ii). (ii) Solve the regular ODE (2.24) with the initial condition Using the solution u we construct the function x 2 according to formula (2.23). Obviously, 2 q is continuously dierentiable, thus x 2 C 1 D we check that x satises (2.1) indeed. The homogeneous IVP has only the trivial solution. (iii) and (iv) are due to the construction. Remark 2.5 One could relax the constant rank conditions in the Denitions 2.1 and 2.2 to characterize a class of DAEs (2.1) that have index ; 2, but undergo certain nondangerous index changes. If the interval I consists of a number of subintervals where all constant rank conditions are satised, we realize the decoupling on all these subintervals simulaneously and ask then for a continuously dierentiable extension of DP 1 D on the whole interval. If it exists, we may take advantage of (2.23), (2.24) again. However, we have to consider that some terms may have discontinuities now. Example 2.1 The system x 0 has the solution x independently whether the coe-cient function 2 C(I; R) vanishes. If we put this system in the form (2.1), we choose further, if (t) does not vanish, otherwise Then, on subintervals with (t) 6= 0, we have , i. e., the index is two there, further and points where (t) is zero we have G I. Trivially the problem has index 1 on those intervals. Then G In both cases we have DP 1 Remark 2.6 In [2] we aimed at a theory to treat both an original DAE and its adjoint equation in the same way . The adjoint equation of is now Both equations have properly stated leading terms at the same time. More- over, they have index ; 2 f1; 2g, simultaneously. Nice relations between the fundamental solution matrices are shown in [2]. Remark 2.7 The index notion given by Denition 2.2 resp. Remark 2.3 is approved for creating a practical index monitor ([6]). 3 Numerical integration methods and numerically well formulated DAEs Given a DAE (2.1) with a properly formulated leading term and having index 2g. The natural modication of the k-step BDF applied to (2.1) is to provide an approximation x n to x(t n ) at the current step t Let us use shorter denotations A n := A(t n ) and so on for all given coe-cient functions and resulting projectors. Denote the numerical derivative by we proceed in the same way as we did, in Section 2, with the DAE (2.1) itself, i. e. we scale by G 1 2n and then use the decoupling projectors. The resulting system analogous to system (2.20)-(2.22) is, with Denoting we derive nh (D A nj D n j x where A nj := further sjh)ds , so that Insert (3.7) into (3.4) to obtain the recursion formula for A nj Then, by (3.5), (3.6), (3.8), we compute the representation of x n in the form with ~ On the other hand, if we decoupled the DAE (2.1) rst, applied the same BDF to the inherent regular ODE (2.24) and then used formula (2.23) to compute x n , we would obtain Comparing (3.11), (3.12) and (3.9), (3.10) we observe that, except for the terms these formulas coincide, supposed the subspaces D(t)S 1 (t); D(t)N 1 (t) do not vary with t (cf. Lemma 2.2 above). Namely, in (3.11) and (3.12), we have then but in (3.9), (3.10) we use u and, analogously, A nj Denition 3.1 We will say that a discretization method and the DAE-decoupling commute, if for the approximation x n obtained by applying the method to the original DAE, and the further approximation ^ obtained by applying the method to the inherent regular ODE and then using formula (2.23), it holds that and [DQ 1 G 1 n denotes a numerical approximation of (DQ 1 G 1 Let us stress that in the index-1 case, commutativity means simply x because of Q Theorem 3.1 Given an index- DAE (2.1), 2 f1; 2g; which has constant characteristic subspaces DS 1 and DN 1 . Then the BDF-discretization and the decoupling commute. Remark 3.1 Recall once more that one has for the index-1 case. Trivially, DS 1 and DN 1 are constant i im being constant is a necessary condition for DS 1 and DN 1 to be constant. Remark 3.2 Here, we have chosen the BDF only for brevity. Of course, similar results can be proved for Runge-Kutta methods. Also, nonlinear DAEs may be considered ([14], [15]). Remark 3.3 In particular, Theorem 3.1 generalizes the respective results obtained for DAEs with constant leading nullspace (cf. (1.3) with P 0 e. g. in [8] for in [10] for 2. On the other hand, Theorem 3.1 covers the situation studied in [12], i. e., an equation (1.1) rewritten as time-invariant. Remark 3.4 It is a remarkable benet of commutativity that we can now prove nice assertions on contractive or dissipative ows and the respective discretized versions ([14]). Having modied the respective notions (contrac- tivity and dissipativity inequalities, adsorbing sets etc.) in such a way that the standard ODE notions apply to the inherent regular ODE on its invariant subspace DS 1 , we may use the well-known techniques and results approved in the regular ODE case. In particular ([14]), a stiy accurate and algebraically stable Runge-Kutta method applied to a contractive index-1 DAE yields for each two sequences of approximations starting with consistent x Thereby, is a bound of the canonical version of D corresponding to the (canonical) projector onto S 0 along N 0 . It is also shown in [14] that the Euler backward method re ects the dissipativity behaviour properly without any stepsize restriction. Example 3.1 Rewrite the equation (cf. [20]) E(t)x 0 (t)+F with as or as where Both reformulated versions (3.15) and (3.16) have properly formulated leading Now, we observe that we have index-2 DAEs with constant subspaces as supposed in Theorem 3.1. Thus, for both versions (3.15) and (3.16), the BDF and the decoupling commute. In both cases, the Euler backward yields simply Recall that the Euler backward directly applied to (3.14) gets into great diculties for jj > j1 g. [4], [20]). Example 3.2 The standard form DAE E(t)x has index-1 on I = R for arbitrary values 6= 0; - 6= 1. The solution is For - < , one would expect the backward Euler method to generate a sequence fx n;2 g n such that x n;2 ! 0(n !1) without any stepsize restriction. However, the formula reads in detail While for the constant coe-cient case works ne, for - 6= 0, the numerical solution may explode close to h 1. To realize j1 j1 hj one has to accept strong extra stepsize restriction. On the other hand, if we apply the Euler method to the reformulation we arrive at and things work well. No stepsize restrictions are caused by stability. Example 3.3 Consider the seemingly harmless Hessenberg index-2 DAE with The leading term may be rewritten in a proper form by choosing (cf. (2.9)) ~ A ~ With ~ The subspaces but also ~ ~ move with time. We refer to [10] for the sobering eect of the numerical tests. Recall that this problem was introduced in [10] to demonstrate that just a Hessenberg form DAE may get into dieculties with extra stepsize restrictions Observe, on the other hand, that there are more possibilities in factorizing E. If we nd a further well-matched factorization AD such that appear to be time-invariant, we are done since Theorem 3.1 applies. Such a much more comfortable factorization is given by ~ D, The reformulated DAE is D, and the subspaces are constant such that Theorem 3.1 applies in fact. As we could see above, the problem whether discretization and decoupling commute depends primarily on how the DAE is formulated. Denition 3.2 A DAE (2.1) with index 1 is said to be numerically well formulated if im D(t) is time-invariant (cf. [13]). A DAE (2.1) with index- 2 is said to be numerically well formulated if D(t)S 1 (t) and D(t)N 1 ) are constant. This new possibility to rearrange subspaces for better numerical properties is a surprise. One should think further on how to exploit this idea best. Fortunately, e. g. in circuit simulation, the relevant subspaces could be shown to be constant for a large class of problems ([7], [5]), hence there is no need for reformulations. Special questions concerning necessary reformulations will be discussed in [14], Abstract DAEs Nowadays, e. g. in circuit simulation and simulation of multibody dynam- ics, there is a remarkable interest in complex systems that consist of coupled systems of DAEs and PDEs (cf. [21], [9]). There are also certain proposals for treating so-called PDAEs (= partial dierential algebraic equations, e. g. [17]). Thereby, one of the questions to be considered is how to formulate initial and boundary conditions in an appropriate way. In the following we try to show the usefulness of the decoupling procedure given in Section 2 for the nite dimensional case, now in an abstract modi- cation. Being able to understand those abstract DAEs, a more systematical constructive and numerical treatment can be started on. In this section we deal with abstract DAEs are linear operators acting in the real Hilbert spaces X; Y; Z. For all t 2 I, let A(t) and D(t) be bounded and normally solvable. More- over, let A(:) and D(:) depend continuously (in the norm sense) on t. As bounded maps, A(t) and D(t) have nullspaces ker A(t) and ker D(t), respec- tively, which are closed linear manifolds, i. e., subspaces. Due to the normal solvability, im A(t) and im D(t) are subspaces (e. g. [1]), too. For all t 2 I, let the operator B(t) be dened on a dense subset DB X, and let B(:)x 2 C(I; Y ) for all x 2 DB . A continuous path x 2 C(I; X) with is called a solution of (4.1) if Dx 2 C 1 (I; Z) and equation (4.1) is satised pointwise. Denition 4.1 The leading term of (4.1) is properly stated if the operators A(t) and D(t); t 2 I, are well matched in the following sense: ker A(t) im (ii) The projector R(t) 2 L b (Z) that realizes this decomposition of Z (i. e., depends continuously dierentiably on t. Remark 4.1 Since R(t) depends smoothly on t, the subspaces im D(t) are isomorphic for dierent values of t. For the same reason, the nullspaces ker A(t) are also isomorphic. Let L b (X) denote the linear space of linear bounded maps on X etc. For bounded maps, we alway assume their denition region to be the respective whole space. Let clM denote the closure of the set M in the respective space. Next we generalize the matrix- and subspace-chain given in (2.6) by introducing the following further linear maps, linear manifolds and subspaces for cl ker G 0 (t), cl im G 0 (t), cl ker G 1 (t), cl im G 1 (t), By construction, the operators G 1 (t) and G 2 (t) are at least densely dened in X. In our context, operator products (in particular those with projectors) are often dened on a larger region by trivial reasons. We will always use the maximal trivial extensions. denote the re exive generalized inverse of D(t) such that Denition 4.2 Equation (4.1) with a properly stated leading term is said to be (i) an abstract index-1 DAE if, for all t 2 I; dim(im W 0 (t)) > 0 and injective and densely solvable, (ii) an abstract index-2 DAE if, for all t 2 I; depends continuously on t, and injective and densely solvable. If also Y is a bounded map that is continuous with respect to t in norm sense, then the assertions in Section 2, in particular Theorem 2.3, can be modied to hold true for the abstract DAE in a straightforward way. The more challenging problem is an only densely dened B(t). Hence, we are going to study three dierent cases of this type. Case 1: A coupled system of a PDE and Fredholm integral equations (on a special application of this type I was kindly informed by Hermann Brunner). Given a linear Fredholm integral operator a linear dierential operator 4 with c 0, and linear bounded coupling operators s . The system to be considered is Using the corresponding matrix representations for A; D; B; with s L 2 we rewrite (4.2) in the form (4.1). Namely, we have K L is dened on s . Clearly, it holds that N I I are dened on X (as trivial extensions of bounded maps). G 1 is a bijection such that this abstract DAE has index 1. we nd DG 1 dened on _ Each solution of the DAE is given by the expression where u(t) is a solution of the abstract regular dierential equation Obviously, one has to state an appropriate initial condition for (4.3), i. e., Case 2: A special linear constant coe-cient PDAE discussed in [17]. Consider the PDAE4 1 Suppose 3. How should we formulate boundary and initial conditions? Rewrite (4.4) in abstract form with x(t) := w(; t). Choose L L L and use again matrix representaions for our coe-cients, i. e., 0 a is the Laplacian. Let us start with The operator is dened on X and bounded. Obviously, ker G is a nontrivial closed subspaces and im G L 0 is closed. is dened on is a subspace, and obviously 0 a 5 represents a projector onto N 1 along S 1 , thus . The map is dened on L C L 8 further L C densely solvable. The injectivity of G 2 is immediately checked, therefore, this DAE has index 2. The inherent regular dierential equation is now with and dened on L dened on C L i. e., (4.6) is in fact nothing else but Now it becomes clear that, for (4.6), the initial condition as well as boundary conditions should be given. We take homogeneous Dirichlet conditions and put them into the denition region of B, i. e., we restart our procedure with instead of (4.5) or, in spite of more general solvability, with Using (4.8), G 2 is dened on L L 6 but G 1 2 on L Consequently, admissible right-hand sides are q(t) with 1(Computing 0 a we nd the solution representation (cf. (2.23) A q(t) +@ 0a c3 q(t) a while x 1 (t) solves (4.7). No further initial or boundary conditions should be given. With (4.8) we obtain the unique solvability of the IVP Case 3: A PDAE and a DAE coupled by a restriction operator. Consider the system ~ uj @ ~ Assume the linear restriction map R m to be bounded and to depend continuously on t. Rewrite the system (4.9), (4.10) as an abstract DAE for ~ u(; t) ~ and choose A := A A ~ R Restrik ~ where DB := _ is dened on X, im ~ Supposing that the operator Restrik(t) maps into im ~ holds that ~ is the projector onto N 1 along S 1 , G In the general case the projector Q 1 onto N 1 along S 1 is more di-cult to construct. Obviously, we have N 1 \ G 2 is a bijection if ~ G 2 is so. Consequently, the coupled system (4.9) (4.10) interpreted as an abstract DAE has the same index as the DAE (4.10). 5 Linear-quadratic control problems Now, the quadratic cost functional is to be minimized on solutions of the DAE which satisfy the initial condition real Hilbert spaces, Let all these operators depend continuously in the norm sense on t. Assume further: (ii) V and the maps are positive semidenite. (iii) A(t) and D(t) are normally solvable and well matched for t 2 I. Admissible controls are those functions u 2 C(I; U) for which a solution D (I; X) of the IVP (5.2), (5.3) exists. Consider the boundary value problem The system (5.4) is a DAE with a properly statet leading term. D(t) and A(t) are normally solvable and well matched at the same time as A(t) and are so. Theorem 5.1 If the triple x 2 C 1 A (I; Y ); u 2 C(I; U) solves the BVP (5.4), (5.5), then u is an optimal control for the problem (5.1)-(5.3). Proof. By straightforward calculations we prove that the expression vanishes. Then the assertion follows immediately from R x(t) x (t) u(t) u (t) x(t) x (t) u(t) u (t) dt In Theorem 5.1, no assumption on the index of the DAE (5.2) is made. How- ever, for obtaining a theoretically and practically solvable BVP (5.4), (5.5), additional conditions ensuring the index-1 property of (5.4) should be satised denote the orthoprojectors onto ker AD and along im AD, respectively (cf. Section 4). Denote Lemma 5.2 The DAE (5.4) has index 1 i, for t 2 I, Proof. Put ~ The respective maps for (5.4) rewritten as ~ are ~ ~ ~ ~ Y is a bijection i acts bijectively on ker D(t) ker A(t) U onto ker A(t) ker D(t) U . Further, ~ is bijective at the same time as G 1 (t) is so (cf. Remark 2.3). Remark 5.1 In [16], a linear-quadratic control problem of a similar form is studied. Instead of the DAE (5.2) the equation is considered. Because of the somehow incomplete leading term, the resulting system corresponding to (5.4) looks less ne and transparent. Remark 5.2 Since the operator F has the same structure as its counterpart in [16], all su-cient conditions for bijectivity proved in [16] hold true also here. Lemma 5.3 Let, for t 2 I, be bijective. Then the inherent regular dierential equation of (5.4) is of the Dx A R Dx A are positive semidenite. Proof. This assertion can be proved following the lines of [16], Theorem 1. Obviously, (5.8) is a non-negative Hamiltonian system RR but this is exactly the case if (5.4) is numerically well formulated. For Hamiltonian systems, we know the respective BVPs to be solvable. Theorem 5.4 Let F(t) in (5.7) be bijective, R 0 is at least one optimal control for the problem (5.1), (5.2), (5.3). If Z is nite dimensional, there is exactly one optimal control of (5.1), (5.2), (5.3). Proof. It remains to show uniqueness in case of nite dimensional Z, but this can be done in the same way as [16], Theorem 4, is proved. --R Theorie der linearen Opera- toren im Hilbertraum Numerical solution of initial value problems in di Indexes and special discretization methods for linear partial di Simulation gekoppelter Systeme von partiellen und di --TR Difference methods for the numerical solution of time-varying singular systems of differential equation Stability of computational methods for constrained dynamics systems On Asymptotics in Case of Linear Index-2 Differential-Algebraic Equations Runge-Kutta methods for DAEs. A new approach A Differentiation Index for Partial Differential-Algebraic Equations Analyzing the stability behaviour of solutions and their approximations in case of index-2 differential-algebraic systems --CTR G. A. Kurina, Linear-Quadratic Discrete Optimal Control Problems for Descriptor Systems in Hilbert Space, Journal of Dynamical and Control Systems, v.10 n.3, p.365-375, July 2004 I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-1 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.175-200, May I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-2 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.201-229, May

numerical integration methods;differential algebraic equations;abstract differential algebraic equations;partial differential algebraic equations

635794

Non-nested multi-level solvers for finite element discretisations of mixed problems.

We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations.

Introduction Multi-grid methods are among the most ecient and most popular solvers for nite element discretisations of elliptic partial dierential equations. Their convergence theory in the case of symmetric operators and nested conforming nite element methods is well-established; see for example the books [7, 17, 30] and the bibliographies therein. Multi-grid methods for nonconforming nite element approximations have been also studied in a number of papers, e.g. see [3, 4, 6, 8, 9, 10, 11, 12, 13, 19, 31]. In this case, the nite element space of a coarser level is in general not a subspace of the nite element space of a ner level; Institut fur Analysis und Numerik, Otto-von-Guericke-Universitat Magdeburg, PF 4120, D-39016 Magdeburg, Germany y Institute of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, the resulting multi-grid method is called non-nested. The general framework of analysing the two-level convergence of non-nested multi-grid methods, developed in [4] for elliptic problems, will be the starting point for the methods studied herein. Multi-grid methods for mixed problems, arising in the discretisation of the Stokes equations, are analysed in [3, 5, 9, 19, 27, 31, 34]. The crucial point in the investigation of multi-grid methods for mixed problems is the denition and analysis of the smoother. The currently best understood type of smoothers, here called Braess-Sarazin type smoother, is a class of symmetric incomplete Uzawa iteration proposed in [1] and analysed on its smoothing properties in [5, 27, 34]. The Braess-Sarazin type smoother has a convergence rate of O(1=m) with respect to the number of smoothing steps. By far the most analytical studies are applied to standard multi-grid methods where on each multi-grid level the same discretisation of the partial dierential equation is used. In this paper, we investigate multi-level solvers for nite element discretisations of mixed problems which allow dierent discretisations, in particular the use of dierent - nite element spaces, on each level of the multi-grid hierarchy. The motivation for using this type of multi-level solvers comes from general experiences that standard multi-level solvers are very ecient for low order discretisations. But higher order discretisations might lead to an overwhelming gain of accuracy of the computed solution so that their use should be preferred for this reason. The multi-level solvers investigated in this paper allow an accurate discretisation on the nest level and low order discretisations on all coarser levels. In this way, an accurate solution can be obtained for whose computation the eciency of multi-level solvers for low order discretisations is exploited. The crucial point in the construction of such multi-level methods is the transfer operator between the nite element spaces dened on dierent levels. We show that rather simple L 2 -stable prolongations guarantee already the convergence of the two-level method for a suciently large number of smoothing steps with a Braess-Sarazin type smoother. Our approach allows to handle conforming and nonconforming nite element spaces in a general framework. As a concrete application of the general theory developed in this paper, we have in mind in particular the Stokes and Navier-Stokes equations. However, it is clear that the same ideas can be also applied to selfadjoint elliptic equations. The eciency of multi-grid solvers for lowest order non-conforming discretisations of these equations has been demonstrated, e.g., in [20, 32] and the gain of accuracy of higher order discretisations in a benchmark problem in [18]. Dierent discretisations on the nest and on coarser grids have been already used in the convergence theory of nonconforming nite element discretisations of the Possion equation in [12, 13]. However, the motive of the approach in [12, 13] is completely dierent to ours. The replacement of the P 1 -nonconforming coarse grid correction by a conforming P 1 coarse grid correction enables the authors to apply the well-developed theory of multi-grid solvers for conforming discretisations on the coarser levels. The plan of the paper is as follows. In Section 2, we investigate the convergence properties of a multi-level method for solving mixed nite element discretisations in an abstract way. First, we introduce the variational and the discrete mixed problem and we describe the matrix representation. Then, based on abstract mappings between nite element spaces, the prolongation and restriction operators are dened. The smoothing property of the basic iteration proposed in [5] is shown for symmetric positive denite spectral equivalent pre-conditioners and without an additional projection step. Together with the approximation property, the convergence of a two-level method and the W-cycle of a multi-level method are established. Section 3 is devoted to the construction of a general mapping between two nite element spaces. We show that this general transfer operator satises all assumptions which are essential for our theory. As applications, we show in Section 4 how various discretisation concepts for solving the Stokes problem t in our general theory. Finally, we present numerical results to verify the theoretical predictions on the multi-level solvers. Throughout this paper, we denote by C a \universal" constant which is independent of the mesh size and the level but whose value can dier from place to place. Multi-level Approach 2.1 Variational Problem We consider a variational problem of the following form. Let V and Q be two Hilbert spaces and let V H V 0 be the Gelfand triple. Given the symmetric bilinear form R and a functional f 2 V 0 , we look for a solution (u; We assume: (Solvability) For all f 2 V 0 , the problem (1), (2) admits a unique solution (u; One example for this type of problems is the weak formulation of the Stokes problem in d space dimensions in r in @ where is a bounded domain with Lipschitz continuous boundary. Here we set and a and b are given by Z Z pr v dx: (4) Note that in this case (H1) is satised [16] since a is V -elliptic and b satises the Babuska- Brezzi condition, i.e., there is a positive constant such that sup 2.2 Discretisation be sequences of (possibly nonconforming) nite element spaces approximating V and Q, respectively. Instead of the continuous bilinear forms a and b we use the discrete versions a l : again assume that the bilinear forms a l are symmetric. For f 2 H the discrete problem corresponding to (1), (2) reads Find such that a l (u l ; v l where (:; :) denotes the inner product in H. (Solvability and convergence) We assume that the problem (6), (7) admits a unique solution and that the error estimate l kfkH (8) holds, where h l characterises how ne is the nite element mesh on which V l and Q l are dened. Remark 2.1 Typically, the convergence estimate (8) is established using a regularity property of the problem (1), (2). Such a regularity property usually states that if f 2 H, then the solution (u; p) of (1), (2) belongs to a 'better' space W R V Q and satises For example, for the Stokes problem mentioned above, one has 0(d , , and the regularity property holds if the boundary of is of class C 2 or is a plane convex polygon. For (u; p) 2 W R one can often prove that the solution of (6), (7) satises l and taking into considerations the estimate in the regularity property, one obtains (8). 2.3 Matrix Representation be bases of the spaces V l and Q l , respectively, where I l , J l denote the corresponding index sets. The unique representations i2I l dene the nite element isomorphisms between the vector spaces of coecient vectors u and the nite element spaces V l and Q l , respectively. We introduce the nite element matrices A l and B l having the entries a Now the discrete problem (6), (7) is equivalent to l u l l with f that A l is a symmetric matrix. In the vector spaces U l and P l , we will use the usual Euclidean norms scaled by suitable factors such that the following norm equivalences are satised with a mesh- and level-independent constant C. 2.4 Prolongation and Restriction Essential ingredients of a multi-level algorithm for mixed problems are the prolongations and the restrictions R u l 1;l l 1;l In case of a nested nite element hierarchy V 0 V l 1 V l , the canonical prolongation P u l 1;l is obtained by the nite element isomorphisms between U l 1 and V l 1 , U l and V l and the embedding V l 1 V l , thus l 1;l := 1 l l Similarly, we would have l 1;l := 1 l l 1 for assumed inclusion property Q 0 Q l 1 Q l is often but not always satised in applications. For example, this inclusion property is violated if the spaces Q l are constructed using the nonconforming piecewise linear element. The corresponding velocity spaces V l , such that the pair (V l ; Q l ) satises the discrete version of (5), may then be constructed using the bubble enriched nonconforming piecewise linear element [33] or the modied nonconforming piecewise linear element [22]. As a dierent example one could also think of using a continuous pressure space Q l on the level l and a discontinuous space Q l 1 on the level l 1. We emphasise that Q l 1 can be dened on the same mesh as Q l . In the general case of non-nested velocity spaces, when V l 1 6 V l , we have to replace the natural embedding by a suitable mapping l , which results in the following prolongation and restriction: l l u l 1 and R u l 1 l It turns out that the convergence analysis requires estimates for u on the sum V l 1 +V l , thus we will dene l on the (possibly larger) space l with V l 1 +V l l H. In the case we introduce a mapping giving the following prolongation and restriction, respectively, l l p l 1 and R p l 1 l We assume that the following properties of the mapping u hold: Remark 2.2 In [4], the mapping u has been assumed to be the product l to allow more exibility in constructing a suitable u . In many cases S can be chosen to coincide with V l or U l : 2.5 Smoothing Property In this section, we will omit the index l indicating the current level and the underline symbol for noting coecient vectors in U and P. For smoothing the error of an approximate solution of (10), we take the basic iteration 0, which can be considered as a special case of the symmetric incomplete Uzawa algorithm proposed by Bank, Welfert and Yserentant in [1]. The smoothing properties of have been studied in [5] for the special case in [27] for the general case provided that an additional projection step is performed, and in [34] for a more general setting. For completeness, we give a proof of the smoothing property without an additional projection step using partially results of [27]. The matrix D is a pre-conditioner of A such that the linear system (13) is more easily solvable than (10). Note that we have implying that after one smoothing step the iterate u j+1 , j 0, is discretely divergence-free, i.e., Bu Here and in the following, we assume that (H5) D is symmetric and positive denite and BB T is regular. The matrix BB T is regular if the bilinear form b l , generating the matrix B satises a discrete version of the Babuska-Brezzi condition (5). From the regularity of BB T it follows that S := BD 1 B T is also regular. The iteration is a so-called u-dominant method since the new iterate (u on u j but not on p j . Indeed, let (u; p) be the solution of (10). Then, we have u u j+1 (D A)(u It is easy to verify that and hence we particularly obtain for j 0 a projector, we can also write where M is given by From (14) and (15) it follows for m 3 In the case of D = I, both P and M are symmetric. Choosing max (A) the matrix I 1 A is a contraction. Moreover, the spectrum of M is contained in [0; 1] and the usual spectral decomposition argument (see [3]) results in the smoothing property 3: (19) For proving (19) it is essential that the basis f' chosen in such a way that holds. The proof of the smoothing property for D 6= I is more tricky. In [27] an additional pressure update has been introduced to guarantee the smoothing property. However, as already noticed in [3], a careful inspection shows that this additional step can be omitted. Lemma 2.1 Let be chosen such that for some level- and mesh-independent constants 2 [1; 2) and > 0. Moreover, let the basis be chosen such that Then, the basic iteration satises the smoothing property Proof. The projector P is similar to the matrix D 1=2 PD 1=2 . Since D 1=2 PD 1=2 is symmetric it follows that D 1=2 MD 1=2 is also symmetric. In [27], it has been shown (see proof of Lemma 3.9) that the spectrum of M satises for some 2 [1; 2). This assumption holds since > max (A)= min (D) 2 . Then, from the representation we get by a spectral decomposition argument applied to the symmetric matrix D 1=2 MD 1=2 the existence of a (level independent) constant C Since I 1 1 with the mapping I is a contraction for 2 =2. Thus, from (18) and (23), we get for m 3 C The cases have to be considered separately. Starting with (14), we have for A(u (D A)(u j+1 Taking into consideration that I 1 D 1=2 AD 1=2 is contractive for 2 =2, we obtain for kA(u From (16), we get and because is the product of the projector D 1=2 PD 1=2 and the contraction (I 1 D 1=2 AD 1=2 ), we conclude for all j 0 Using the triangular inequality, we nally obtain from (27) for all j 0 kA(u which proves (20) for 2. | 2.6 Approximation Property Let an approximation (~u l ; ~ of the solution of the problem (6), (7) be given. We can think of (~u l ; ~ l ) as being the result after some smoothing steps, and consequently assume that b l (~u l ; q l Then, the coarse-level correction is dened as the solution of the following problem: Find 1 such that for all (v l 1 a l 1 (u The coarse-level correction yields via the transfer operators u , p from Section 2.4 the new approximation new new Now, the basic idea for proving the approximation property is to construct an auxiliary (continuous) problem such that (u l ) are nite element solutions of the corresponding discrete problems on the spaces V l 1 Q l 1 and V l Q l , respectively. This idea has been used for scalar elliptic equations already in [6] and has been applied to more general situations in [4], [19]. We dene the Riesz representation F l 2 l of the residue by Then, the auxiliary problem will be Find (z; w) 2 V Q such that Indeed, for s 2 V l we have which means that (u l ~ l ) is a nite element approximation of (z; w) in the space l Q l . On the other hand, becomes just the right-hand side of (28) if s i.e., l 1 ) is the nite element approximation of (z; w) in the space V l 1 Q l 1 . Lemma 2.2 Let h l 1 Ch l with a mesh- and level-independent constant C. Then, the approximation property l u new l kH Ch 2 l l (p l ~p l holds. Proof. Applying (H3), (H4), the triangular inequality, and (8), we get l u new l l l Cku l ~ l 1 kH C(ku l ~ It remains to state a relation between the residue F l 2 l and the \algebraic" residue d l 2 U l given by d l := A l (u l ~ l (p l ~p l For this, we rst consider the Riesz representation of the residue in V l , i.e., r l 2 V l dened by Since A l (u l ~ l (p l ~p l U l for all v l 2 V l , we have l r l ; v l ) U l from which the representation l r l (33) follows. Now, using l l u F l ) U l we get from (11) and (H4) H kd l k U l l u F l k U l Ckd l k U l k u F l k H Ckd l k U l Dividing by kF l k H , we obtain which together with (32) yields the statement of the lemma. | 2.7 Multi-level Convergence We shortly describe the two-level algorithm using m smoothing steps on the level l, l and the coarse-level correction (28), (29). Let (u 0 l ) be an initial guess for the solution l ) of (6), (7). We apply m steps of the basic iteration l ). Now, the coarse-level correction (28), (29) is performed using l as an approximate solution of the discrete problem (6), (7). Finally, the new approximation is dened as new l new l Combining the smoothing and approximation property, we obtain Theorem 2.1 Under the assumptions of Lemma 2.1 and Lemma 2.2, the two-level method converges for suciently many smoothing steps with respect to the H- and U l -norm. In particular, there are level- and mesh-independent constants C and ~ C such that l u new l k U l l k U l and l u new l k H ~ l Proof. Applying Lemma 2.1 we have l l (p l l C l l k U l Taking into consideration the norm equivalence (11) and Lemma 2.2, we conclude l u new l k U l Cku l u new l k H l kA l (u l ~ l (p l ~p l l k U l which proves convergence in the U l -norm for suciently many smoothing steps m. The convergence in the H-norm follows from the norm equivalence (11). | Remark 2.3 Note that in our context the notation \two-level" does not necessarily mean that we consider two levels of mesh renements. Thus, Theorem 2.1 is also applicable in cases where we have two dierent nite element discretisations on the same mesh (see Sections 4.2 and 4.3). Once proven the convergence of the two-level method, the convergence of the W-cycle multi-level method follows in a standard way (see e.g. [4], [17]). This is also true for a combination of a nite number of dierent two-level algorithms provided that Theorem 2.1 holds true for each of these two-level methods. 3 A General Transfer Operator In this section we will describe general nite element spaces V l and we will show how then a space l and a transfer operator and the assumptions (H3) and (H4) can be constructed. In this way, the transfer operator can be applied on a large class of nite elements, including in case of the Stokes problem, for example, also velocity spaces generated by vector-valued basis functions [16]. We denote by fT l g l0 a family of triangulations of the domain Each triangulation l consists of a nite number of mutually disjoint simply connected open cells K so that we have K2T l We assume that (H6) For any l > 0, the triangulation T l is obtained from T l 1 by some \renement", i.e., each cell K 2 T l 1 is either a member of T l or it has been rened into child cells In particular, we allow T l l , which gives us the additional possibility to dene dierent nite element spaces V l 1 and V l on the same mesh. We assume that there exists a nite number of reference domains b Kc M such that, for any level l 0 and for any cell K 2 T l , there exists Mg and a one-to-one mapping FK 2 W 1;1 ( b We assume that are independent of K and l. In addition, we suppose that and BK is a ball again with C independent of K, K 0 and l. The validity of (38)-(40) usually follows from some shape-regularity assumption on the triangulations. Particularly, for simplicial trian- gulations, the condition (40) already guarantees a shape-regularity of the cells and implies (38). Moreover, if the triangulations satisfy usual compatibility assumptions (cf. e.g. [14]), then (39) also follows. In case of hanging nodes, the dierence in the renement levels of neighbouring elements is limited by (39). On each reference cell b M , we introduce a nite-dimensional space b . Employing the mappings FK from (37), we introduce, for any cell K 2 T l , a nite{dimensional space P l (K) H 1 (K) d such that We denote by 'K;j , functions in P l (K) satisfying Usually, the coecients aK;jk are zeros or unit vectors in the direction of coordinate axes, and one takes only one non-vanishing coecient for each j. However, in some cases, also other choices of the coecients aK;jk may be of use. Particularly, this is the case if the space P l (K) contains vector-valued basis functions which cannot be obtained by transforming xed basis functions from the space [ b K. As an example may serve the Bernardi/Raugel element (cf. [2]) which contains vector-valued basis functions perpendicular to element faces. We introduce linear functionals fNK;i g dimP l (K) dened on P l (K), which we will call local nodal functionals in the following. We assume that these functionals possess the usual duality property with respect to the basis functions, i.e., denotes the Kronecker symbol. Examples of such nodal functionals can be found in Section 4. Now, for each level l, we introduce a nite element space V l satisfying This space is smaller than the piecewise discontinuous space on the right-hand side of (41) since we assume that there is some connection between functions on neighbouring cells. This connection can be enforced by choosing pairs of nodal functionals from neighbouring cells and by requiring that the two functionals from any pair are equal for any function from the space V l . We denote by f' l;i g i2I l a basis of the space V l obtained in this way, where the set I l is an index set whose elements will be called nodes in the following. We assume that the mentioned pairs of local nodal functionals were chosen in such a way that, as usual, these basis functions have \small" supports. Precisely, we require that, for any l , there exists K 2 T l such that Further, we assume that, on any cell K 2 T l , the basis functions ' l;i coincide with the local basis functions 'K;j . Thus, denoting by I l the set of all nodes which are associated with a cell K, we can introduce a one-to-one mapping such that Using the mappings K we can renumber the local nodal functionals, i.e., we set Then Furthermore, we dene for any node j 2 I l which is the set of all cells K 2 T l which are connected with the node j. Then we can give a precise characterization of the space V l , namely, This shows that a natural choice for a global nodal functional is the arithmetic mean of the local nodal functionals K2T l;i NK;i (wj K Note that we again have the duality relation which implies that i2I l having described the spaces V l , we can turn to the question how the spaces l and the mappings satisfying (36), (H3) and (H4) can be dened. Following the ideas developed in [29], where the construction of general transfer operators for nite element spaces has been investigated, we construct l as a discontinuous nite element space l with a nite dimension. To guarantee (36), we furthermore assume that (H7) The local function space S l (K) is constructed such that is the parent cell of a child cell K 2 T l for We suppose that, for any K 2 T l , the local nodal functionals NK;i are well dened on l (K), which usually means that the functions in S l (K) are of the same type as those in l (K) (e.g., continuous). Then we can dene the transfer operator simply by i2I l In view of (44) we immediately obtain the validity of (H3). To be able to prove (H4), we assume that each of the spaces S l (K) can be obtained by transforming functions from one of the reference cells onto K. Thus, we introduce nite element spaces b M , with bases f b and we assume that, for any K 2 T l , there exists Mg such that l where FK is the mapping from (37) and C is independent of K and l. The last assumption is automatically satised if the local nodal functionals NK;i are dened by means of a nite number of functionals dened on the reference spaces b which is often the case. Lemma 3.1 The operator u dened by (45) is uniformly L 2 stable, i.e., it satises (H4) with a constant C independent of l and with k k Proof. Consider any w 2 l and any K 2 T l and let FK and b S i be the mapping and the space from (46), respectively. Then for some real numbers w j and it follows from (47), from the equivalence of norms on nite-dimensional spaces and from (38) that, for any m 2 I l (K), jNK;m (wj K )j C Thus, we see that, for any i 2 I l , we have jN l;i (w)j C K2T l;i For any cell e l , we derive using (38), (48), (42) and (39) that k u wk 0; e i2I l ( e e i2I l ( e jN l;i (w)j e where we denoted by i2I l ( e K2T l;i the vicinity of any cell e l . In view of (39), (40) and (42), the number of cells in ( e is bounded independently of e K and l and hence we obtain which expresses a local L 2 stability of the operator u . Again, according to (39), (40) and (42), the number of the sets ( e K) which contain any xed cell K 2 T l is bounded independently of l and hence the local L 2 stability of u immediately implies the global L 2 stability (H4). | A signicant step in the above considerations was the assumption that there exist spaces l (K) satisfying (H7) and (46). In the remaining part of this section, we will prove that this is true for simplicial nite elements. Thus, from now on, we assume that there is only one reference cell b K, which is a d-simplex. It is essential for our further proceeding that, for any simplicial cell K, there exists a regular ane mapping K onto K. We denote the set of all these mappings by LR ( b K;K). above, we introduce a xed nite{dimensional space b K) d and we assume that, for any K 2 T l and any FK 2 LR ( b K;K), we have l (K) fbv F 1 We assume that, as usual, the cells of the triangulations are rened according to a nite number of geometrical rules. Therefore, the renements of all cells K can be mapped by the linear mappings F 1 K onto a nite number b L of renements of the reference cell b K into child cells b L. That means that, for any cell K 2 T l obtained by rening a parent cell F(K) 2 T l 1 and for any FF(K) 2 LR ( b K;F(K)), there exist indices such that We denote Then, for any v 2 P l 1 there exists b ij such that P and hence Thus, for any K 2 T l and any FK 2 LR ( b K;K), we have l 1 for some indices i g. Now, we dene c and, for any K 2 T l , we choose some FK 2 LR ( b K;K) and set l It is easy to see that these spaces S l (K) satisfy the assumption (H7). 4 Applications to the Stokes Equations In this section we give details how the general assumptions made in Section 3 can be fullled for the Stokes problem. In particular, we will show that the usually used multi-grid technique for the nonconforming nite elements of lowest order coincides with the use of the general transfer operator described in the preceding section. 4.1 Lowest Order Nonconforming Elements Our rst examples are the nonconforming nite elements of rst order on triangles and quadrilaterals. The triangular element was introduced by Crouzeix and Raviart and analysed in [15]. The element on quadrilaterals was established by Rannacher and Turek and analysed in [24, 28]. We consider a hierarchy of uniformly rened grids. Let T 0 be a regular triangulation of triangles or into convex quadrilaterals. The mesh T l is obtained from T l 1 by subdividing each cell of T l 1 into four child cells. For triangles, we connect the midpoints of the edges. In the quadrilateral case, we connect the midpoints of opposite edges. Now we construct the nite element spaces V l . Let P 1 (K) be the space of linear polynomials on the triangle K. The space of rotated bilinear functions on a quadrilateral K is dened by is the bilinear reference transformation from the reference cell ^ onto the cell K, see [24, 28]. Let E(K) denote the set of all edges of the element K. We dene for any E 2 E(K) the nodal functional Z vj K ds: K2T l E(K) be the set of all edges of T l . The set E l @ @ contains all boundary edges. The nite element space V l is given by @ where P (K) is P 1 (K) on triangles and Q rot 1 (K) on quadrilaterals. Since the triangulation T l is obtained from T l 1 by regular renement of all cells, we have us show that the assumptions (H7) and (46) hold. In the case of triangles, @ @ @ @ FK Figure 1: Renement of original and reference cell the inclusion P l 1 directly from the ane reference mapping and we have P l 1 . Thus, we can choose S l The situation is more complicated for the Q rot element. Let us consider K l be the child cells of K. We set ^ Fig. 1). Let for example which means that there is a function K1 The mapping G and FK are bijective. Thus, we have FK 1 K1 Since the local space span(1; ^ 2 ) is invariant with respect to the mapping G 1 we conclude P l (K 1 ). The same arguments can be applied to K i which results in l l (K) Q rot This allows us to choose S l in the denition of the space l . Note that the assumptions (46) and (47) are then fullled. The nite element spaces V l 1 and V l are non{nested. In order to get a suitable prolongation we will use the general transfer operator which was introduced in Section 3. The denition of the global nodal functionals (43) simplies for the above spaces to for all inner edges E with the neighbouring elements K and K 0 . For boundary edges we get E(K). The resulting mapping u is the same as in [19]. The space Q l which approximates the pressure consists of piecewise constant functions, i.e., For the proof of assumption (H2) we refer to [15, 24]. Since Q l 1 Q l the transfer operator p is the identity. For numerical experiments with the Crouzeix-Raviart element we refer to [19]. 4.2 Modied Crouzeix-Raviart Element It is sometimes necessary to use other types of boundary conditions than the Dirichlet boundary condition considered in (3). For example, if a part N of @ represents a free surface of a uid, then one can use the boundary conditions where I is the identity tensor and n is the outer normal vector to N . The rst condition in (50) states that zero surface forces act in the tangential direction to N . The boundary conditions (50) generally do not allow to use the bilinear form a dened in (4) and instead we have to consider Z If meas d 1 (@ then the ellipticity of this bilinear form for functions from H d vanishing on @ n N is assured by the Korn inequality. However, the discrete Korn inequality does not hold for most rst order nonconforming nite element spaces (cf. [21]), particularly, it fails for the elements investigated in the preceding section. Consequently, in these cases, the validity of (H2) cannot be shown. One of the few rst order nonconforming elements which do not violate the discrete Korn inequality is the modied Crouzeix-Raviart element P mod which was developed in [23] for solving convection dominated problems. Here we conne ourselves to a particular example of this element for which the space of shape functions on the reference triangle b is are the barycentric coordinates on b K. To each edge b K, we assign two nodal functionals, b E;1 and b E;2 , dened by Z Z E2 Figure 2: Coarsest grids (level 0). E is a barycentric coordinate on b E. The space V l is now obtained by transforming the space b P and the six nodal functionals b K, onto the cells of the triangulation by means of regular ane mappings (cf. the preceding section). In this way, we get the space Z denotes the jump of v across the edge E, and 1 , 2 , 3 are the barycentric coordinates on the cell K. A nice property of the P modelement is that Z It is easy to verify that the assumptions made in Section 3 are satised. The assumption holds if the space Q l consists of discontinuous piecewise linear functions from L 2 and each cell has at least one vertex in since then an inf-sup condition holds (cf. [22] and [15]). Consequently, (H2) is also satised if Q l is a subspace of L 2 consisting e.g. of piecewise constant functions, continuous piecewise linear functions or nonconforming piecewise linear functions. Here we present numerical results for Q l consisting of piecewise constant functions so that the transfer operator p can be replaced by the identity. The discretisations are dened on a sequence of uniformly rened triangular grids starting with the triangular grid from Fig. 2 which we denote as T 0 (level 0). We shall consider two kinds of multi-level solvers for solving the Stokes equations on a given geometrical mesh level. The rst one is a classical multi-level method where each mesh corresponds to one level of the algorithm and on each mesh we consider a discrete problem of the same type, i.e., the Stokes equations discretized using the P mod element. In the second multi-level solver, the discretisation using the element is used on the nest mesh T L only and on all other mesh levels a 'cheaper' discretisation is employed, namely the Crouzeix-Raviart element with piecewise constant pressure denoted as P nc In addition, the P nc discretisation is also applied on the nest mesh level T L so that two dierent discretisations corresponding to two levels of the multi-level method are considered on the nest geometrical mesh. We refer to the next section for more details on this technique. Note that both kinds of multi-level solvers t into the framework of this paper and the prolongations and restrictions can be dened using the general transfer operator u described above. The numbers of degrees of freedom to which the mentioned discretisations lead on dierent meshes are given in Table 3. As a smoother, we use the basic iteration described in Section 2.5. Therefore, the system (13) has to be solved in each smoothing step which implies that the ecient solution of is essential for the eciency of the multi-level solver. In view of an application of our procedure to the Navier-Stokes equations, where A and D are no longer symmetric, we solve (13) iteratively by a preconditioned exible GMRES method, see e.g. [25]. The preconditioner is dened via the pressure Schur complement equation of (13) is the right-hand side of (13). First, (51) is solved approximately by some steps of the standard GMRES method by Saad and Schultz [26] providing the approximation ~ p of Second, an approximation ~ u of u j+1 u j is computed by ~ The solution of (13) is the most time consuming part of the multi-grid iteration. It was shown by Zulehner [34] that the smoothing property of the Braess{Sarazin smoother is maintained if (13) is solved only approximately as long as the approximation is close enough to the solution. Numerical experiments show that, in general, one obtains similar rates of convergence of the multi-level algorithm like with an exact solution of (13). Considering the ecency of the multi-level solver measured in computing time, the variant with the approximate solution is in general considerably better and therefore it is used in practice. Thus, in the computations presented below we stopped the solution of (13) after the Euclidean norm of the residual has been reduced by the factor 10. The systems on level 0 were solved exactly. We shall present numerical results for the following Example 4.1 W-cycle in a 2d test case. We consider the Stokes problem in with the prescribed solution This example is taken from the paper [5] by Braess and Sarazin. The computations were performed using the W (m; m)-cycle, 1:0. Table 1 shows the geometric means of the error reduction rates l u new l k 0 l k 0 for the multi-level solver which uses the P mod discretisations on all levels. Table 2 shows the averaged error reduction rates for the multi-level solver which combines the use of the P mod discretisations on the nest level with the use of P nc discretisations on all lower levels. We observe that, for each m, the error reduction rates can be bounded by a level-independent constant as predicted by Theorem 2.1. In addition, it can be clearly seen from the two tables that, for m > 1, the multi-level method which uses the Crouzeix- Raviart element on lower levels converges faster than the standard multi-level method. Table 1: Example 4.1, P mod averaged error reduction rates. mesh level 5 2.20e-1 2.87e-1 3.13e-1 3.13e-1 3.08e-1 6 1.70e-1 2.36e-1 2.62e-1 2.62e-1 2.59e-1 Table 2: Example 4.1, P mod nest level combined with P nc lower levels, averaged error reduction rates. mesh level 6 6.17e-2 1.27e-1 1.06e-1 1.01e-1 1.02e-1 4.3 A Multi-grid Method for Higher Order Discretisations Based on Lowest Order Nonconforming Discretisations As the last and certainly most important application, we consider a multi-grid method for higher order discretisations which is based on lowest order nonconforming discretisations on the coarser multi-grid levels. Like in the previous section, the hierarchy of this multi-grid has two dierent discretisations on the nest geometric mesh level L, i.e. the spaces for the multi-grid levels L and L+1 are dened both on T L , see Figure 3. The higher order discretisation is used on multi-grid level on all coarser levels l, 0 l L, a nonconforming discretisation of lowest order is applied. level geometry multigrid discretization higher order low order nonconforming low order nonconforming low order nonconforming low order nonconforming Figure 3: The multi-level approach for higher order discretisations. Remark 4.1 The construction of this multi-grid method was inspired by a numerical study of a benchmark problem for the steady state Navier-Stokes equations in [18]. This study shows on the one hand a dramatic improvement of benchmark reference values using higher order discretisations in comparison to lowest order nonconforming discretisations. On the other hand, the arising systems of equations for the lowest order nonconforming discretisations could be solved very fast and eciently with the standard multi-grid approach. This standard multi-grid approach showed a very unsatisfactory behaviour for all higher order discretisations. Often, it did not converge at all. Sometimes, convergence could be achieved by heavily damping, leading to a bad rate of convergence and a very inecient solver. These diculties could be overcome by applying the multi-grid method described in this section. Thus, this method has been proved already to be a powerful tool in the numerical solution of Navier-Stokes equations combining the superior accuracy of higher order discretisations and the eciency of multi-grid solvers for lowest order nonconforming discretisations. In case of the Stokes problem, the proposed multi-grid method ts into the framework of this paper. Between the lower levels l and l +1 of the multi-grid hierarchy, 0 l < L, we Table 3: Degrees of freedom for the discretisations in the 2d tests cases, Example 4.1 and 4.2. disc. mesh level use the transfer operators which have been dened in Section 4.1. To dene the transfer operator between the levels L and L have to be constructed since in general V L 6 V L+1 and for some higher order discretisations also Q L 6 Q L+1 . The construction can be done by applying the techniques from Section 3. We choose where P L+1 (K) is the space of local shape functions of the corresponding nite element spaces V L+1 and Q L+1 , respectively. We will present numerical results obtained with this multi-grid technique for a number of higher order nite element discretisations. Let P 0 and Q 0 denote the spaces of piecewise constant functions on simplicial mesh cells and quadrilateral/hexahedral mesh cells, respectively. By we denote the well-known nite element spaces of continuous functions of piecewise k-th degree. The notation P disc k is used for spaces of discontinuous functions whose restriction to each mesh cell is a polynomial of degree k. The Crouzeix-Raviart nite element space is again denoted by P nc 1 . Three examples for the multi-level approach for higher order discretisations will be considered. The rst example conrms the theoretical results of Theorem 2.1, i.e., we consider a two-level method and solve the systems (13) arising in the smoothing process exactly. The solution procedure was decribed in the preceding section. Then we consider Example 4.1 from the previous section and Example 4.4 dened below to demonstrate the behaviour of the multi-level W-cycle with approximated solutions of (13) for test problems in 2d and 3d. Example 4.2 Two-level method. This example has been designed to check the theoretically predicted results with respect to the two-level method. We consider the Stokes equations (3) in 0, such that is the solution of (3). The computations were carried out on a sequence of meshes starting with level 0 (see Fig. 2), for which the corresponding numbers of the degrees of freedom are given in Table 3. The discrete solution on each level is u we consider a two-level method, there is only one mesh level in a specic computation. On this geometric mesh level, the lowest order nonconforming discretisation of (3) denes the coarse level in the two-level method and a higher order discretisation the ne level. As initial guess of the two-level method for each computation, we have chosen u for all interior degrees of freedom and p Thus, the initial error is smooth. The results presented in Table 4 were obtained with 3 pre-smoothing steps, without post-smoothing, and with Thus, we have exactly the situation investigated in Section 2. The results of the numerical studies are given in Table 4 and Figure 4. Table 4 shows the averaged error reduction rate for the two-level method applied to a number of higher order discretisations. It can be clearly seen that the error reduction rate is for all discretisations independent of the level as stated in Theorem 2.1. The second statement of Theorem 2.1, the decrease of the error reduction rate by O(1=m), where m is the number of pre-smoothing steps, is illustrated for the P 3 =P 2 discretisation on mesh level 4 in Figure 4. The situation is quite similar for the other higher order discretisations. Table 4: Example 4.2, pre-smoothing steps, no post-smoothing, averaged error reduction rates. disc. mesh level Example 4.3 Example 4.1 continued Now let us turn back to Example 4.1 formulated in the preceding section. Table 5 shows the averaged error reduction rates obtained for higher order discretisations with the W (1; 1)-cycle, solving by the exible GMRES method the reduction of the Euclidean norm of the residual by the achieved with the rst exible GMRES steps. We applied 10 iteration steps each time in the GMRES method for solving the pressure Schur complement equation (51). Although system (13) is solved only approximately in each smoothing step, the results in Table 5 show averaged error reduction rates which are independent of the level. number of pre smoothing steps averaged error reduction rate Figure 4: Example 4.2, averaged error reduction rate for dierent numbers of pre-smoothing steps, P 3 =P 2 discretisation, mesh level 4. Table 5: Example 4.3, averaged error reduction rates. disc. mesh level 3 2.29e-1 2.27e-1 2.25e-1 2.21e-1 2.18e-1 Example 4.4 W-cycle in a 3d test case. This example demonstrates the behaviour of the multi-level solver applied to a three-dimensional problem. We consider the prescribed solution given by The constant c is chosen such that p 2 L 2 and the right hand side f is chosen such that (u; p) full (3). For discretisations based on hexahedra, the initial grid (level 0) was obtained by dividing the unit cube into eight cubes of edge length 1=2 as indicated in Figure 5. The initial Figure 5: Meshlevel 0 (left) and 1 (right) grid for discretisations based on simplicial mesh cells consists of 48 tetrahedrons. The corresponding numbers of degrees of freedom are given in Table 6. It is noteworthy that sometimes a lower order nite element space possesses more degrees of freedom than a higher order space on the same mesh level, e.g. compare P nc Table Degrees of freedom for the discretisations in the 3d test case, Example 4.4. disc. mesh level Table 7 presents results obtained with the W (3; 3)-cycle, The saddle point problems in the smoothing process (13) were solved up to a reduction of the Euclidean norm of the residual by a factor 10 4 or at most 20 exible GMRES iterations. Computations with weaker stopping criteria did not lead to such clearly level-independent rates of convergence as presented in Table 7. This behaviour corresponds to the theory by Zulehner [34] which is valid if the approximation of the solution of (13) is close enough to the solution. 4.4 Acknowledgement The research of Petr Knobloch has been supported under the grants No. 201/99/P029 and 201/99/0267 of the Czech Grant Agency and by the grant MSM 113200007. The research Table 7: Example 4.4, averaged error reduction rate disc. mesh level of Gunar Matthies has been partially supported under the grant FOR 301 of the DFG. --R A class of iterative methods for solving saddle point problems. Analysis of some A multigrid algorithm for the mortar A multigrid method for nonconforming fe- discretisations with applications to non-matching grids Multigrid Methods An optimal-order multigrid method for P 1 nonconforming nite elements A nonconforming multigrid method for the stationary Stokes equations. On the convergence of Galerkin-multigrid methods for nonconforming nite ele- ments On the convergence of nonnested multigrid methods with nested spaces on coarse grids. Multigrid algorithms for the nonconforming and mixed methods for nonsymmetric and inde Multigrid and multilevel methods for nonconforming Q 1 elements. Conforming and nonconforming Finite element methods for Navier-Stokes equations Higher order A coupled multigrid method for nonconforming Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations On Korn's inequality for nonconforming On the inf-sup condition for the P mod 1 element The new nonconforming Simple nonconforming quadrilateral Stokes element. A exible inner-outer preconditioned GMRES algorithm GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Eine Klasse von e Parallele L A general transfer operator for arbitrary Multigrid Methods for Finite Elements. Multigrid techniques for a divergence-free nite element discretization The derivation of minimal support basis functions for the discrete divergence operator. A class of smoothers for saddle point problems. --TR GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems Multigrid methods for nonconforming finite element methods A flexible inner-outer preconditioned GMRES algorithm The derivation of minimal support basis functions for the discrete divergence operator An efficient smoother for the Stokes problem Multigrid and multilevel methods for nonconforming <italic>Q</italic><subscrpt>1</subscrpt> elements Multigrid Algorithms for Nonconforming and Mixed Methods for Nonsymmetric and Indefinite Problems Convergence of nonconforming multigrid methods without full elliptic regularity A multigrid method for nonconforming FE-discretisations with application to non-matching grids A Multigrid Algorithm for the Mortar Finite Element Method A coupled multigrid method for nonconforming finite element discretizations of the 2D-Stokes equation A class of smoothers for saddle point problems Finite Element Method for Elliptic Problems --CTR M. Jung , T. D. Todorov, Isoparametric multigrid method for reaction-diffusion equations on two-dimensional domains, Applied Numerical Mathematics, v.56 n.12, p.1570-1583, 12 December 2006

finite element discretisation;multi-level method;stokes problem;mixed problems

635802

Approximating most specific concepts in description logics with existential restrictions.

Computing the most specific concept (msc) is an inference task that allows to abstract from individuals defined in description logic (DL) knowledge bases. For DLs that allow for existential restrictions or number restrictions, however, the msc need not exist unless one allows for cyclic concepts interpreted with the greatest fixed-point semantics. Since such concepts cannot be handled by current DL-systems. we propose to approximate the msc. We show that for the DL ALE, which has concept conjunction, a restricted form of negation, existential restrictions, and value restrictions as constructors, approximations of the msc always exist and can effectively be computed.

Introduction The most specic concept (msc) of an individual b is a concept description that has b as instance and is the least concept description (w.r.t. subsump- tion) with this property. Roughly speaking, the msc is the concept description that, among all concept descriptions of a given DL, represents b best. Closely related to the msc is the least common subsumer (lcs), which, given concept descriptions is the least concept description (w.r.t. subsumption) subsuming Thus, where the msc generalizes an individual, the lcs generalizes a set of concept descriptions. In [2{4], the msc (rst introduced in [15]) and the lcs (rst introduced in [5]) have been proposed to support the bottom-up construction of a knowledge base. The motivation comes from an application in chemical process engineering [17], where the process engineers construct the knowledge base (which consists of descriptions of standard building blocks of process models) as follows: First, they introduce several \typical" examples of a standard building block as individuals, and then they generalize (the descriptions of) these individuals into a concept description that (i) has all the individuals as instances, and (ii) is the most This work was carried out while the author was still at the LuFG Theoretische Informatik, RWTH Aachen, Germany. specic description satisfying property (i). The task of computing a concept description satisfying (i) and (ii) can be split into two subtasks: computing the msc of a single individual, and computing the lcs of a given nite number of concepts. The resulting concept description is then presented to the knowledge engineer, who can trim the description according to his needs before adding it to the knowledge base. The lcs has been thoroughly investigated for (sublanguages of) Classic [5, 2, 12, 11], for DLs allowing for existential restrictions like ALE [3], and most recently, for ALEN , a DL allowing for both existential and number restrictions [13]. For all these DLs, except for Classic in case attributes are interpreted as total functions [12], it has turned out that the lcs always exists and that it can eectively be computed. Prototypical implementations show that the lcs algorithms behave quite well in practice [7, 4]. For the msc, the situation is not that rosy. For DLs allowing for number restrictions or existential restrictions, the msc does not exist in general. Hence, the rst step in the bottom-up construction, namely computing the msc, cannot be performed. In [2], it has been shown that for ALN , a sublanguage of Classic, the existence of the msc can be guaranteed if one allows for cyclic concept descriptions, i.e., concepts with cyclic denitions, interpreted by the greatest xed-point semantics. Most likely, such concept descriptions would also guarantee the existence of the msc in DLs with existential restrictions. However, current DL-systems, like FaCT [10] and RACE [9], do not support this kind of cyclic concept descriptions: although they allow for cyclic denitions of concepts, these systems do not employ the greatest xed-point semantics, but descriptive semantics. Consequently, cyclic concept descriptions returned by algorithms computing the msc cannot be processed by these systems. In this paper, we therefore propose to approximate the msc. Roughly speak- ing, for some given non-negative integer k, the k-approximation of the msc of an individual b is the least concept description (w.r.t. subsumption) among all concept descriptions with b as instance and role depth at most k. That is, the set of potential most specic concepts is restricted to the set of concept descriptions with role depth bounded by k. For (sublanguages of) ALE we show that k-approximations always exist and that they can eectively be computed. Thus, when replacing \msc" by \k-approximation", the rst step of the bottom-up construction can always be carried out. Although the original outcome of this step is only approximated, this might in fact suce as a rst suggestion to the knowledge engineer. While for full ALE our k-approximation algorithm is of questionable practical use (since it employs a simple enumeration argument), we propose improved algorithms for the sublanguages EL and EL: of ALE . (EL allows for conjunction and existential restrictions, and EL: additionally allows for a restricted form of negation.) Our approach for computing k-approximations in these sublanguages is based on representing concept descriptions by certain trees and ABoxes by certain (systems of) graphs, and then characterizing instance relationships by homomorphisms from trees into graphs. The k-approximation operation then consists in unraveling the graphs into trees and translating them back into con- Table 1. Syntax and semantics of concept descriptions. Construct name Syntax Semantics EL EL: ALE top-concept > x x x conjunction C u D C I \ D I x x x existential restriction 9r:C fx 2 primitive negation :P n P I x x value restriction 8r:C fx 2 cept descriptions. In case the unraveling yields nite trees, the corresponding concept descriptions are \exact" most specic concepts, showing that in this case the msc exists. Otherwise, pruning the innite trees on level k yields k- approximations of the most specic concepts. The outline of the paper is as follows. In the next section, we introduce the basic notions and formally dene k-approximations. To get started, in Section 3 we present the characterization of instance relationships in EL and show how this can be employed to compute k-approximations or (if it exists) the msc. In the subsequent section we extend the results to EL: , and nally deal with ALE in Section 5. The paper concludes with some remarks on future work. Due to space limitations, we refer to [14] for all technical details and complete proofs. Preliminaries Concept descriptions are inductively dened with the help of a set of construc- tors, starting with disjoint sets NC of concept names and NR of role names. In this work, we consider concept descriptions built from the constructors shown in Table 1, where r 2 NR denotes a role name, P 2 NC a concept name, and C; D concept descriptions. The concept descriptions in the DLs EL, EL: , and ALE are built using certain subsets of these constructors, as shown in the last three columns of Table 1. An ABox A is a nite set of assertions of the form (a; (role assertions) or are individuals from a set N I (disjoint from role name, and C is a concept description. An ABox is called L-ABox if all concept descriptions occurring in A are L-concept descriptions. The semantics of a concept description is dened in terms of an interpretation The domain of I is a non-empty set of objects and the interpretation function I maps each concept name P 2 NC to a set P I , each role name r 2 NR to a binary relation r I , and each individual I to an element a I 2 such that a 6= b implies a I 6= b I (unique name assumption). The extension of I to arbitrary concept descriptions is inductively dened, as shown in the third column of Table 1. An interpretation I is a model of an ABox A i it satises (a I ; b I I for all role assertions (a; and a I 2 C I for all concept assertions a : C 2 A. The most important traditional inference services provided by DL-systems are computing the subsumption hierarchy and instance relationships. The concept description C is subsumed by the concept description D (C v D) i C I D I for all interpretations I; C and D are equivalent (C D) i they subsume each other. An individual a 2 N I is an instance of C w.r.t. A (a 2A C) i a I 2 C I for all models I of A. In this paper, we are interested in the computation of most specic concepts and their approximation via concept descriptions of limited depth. The depth depth(C) of a concept description C is dened as the maximum of nested quantiers in C. We also need to introduce least common subsumers formally. Denition 1 (msc, k-approximation, lcs). Let A be an L-ABox, a an individual in A, C; C { C is the most specic concept (msc) of a w.r.t. A (mscA (a)) i a 2A C, and for all L-concept descriptions C 0 , a 2A C 0 implies C v C { C is the k-approximation of a w.r.t. A (msc k;A (a)) i a 2A C, depth(C) k, and for all L-concept descriptions C 0 , a 2A C 0 and depth(C 0 ) k imply { C is the least common subsumer (lcs) of C descriptions C for all Note that by denition, most specic concepts, k-approximations, and least common subsumers are uniquely determined up to equivalence (if they exist). The following example shows that in DLs allowing for existential restrictions the msc of an ABox-individual b need not exist. Example 1. Let L be one of the DLs EL, EL: , or ALE . Consider the L-ABox rg. It is easy to see that, for each n 0, b is an instance of the L-concept description Cn := 9r: 9r | {z } times The msc of b can be written as the innite conjunction un0 Cn , which, however, cannot be expressed by a (nite) L-concept description. As we will see, the k- approximation of b is the L-concept description u 0nk Cn . 3 Most specic concepts in EL In the following subsection, we introduce the characterization of instance relationships in EL, which yields the basis for the algorithm computing k-approxi- mations (Section 3.2). All results presented in this section are rather straightfor- ward. However, they prepare the ground for the more involved technical problems one encounters for EL: . 3.1 Characterizing instance in EL In order to characterize instance relationships, we need to introduce description graphs (representing ABoxes) and description trees (representing concept descriptions). For EL, an EL-description graph is a labeled graph of the form whose edges vrw 2 E are labeled with role names r 2 NR and whose nodes v 2 V are labeled with sets '(v) of concept names from NC . The empty label corresponds to the top-concept. An EL-description tree is of the is an EL-description graph which is a tree with root v EL-concept descriptions can be turned into EL-description trees and vice versa [3]: Every EL-concept description C can be written (modulo equivalence) as f>g. Such a concept description is recursively translated into an EL-description tree follows: The set of all concept names P i occurring on the top-level of C yields the label '(v 0 ) of the root v 0 , and each existential restriction yields an r i -successor that is the root of the tree corresponding to C i . For example, the concept description yields the description tree depicted on the left hand side of Figure 1. Every EL-description tree translated into an EL-concept description CG as follows: the concept names occurring in the label of v 0 yield the concept names in the top-level conjunction of C G , and each v 0 an existential restriction 9r:C, where C is the EL-concept description obtained by translating the subtree of G with root v. For a leaf v 2 V , the empty label is translated into the top-concept. Adapting the translation of EL-concept descriptions into EL-description trees, an EL-ABox A is translated into an EL-description graph G(A) as follows: Let Ind(A) denote the set of all individuals occurring in A. For each a 2 Ind(A), let C a := u a:D2A D, if there exists a concept assertion a : D 2 A, otherwise, C a := >. Let G(C a denote the EL-description tree obtained from C a . (Note that the individual a is dened to be the root of G(C a ); in particular, a 2 V a .) W.l.o.g. let the sets V a , a 2 Ind(A) be pairwise disjoint. Then, is dened by { V := { { '(v) := ' a (v) for v 2 V a . As an example, consider the EL-ABox The corresponding EL-description graph G(A) is depicted on the right hand side of Figure 1. r s s r r r s s r r s Fig. 1. The EL-description tree of C and the EL-description graph of A. Now, an instance relationship, a 2A C, in EL can be characterized in terms of a homomorphism from the description tree of C into the description graph of A: a homomorphism from an EL-description tree into an EL-description graph VH such that Theorem 1. [14] Let A be an EL-ABox, a 2 Ind(A) an individual in A, and C an EL-concept description. Let G(A) denote the EL-description graph of A and G(C) the EL-description tree of C. Then, a 2A C i there exists a homomorphism ' from G(C) into G(A) such that '(v 0 is the root of G(C). In our example, a is an instance of C, since mapping v 0 on a, v i on w i , and v 3 on b and v 4 on c yields a homomorphism from G(C) into G(A). Theorem 1 is a special case of the characterization of subsumption between simple conceptual graphs [6], and of the characterization of containment of conjunctive queries [1]. In these more general settings, testing for the existence of homomorphisms is an NP-complete problem. In the restricted case of testing homomorphisms mapping trees into graphs, the problem is polynomial [8]. Thus, as corollary of Theorem 1, we obtain the following complexity result. Corollary 1. The instance problem for EL can be decided in polynomial time. Theorem 1 generalizes the following characterization of subsumption in EL introduced in [3]. This characterization uses homomorphisms between description trees, which are dened just as homomorphisms from description trees into description graphs, but where we additionally require to map roots onto roots. Theorem 2. [3] Let C, D be EL-concept descriptions, and let G(C) and G(D) be the corresponding description trees. Then, C v D i there exists a homomorphism from G(D) into G(C). 3.2 Computing k-approximations in EL In the sequel, we assume A to be an EL-ABox, a an individual occurring in A, and k a non-negative integer. Roughly, our algorithm computing msc k;A (a) works as follows: First, the description graph G(A) is unraveled into a tree with root a. This tree, denoted T (a; G(A)), has a nite branching factor, but possibly innitely long paths. Pruning all paths to length k yields an EL-description tree T k (a; G(A)) of depth k. Using Theorem 1 and Theorem 2, one can show that the EL-concept description C Tk (a;G(A)) is equivalent to msc k;A (a). As an immediate consequence, in case T (a; G(A)) is nite, C T (a;G(A)) yields the msc of a. In what follows, we dene T (a; G(A)) and T k (a; G(A)), and prove correctness of our algorithm. First, we need some notations: For an EL-description graph is a path from v 0 to vn of length and with label n. The empty path in which case the label of p is the empty word ". The node v n is an r 1 r n - successor of v 0 . Every node is an "-successor of itself. A node v is reachable from there exists a path from v 0 to v. The graph G is cyclic, if there exists a non-empty path from a node in G to itself. Denition 2. Let and a 2 V . The tree T (a; G) of a w.r.t. A is dened by T (a; { V T := fp j p is a path from a to some node in Gg, { { ' T (p) := '(v) if p is a path to v. For IN, the tree T k (a; G) of a w.r.t. G and k is dened by T k (a; G) := { V T { { ' T k . Now, we can show the main theorem of this section. Theorem 3. Let A be an EL-ABox, a 2 Ind(A), and k 2 IN. Then, C is the k-approximation of a w.r.t. A. If, starting from a, no cyclic path in A can be reached (i.e., T (a; G(A)) is nite), then C T (a;G(A)) is the msc of a w.r.t. A; otherwise no msc exists. Proof sketch. Obviously, there exists a homomorphism from T k (a; G(A)), a tree isomorphic to G(C with a mapped on a. By Theorem 1, this implies a 2A C Let C be an EL-concept description with a 2A C and depth(C) k. Theorem 1 implies that there exists a homomorphism ' from G(C) into G(A). Given ', it is easy to construct a homomorphism from G(C) into T k (a; G(A)). Thus, with Theorem 2, we conclude C C. Altogether, this shows that C is a k-approximation as claimed. Now, assume that, starting from a, a cycle can be reached in A, that is, (a; G(A)) is innite. Then, we have a decreasing chain C k-approximations C k ( C increasing depth k, k 0. From Theorem 2, we conclude that there does not exist an EL-concept description subsumed by all of these k-approximations (since such a concept description only has a xed and nite depth). Thus, a cannot have an msc. Conversely, if T (a; G(A)) is nite, say with depth k, from the observation that all k 0 -approximations, for k 0 k, are equivalent, it immediately follows that C T (a;G(A)) is the msc of a. ut Obviously, there exists a deterministic algorithm computing the k-approximation (i.e., C The size jAj of A is dened by a:C2A where the size jCj of C is dened as the sum of the number of occurrences of concept names, role names, and constructors in C. Similarly, one obtains an exponential complexity upper bound for computing the msc (if it exists). Corollary 2. For an EL-ABox A, an individual a 2 Ind(A), and k 2 IN, the k- approximation of a w.r.t. A always exists and can be computed in time O(jAj k ). The msc of a exists i starting from a no cycle can be reached in A. The existence of the msc can be decided in polynomial time, and if the msc exists, it can be computed in time exponential in the size of A. In the remainder of this section, we prove that the exponential upper bounds are tight. To this end, we show examples demonstrating that k-approximations and the msc may grow exponentially. Example 2. Let sg. The EL-description graph G(A) as well as the EL-description trees T 1 (a; G(A)) and T 2 (a; G(A)) are depicted in Figure 2. It is easy to see that, for k 1, T k (a; G(A)) yields a full binary tree of depth k where { each node is labeled with the empty set, and { each node except the leaves has one r- and one s-successor. By Theorem 3, C Tk (a;G(A)) is the k-approximation of a w.r.t. A. The size of Moreover, it is not hard to see that there does not exist an EL-concept description C which is equivalent to but smaller than C The following example illustrates that, if it exists, also the msc can be of exponential size. Example 3. For n Obviously, An is acyclic, and the size of An is linear in n. By Theorem 3, is the msc of a 1 w.r.t. An . It is easy to see that, for each n, coincides with the tree obtained in Example 2. As before we obtain that { C T (a1 ;G(A)) is of size exponential in jAn j; and { there does not exist an EL-concept description C equivalent to but smaller than C T (a1 ;G(A)) . r s r s Fig. 2. The EL-description graph and the EL-description trees from Example 2. Summarizing, we obtain the following lower bounds. Proposition 1. Let A be an EL-ABox, a 2 Ind(A), and k 2 IN. { The size of mscA;k (a) may grow with jAj k . { If it exists, the size of mscA (a) may grow exponentially in jAj. Most specic concepts in EL: Our goal is to obtain a characterization of the (k-approximation of the) msc in EL: analogously to the one given in Theorem 3 for EL. To achieve this goal, rst the notions of description graph and of description tree are extended from EL to EL: by allowing for subsets of NC [f:P j P 2 NC g[f?g as node labels. Just as for EL, there exists a 1{1 correspondence between EL: -concept descriptions and EL: -description trees, and an EL:-ABox A is translated into an EL: -description graph G(A) as described for EL-ABoxes. The notion of a homomorphism also remains unchanged for EL: , and the characterization of subsumption extends to EL: by just considering inconsistent EL: -concept descriptions as a special case: there exists a homomorphism ' from G(D) into G(C). Second, we have to cope with inconsistent EL:-ABoxes as a special case: for an inconsistent ABox A, a 2A C is valid for all concept descriptions C, and hence, mscA (a) ?. However, extending Theorem 1 with this special case does not yield a sound and complete characterization of instance relationships for EL: . If this was the case, we would get that the instance problem for EL: is in P, in contradiction to complexity results shown in [16], which imply that the instance problem for EL: is coNP-hard. The following example is an abstract version of an example given in [16]; it illustrates incompleteness of a nave extension of Theorem 1 from EL to EL: . Example 4. Consider the EL: -concept description the EL:-ABox rg; G(A) and G(C) are depicted in Figure 3. Obviously, there does not exist r r r r r r Fig. 3. The EL:-description graph and the EL:-description tree from Example 4. a homomorphism ' from G(C) into G(A) with '(w 0 neither For each model I of A, however, either b I I or I I , and in fact, a I 2 C I . Thus, a is an instance of C w.r.t. A though there does not exist a homomorphism ' from G(C) into G(A) with '(w 0 In the following section, we give a sound and complete characterization of instance relationships in EL: , which again yields the basis for the characterization of k-approximations given in Section 4.2. 4.1 Characterizing instance in EL: The reason for the problem illustrated in Example 4 is that, in general, for the individuals in the ABox it is not always xed whether they are instances of a given concept name or not. Thus, in order to obtain a sound and complete characterization analogous to Theorem 1, instead of G(A), one has to consider all so-called atomic completions of G(A). Denition 3 (Atomic completion). Let be an EL: -description graph and N '(v)g. An EL: -description graph G ) is an atomic completion of G if, for all 1. '(v) ' (v), 2. for all concept names P 2 N Note that by denition, all labels of nodes in completions do not contain a con- ict, i.e., the nodes are not labeled with a concept name and its negation. In particular, if G has a con icting node, then G does not have a completion. It is easy to see that an EL:-ABox A is inconsistent i G(A) contains a con icting node. For this reason, in the following characterization of the instance relation- ship, we do not need to distinguish between consistent and inconsistent ABoxes. Theorem 4. [14] Let A be an EL:-ABox, the corresponding description graph, C an EL: -concept description, the corresponding description tree, and a 2 Ind(A). Then, a 2A C i for each atomic completion G(A) of G(A), there exists a homomorphism ' from G(C) into G(A) with '(w 0 The problem of deciding whether there exists an atomic completion G(A) such that there exists no homomorphism from G(C) into G(A) is in coNP. Adding the coNP-hardness result obtained from [16], this shows Corollary 3. The instance problem for EL: is coNP-complete. 4.2 Computing k-approximations in EL: Not surprisingly, the algorithm computing the k-approximation/msc in EL does not yield the desired result for EL: . For instance, in Example 4, we would get But as we will see, mscA (a) As in the extension of the characterization of instance relationships from EL to EL: , we have to take into account all atomic completions instead of the single description graph G(A). Intuitively, one has to compute the least concept description for which there exists a homomorphism into each atomic completion of G(A). In fact, this can be done by applying the lcs operation on the set of all concept descriptions C obtained from the atomic completions G(A) of G(A). Theorem 5. Let A be an EL:-ABox, a 2 Ind(A), and k 2 IN. If A is inconsis- tent, then msc k;A (a) mscA (a) ?. Otherwise, let be the set of all atomic completions of G(A). Then, lcs(C If, starting from a, no cycle can be reached in A, then lcs(C T otherwise the msc does not exist. Proof sketch. Let A be a consistent EL:-ABox and the atomic completions of G(A). By denition of C there exists a homomorphism i from C denote the lcs of g. The characterization of subsumption for EL: yields homomorphisms ' i from G(C k ) into G(C Now it is easy to see that i ' i yields a homomorphism from G(C k ) into mapping the root of G(C k ) onto a. Hence, a 2A C k . Assume C 0 with depth(C 0 ) k and a 2A C 0 . By Theorem 4, there exist homomorphisms i from G(C 0 ) into G(A) i for all 1 i n, each mapping the root of G(C 0 ) onto a. Since depth(C 0 ) k, these homomorphisms immediately yield homomorphisms 0 i from G(C 0 ) into G(C the characterization of subsumption yields C and hence C k v C 0 . Thus, C k msc k;A (a). Analogously, in case starting from a, no cycle can be reached in A, we conclude (a). Otherwise, with the same argument as in the proof of Theorem 3, it follows that the msc does not exist. ut In Example 4, we obtain two atomic completions, namely fPg, and G(A) 2 with ' 2 (b 2 f:Pg. Now Theorem 5 implies mscA (a) which is equivalent to The examples showing the exponential blow-up of the size of k-approx- imations and most specic concepts in EL can easily be adapted to EL: . However, we only have a double exponential upper bound (though we strongly conjecture that the size can again be bounded single-exponentially): the size of each tree (and the corresponding concept descriptions) obtained from an atomic completion is at most exponential, and the size of the lcs of a sequence of EL: -concept descriptions can grow exponentially in the size of the input descriptions [3]. Moreover, by an algorithm computing the lcs of the concept descriptions obtained from the atomic completions, the k-approximation (the msc) can be computed in double exponential time. Corollary 4. Let A be an EL:-ABox, a 2 Ind(A), and k 2 IN. { The k-approximation of a always exists. It may be of size jAj k and can be computed in double-exponential time. { The msc of a exists i A is inconsistent, or starting from a, no cycle can be reached in A. If the msc exists, its size may grow exponentially in jAj, and it can be computed in double-exponential time. The existence of the msc can be decided in polynomial time. 5 Most specic concepts in ALE As already mentioned in the introduction, the characterization of instance relationships could not yet be extended from EL: to ALE . Since these structural characterizations were crucial for the algorithms computing the (k-approximation of the) msc in EL and EL: , no similar algorithms for ALE can be presented here. However, we show that 1. given that NC and NR are nite sets, the msc k;A (a) always exists and can eectively be computed (cf. Theorem 6); 2. the characterization of instance relationships in EL is also sound for ALE (cf. Lemma 1), which allows for approximating the k-approximation; and 3. we illustrate the main problems encountered in the structural characterization of instance relationships in ALE (cf. Example 5). The rst result is achieved by a rather generic argument. Given that the signa- ture, i.e., the sets NC and NR , are xed and nite, it is easy to see that also the set of ALE-concept descriptions of depth k built using only names from NC [NR is nite (up to equivalence) and can eectively be computed. Since the instance problem for ALE is known to be decidable [16], enumerating this set and retrieving the least concept description which has a as instance, obviously yields an algorithm computing msc k;A (a). Theorem 6. Let NC and NR be xed and nite, and let A be an ALE-ABox built over a set N I of individuals and NC [NR . Then, for k 2 IN and a 2 Ind(A), the k-approximation of a w.r.t. A always exists and can eectively be computed. Note that the above argument cannot be adapted to prove the existence of the msc for acyclic ALE-ABoxes unless the size of the msc can be bounded appropriately. Finding such a bound remains an open problem. The algorithm sketched above is obviously not applicable in real applications. Thus, in the remainder of this section, we focus on extending the improved algorithms obtained for EL and EL: to ALE . 5.1 Approximating the k-approximation in ALE We rst have to extend the notions of description graph and of description tree from EL: to ALE : In order to cope with value restrictions occurring in ALE-concept descriptions, we allow for two types of edges, namely those labeled with role names r 2 NR (representing existential restrictions of the form 9r:C) and those labeled with 8r (representing value restrictions of the form 8r:C). Again, there is a 1{1 correspondence between ALE-concept descriptions and ALE- description trees, and an ALE-ABox A is translated into an ALE-description graph G(A) just as described for EL-ABoxes. The notion of a homomorphism also extends to ALE in a natural way. A homomorphism ' from an ALE-description into an ALE-description graph is a mapping satisfying the conditions (1) and (2) on homomorphisms between EL-description trees and EL-description graphs, and additionally (3) We are now ready to formalize soundness of the characterization of instance relationships for ALE . Lemma 1. [14] Let A be an ALE-ABox with C an ALE-concept description with If there exists a homomorphism ' from GC into G(A) with '(v 0 a 2A C. As an immediate consequence of this lemma, we get a 2A C k 0, where the trees T (a; G(A)) and T k (a; G(A)) are dened just as for EL. This in turn yields msc k;A (a) v C hence, an algorithm computing an approximation of the k-approximation for ALE . In fact, such approximations already turned out to be quite usable in our process engineering application [4]. The following example now shows that the characterization is not complete for ALE , and that, in general, C In particular, it demonstrates the diculties one encounters in the presence of value restrictions. Example 5. Consider the ALE-ABox A and the ALE-concept description are depicted in Figure 4. Note that G(A) is the unique atomic completion of itself (w.r.t. r r r r s r s r Fig. 4. The ALE-description graph and the ALE-description tree from Example 5. It is easy to see that there does not exist a homomorphism ' from G(C) into However, a 2A C: For each model I of A, either b I does not have an s-successor, or at least one s-successor. In the rst case, b I and hence b I 2 yields the desired r-successor of a I in (8s:P u 9r:9s:>) I . In the second case, it is b I I , and hence b I 1 yields the desired r-successor of a I . Thus, for each model I of A, a I 2 C I . Moreover, for 9r:(P u 9s:P ))) u 9r:(P u 9r:(P u 9s:P )). It is easy to see that C T4 (a;A) 6v C. Hence, C T4 (a;A) u C < C T4 (a;A) , which implies msc 4;A (a) < C T4 (a;A) . Intuitively, the above example suggests that, in the denition of atomic com- pletions, one should take into account not only (negated) concept names but also more complex concept descriptions. However, it is not clear whether an appropriate set of such concept descriptions can be obtained just from the ABox and how these concept descriptions need to be integrated in the completion in order to obtain a sound and complete structural characterization of instance relationships in ALE . 6 Conclusion Starting with the formal denition of the k-approximation of msc we showed that, for ALE and a nite signature (NC ; NR ), the k-approximation of the msc of an individual b always exists and can eectively be computed. For the sublanguages EL and EL: , we gave sound and complete characterizations of instance relationships that lead to practical algorithms. As a by-product, we obtained a characterization of the existence of the msc in EL-/EL:-ABoxes, and showed that the msc can eectively be computed in case it exists. First experiments with manually computed approximations of the msc in the process engineering application were quite encouraging [4]: used as inputs for the lcs operation, i.e., the second step in the bottom-up construction of the knowledge base, they led to descriptions of building blocks the engineers could use to rene their knowledge base. In next steps, the run-time behavior and the quality of the output of the algorithms presented here is to be evaluated by a prototype implementation in the process engineering application. --R Foundations of Databases. Building and structuring description logic knowledge bases using least common subsumers and concept analysis. Computing least common subsumers in description logics. Conceptual graphs: Fundamental notions. Learning the classic description logic: Theoretical and experimental results. Computers and Intractability: A Guide to the Theory of NP-Completeness Using an expressive description Logic: FaCT or Reasoning and Revision in Hybrid Representation Systems. On the complexity of the instance checking problem in concept languages with existential quanti ROME: A repository to support the integration of models over the lifecycle of model-based engineering processes --TR On the complexity of the instance checking problem in concept languages with existential quantification Foundations of Databases Computers and Intractability Building and Structuring Description Logic Knowledge Bases Using Least Common Subsumers and Concept Analysis Computing Least Common Subsumers in Description Logics with Existential Restrictions Computing the Least Common Subsumer and the Most Specific Concept in the Presence of Cyclic ALN-Concept Descriptions

description logics;non-standard inferences;most specific concepts

635905

An efficient direct solver for the boundary concentrated FEM in 2D.

The boundary concentrated FEM, a variant of the hp-version of the finite element method, is proposed for the numerical treatment of elliptic boundary value problems. It is particularly suited for equations with smooth coefficients and non-smooth boundary conditions. In the two-dimensional case it is shown that the Cholesky factorization of the resulting stiffness matrix requires O(Nlog4N) units of storage and can be computed with O(Nlog8N) work, where N denotes the problem size. Numerical results confirm theoretical estimates.

Introduction The recently introduced boundary concentrated nite element method of [8] is a numerical method that is particularly suited for solving elliptic boundary value problems with the following two properties: a) the coe-cients of the equations are analytic so that, by elliptic regularity, the solution is globally, the solution has low Sobolev regularity due to, for example, boundary conditions with low regularity or non-smooth geometries. The boundary concentrated FEM exploits interior regularity in the framework of the hp-version of the nite element method (hp-FEM) by using special types of meshes and polynomial degree distributions: meshes that are strongly rened toward the boundary (see Figs. 5, 6 for typical examples) are employed in order to cope with the limited regularity near the boundary; away from the boundary where the solution is smooth, high approximation order is used on large elements. In fact, judiciously linking the approximation order to the element size leads to optimal approximation results (see Theorem 2.4 and Remark 2.6 for the precise notion of optimality). In the present paper, we focus on the boundary concentrated FEM in two space dimensions and present a scheme for the Cholesky factorization of the resulting stiness matrix that requires O(N log 4 N) units of storage and O(N log 8 N) work; here, N is the problem size. The key to this e-cient Cholesky factorization scheme is an algorithm that numbers the unknowns such that the prole of the stiness matrix is very small (see Fig. 1 for a typical sparsity pattern of the Cholesky factor). Numerical examples conrm our complexity estimates. The boundary concentrated FEM can be used to realize a fast (i.e., with linear-logarithmic complexity) application of discrete Poincare-Steklov and Steklov-Poincare operators as we will discuss in Section 2.5. This use of the boundary concentrated FEM links it to the classical boundary element method (BEM). Indeed, it may be regarded as a generalization of the BEM: While the BEM is eectively restricted to equations with constant coe-cients, the boundary concentrated FEM is applicable to equations with variable coe-cients yet retains the rate of convergence of the BEM. Since the Cholesky factorization of the stiness matrix allows for an exact, explicit data-sparse representation of boundary operators such as the Poincare-Steklov operator with linear-logarithmic complexity, the boundary concentrated FEM provides a sparse direct solver for the 2D BEM (because it directly computes the full set of Cauchy data on the boundary corresponding to L-harmonic functions) that has almost linear complexity. It is also a new alternative to modern matrix compression techniques now used in BEM. We present a Cholesky factorization scheme for the boundary concentrated FEM in two dimen- sions, that is, a direct method. We mention that iterative methods for solving the system of linear equations arising in the boundary concentrated FEM are considered in [8]. Depending on the boundary conditions considered, dierent preconditioners are required for an e-cient iterative solution method. For example, while for Dirichlet problems the condition number of the stiness matrix grows only polylogarithmically with the problem size, [8], Neumann problems require more eective preconditioning. A suitable preconditioner, which has block-diagonal structure, was proposed in [8]. We point out, however, that an application of this preconditioner requires an inner iteration making our direct solver an attractive option. We conclude this introduction by mentioning that, although we restrict our exposition to the case of symmetric positive denite problems, our procedure can be extended to non-symmetric problems by constructing the LU-decomposition rather than the Cholesky factorization. Boundary Concentrated FEM In this section, we present a brief survey of the boundary concentrated FEM. We refer to [8] for a detailed description of this technique and complete proofs. For the sake of concreteness, we discuss Dirichlet problems although other types of boundary conditions such as Neumann or mixed boundary conditions can be treated analogously. 2.1 Problem class and abstract Galerkin FEM For a polygonal Lipschitz domain we consider the Dirichlet problem in @ where the dierential operator L is given by with uniformly (in x 2 symmetric positive denite matrix i;j=1 . Moreover, in the boundary concentrated FEM, we assume that A and the scalar functions a are analytic on The operator is the trace operator that restricts functions on to the boundary @ We assume that the operator L generates an H i.e., The boundary value problem (2.1) is understood in the usual, variational sense. That is, solving is equivalent to the problem: Find with Z The standard FEM is obtained from the weak formulation (2.5) by replacing the space H withanitedimensionalsubspace VN H 8 For the Dirichlet problem (2.1), we introduce the trace space @ For an approximation N 2 YN to we can then dene the FEM for (2.5) as follows: Find uN 2 VN s.t. Z The coercivity assumption (2.4) ensures existence of the nite element approximation uN . Furthermore, by Cea's Lemma there is a C > 0 independent of VN such that uN satises C inf In practice, the approximations N are obtained with the aid of the L 2 -projection operator YN by setting the function QN is dened by In the next section, we specify the approximation spaces VN , the proper choice of which is intimately linked to the regularity properties of the solution u of (2.1). The analyticity of the data A, a 0 , and f implies by interior regularity that the solution u is analytic on if furthermore for some - 2 (0; 1], then the blow-up of higher order derivatives near the boundary can be characterized precisely in terms of so-called countably normed spaces (see [8] for the details). This regularity allows us to prove an optimal error estimate for the boundary concentrated FEM in Theorem 2.4 below. 2.2 Geometric meshes and linear degree vectors For ease of exposition, we will restrict our attention to regular triangulations (i.e., no hanging nodes) consisting of a-ne triangles. (We refer, for example, to [?] for the precise denition of regular triangulations.) We emphasize, however, that an extension to quadrilateral and curvilinear elements is possible. The triangulation of the domain consists of elements K, each of which is the image FK K) of the equilateral reference triangle under the a-ne map FK . We furthermore assume that the triangulation T is -shape-regular, i.e., Here, hK denotes the diameter of the element K. Of particular importance to us will be geometric meshes, which are strongly rened meshes near the boundary @ Denition 2.1 (geometric mesh) A -shape-regular (cf. (2.10)) mesh T is called a geometric mesh with boundary mesh size h if there exist c 1 , c 2 > 0 such that for all K 1. if K \ @ 2. if K \ @ dist (x; @ hK c 2 sup dist (x; @ Typical examples of geometric meshes are depicted in Figs. 5, 6. Note that the restriction to the boundary @ of a geometric mesh is a quasi-uniform mesh, which justies speaking of a \boundary mesh size h". In order to dene hp-FEM spaces on a mesh T , we associate a polynomial degree p K 2 N with each element K, collect these p K in the polynomial degree vector p := (p K ) K2T and set where for p 2 N we introduce the space of all polynomials of degree p as The linear degree vector is a particularly useful polynomial degree distribution in conjunction with geometric meshes: Denition 2.2 (linear degree vector) Let T be a geometric mesh with boundary mesh size h in the sense of Denition 2.1. A polynomial degree vector is said to be a linear degree vector with slope > 0 if An important observation about geometric meshes and linear degree vectors is that the dimension dimS of the space S proportional to the number of points on the boundary: Proposition 2.3 ([8]) Let T be a geometric mesh with boundary mesh size h. Let p be a linear degree vector with slope > 0 on T . Then there exists C > 0 depending only on the shape-regularity constant and the constants c 1 , c 2 , of Denitions 2.1, 2.2 such that dimS K2T K2T 2.3 Error and complexity estimates We formulate an approximation result for the hp-FEM on geometric meshes applied to (2.1): Theorem 2.4 ([8]) Let u be the solution to (2.1) with coe-cients A, a 0 , and right-hand side f analytic on . Assume additionally that u 2 H for some - 2 (0; 1). Let T be a geometric mesh with boundary mesh size h and let p be a linear degree vector on T with slope > 0. Then the FE solution uN given by (2.7) satises C The constants C, b > 0 depend only on the shape-regularity constant , the constants c 1 , c 2 appearing in Denition 2.1, the data A, c, f , and -, kuk H . For su-ciently large the boundary concentrated FEM achieves the optimal rate of convergence number of boundary points: Remark 2.5 Theorem 2.4 is formulated for the Dirichlet problem (2.1). Analogous approximation results hold for Neumann or mixed boundary conditions as well. Remark 2.6 Theorem 2.4 asserts a rate O(n - ) for the boundary concentrated FEM, where This rate is optimal in the following sense: Setting for - 2 (0; 1) on we can introduce the n-width En sup v2En where the rst inmum is taken over all subspaces E n H of dimension n. It can then be shown that Cn - d n for some C > 0 independent of n. 2.4 Shape functions and stiness matrix In order to convert the variational formulation (2.7) into a system of linear equations, a basis of the nite element space S has to be chosen. Several choices of basis functions (\shape are standard in hp-FEM, [2, 12, 7]. Their common feature is that the shape functions can be associated with the topological entities \vertices," \edges," and \elements" of the triangulation T . This motivates us to introduce the following notion of \standard" bases: Denition 2.7 A basis B of S P( said to be a standard hp-FEM basis if each shape function exactly one of the following three categories: 1. vertex shape functions: ' is a vertex shape function associated with vertex V if supp ' consists of all elements that have V as a vertex; 2. edge shape functions: ' is an edge shape function associated with edge e of T if supp ' consists of the (at most two) elements whose edge includes e; 3. internal shape functions: ' is an internal shape function associated with element K if For a standard basis in this sense, we assign spatial points, called nodes, to degrees of freedom as follows: 1. we assign to the shape function associated with vertex V the point 2. we assign to the side shape functions associated with edge e the midpoint of e; 3. we assign to the internal shape functions associated with element K the barycenter of K. One example of a standard basis in the sense of Denition 2.7 is obtained by assembling (see, e.g., [2, 12, 7]) the so-called hierarchical shape functions: Example 2.8 We construct a basis of the space S in two steps: In the rst step, we dene shape functions on the reference element ^ K. In the second step, we dene the basis of by an assembling process. 1. step: Dene one-dimensional shape functions i on the reference interval ( 1; 1) by where the functions L i are the standard Legendre polynomials and the scaling factors c i are given by c Denote by v i , the three vertices of ^ K and by i , the three edges (we assume polynomial degrees that we associate with the edges i and let p 2 N be the polynomial degree of the internal shape functions. We then dene vertex shape functions V, side shape functions functions I as follows: V := the usual linear nodal shape functions n i with y I The side shape functions S 2 , S 3 are obtained similarly with p 1 replaced with suitable coordinate transformation. Note that internal shape functions vanish on @ ^ K and that edge shape functions vanish on two edges. 2. step: Shape functions as dened on the reference element ^ K are now assembled to yield a basis of S First, the standard piecewise linear hat functions are obtained by simply assembling the shape functions V of each element. The internal shape functions are simply taken as else I is the internal shape function on the reference element ^ K dened above. It remains to assemble the side shape functions. To that end, we associate with each edge e a polynomial degree p e := min fp K j e is edge of Kg. Let e be an edge shared by two elements K 0 . For simplicity of notation assume that the element maps FK , FK 0 are such that FK( 1 and that additionally FK (x; (we refer to [2, 7] for details on how to treat the general case). We then dene p e 1 edge shape functions ' i;e associated with edge e by setting else An analogous formula holds for edges e with e @ Once a basis of chosen, the hp-FEM (2.7) can be formulated as seeking the solution U of a system of linear equations We mention in passing that computing the stiness matrix A and the load vector F to su-cient accuracy can be accomplished with work O(dimVN ), [8]. 2.5 Sparse Factorization of the Schur complement We discuss how the Cholesky factorization of the stiness matrix A leads to the explicit sparse representation of discrete Poincare-Steklov operators. Let T be the Poincare{Steklov operator (Dirichlet-Neumann map) where 1 u is the co-normal derivative of the solution u to the equation condition and the trace space YN := VN , the discrete approximation N is dened as follows: For 2 YN , the approximation TN 2 Y 0 is given by where ev 2 VN is an arbitrary extension of v satisfying An analysis of the error T TN was presented in [8]: Theorem 2.9 Let be a polygon. Then the following two statements hold: 1. There exists - 0 > 1=2 such that the Poincare-Steklov operator T maps continuously from 2. Under the hypotheses of Theorem 2.4 (with - 2 (0; 1) as in the statement of Theorem 2.4) there holds for arbitrary - 2 [0; -] \ [0; - 0 ) 1=2;@ If a standard basis in the sense of Denition 2.7 of the space S chosen, then the shape functions can be split into \interior" and \boundary" shape functions. A shape functions is said to be \interior" if its node (see Denition 2.7) lies in it is a \boundary" shape function if its node lies on @ To this partitioning of basis functions corresponds a block partitioning of the stiness matrix AN 2 R dim VN dimVN for the unconstrained space VN of the following form: A A I A I A II The subscript I indicates the interior shape functions and marks the boundary shape func- tions. If we choose Y 0 then the matrix representation of the operator TN is given by the Schur complement TN := A A I A 1 II A I : Inserting the Cholesky factorization LL leads to the desired direct FE method for the Because the Cholesky factor L can be computed with linear-logarithmic complexity (see Section provides an e-cient representation of the Poincare-Steklov operator. We nally mention that our factorization scheme for the Schur complement carries over verbatim to the case of the Neumann-Dirichlet map and also to the case of mixed boundary conditions. 3 Cholesky factorization of the stiness matrix This section is devoted to the main result of the paper, the development of an e-cient Cholesky factorization scheme for the stiness matrix arising in the 2D-boundary concentrated FEM. The key issue is the appropriate numbering of the degrees of freedom. First, we illustrate this numbering scheme for the case of geometric meshes and constant polynomial degree (Sections 3.2, 3.3). We start with this simpler case because our numbering scheme for the degrees of freedom (Algorithm 3.10) is based on a binary space partitioning; in the case the degrees of freedom can immediately be associated with points in space, namely, the vertices of the mesh. The general case of linear degree vectors is considered in Section 3.4. For the case we recall that by Denition 2.7, the vertices of the mesh are called nodes. We denote by V the set of nodes and say that a node V 0 is a neighbor of a node V if there exists an element K 2 T such that V and V 0 are vertices of K. It will prove useful to introduce the set of neighbors of a node V 2 V as is a neighbor of V because the sparsity pattern of the stiness matrix can be characterized with the aid of the sets N . 3.1 Nested dissection in Direct Solvers Let A 2 R NN be a symmetric positive denite matrix. We denote by L its Cholesky factor, i.e., the lower triangular matrix L with LL A. For sparse, symmetric positive matrices A 2 R NN it is customary, [5], to introduce the i-th bandwidth i and the j-th frontwidth ! j of A as It can be shown (see Proposition 3.1(i)) that in each row i, only the entries L ij with i i j i are non-zero. The j-th frontwidth ! j measures the number of non-zero subdiagonal entries in column j of L, i.e., The frontwidth ! is given by The cost of computing the Cholesky factor L can then be quantied in terms of the numbers Proposition 3.1 Let A 2 R NN be symmetric, positive denite. Then (i) the storage requirement for L is (ii) the number of oating point operations to compute L is N square roots for the diagonal entries L ii , 1P N 1 multiplications. In particular, the storage requirement nnz and the number of oating point operations W can be bounded by Proof. See, for example, [5, Chapters 2, 4]. In view of the estimate (3.5), various algorithms have been devised to number the unknowns so as to minimize the frontwidth !; the best-known examples include the Reverse Cuthill-McKee algorithm, [4], the algorithm of Gibbs-Poole-Stockmeyer, [6], and nested dissection. For stiness matrices arising in the 2D-boundary concentrated FEM, we will present an algorithm based on nested dissection in Section 3.3 that numbers the nodes such that The basic nested dissection algorithm in FEM reads as follows: Algorithm 3.2 (nested dissection) nested dissection(V; N 0 ) input: Set of nodes V, starting number N 0 output: numbering of the nodes V starting with N 0 label the element of V with the number N 0 else f 1. partition the nodes V into three mutually disjoint sets V left , V right , V bdy such that (a) jV left j jV right j (b) jV bdy j is \small" (c) right [V bdy for all right 2. if V left nested dissection(V left 3. if V right nested dissection(V right 4. enumerate the elements of V bdy starting with the number return The key property is (1c). It ensures that the stiness matrix A has the following block structure: A =@ A lef t;left 0 A > bdy;left bdy;right A bdy;left A bdy;right A bdy;bdyA (3.6) sparsity pattern of L Figure 1: Left: mesh and initial geometric partitioning (thick line). Right: sparsity pattern of Condition (1b) is imposed in view of the fact that the frontwidth !(A) of A can be bounded by where are the frontwidths of the submatrices A left , A right . Thus, if the recursion guarantees that jV bdy j is small and the frontwidths of these submatrices are small, then the numbering scheme is e-cient. Since nested dissection operates recursively on the sets V left , V right , its eectiveness hinges on the availability of partitioning strategies for Step 1 of Algorithm 3.2 that yield small jV bdy j. The particular node distributions appearing in the boundary concentrated FEM will allow us to devise such a scheme in Section 3.3. 3.2 Nested dissection: an Example We show that for meshes that arise in the boundary concentrated FEM, it is possible to perform the partitioning of Step 1 in Algorithm 3.2 such that the set V bdy is very small compared to V. We illustrate this in the following example. Example 3.3 Consider meshes that are rened toward a single edge as shown in Fig. 1. The thick line partitions R 2 into two half-spaces H< , H> , and the nodes are partitioned as follows: has a neighbor in H> g [ has a neighbor in H< Note that due to choosing the partitioning line as the center line, we have (Estimates of this type are rigorously established in Lemma 3.4 below.) We then proceed as in Algorithm 3.2 by partitioning along a \center line" of the subsets of nodes (a more rigorous realization of this procedure is Algorithm 3.9 below that is based on binary space partitioning). We note that the subsets V left , V right have a similar structure as the original set V; thus, they are partitioned satisfying an estimate analogous to (3.8). Using this partitioning scheme in Algorithm 3.2 leads to very small frontwidths: For a mesh with nodes, a frontwidth obtained (see Fig. 1 for the actual sparsity pattern). We analyze this example in more detail in Examples 4.1, 4.2 below. The bounds in (3.8) are \geometrically clear." A more rigorous proof is established in the following lemma. Lemma 3.4 Let T be a geometric mesh with boundary mesh size h on a domain . Fix b 2 @ and choose a partitioning vector t 6= 0 such that the following cone condition is satised (cf. Fig. ni > -jx bj j~njg \ B (b) Dene the half-spaces and set is a node of T and has a neighbor in H> g [ has a neighbor in H< Then log jV j; where depends only on -, , and the constants describing the geometric mesh T . In particular, is independent of the point b. Proof. Let l be the line passing through the point b with direction ~ n, 0g. Next, dene the function dist The key property of d is that Denoting by K bdy the set of all elements that intersect the line l, K bdy we can bound K2K bdy Z Z In order to proceed, we need two assertions: Assertion 1: For - 2 (0; 1) given by the cone condition (3.9) there holds dist (x; @ dist (x; b) p dist (x; @ 8x\ l: (3.12) PSfrag replacements @ H< H> Figure 2: Notation for partitioning at a boundary point b. The rst estimate of (3.12) is obvious. For the second one, geometric considerations show for x\ l dist dist dist where C 0 is the lateral part of @C t . This proves (3.12). Assertion 2: There exists C > 0 depending only on the parameters describing the geometric mesh and the parameters of the cone condition such that Again, the rst bound is obvious. For the second bound, let x 2 K for some K 2 K bdy and choose xK 2 K \ l. Then dist dist dist where we used (3.12). Next, we use the properties of geometric meshes and (3.10) to get dist (x; b)g max fh; hK g +p dist Inserting this bound in (3.11) gives Z dx C Z r dr Cj log hj: Since jV j h 1 (cf. [8, Prop. 2.7]), we have proved the rst estimate of the lemma. For the second estimate of the lemma, we note that the boundary parts < := @ \H< \B (b) and > @ \H> \B (b) have positive lengths. Thus we have jV \ < j h 1 and jV \ > j h 1 and a fortiori jV left j jV j jV right j; the proof of the lemma is now complete. The reason for the eectiveness of the partitioning strategy in Example 3.3 is that at each stage of the recursion, Algorithm 3.2 splits the set of nodes into two sets of (roughly) equal size, and a set of boundary nodes V bdy that is very small. The property (3.8), proved in Lemma 3.4, motivates the following denition: Denition 3.5 (( q)-balanced partitioning) The nested dissection Algorithm 3.2 is said to be ( q)-balanced for a set V if at each stage i of the recursion there holdsjV (i) right j left Here, the superscripts i indicate the level of the recursion. For a ( q)-balanced nested dissection algorithm, we can then show that the frontwidth grows only moderately with the problem size: Proposition 3.6 If the nested dissection Algorithm 3.2 is ( q)-balanced for V, then the numbering generated by Algorithm 3.2 leads to a frontwidth !(A) of the stiness matrix A with Proof. The assumption that the algorithm is ( q)-balanced implies easily right j Thus, jV (i) j jVj, and the depth of the recursion is at most since the recursion stops if jV (i) 1. The bound (3.7) then implies where we used the denition of C of the statement of the proposition. In Example 3.3, we studied the model situation of meshes rened toward a straight edge. In view of Lemma 3.4, we expect the partitioning strategy of be ( 1)-balanced. Hence, we expect the frontwidth to be of the order O(log 2 jVj). In Examples 4.1, 4.2, we will conrm this numerically. 3.3 Node Numbering for geometric meshes: the case We now present a partitioning strategy that allows Algorithm 3.2 to be ( 1)-balanced for node sets that arise in the boundary concentrated FEM. The partitioning rests on the binary space partitioning (BSP), [3], which is reproduced here for convenience's sake: Algorithm 3.7 (BSP) input: Set of points X , partitioning vector t output: partitioning of X into X< , X> , X= with jX < j jX > j and 1. determine the median m of the set fhx; ti 2. X< := fx return Remark 3.8 Since the median of a set can be determined in optimal (i.e., linear in the number of elements) complexity (see, e.g., [1, 9]), Algorithm 3.7 can be realized in optimal complexity. The next algorithm formalizes our procedure of the example in Section 3.2. Algorithm 3.9 (subdomain numbering) nodes V, vector t, starting number N 0 %output: numbering of nodes V label the element of V with the number N 0 else f 1. 2. right 3. if V left 4. if V right 5. enumerate the elements of V bdy starting with the number return Algorithm 3.9 allows us to number e-ciently nodes of a mesh that is rened toward a line as in Fig. 1. Our nal algorithm splits the domain into subdomains, each of which can be treated e-ciently by Algorithm 3.9. Algorithm 3.10 (node numbering) 1. split the domain into subdomains choose vectors t i 2. 3. 4. N := 1 5. for do f call numbering 6. number the nodes of V bdy starting at N The subdomains i and the partitioning vectors t i should be chosen such that (a) jV bdy (b) the partitioning in the subsequent calls of 1)-balanced in the sense of Denition 3.5. To obtain guidelines for the selection of subdomains i and partitioning vectors t i , it is valuable to study examples where Condition (a) or Condition (b) are not satised. This is the purpose of the next example. Example 3.11 The left and center pictures in Fig. 3 illustrate situations in which Conditions (a), (b) are violated: In the left picture of Fig. 3, the common boundary @ @ j is tangential to @ at b, and thus we cannot expect jV bdy (cf. the cone condition (3.9)). In the center picture of Fig. 3, the partitioning vector t i is parallel to the outer normal vector n(x) at the boundary point x 2 @ This prevents the partitioning from being 1)-balanced, since at some stage of the recursion, Condition (3.15) will be violated (note again the cone condition (3.9)). We refer to Example 4.5 below, where this kind of failure is demonstrated numerically. Finally, we point out that in the center picture of Fig. 3 the vector for the subdomain @ >From the two cases of failure in Example 3.11, we draw the following guidelines: 1. The subdomains should be such that @ @ j is non-tangential to @ 2. For each subdomain i , the partitioning vector t i should be chosen such that the cone condition holds uniformly in b 2 @ are independent of b. A partition chosen according to these rules is depicted in the right part of Fig. 3. Remark 3.12 The guideline for choice t i such that the cone condition (3.9) is satised at each point @ \@ guarantees that in the partitioning, jV (i) O(log jVj) at each stage i of the recursion (see Lemma 3.4). Thus, the partitioning is ( 1)-balanced if we can ensure (3.14), that is, that V left and V right are comparable in size. Note that this could be monitored during run time. Remark 3.13 In all steps of the recursion in Algorithm 3.9, we use a xed partitioning vector . This is done for simplicity of exposition. In principle, it could be chosen dierently in each step of the recursion depending on the actual set to be partitioned. Since the partitioning strategy should be ( 1)-balanced, one could monitor this property during run time and adjust the vector t as necessary. We conclude this section with a work estimate for the case PSfrag replacements @ @ @ PSfrag replacements @ @ @ @ x PSfrag replacements @ @ @ @ x Figure 3: Choosing subdomains and partitioning vectors. Left and center: cases of failure. Right: possible partitioning with arrows indicating a good choice of vector t i ; in the three subdomains without an arrow, t i may be chosen arbitrary. Proposition 3.14 Let V be the set of nodes corresponding to a geometric mesh on a domain . Assume that the subdomains Algorithm 3.10 are chosen such that a) jV bdy the partitioning in each call 1)-balanced. Then the frontwidth !(A) of the stiness matrix on geometric meshes with bounded by where the constant C is independent of jVj. The storage requirements nnz and work W for the Cholesky factorization are bounded by nnz CjVj log 2 jVj; W jVj log 4 jVj: Proof. The hypothesis that Algorithm 3.10 is based on a ( 1)-balanced partitioning together with Proposition 3.6 implies jVj). The estimates concerning storage requirement and work then follow from Proposition 3.1. Remark 3.15 On each level the nodes of V (i) bdy are numbered arbitrarily. Suitable numbering strategies of these sets could further improve the frontwidth !(A). 3.4 Node Numbering: geometric meshes and linear degree vectors We now consider the case of geometric meshes and linear degree vectors. We proceed as in Section 3.3 for the case identifying degrees of freedom with points in space. We use the notion of nodes introduced in Denition 2.7 and denote by V the set of all nodes. We count nodes according to their multiplicity, that is, the number of shape functions corresponding to that node. This procedure is justied by the fact that shape functions associated with the same node have the same support and therefore the same neighbors. As in the case that node V 0 is the neighbor of a node V , if the intersection of the supports of the associated shape functions has positive measure. The set of neighbors of a node is dened as in (3.1). Remark 3.16 Nodes are counted according to their multiplicity. If a one-to-one correspondence between points in space and degrees of freedom is desired, one could choose distinct nodes on an edge (e.g., uniformly distributed) to be assigned to the shape functions associated with that edge; likewise distinct nodes in an element could be selected to be assigned to shape functions associated with that element. The performance of the algorithms below will be very similar. To this set of nodes and this notion of neighbors, we can apply Algorithm 3.10. In order to estimate the resulting frontwidth, we need the analog of Lemma 3.4. Lemma 3.17 Let T be a geometric mesh on a domain , p be a linear degree vector, and assume the cone condition (3.9). Dene the half-spaces and set is a node and has a neighbor in H> g [ has a neighbor in H< Then log 3 jV j; where depends only on -, , and the constants describing the geometric mesh T and the linear degree vector p. Proof. The proof of this lemma is very similar to that of Lemma 3.4. For the bound on jV bdy j we have to estimate (using the notation of the proof of Lemma 3.4) K2K bdy The desired bound then follows as in the proof of Lemma 3.4 if we observe that In view of the appearance of the exponent 3 in Lemma 3.17, we expect Algorithm 3.10 to be 3)-balanced. In this case, we can obtain the following result for the performance of the numbering obtained by Algorithm 3.10: Theorem 3.18 Let T be a geometric mesh and p be a linear degree vector. Set . Assume that subdomains i and partitioning vectors t i , Algorithm 3.10 are chosen such that a) jV bdy the partitioning in each call is ( 3)-balanced. Then the frontwidth !(A) of the stiness matrix is bounded by The storage requirements nnz and work W for the Cholesky factorization are bounded by nnz CN log 4 N; W N log 8 N: Remark 3.19 In view of Remark 3.8, Algorithm 3.10 requires O(N log N) work (i.e., optimal complexity) to compute the numbering. Remark 3.20 We assumed that the mesh consists of triangles only. However, Algorithm 3.10 can be applied to meshes containing quadrilaterals and curved elements. Theorem 3.18 holds verbatim in these cases as well. frontwidth for refinement towards a single edge log(N) frontwidth frontwidth, t=(1,1) log 2 (N) Figure 4: Examples 4.1, 4.2: in uence of partitioning vector in BSP on frontwidth. In this section, we conrm that the numbering obtained by Algorithm 3.10 allows for computing the Cholesky factorization with O(N log q N ), q 2 f4; 8g, work. We restrict ourselves to the case that is, we illustrate Proposition 3.14. In all examples, the nodes on the boundary correspond to unknowns, i.e., we consider Neumann problems. In all examples, we use Algorithms 3.10, 3.9 to obtain the numbering of the nodes. In our computational experiments, the meshes are generated with the code Triangle of J.R. Shewchuck, [11]. Triangle is a realization of Ruppert's Algorithm, [10], which creates triangulations with a guaranteed minimum angle of 20 - . Our reason for working with this particular triangulation algorithm is that it automatically produces meshes the desired property diamK dist (K; @ if its input consists of quasi-uniformly distributed boundary nodes only (the meshes in Figs. 5, 6, for example, are obtained with Triangle from 200 uniformly distributed points on the boundary). In all tables and gures, N stands for the number of nodes of the mesh generated by Triangle, ! is the frontwidth of the stiness matrix, and nnz the storage requirement for the Cholesky ops is the number of multiplications, and t chol the CPU-time required to perform the We implemented the Cholesky factorization in \inner product form," where L is computed columnwise and the sparsity pattern of L is exploited. The basic building block of our procedure is Algorithm 3.9. Our rst example, therefore, analyzes in detail the situation already discussed in Example 3.3. Example 4.1 Let be the unit square. For n 2 N the initial input for Triangle are the points f(i=n; ng [ f(1; 0:3); (1; 1); (0:7; 1); (0; 1); (0; 0:7)g (see Fig. 4 for Triangle's output for 50). The node numbering is then obtained by applying Algorithm 3.9 with the partitioning vector . The results are collected in Table 1 and Fig. 4. In view of Proposition 3.6 we expect the frontwidth ! to be O(log 2 N ). The results of Table are plotted in Fig. 4, and the observed growth is indeed very close to O(log 2 N ). In ops t chol [sec] ! nnz Table 1: Examples 4.1, 4.2: points on edge, view of Proposition 3.1, ops t chol these estimates. Example 4.2 The choice in Example 4.1 of the partitioning vector well-suited to the case of renement toward a straight edge. In view of Lemma 3.4, we expect Algorithm 3.9 to be still ( 1)-balanced for partitioning vectors t that are not parallel to the normal vector. In order to illustrate that \non-optimal" choices of partitioning vectors t still lead to ( 1)-balanced nested dissection, we consider the same meshes as in Example 4.1 but employ Algorithm 3.9 with the vector . The results are presented in the right part of Table 1 and in Fig. 4. We observe that this choice of partitioning strategy leads to very similar results as in Example 4.1, showing robustness of our algorithm with respect to the choice of partitioning vector t. Example 4.3 In this example, the domain is the unit square input are n uniformly distributed nodes on the boundary @ (see Fig. 5 for Triangle's output 200). The node numbering is achieved with Algorithm 3.10 for subdomains corresponding vectors t i given by: The numerical results are collected in Table 2. We expect the frontwidth to be O(log 2 N ), which is visible in Fig. 5. refinement towards bdy of square and clover leaf log(N) frontwidth square clover leaf log 2 (N) Figure 5: Examples 4.3, 4.4: meshes with 200 points on boundary (left) and frontwidth vs. log N (right). Example 4.4 In this example, we replace the square of Example 4.3 with (See Fig. 5 for Triangle's output for boundary points.) The partitioning into four subdomains and the choice of the partitioning vectors t i is given by (4.1). The numerical results are collected in Table 3. The expected relation visible in Fig. 5. Example 4.5 The key feature of the choice of the partitioning vectors t i in Examples 4.3, 4.4 is that, for each i, sup x 2 @ @ is the normal at a boundary point x). (4.2) This condition was identied in Example 3.11 as necessary for the binary space partitioning strategy with (xed) vector t i to be ( 1)-balanced. In this last example, we illustrate that (4.2) is indeed necessary. To that end, we replace the square of Example 4.3 with The subdomains i and the partitioning vectors t i are chosen as in Example 4.3 and given by (4.1). A calculation shows that condition (4.2) is satised for c 2 (0; 2=3) and is violated for subdomains, 4 in Fig. 6). For c close to 2=3 we therefore expect our binary space partitioning to perform poorly in the sense that the sets V (i) bdy become large, thus resulting in large frontwidths. This is illustrated in Table 4, where the frontwidths for dierent values of c are shown in dependence on n, the number of points on the boundary. Fig. 6 shows that in the limit 2=3, the frontwidth ! does not grow polylogarithmically in N . In Table 4, we reported the number N of nodes for the case however, that the meshes produced by Triangle for the three cases are very similar. Acknowledgments : The authors would like to thank Profs. W. Hackbusch and L.N. Trefethen for valuable comments on the paper. ops t chol [sec] Table 2: Example 4.3: ops t chol [sec] Table 3: Example 4.4: -0.4 c=2/3 c=0.3 effect of subdomain choice on frontwidth; domain: dumbbell log(N) frontwidth c=2/3 log 2 (N) Figure Example 4.5: mesh with 200 points on boundary (top left), the four subdomains for (bottom left) and frontwidth vs. log N (right). 1024 2022 107 118 118 2048 4079 128 159 155 131072 264600 370 577 768 262144 529340 423 723 1043 Table 4: Example 4.5: --R Time bounds for selection. A general and exible fortran 90 hp-FE code On visible surface generation by a priori tree structures. Computer implementation of the Computer solution of large sparse positive de An algorithm for reducing the bandwidth and pro Element Methods for CFD. Boundary concentrated The Art of Computer Programming A Delaunay re 2d mesh generator --TR A Delaunay refinement algorithm for quality 2-dimensional mesh generation The art of computer programming, volume 3 Computer Solution of Large Sparse Positive Definite Finite Element Method for Elliptic Problems Computer implementation of the finite element method

hp-finite element methods;direct solvers;schur complement;meshes refined towards boundary

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Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling.

Normal form theory is one of the most power tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Recently, many researchers have paid attention to further reduction of conventional normal forms (CNF) to so called the simplest normal form (SNF). However, the computation of normal forms has been restricted to systems which do not contain perturbation parameters (unfolding). The computation of the SNF is more involved than that of CNFs, and the computation of the SNF with unfolding is even more complicated than the SNF without unfolding. Although some author mentioned further reduction of the SNF, no results have been reported on the exact computation of the SNF of systems with perturbation parameters. This paper presents an efficient method for computing the SNF of differential equations with perturbation parameters. Unlike CNF theory which uses an independent nonlinear transformation at each order, this approach uses a consistent nonlinear transformation through all order computations. The particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth-order equations containing the nth-order terms only. This greatly saves computational time and computer memory. The recursive formulations have been implemented on computer systems using Maple. As an illustrative example, the SNF for single zero singularity is considered using the new approach.

Introduction Normal form theory for dierential equations can be traced back to the original work of one hundred years ago, and most credit should be given to Poincare [1]. The theory plays an important role in the study of dierential equations Preprint submitted to Elsevier Preprint 25 May 2001 related to complex behavior patterns such as bifurcation and instability [2{4]. The basic idea of normal form theory is employing successive, near-identity nonlinear transformations to obtain a simple form. The simple form is qualitatively equivalent to the original system in the vicinity of a xed point, and thus greatly simplify the dynamical analysis. However, it has been found that conventional normal forms (CNF) are not the simplest form which can be obtained, and may be further simplied using a similar near-identity non-linear transformation. (e.g., see [5{13]). Roughly speaking, CNF theory uses the kth-order nonlinear transformation to possibly remove the kth-order non-linear terms of the system, while in the computation of the simplest normal form (SNF) the terms in the kth-order nonlinear transformation are not only used to simplify the kth-order terms of the system, but also used to eliminate higher order nonlinear terms. Since the computation for the SNF is much more complicated than that of CNFs, computer algebra systems such as Maple, Mathematics, Reduce, etc. have been used (e.g., see [12,14{18]). Recently, researchers have paid particular attention to the development of efcient computation methodology for computing the SNF [18]. The computation of normal forms has been restricted to systems which do not contain perturbation parameters (unfolding). However, in general a physical system or an engineering problem always involves some system parameters, usually called perturbation parameter or unfolding. In practice, nding such a normal form is more important and applicable. There are two ways for nd- ing such a normal form. One way is to extend the dimension of a system by including the dimension of the parameter and then apply normal form theory to the extended system. The other way employs normal form theory directly to the original system. The former may be convenient for proving theorems while the later is more suitable for the computation of normal forms, which is particularly useful when calculating an explicit normal form for a given sys- tem. However, in most cases of computing such CNFs with unfolding, people are usually interested in the normal form only. Thus one may rst ignore the perturbation parameter and compute the normal form for the corresponding \reduced" system (by setting the parameters zero), and then add unfolding to the resulting normal form. In other words, the normal form of the original system with parameters is equal to the normal form of the \reduced" system plus the unfolding. This way it greatly reduces the computation eort, with the cost that it does not provide the nonlinear transformation between the original system and the normal form. This \simplied" approach is based on the fact that the normal form terms (besides the unfolding) for the original system (with perturbation parameters) are exactly same as that of the \re- duced" system, implying that all higher order nonlinear terms consisting of the parameters can be eliminated by nonlinear transformations. The computation of the SNF is more involved than that of CNFs, and the computation of the SNF with unfolding is even more complicated than the SNF without unfolding. Although some authors mentioned further reduction of the SNF, no results have been reported on the exact computation of the SNF of systems with perturbation parameters. One might suggest that we may follow the \simplied" way used for computing CNFs. That is, we rst nd the SNF for the \reduced" system via a near-identity nonlinear transfor- mation, and then add an unfolding to the SNF. However, it can be shown that this \simplied" way is no longer applicable for computing the SNF of systems with perturbation parameters. In other words, in general it's not possible to use only near-identity transformations to remove all higher order terms which involve the perturbation parameters. In this paper, we propose, in addition to the near-identity transformation, to incorporate the rescaling on time to form a systematic procedure. The particular advantage of the method is to provide an ecient recursive formula which can be used to obtain the nth-order equations containing the nth-order terms only. This greatly saves computational time and computer memory. The recursive formulation can be easily implemented using a computer algebra system such as Maple. Moreover, unlike CNF theory which uses an independent nonlinear transformation at each order, this approach uses a consistent nonlinear transformation through all order compu- tations. This provides a one step transformation between the original system and the nal SNF, without the need of combining the multiple step nonlinear transformations at the end of computation. In the next section, the ecient computation method is presented and the general explicit recursive formula is derived. Section 3 applies the new approach to derive the SNF for single zero singularity, and conclusions are drawn in Section 4. 2 Computation of the SNF using Lie transform Consider the general nonlinear dierential equation, described by dx dt where x and are the n-dimensional state variable and m-dimensional parameter variable, respectively. It is assumed that is an equilibrium of the system for any values of , i.e., f(0; ) 0. Further, we assume that the nonlinear function f(x; ) is analytic with respect to x and , and thus we may expand equation (1) as dx dt where Lx 4 represents the linear part and L is the Jacobian matrix, Dxf , evaluated on the equilibrium at the critical point It is assumed that all eigenvalues of L have zero real parts and, without loss of generality, is given in Jordan canonical form. (Usually J is used to indicate Jacobian matrix. Here we use L in order to be consistent with the Lie bracket denotes the kth-degree vector homogeneous polynomials of x and . To show the basic idea of normal form theory, we rst discuss the case when system (1) does not involve the perturbation parameter, , as normal forms are formulated in most cases. In such a case f k (x; ) is reduced to f k (x). The basic idea of normal form theory is to nd a near-identity nonlinear transformation such that the resulting system dy dt becomes as simple as possible. Here h k (y) and g k (y) denote the kth-degree vector homogeneous polynomials of y. According to Takens' normal form theory [19], we may rst dene an operator as follows: denotes a linear vector space consisting of the kth-degree homogeneous vector polynomials. The operator called Lie bracket, dened by Next, dene the space R k as the range of L k , and the complementary space of R k as K and we can then choose bases for R k and K k . Consequently, a vector homogeneous polynomial f k (x) 2 H k can be split into two parts: one of them can be spanned by the basis of R k and the other by that of K k . Normal form theory shows that the part belonging to R k can be eliminated while the part belonging to K k must be retained in the normal form. It is easy to apply normal form theory to nd the \form" of the CNF given by equation (4). In fact, the coecients of the nonlinear transformation h k (y) being determined correspond to the terms belonging to space R k . The \form" of the normal form g k (y) depends upon the basis of the complementary space which is induced by the linear vector v 1 . We may apply the matrix method [2] to nd the basis for space R k and then determine the basis of the complementary space K k . Since the main attention of this paper is focused on nding further reduction of CNFs and computing the explicit expressions of the SNF and the nonlinear transformation, we must nd the \form" of g k (y). Similar to nding CNFs, the SNFs have been obtained using near-identity nonlinear transformations. It should be mentioned that some author has also discussed the use of \rescal- ing" to obtain a further reduction (e.g., see [14,15]). However, no results have been reported on the study of the SNF of system (1). In this paper, we present a method to explicitly compute the SNF of system (1). The key idea is still same as that of CNFs: nding appropriate nonlinear transformations so that the resulting normal form is the simplest. The simplest here means that the terms retained in the SNF is the minimum up to any order. The fundamental dierence between the CNF and the SNF is explained as fol- lows: Finding the coecients of the nonlinear transformation and normal form requires for solving a set of linear algebraic equations at each order. Since in general the number of the coecients is larger than that of the algebraic equa- tions, some coecients of the nonlinear transformation cannot be determined. In CNF theory, the undetermined coecients are set zero and therefore the nonlinear transformation is simplied. However, in order to further simplify the normal form, one should not set the undetermined coecients zero but let them carry over to higher order equations and hope that they can be used to simplify higher order normal form terms. This is the key idea of the SNF com- putation. It has been shown that computing the SNF of a \reduced" system (without perturbation parameters) is much complicated than that for CNFs. Therefore, it is expected that computing the SNF of system (1) is even more involved than nding the SNF of the \reduced" system. Without a computer algebra system, it is impossible to compute the SNF. Even with the aid of symbolic computation, one may not be able to go too far if the computational method is not ecient. To start with, we extend the near-identity transformation (3) to include the parameters, given by and then add the rescaling on time: Further we need to determine the \form" of the normal form of system (1). In generic case, we may use the basis of K k (see equation (7)) to construct k (y), which is assumed to be the same as that for the CNF of system (1), plus the unfolding given in the general form, L 1 () y, so that equation (4) becomes dy d is an nn matrix linear function of , to be determined in the process of computation, representing the unfolding of the system. Now dierentiating equation (8) with respect to t and then applying equations (2), and (8){(10) yields a set of algebraic equations at each order for solving the coecients of the SNF and the nonlinear transformation. A further reduction from a CNF to the SNF is to nd appropriate h k (y; )'s such that some coecients of g k (y)'s can be eliminated. When one applies normal form theory (e.g., Takens normal form theory) to a system, one can easily nd the \form" of the normal form (i.e., the basis of the complementary space K k ), but not the explicit expressions. However, in practical applications, the solutions for the normal form and the nonlinear transformation are both important and need to be found explicitly. To do this, one may assume a general form of the nonlinear transformation and substitute it back to the original dierential equation, with the aid of normal form theory, to obtain the kth-order equations by balancing the coecients of the homogeneous polynomial terms. These algebraic equations are then used to determine the coecients of the normal form and the nonlinear transformation. Thus, the key step in the computation of the kth-order normal form is to nd the kth-order algebraic equations, which takes the most of the computation time and computer memory. The solution procedures given in most of normal form computation methods contain all lower order and many higher order terms in the kth-order equations, which extremely increases the memory requirement and the computation time. Therefore, from the computation point of view, the crucial step is rst to derive the kth-order algebraic equations which only contain the kth-order nonlinear terms. The following theorem summarizes the results for the new recursive and computationally ecient approach, which can be used to compute the kth-order normal form and the associated nonlinear transformation. Theorem 1: The recursive formula for computing the coecients of the simplest normal form and the nonlinear transformation is give by Tm h l i are all jth-degree vector homogeneous polynomials in their arguments, and T j (y; ) is a scalar function of y and . The variables y and have been dropped for simplicity. The notation h l i denotes the ith order terms of the Taylor expansion of y. More precisely, where each dierential operator D aects only function f j , not h l m (i.e., h l m is treated as a constant vector in the process of the dierentiation), and thus j. Note that at each level of the dierentiation, the D operator is actually a Frechet derivative to yield a matrix, which is multiplied with a vector to generate another vector, and then to another level of Frechet derivative, and so on. Proof: First dierentiating equation (8) results in dx dt dy dt dy dt dy d d dt then substituting equations (2), (9) and (10) into equation (13) yields Note that the T 0 can be used for normalizing the leading non-zero nonlinear coecient of the SNF. Since this normalization may change stability analysis if time is reversed, we prefer to leave the leading non-zero coecient unchanged and thus set T Next substituting equation (8) into equation (14) and re-arranging gives Then we use Taylor expansions of f near to rewrite equation (15) as Further, applying equations (8) and (9) yields representing the linear part of the system, and the Lie operator with respect to y has been used. Note in equation (17) that the expansion of f and h do not have the purely parameter terms since f(0; for any values of . Finally comparing the same order terms in equation (17) results in . (18) y, and the variables in g i have been dropped for simplicity. For a general k, one can obtain the recursive given in equation (11), and thus the proof is completed. It has been observed from equations (11) and (18) that (i) If system (1) does not have the parameter , then thus equation (17) is reduced which has been obtained in [18] for the \reduced" system. This indicates that computing the SNF with unfolding needs much more computation eort than that for computing the SNF without unfolding. (ii) The only operation involved in the formula is the Frechet derivative, involved in Dh i , D i f j and the Lie bracket [; ]. This operation can be easily implemented using a computer algebra system. (iii) The kth-order equation contains all the kth-order and only the kth-order terms. The equation is given in a recursive form. (iv) The kth-order equation depends upon the known vector homogeneous polynomials , and on the results for obtained from the lower order equations. (v) The equation involves coecients of the nonlinear transformation h, rescaling T and the coecients of the kth-order normal form g k . If the jth-order (j < coecients of h j and T j are completely determined from the jth-order equation, then the kth-order equation only involves the unknown coecients of h k ; T k and g k , which yields a CNF. (vi) If the kth-order equation contains lower order coecients of h (j < which have not been determined in the lower order (< tions, they may be used to eliminate more coecients of g k , and thus the CNF can be further simplied. (vii) In most of the approaches for computing the SNFs without unfolding (e.g., see [7,8,11{17]), the nonlinear vector eld f is assumed to be in a CNF for the purpose of simplifying symbolic computations. Since for such an approach, the kth-order equations usually include all the lower order terms as well as many higher order terms, it is extremely time consuming for symbolic computation and takes too much computer memory. For the approach proposed in this paper, the kth-order equation exactly only contains the kth-order terms, which greatly saves computer memory and computational time. This is particularly useful for computing the SNF with unfolding. Therefore, for our approach, the vector eld f(x; ) can be assumed as a general analytic function, not necessary a CNF form. Now we can use equation (18) to explain the idea of SNF. Consider the rst equation of (18) for L 2 and g 2 , we can split the right-hand side into two parts: one contains y only while the other involves both y and . The part containing only y can determine g 2 . That is, the part from f 2 that cannot be eliminated by h 2 and T 1 is the solution for . Similarly, the other part containing both y and can be used to nd the unfolding L 1 (). However, it can be seen that some coecients of h 2 and T 1 are not used at this order. Setting these \unnecessary" coecients zero results in the next equation of in the same situation: it only requires to use h 3 and T 2 to remove terms from f 3 as much as possible, since other terms h been solved from the second order equations. This procedure can be continued to any higher order equations. This is exactly CNF theory. However, when we solve for g 2 and L 2 let the \unnecessary" coecients of h 2 and T 1 be carried over to next step equation, then it is clear to see from the second equation of (18) that four terms may contain these \unnecessary" coecients. These \unnecessary" coecients can be used to possibly further eliminate a portion or the whole part of f 3 which cannot be eliminated by the CNF approach. If these \unnecessary" coecients are not used at this step, they can be carried over further to higher order equations and may be used to remove some higher order normal form terms. 3 The SNF for the single zero singularity In this section, we shall apply the results and formulas obtained in the previous section as well as the Maple program developed on the basis of recursive formula (11) to compute the SNF of single zero singularity. We use this simple example to demonstrate the solution procedure in nding the explicit SNF of a general system. For the single zero singularity, the linear part, L x, becomes zero, and we may put the general expanded system in a slightly dierent form for convenience: dx dt a 1i i x +X a 2i i x 2 +X a Similarly, the near-identity nonlinear transformation and the time scaling are, respectively, given by and with In order to have a comparison between system (20) and its \reduced" system, given by (obtained by setting (20)), dx dt we list the SNF of the \reduced" system below, which were obtained only using a near-identity transformation [17]: (i) If a 20 6= 0, then the SNF is dy dt (ii) If a then the SNF is dy dt where the coecient b (2k 1)0 is given explicitly in terms of a j0 's. This shows that the SNF of the \reduced" system (23) contains only two terms up to any order. To obtain the unfolding, we assume a 11 6= 0. Other cases can be similarly discussed. Since equation (20) describes a codimension one system, we expect that the nal SNF should have only one linear term for the unfolding, and all higher order terms in equation (20) are eliminated except for, at most, the terms with the coecients a 20 and a 30 . In other words, the SNF of equation (20) is expected to have the form: dy d We start from the case a 20 6= 0, and then discuss the general case. Generic case: Suppose a 20 6= 0, and in addition a 11 6= 0. This is a generic case. Applying the rst equation of (18) gives which in turn results in as expected, i.e., no second order terms can be removed. Similarly, from the second equation of (18) one can nd the following equations: a a 20 a 21 a First it is observed from equation (29) that if we only use the near-identity transformation (21), then only the coecient h 11 can be used in equation (29), and thus a must be retained in the normal form as expected. However, an additional term a 12 must be retained too in the normal form. More terms (in addition to a 11 ; a 20 and a will be found in higher order normal forms. This can also happen in other singularities. This shows that one cannot apply the \simplied" way to nd SNF of a system with perturbation parameters. In other words, unlike CNFs, the SNF of system (20) is not equal to the SNF of the \reduced" system (23), given by equation (24) or (25), plus an unfolding (which is a 11 for this case). Secondly, note from equation (29) that with the aid of rescaling, we can remove g 3 which appears in the SNF of the \reduced" system. There are three coecients three equations in (29), and thus the three equations can be solved when It is further seen from equation (29) must be chosen for the rst equation, indicating that the rescaling must include the state variable in order to obtain the SNF of the system. h 11 and T 01 are used for the second and the third equations of (29), respectively. Before continuing to the next order equation, we summarize the above results as the following theorem. Theorem 2: The general SNF of system (20) for the single zero singularity cannot be obtained using only a near-identity transformation. When the rescaling on time is applied, the SNF of the system can be even further simplied. However, the rescaling must contain the state variable. By repeatly applying the recursive formula (11), one can continue the above procedure to nd the algebraic equations for the third, fourth, etc. order equations and determine the nonlinear coecients. The recursive algorithm has been coded using Maple and executed on a PC. The results are summarized in Table 1, where NT stands for Nonlinear Transformation. The table shows the coecients which have been computed. The coecients in the rst row, are actually the two coecients of the SNF. Table 1. NT coecients for a 20 6= 0. l 70 h 61 h 52 h 43 h 34 T 15 h It is observed from Table 1 that except for the two coecients g 2 and L 1 , all other coecients are lined diagonally in the ascending order, according to one of the subscripts of h ij or T ij . The Maple program has been executed up to 10th order. But the general rule can be easily proved by the method of mathematical induction. Note that the coecients h 2j are not presented in Table 1, instead the coecients T 1j are used. In fact, the coecients h 2j do not appear in the algebraic equations and that's why the coecients T 1j have to be introduced, which causes the state variable involved in the time rescaling. Further it is seen that the coecients h 0j follows L 1 , the coecients h 1j follows g 2 , and the coecients T 1j are below g 2 . After this, h are followed. This rule will be seen again in other non-generic cases discussed later. Each row of the table corresponds to a certain order algebraic equations obtained from the recursive equation (11). For example, the two coecients in the top row are used for solving the second algebraic equations, corresponding to the coecients of y 2 and y , and thus the SNF of system (20) for a 11 6= 0 and a 20 6= 0 is given by dy d up to any order. The three coecients, in the second row are used for solving the third order algebraic equations (see equation (11)) corresponding to the coecients y 3 ; y 2 and y 2 . In other words, all the three third order nonlinear terms in system (20) can be removed by the three coecients, and so on. All the coecients listed in Table 1 are explicitly expressed in terms of the original system coecients a ij 's. Therefore, the two nonlinear transformations given by equations (21) and (22) are now explicitly obtained. Simple non-general case: Now suppose a we assume a 11 6= 0. This is a non-generic case. Then the coecient of the in the second algebraic equation is identically equal to zero due to a the solution procedure is similar to Case 1, we omit the detailed discussion and list the results in Table 2, where Table 2. NT coecients for a Thus the SNF of system (20) for this case is given by dy d Note from Table 2 that the top left entry is empty due to a 22 = 0, while g 3 moves downwards by one row. The coecients h 1j still follow g coecient, and the T ij coecients are still below the g coecients. T 0j coecients do not change. However, comparing Table 2 with Table 1 shows that Table 2 has one more line of T 2j coecients, in addition to T 1j . Moreover, there is a new line given by the coecients h 2j following the empty box where One may continue to apply the above procedure and execute the Maple program to compute the SNF of the single zero singularity for the case a a Tables similar to Tables 1 and 2 can be found. In gen- eral, we may consider the following non-generic case. General non-generic case: a As usual, we assume a 11 6= 0. The results are listed in Table 3, where and a k0 . This indicates that the SNF of system (20) for the general non-generic case is dy d The general table has the similar rule as that of Table 1 and 2: T 0j 's follow follow the empty box where 's follow the non-zero g k , and lines of coecients Table 3. NT coecients for a k0 6= Summarizing the above results yields the following theorem. Theorem 3: For the system dx dt a 1i i x +X a 2i i x 2 +X a which has a zero singularity at the equilibrium the rst non-zero coecients of a j0 's is a k0 , then the SNF of the system is given by dy d up to any order. In the above we only discussed the case a 11 6= 0 which results in the unfolding a 11 y. Other possible unfolding may not be so simple as this case. However, they can be easily obtained by executing the Maple program. For example, suppose a but a 13 6= 0 and a 21 6= 0, then the SNF is found to be dy d up to any order. Conclusions An ecient method is presented for computing the simplest normal forms of dierential equations involving perturbation parameters. The main advantages of this approach are: (i) it provides an algorithm to compute the kth-order algebraic equations which only contain the kth-order terms. This greatly save computational time and computer memory. (ii) The nonlinear transformation is given in a consistent form for the whole procedure. The zero singularity is particularly considered using the new approach. It is shown that the SNF for the single zero singularity with unfolding has a generic form which contains only two terms up to any order. Acknowledgment This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC). --R Normal forms for singularities of vector Unique normal forms for planar vector Normal forms for nonlinear vector Normal forms for nonlinear vector Unique normal form of the Hamiltonian 1:2-resonance Further reduction of the Takens-Bogdanov normal forms Linear grading function and further reduction of normal forms Simplest normal forms of Hopf and generalized Hopf bifurcations Unique normal form of Bogdanov- Takens Singularities Hypernormal forms for equilibria of vector Hypernormal forms calculation for triple zero degeneracies The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue Computation of simplest normal forms of di Singularities of vector --TR --CTR Pei Yu , Yuan Yuan, A matching pursuit technique for computing the simplest normal forms of vector fields, Journal of Symbolic Computation, v.35 n.5, p.591-615, May

nonlinear transformation;the simplest normal form SNF;rescaling;computer algebra;differential equation;lie transform;unfolding

636361

Approximate Nonlinear Filtering for a Two-Dimensional Diffusion with One-Dimensional Observations in a Low Noise Channel.

The asymptotic behavior of a nonlinear continuous time filtering problem is studied when the variance of the observation noise tends to 0. We suppose that the signal is a two-dimensional process from which only one of the components is noisy and that a one-dimensional function of this signal, depending only on the unnoisy component, is observed in a low noise channel. An approximate filter is considered in order to solve this problem. Under some detectability assumptions, we prove that the filtering error converges to 0, and an upper bound for the convergence rate is given. The efficiency of the approximate filter is compared with the efficiency of the optimal filter, and the order of magnitude of the error between the two filters, as the observation noise vanishes, is obtained.

Introduction Due to its vaste application in engineering, the problem of filtering a random signal X t from noisy observations of a function h(X t ) of this signal has been considered by several authors. In particular, the case of small observation noise has been widely studied, and several articles are devoted to the research of approximate filters which are asymptotically efficient when the observation noise vanishes. Among them, one notices a first group in which a one-dimensional system is observed through an injective observation function h (see [4, 5, 1]); in this case, the filtering error is small when the observation noise is small, and one can find efficient suboptimal finite-dimensional filters. The multidimensional case appears later on with [6, 7], but an assumption of injectivity of h is again required; in particular, the extended Kalman filter is studied in When h is not injective, the process fX t g cannot always be restored from the observation of so the filtering error is not always small; such a case is studied in [3]. However, there are some classes of problems in which fX t g can be restored from fh(X t )g; in these cases, the filtering error is small and one again looks for efficient suboptimal filters. For instance, fX t g is sometimes obtained from fh(X t )g and its quadratic variation, see [2, 8, 9, 11]. Here, we are interested in another case in which h(X t ) is differentiable with respect to the time t, and fX t g is obtained from fh(X t )g and its derivative. More precisely, we consider the framework of [10] which we now describe. We consider the two-dimensional process X given by the It"o equation dx (1) dx (2) (1) with initial condition X 0 and we are concerned by the problem of estimating the signal when the observation process is modelled by the equation dy t and f are standard independent real-valued Wiener processes, and " is a small nonnegative parameter. In particular, if f 1 , then x (1) is the position of some moving body on R, x (2) is its speed, the body is submitted to a dynamical force described by f 2 and to a random force described by oe, and one has a noisy observation of the position. if the functions h and x 2 are injective, then the signal X t can (at least theoretically) be exactly restored from the observation; we are here interested by the asymptotic case " ! 0, and we look for a good approximation of the optimal filter This approximation should be finite dimensional (solution of a finite dimensional equation driven by y t ). The same problem has been dealt with in [10] (with oe constant), by means of a formal asymptotic expansion of the optimal filter in a stationary situation. Our aim is to work out a rigorous mathematical study of the filter proposed by [10], namely the solution M of with F 12 and with initial condition M 0 This filter does in fact correspond to the extended Kalman filter with stationary gain if one neglects the contribution of the derivatives of f other than @f 1 . The stability of this filter is not evident and requires some assumptions. When it is stable, we prove in this work that x (1) and x (2) O( We also verify that (6) can be improved when oe is constant, h is linear, and f 2 is linear with respect to x 1 (this case will be referred to as the almost linear case). The proofs follow the method of [7]. The contents are organized as follows. In Section 2, we introduce the assumptions which will be needed in the sequel, and we study the filtering error as " converges to zero; more precisely, we obtain the rate (5). In Section 3 the error between the approximate filter and the optimal filter is studied, and we prove (6). Section 4 is devoted to the almost linear case. Notations The following notations are used: "oe F 22 and are the Jacobian matrices of f and \Sigma; is either a 2 \Theta 2 matrix (if \Phi is R 2 -valued) or a line-vector (if \Phi is real-valued), see Section 3; The symbol is used for the transposition of matrices. When describing the behaviour of approximate filters, we will write asymptotic expressions with the meaning given by the following definition. Definition 1.1 Consider a real or vector-valued stochastic process f t g. If fi is real and p 1, we will write that when for some q 0, ff ? 0 and some positive constants C 1 , C 2 , c 3 , e \Gammac 3 t=" ff for t 0 and " small. In this situation the process f t g is usually said to converge to zero with rate of order " fi , in a time scale of order " ff . 2 Estimation of X The following assumptions will be used throughout this article. The last one depends on a parameter ffi 1. is a random variable, the moments of which are finite; are standard independent Wiener processes independent from X 0 (H3) the function h is C 3 with bounded derivatives, and h 0 is positive; (H4) the function f is C 3 with bounded partial derivatives and F 12 is positive; (H5) the function oe is C 2 with bounded partial derivatives; (H6.ffi) one hasffi oe F for any x = ), and for some positive oe, H and F . We consider the system (1)-(2) and the filter (3). We let F t be the filtration generated by t be the filtration generated by (y t ). Assumption (H6.ffi) says that the system does not contain too much nonlinearity; when it is not satisfied, there may be a small positive probability for the filter to loose the signal (see [8] for a similar problem). This is a rather restrictive condition, so we discuss at the end of the section the general case where it does not hold. Theorem 2.1 Assume (H1) to (H5). There exists some universal ffi ? 1 such that if (H6.ffi) holds, then one has x (1) in L p for any p 1. Before entering the proof of this result, we re-scale the system in order to reduce the notation in (H6.ffi). If we replace the processes x (1) t and y t by x (1) t =oe and y t =(oe F the functions f 1 , oe and h are respectively replaced by (oe (oe oe(oe F and " is replaced by " / (oe F H). We can apply the filter (3) to this new system, and we obtain F t =oe. This shows that the problem can be reduced to the case Now consider a change of basis defined by a matrix T and its inverse T \Gamma1 , where "=2 "=2 Then consider the process We are going to check that Z t is the solution of a linear stochastic differential equation; the study of the exponential stability of this equation will enable the estimation of both components of Z t , and the theorem will immediately follow. An equation for Z t From equations (1) to (3), we have ))dt d In this equation, we introduce the Taylor expansions for the functions f and h and h(x (1) are R 2 and R valued processes depending on fX t g and fM t g, and We obtain a linear equation for X t . By applying the transformation (7), we deduce for Z t an equation of the type d The precise computation shows that A t with A (11) A (12) A (11) A (21) A (22) and where e A t is a 2 \Theta 2 matrix valued process which is uniformly bounded as " converges to 0; similarly, the matrix-valued process U t is also uniformly bounded. Stability of A t A t is the constant matrix and A t A In the general case ffi ? 1, the coefficients of A t A can be controlled so that this matrix is uniformly close to \Gamma2I if ffi is close to 1; in particular, if 2, one can choose ffi so that A I and therefore I if " is small. End of the proof of Theorem 2.1 Our goal is now to deduce an estimate of Z t in L 2p for p integer. From It"o's Formula and (8), the process kZ t is solution of d We deduce that the moment of order p of kZ t k 2 is finite, and that d dt From (9), one has Z As a consequence of the Cauchy-Schwarz inequality, one has kU Thus we obtain the inequality d dt \Gammap ff Moreover, there exists C 0 such that so d dt By solving this differential inequality, one obtains that for some C 00 Thus Z t is O(" 1=4 ), and the order of magnitude of the components of X follows from (7) and the form of T \Gamma1 . \Pi We remark in (10) that the time scale of the estimation is of order "; one can compare it with the time scale " obtained when the observation function is injective (see for instance [5]). This means that it takes here more time to estimate the signal, and this is not surprising since the second component of the signal is not well observed. There are also other systems where the time scale is not the same for the different components of the signal (see [8]). In Theorem 2.1, we need the assumption (H6.ffi) which is a restriction to the non linearity of the system; otherwise it is difficult to ensure that the filter does not loose the signal (this problem also occurs in [8]). Actually, we have chosen the filter (3) because it gives a good approximation of " next section), but it is not the most stable one. If in (4) we replace the processes by constant numbers oe, F and H, we obtain a filter with constant gain; we can again work out the previous estimations and prove that the result of Theorem 2.1 holds for this filter without (H6.ffi), as soon as F Thus we have two filters: a filter which is stable and tracks the signal under rather weak assump- tions, and the filter (3) which seems more fragile but gives (under good stability assumptions) a better approximation of the optimal filter. 3 Estimation of " The main result contained in this section is Theorem 3.1, which states the rate of convergence of the approximate filter considered in this paper towards the optimal filter. In order to give a proof of this theorem a sequence of steps are needed: a change of probability measure, the differentiation with respect to the initial condition and an integration by parts formula. A similar method of proof is adopted in [7]. As in Theorem 2.1, we may have a problem of stability if the general non linear case. Theorem 3.1 Assume (H1) to (H6.ffi) and (H7) The law of X 0 has a C 1 positive density p 0 with respect to the Lebesgue measure and rp 0 ) is in L 2 . If ffi in (H6.ffi) is close enough to 1, then the filter M t given by equation (3) satisfies x (1) O( in L 2 . The rest of this section is devoted to the proof of this theorem. We suppose as in Theorem 2.1 that 1. Consider the matrix which depends only on M t . Notice that P t is the solution of the stationary Riccati equation e F (M t )\Sigma (M t with e and that the process R t of (4) is We will also need the inverse of P t namely Change of probability measure Our random variables can be viewed as functions of the initial condition X 0 and of the Wiener processes w and w. We are going to make a change of variables; in view of the Girsanov theorem, this can be viewed as a change of probability measure; however, all the estimations will be made under the original probability P . Thus consider the new probability measure which is given on F t by d dP where ae \Gamma" Z th(x (1) s Z th 2 (x (1) s ds oe The probability P is the so-called reference probability, and one checks easily from the Girsanov Theorem that y t =" and w t are standard independent Wiener processes under . Let us define now the probability measure e P on F t by d where s s ds oe Then, the processes e Z t\Sigma (M s s ds and y t =" are standard independent Wiener processes under e P . On the other hand, one has and log(L t Z th(x (1) s Z th 2 (x (1) s ds \Gamma Z t\Sigma (M s s s Differentiation with respect to the initial condition and an estimation The random variables involved in our computation can now be viewed as functions of and fy t g; let us denote by r 0 the differentiation with respect to the initial condition X 0 puted in L p ). In particular, we can see on (13) and (14) that the processes X t and log(L t are differentiable, and we obtain respectively matrix and vector-valued processes. Our aim is to estimate the process log(L t U with Then an integration by parts will enable to conclude. By applying the operator r 0 to (14), one gets log(L t Z th 0 (x (1) s x (1) s (dy s s s s =" Z th 0 (x (1) s x (1) s d s We can also differentiate (13), and if \Sigma 0 is the Jacobian matrix of \Sigma, we obtain The matrix r 0 X t is invertible and It"o's calculus shows that From this equation and (16) one can write that: d log(L t t )(r 0 =" log(L t t )(r 0 log(L t t )(r 0 since one has h 0 (x (1) x (1) On the other hand, from the equations of X t and M t (equations (1) and (3), respectively), one has By writing the differential of P in the form we obtain d dt d d J (2) One can write the Taylor expansions for f and h h(x (1) with By using these expansions together with the consequence of (12) in equation (19), we obtain d dt d d J (2) dt By adding this equation to (18), we obtain that the process V t of (15) satisfies )d d d J (2) dt (20) is the matrix given by Consider also the matrix-valued process A t )\Sigma (M t Then the equation (20) can be written in the form where U \Gamma"R J (2) =" U U (apply (12) for the last line). We deduce that E[kV 0 is of order " \Gamma3 , and that d dt We have to estimate the terms of the right-hand side. By computing the matrix A t , we obtain that A t A t with A (11) A (21) A (12) A (22) and e A t is uniformly bounded. As in the proof of Theorem 2.1, we see that if then the matrix A t is simply A t which satisfies A Thus, for 2, when ffi is close enough to 1, and when " is small enough, we have ff We also notice that ffp and that U )U is bounded. Thus (24) implies that for " small, d dt Let us first estimate J (3) . We deduce from the Riccati equation (11) satisfied by P t that the process S t defined in (21) satisfies By computing this matrix and applying Theorem 2.1, we check that in the spaces L p . Thus The term \Sigma (M t )U is easily shown to have the same order of magnitude. On the other hand, by looking at the equation of M t and by applying It"o's formula, we can prove that for any C 2 function ae with bounded derivatives, one has By applying this result to the functions involved in P \Gamma1 , it appears that J (1) We deduce that the terms of J (3) involving J (1) and J (2) are also of order " \Gamma1 . Finally, OE t and are respectively of order " 1=2 and " 3=2 , so the last term is of order " \Gamma1 , and we deduce that We can also estimate J (4) and J (5) and check that they are of order " \Gamma3=4 . Thus (26) enables to conclude that in L 2 . We can take the conditional expectation with respect to Y t in this estimation, because the conditional expectation is a contraction in L 2 ; thus E[V t is O(1= ") in L 2 , and therefore, we obtain from the definition (15) that log(L t t )(r 0 Application of an integration by parts formula The estimation of the right-hand side of (27) can be completed by means of an integration by parts formula. It is proved in Lemma 3.4.2 of [7] that if w; y) is a functional defined on the probability space which is differentiable with respect to the initial condition (in the spaces iis the differentiation with respect to the ith component of X 0 , then 0: (28) We can write equation (17) in the form )\Sigma (M t dt )d e with (r 0 This equation can be differentiated with respect to X 0 , so we can apply the integration by parts formula (28) to the coefficients of the matrix (r 0 its ith line. Then (p \Gamma1@p 0 0: By summing on i and multiplying by U , we have log(L t r i(r 0 U 0: (30) The first term of (30) is exactly the term that we want to estimate in (27). For the second term of (30), if we have from (17) and (22) that We proceed as in the study of (23). The stability of the matrix A t which has been obtained in (25) and the boundedness of U implies that (r 0 exponentially small in L 2 , so the second term is negligible. Let us study the third term of (30). If then by differentiating (29) and transforming back e w into w, we get U \Gamma\Sigma (M t )P \Gamma1 with By summing on i and using @ae is the jth line of ae, we obtain that \Phi t is solution of \Gamma\Sigma (M t where oe 0 is the Jacobian of oe. A computation shows that The multiplication on the right by U yields a process of is also O(" \Gamma1 ), and the term involving the second derivative of oe is O(" \Gamma1=2 ). By proceeding again as in the study of (23), we deduce that \Phi t is of order " \Gamma1=2 . Thus (27), (30) and the estimation of \Psi t and \Phi t yield We multiply on the right by the matrix U , the coefficients of which are of order " 3=2 for the first column and " for the second column, and we deduce the order of " which was claimed in the Theorem. \Pi 4 An almost linear case It is interesting to consider a particular case in which oe, h 0 and F 12 are constant, so that the system (1)-(2) is 8 dx (1) x (2) dx (2) dy In particular, (H6.ffi) holds with it is possible to improve the upper bounds given in Theorem 3.1. The result is stated in the following proposition. Proposition 4.1 Assuming that (H1) to (H7) hold for (32), the filter M t given by equation verifies x (1) in L 2 . Proof The proof follows closely the sequence of steps adopted in Theorem 3.1. The matrices R are now constant; the processes J (1) , J (2) and J (5) are zero. The order of S t is improved into and so that is of order " \Gamma3=4 . Thus V t is O(" \Gamma1=4 ) and we obtain O(" \Gamma1=4 ) in (27). For the end of the proof, we see that so is bounded. Multiplication by U yields a process of order " \Gamma1=2 , so the process \Phi t of (31) is bounded for " small. We can conclude that and deduce the proposition. \Pi --R On some approximation techniques in non linear filtering theory Piecewise monotone filtering with small observation noise Approximate filter for the conditional law of a partially observed process in nonlinear filtering Asymptotic analysis of the optimal filtering problem for one-dimensional diffusions measured in a low noise channel Nonlinear filtering of one-dimensional diffusions in the case of a high signal-to- noise ratio Nonlinear filtering and smoothing with high signal-to-noise ratio Efficiency of the extended Kalman filter for non linear systems with small noise Estimation of the quadratic variation of nearly observed semimartingales with application to filtering Filtrage lin'eaire par morceaux avec petit bruit d'observation Asymptotic analysis of the optimal filtering problem for two dimensional diffusions measured in a low noise channel Nonlinear filtering and control of a switching diffusion with small observation noise --TR

stochastic differential models;approximate filters;nonlinear filtering

636409

An analog characterization of the Grzegorczyk hierarchy.

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n - 3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions.

INTRODUCTION The theory of analog computation, where the internal states of a computer are continuous rather than discrete, has enjoyed a recent resurgence of interest. This stems partly from a wider program of exploring alternative approaches to computation, such as quantum and DNA computation; partly as an idealization of numerical algorithms where real numbers can be thought of as quantities in themselves, rather than as strings of digits; and partly from a desire to use the tools of computation theory to better classify the variety of continuous dynamical systems we see in the world (or at least in its classical idealization). However, in most recent work on analog computation (e.g. [BSS89, Mee93, Sie99, Moo98]) time is still discrete. Just as in standard computation the- ory, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few eorts have been made in the direction of studying computation by continuous-time dynamical systems [Moo90, Moo96, Orp97b, Orp97a, SF98, Bou99, CMC00], no particular set of denitions has become widely accepted, and the various models do not seem to be equivalent to each other. Thus analog computation has not yet experienced the unication that digital computation did through Turing's work in 1936. In this paper, as in [CMC00], we take as our starting point Claude Shan- non's General Purpose Analog Computer (GPAC). This was dened as a mathematical model of an analog device, the Dierential Analyser, the fundamental principles of which were described by Lord Kelvin in 1876 [Kel76]. The Dierential Analyser was developed at MIT under the supervision of Vannevar Bush and was indeed built in for the rst time in 1931 [Bow96]. Its input was the rotation of one or more drive shafts and its output was the rotation of one or more output shafts. The main units were interconnected gear boxes and mechanical friction wheel integrators. Just as polynomial operations are basic to the Blum-Shub-Smale model of analog computation [BSS89], polynomial dierential equations are basic to the GPAC. Shannon [Sha41] showed that the GPAC generates exactly the dierentially algebraic functions, which are unique solutions of polynomial dierential equations. This set of functions includes simple functions like e x and sin x as well as sums, products, and compositions of these, and solutions to dierential equations formed from them such as f Pour-El [Pou74], and later Lipshitz and Rubel [LR87], extended Shannon's work and made it rigorous. The GPAC also corresponds to the lowest level | we denote here by G | in a theory of recursive functions on the reals proposed by Moore [Moo96]. There, in addition to composition and integration, a zero-nding operator analogous to the minimization operator of classical recursion theory is included. In the presence of a liberal semantics that denes even when f is undened on x, this permits contraction of innite computations into nite intervals, and renders the arithmetical and analytical hierarchies computable through a series of limit processes similar to those used by Bournez [Bou99]. However, such an operator is clearly unphysical, except when the function in question is smooth enough for zeroes to be found in some reasonable way. In [CMC00] a new extension of G was proposed. The operators of the GPAC were kept the same | integration and composition | but piecewise- analytic basic functions were added, namely k the Heaviside step function, Adding one of these functions, k for some xed k, as an 'oracle' can be thought of as allowing an analog computer to measure inequalities in a 1)-times dierentiable way. These functions are also unique solutions of dierential equations such as xy two boundary conditions rather than just an initial condition, which is a slightly weaker denition of uniqueness than that used by Pour-El to dene GPAC-computability. By adding these to the set of basic functions, for each k we get a class we denote by G A basic concern of computation theory is whether a given class of functions is closed under various operations. One such operation is iteration, where from a function f(x) we dene a function F (x; applied t times to x, for t 2 N. The main result of [CMC00] is that G is closed under iteration for any k 0, while G is not. (Here we adopt the convention that a function where one or more inputs are integers is in a given analog class if some extension of it to the reals is.) It then follows that G includes all primitive recursive functions, and has other closure properties as well. To rene these results, in this paper we consider a restricted version of Shannon's GPAC. In particular, we restrict integration to linear integra- tion, i.e. solving linear dierential equations. We dene then a class of computable functions L whose operators are composition and linear integration and then add, as before, a basic function k for some xed k > 2. The model we obtain, L+ k , is weaker than G . One of the main results of this paper is that, for any xed k > 0, L+ k contains precisely the elementary functions, a subclass of the primitive recursive functions introduced by Kalmar [Kal43] which is closed under the operations of forming bounded sums and products. Inversely, using Grzegorczyk and Lacombe's denition of computable continuous real function [Grz55, Lac55], we show that all functions in L+ k are elementarily computable for any xed k > 2, and that if a function f 2 L+ k is an extension to the reals of some function 4~ f on the integers, then ~ f is elementary as well. 1 Thus we seem to have found a natural analog description of the elementary functions, the smallest known recursive class closed under time and space complexity [Odi00]. To generalize this further, we recall that Grzegorczyk [Grz53] proposed a hierarchy of computable functions that straties the set of primitive recursive functions. The elementary functions are simply the third level of this hierarchy. We show that if we allow L to solve n 3 non-linear dierential equations of a certain kind, then all functions in the nth level of the Grzegorczyk hierarchy have extensions to the reals in the resulting analog class. A converse result also holds. Therefore, we claim that there is a surprising and elegant connection between classes of analog computers on the one hand, and subclasses of the recursive functions on the other. This suggests, at least in this region of the complexity hierarchy, that analog and digital computation may not be so far apart. The paper is organized as follows. In Section 2 we review classical recursion theory, the elementary functions, and the Grzegorczyk hierarchy. In Section 3 we recall some basic facts about linear dierential equations. Then, in Section 4 we dene a general model of computation in continuous time that can access a set of 'oracles' or basic functions, compose these, and solve linear dierential equations. We call this class L or more generally for a set of oracles '. We then prove bounds on the growth of functions denable in L The existence of those bounds allows us to prove the main lemma of the paper, which shows that L+ k is closed under forming bounded sums and bounded products. With this, we are able to prove that L contains extensions to the reals of all elementary functions. Inversely, we show that all functions in are elementarily computable. This shows that the correspondence between L+ k and the elementary functions is quite robust. Then, in Section 5 we consider the higher levels in the Grzegorczyk hierarchy by dening a hierarchy of analog classes Gn Each one of these classes is dened similarly to L that our model is now allowed to solve up to n 3 non-linear dierential equations, of a certain kind, which produces iterations of functions previously dened. We then show that this hierarchy coincides, level by level, with the Grzegorczyk hierarchy in the same sense that L coincides with the elementary functions. Finally, we end with some remarks and open questions. 1 The approach we follow [Ko91, Pou74, Wei00] to describe the complexity of real functions is eective, in the sense that it extends standard complexity theory and relies on the Turing machine as the model of computation to dene computability and complexity. A distinct approach is to consider reals as basic entities as in [BSS89]. 2. SUBRECURSIVE CLASSES OVER N AND THE In classical recursive function theory, where the inputs and output of functions are natural numbers N, computational classes are often dened as the smallest set containing a set of initial functions and closed under certain operations, which take one or more functions in the class and create new ones. Thus the set consists of all those functions that can be generated from the initial ones by applying these operations a nite number of times. Exemples of common initial functions are zero, successor, projections x y and 0 if x < y. Typical operations include (where ~x represents a vector of variables, which may be absent): 1. Composition: Given an n-ary function f and a function g with n 2. Primitive recursion: Given f and g of the appropriate arity, dene h such that h(~x; 3. Bounded sum: Given f(~x; y), dene h(~x; z<y f(~x; z). 4. Bounded product: Given f(~x; y), dene h(~x; z<y f(~x; z). By starting with various sets of basic functions and demanding closure under various properties, we can dene various natural classes. In partic- ular, we will consider: Primitive recursive functions are those that can be generated from zero, successor, and projections using composition and primitive recursion. Elementary functions are those that can be generated from zero, suc- cessor, projections, addition, and cut-o subtraction using composition and the operation of forming bounded sums and bounded products. In classical recursive function theory more general objects called which take an innite sequence and a nite number of integers as input, can be dened. To dene a functional we add a new operation A(; which accesses the x-th element of a given innite sequence . When A is not used in the recursive denition of , degenerates into a function. In particular we say that a functional is elementary if it can be generated from the basic functionals y, F (; x; using composition, bounded sums and bounded prod- ucts. We will need this notion of elementary functional in Proposition 4.4. The class of elementary functions, which we will call E , was introduced by Kalmar [Kal43]. As examples, note that multiplication and exponentiation over N are both in E , since they can be written as a bounded sum and a bounded product respectively: z<y x and x z<y x. Since E is closed under composition, for each m the m-times iterated exponential In fact, no elementary function can grow faster than 2 [m] for some xed m, and many of our results will depend on the following bound on their growth [Cut80]: Proposition 2.1. If f 2 E, there is a number m such that, for all ~x, The elementary functions are exactly the functions computable in elementary time [Cut80], i.e., the class of functions computable by a Turing machine in a number of steps bounded by some elementary function. The class E is therefore very large, and many would argue that it contains all practically computable functions. It includes, for instance, the connectives of propositional calculus, functions for coding and decoding sequences of natural numbers such as the prime numbers and factorizations, and most of the useful number-theoretic and metamathematical functions [Cut80, Ros84]. However, Proposition 2.1 shows that it does not contain the iterated exponential 2 [m] (x) where the number of iterations m is a variable, since any function in E has an upper bound where m is xed. The iterated exponential is, however, primitive recursive. As a matter of fact, it belongs to one of the lowest levels of the Grzegorczyk hierarchy [Grz53, Ros84], which measures the structural complexity of the class of primitive recursive functions, and we review below. Let's rst recall what a LOOP-program is. It is a program which can be written in a programming language with assignments, conditional and FOR statements, but with no WHILE or GO-TO statements. Notice that a LOOP-program always halts. The primitive recursive functions are precisely the functions that are computed by LOOP-programs. We can stratify them considering the subclasses of functions computable by LOOP- programs with up to n nested FOR statements. For this gives the elementary functions [HW99], and for n 2 this gives precisely E n+1 , the 1-th level of the Grzegorczyk hierarchy. Originally [Grz53], the Grzegorczyk hierarchy was dened recursively. The elementary functions are the third level of the Grzegorczyk hierarchy, . For the following levels we consider a family of functions N. These are, essentially, repeated iterations of the successor function, and each one grows qualitatively faster than the previous one. composing it yields functions as large as 2 [m] for any xed m. Iterating In general, En+1 can be dened with En 1 and closure under bounded sums and products (cf. [Odi00]): (The Grzegorczyk hierarchy) For n 3, E n is the smallest class containing zero, successor, the projections, cut-o subtraction, and En 1 , which is closed under composition, bounded sum, and bounded product. The union of all the levels of the Grzegorczyk hierarchy is the class PR of primitive recursive functions, i.e., We can also generalize Proposition 2.1 and put a bound on the growth of functions anywhere in the Grzegorczyk hierarchy: Proposition 2.2. If n 2 and f 2 E n then there is an integer m such that f(~x) E [m] 3. LINEAR DIFFERENTIAL EQUATIONS An ordinary linear dierential equation is a dierential equation of the where A(t) is a n n matrix whose entries are functions of t and ~ b(t) is a vector of functions of t. If ~ we say that the system is homogeneous. We can reduce a non-homogeneous system to a homogeneous one by introducing an auxiliary variable xn+1 such that xn+1 t, that is, which satises xn+1 The new matrix will just be This matrix is not invertible, which makes (1) harder to solve. However, since we don't need to solve the system explicitly, we prefer to consider the homogeneous equation as the general case in the remainder of the paper. This leads to the original denition of the Grzegorczyk hierarchy (cf. [Clo99]). The fundamental existence theorem for dierential equations guarantees the existence and uniqueness of a solution in a certain neighborhood of an initial condition for the system ~x when f is Lipschitz. For linear dierential equations, we can strengthen this to global existence whenever A(t) is continuous, and establish a bound on ~x that depends on kA(t)k. 3 Proposition 3.1 ([Arn96]). If A(t) is dened and continuous on an interval I = [a; b], where a 0 b, then the solution of Equation 2 with initial condition dened and unique on I. Furthermore, if kA(t)k is non-decreasing this solution satises Therefore, if A(t) is continuous and non-decreasing on R then solutions of linear dierential equations are dened on arbitrarily large domains and have an exponential bound on their growth that depends only on kA(t)k. Proposition 3.1 holds both for the max norm, and for the Euclidean norm [Har82]. If we use the max norm, which satises when the conditions of Proposition 3.1 are fullled. 4. THE ANALOG CLASS L FUNCTIONS In [Moo96, CMC00] a denition of Shannon's General Purpose Analog Computer (GPAC) in the framework of the theory of recursive functions on the reals is given. We denote the corresponding set of functions by G. It is the set of functions that can be inductively dened from the constants 0, 1, and 1, projections, and the operations of composition and integration. Integration generates new functions by the following rule: if f and g have appropriate arities and belong to G then the function h dened by the initial condition h(~x; and the dierential equation @ y h(~x; also belongs to G, over the largest interval containing 0 on which the solution is nite and unique. Thus G has the power 3 By k k we denote both the norm of a vector and the norm of a matrix, with 1g. By A being continuous, respectively increasing, we mean that all entries of A are continuous, respectively increasing, functions. to solve arbitrary initial value problems with unique solutions, constructed recursively from functions already generated. We dene here a proper subclass of G which we call L, by restricting the integration operator to solving time-varying linear dierential equations. To make the denition more general, we add a set of 'oracles' or additional basic functions. Let ' be a set of continuous functions dened on R k for some k. Then L+' is the class of functions of several real variables dened recursively as follows: Definition 4.1. A function h belongs to L its components can be inductively dened from the constants 0, 1, 1, and , the projections U i functions in ', and the following operators: 1.Composition: if a p-ary function f and a function g with p components belong to L+', then dened as belongs to L+'. 2.Linear integration: if f and g are in L+' then the function h satisfying the initial condition h(~x; solving the dierential equation @ y h(~x; belongs to L vector-valued with n components, then f has the same dimension and g(~x; y) is an n n matrix whose components belong to As shorthand, we will write R gh dy. Several notes on this denition are in order. First, note that linear integration can only solve dierential equations @ y gh where the right-hand side is linear in h, rather than the arbitrary dependence @ y which the GPAC is capable. Secondly, using the same trick as in Section 3 we can expand our set of variables, and so solve non-homogeneous linear dierential equations of the form @ y h(~x; Finally, the reader will note that we are including as a fundamental constant. The reason for this will become clear in Proposition 4.7. Unfor- tunately, even though it is easy to show that belongs to G, we have not found a way to derive using this restricted class of dierential equations. Perhaps the reader can do this, or nd a proof that we cannot. We will use the fact that, unlike solving more general dierential equa- tions, linear integration can only produce total functions: Proposition 4.2. All functions in L+' are continuous and are dened everywhere. Let us look at a few examples. Addition, as a function of two variables, is in L since Similarly, multiplication can be dened as is in L since it can be dened as by using either composition or linear integration, with exp [2] (Note that we are now using e rather than 2 as our base for exponentiation.) Thus the iterated exponential exp [m] is in L for any xed m. However, the function exp [n] (x), where the number of iterations is a variable, is neither in L nor in G. We prove this in [CMC00], and use it to show that Shannon's GPAC is not closed under iteration. However, if G is extended with a function k , for k 0, then the resulting class G + k is closed under iteration, where k is dened as follows. is the Heaviside step function Each k (x) can be interpreted as a function which checks inequalities such as x 0 in a (k 1)-times dierentiable way (for k > 1). It was also shown in [CMC00] that allowing those functions is equivalent to relaxing slightly the denition of GPAC by solving dierential equations with two boundary values instead of just an initial condition. In this section we consider L and prove that for any xed k > 2 this class is an analog characterization of the elementary functions. We will start by noting that all functions in L+ k have growth bounded by a nitely iterated exponential, exp [m] for some m. This is analogous to the bound on elementary functions in Proposition 2.1, and can easily be proved by structural induction, using the bound in Proposition 3.1. Proposition 4.3. Let h be a function in L+ k of arity m. Then there is a constant d and constants A; B; C; D such that, for all ~x j. The least d for which there are such A; B; C; D will be called the degree of h or deg h. Propositions 2.1 and 4.3 establish the same kind of bounds for E and . But the relation between those two classes can be shown to be much tighter: namely, all functions in can be approximated by elementary functions, and all extensions to the reals of elementary functions are contained in L Since E is dened over the natural numbers and L is dened over the reals, we rst need to set some conventions. (Convention 1) We say that a function over the reals is elementary if it fullls Grzegorczyk and Lacombe's denition of computable continuous real function [Grz55, Grz57, Lac55, PR89] and if the corresponding functional is elementary. First, we write S ; a if an integer sequence N. Note that, by dividing denition allows sequences of integers to converge to real numbers. To dene computability of real functions that range over R with elementary functions and functionals dened on N we use a simple bijective encoding and we say a real number a 2 R is elementarily computable if there is an elementary function s that Finally, a continuous real function f is elementarily computable if there is an elementary functional which, for all a 2 R and for all sequences : N ! N such that - ; a, denotes the sequence :g. The denition for vector-valued functions and functions of several variables is similar. The denition above can be extended in a straightforward manner to -computability. This was already described in [Zho97], where f is said to be E n -computable if: (1) f maps every E n -computable sequence of reals into an E n -computable sequence of reals; and (2) there is a function d in E n such that jx yj < 1=d(k) implies jf(x) f(y)j < 1=(k+1) for all x; y in any bounded domain of f . We notice that time and space complexity for functions, with n 3, can alternatively be dened using a function oracle Turing machine as the model of computation [Ko91, Wei00]. For example, a function f is elementary if f(x) can be computed with precision 1=n in a number of steps elementary in jxj and n. (Convention Conversely, we say that contains a function f contains some extension of f to the reals, and similarly for functions of several variables. These two conventions allow us to compare analog and digital complexity classes. Proposition 4.4. If f belongs to L f is elementarily computable. Proof. Once again, the proof will be done by structural induction. To keep the notation simpler, we won't include the encoding function in the proof. It is clear that the constants 0, 1, 1 and are elementarily computable (e.c. for short). U i (~x) is simply x i and is obviously e.c., and k (x) is e.c. since and polynomials are in E and the parity of x is computable in E too. For simplicity, we prove that composition of e.c. functions is also e.c. just for real functions of one variable. The proof is similar for the general case. If g is e.c. then there is a functional g such that g (; sequence ; a, and if f is e.c. then there is a functional f such that sequence ' ; b. Setting f(g(a)). Since the composition of two elementary functionals is elementary, we are done. Finally, we have to show that if f and g are e.c. then h such that h(~x; f(~x) and @ y h(~x; is also. This means that we have to show that there is an elementary functional such that (; for all sequences We will do this using standard numerical techniques, namely Euler's method. Let us suppose that h 2 L+ k is twice continuously dierentiable, which is guaranteed if k > 2. 4 Fixing ~x and expanding h we obtain for some where i < < i+1 . Since g is e.c. there is an elementary functional g such that g (; To obtain an estimate of the value of g on (~x; i ), we set 1). The accuracy of this estimate depends on n since k We will set below a lower bound for n. The discrete approximation of h is then simply done by Euler's method, where the i 's are the discretization steps, and we will show that we can make the discretization error su-ciently small with an elementary number of discrete steps. 5 We dene a function by where the step size of the discretization are to be xed by the number of steps of the numerical approximation. We dene now an elementary functional . For each xed ~x and any sequence ; y, (; l) is dened as being the integer closest to (l + 1) N , where N is a suitably increasing elementary function of l and where N is obtained using 4 We don't study the particular cases since we are mainly interested in the properties of L+ k for large k, i.e., when L+ k only contains \smooth" functions. 5 When xed point numerical calculations are used, there is also a round-o error. However, in the worst case, the acumulated round-o error is of the same order of the discretization error (cf. [Har82, 3.4.4] and [VSD86]). Therefore, we will only study the discretization error of the numerical approximation. a discretization step (l in (5). Note that then (; l) is always an integer as required by the denition, and k(; l)=(l We will now show how to choose n, m and N as functions of l such that To prove that this can be done in elementary time, we rst need to set a bound on Since h 00 can be written as the sum and product of functions with bounds of the form A exp [d] (Bk(~x; t)k), from Proposition 4.3, then kh 00 (~x; t)k is bounded by A exp [d] (Bk(~x; t)k) for some d, A and B. We will call this bound and we will denote by a number larger than (~x; y), for instance, y, for any l and any ; y. The discretization error is and satises Furthermore, because f is e.c., where f is the elementary functional that computes f . A little tedious algebra shows then that Therefore, given , which is elementary, it su-ces to set to guarantee that k N l. Note that, since n, m and N are elementary functions of l, can be computed in elementary time. Therefore, by the triangle inequality, Finally, we just have to show how to dene from another elementary functional such that (; be the integer closest to 1). is elementary since it is a composition of and which is an elementary function. Since, for all l and ; y, kh(~x; y) little more algebra allows us to show that This concludes the proof. As a corollary, any function in L that sends integers to integers is elementary on the integers: Corollary 4.5. If a function f 2 L is an extension of a function ~ f is elementary. Proof. Proposition 4.4 shows us how to successively approximate f(x) to within an error in an amount of time elementary in 1= and x. If f is an in- teger, we just have to approximate it to error less than 1=2 to know its value exactly. Next we will prove the converse of this, i.e. that L contains all elementary functions, or rather, extensions of them to the reals. We will rst prove two lemmas. Lemma 4.6. For any xed k > 0, L contains sin, cos, exp, the constant q for any rational q, and extensions to the reals of successor, addition, and cut-o subtraction. Proof. We showed above that L+ k includes addition, and the successor function is just addition by 1. For subtraction, we have x @ y We can obtain any integer by repeatedly adding 1 or 1. For rational constants, by repeatedly integrating 1 we can obtain the function z k =k! and thus k. We can multiply this by an integer to obtain any rational q. For cut-o subtraction x : y, we rst dene a function s(z) such that Z. This can be done in L by setting R 1z k (1 z) k dz is a rational constant depending on k. Then is an extension to the reals of cut-o subtraction. Finally, dened by with exp as proved above. We now show that L has the same closure properties as E , namely the ability to form bounded sums and products. Lemma 4.7. Let f be a function on N and let g be the function on N dened from f by bounded sum or bounded product. If f has an extension to the reals in L does also. Proof. For simplicity, we give the proof for functions of one variable. We will abuse notation by identifying f and g with their extensions to the reals. We rst dene a step function F which matches f on the integers, and whose values are constant on the interval j. F can be dened as F is a function such that and s 0 R 1=2sin k 2t dt is a constant depending only on k. Since c k is rational for k even and a rational multiple of for k odd, s is denable in L (Now our reasons for including in the denition of L The bounded sum of f is easily dened in L by linear integra- tion. Simply write Dening the bounded product of f in L + k is more di-cult. Let us rst set some notation. Let f j denote f(j) for j 2 N, which is also equal to F (t) for t 2 the bounded product we wish to dene. The idea of the proof is to approximate the iteration g using synchronized clock functions as in [Bra95, Moo96, CMC00]. However, since the model we propose here only allows linear integration, the simulated functions cannot coincide exactly with the bounded product. Nevertheless, we can dene a su-ciently close approximation because f and g have bounded growth by Proposition 4.3. Then, since f and g have integer values, the accumulated error resulting from this approximation can be removed with a suitable continuous step function r simply dened by returns the integer closest to t as long as the error is 1=4 or less. Now dene a two-component function ~y(; t) where y 1 (; 1 and where () is an increasing function of . Then we claim that r(y 1 (n; n)). We will see that if grows quickly enough, then by setting we can make the approximation error jy 1 (n; n) g n j as small as we like, and then remove the error by applying r. Again, the idea is that on alternate intervals we hold either y 1 or y 2 constant and update the other one. For integer j, it is easy to see that when the term k ( sin 2t) holds y 2 constant, and y 1 moves toward y 2 f(j). Quantitatively, solving (7) for t 2 Similarly, when held constant and y 2 moves toward y 1 . This gives us the recursion Note that if () is su-ciently small then Now let are bounded from above by A exp [m] (Bn) as in Proposition 4.3. Below we show that we can set for instance adjusting m, and that for all m;n 1. Since this is less than 1=4, we can round the value of y 1 to the nearest integer using r, as claimed. To conclude the proof, we show that jy 1 (n; n) g n j < 1=4 . Without loss of generality, we will set the constants in the bound on f and g to A = 3 since A exp [m] (B) can always be bounded by 3 exp [m 0 ] () with large enough. Also to simplify the notation, we will denote y 1 (n; and y 2 (n; j) by y 1 (j) and y 2 (j), respectively. We will prove that jy 1 (n) for all m;n 1. We will proceed by induction on j for Recall that y 1 Equation 8 shows that jy 1 (1) f 0 j j1 f 0 j and that We will now show that if and then and for all j n 1, when f First, note that from (9), (10) and the triangle inequality, we obtain To prove (11) from (9) and (10) we use the recursion of Equation 8 and the bounds on f j and g j . From (8) we have From (9) we can write y 1 from (13), y 2 . Then (14) can be rewritten as which is Therefore, Since is small and is large then the rst term dominates the others and, consequently, we have jy 1 (j claimed. The proof that (9) and (10) imply (12) is similar. Finally, we brie y show that 2 4n (n which is always positive, is smaller than 32:9:e 4 =e 3e 2 we obtain the previous value. It is easy to verify that decreases when m and n increase. We illustrate this construction in Figure 1. We approximate the bounded product of the identity function, i.e. the factorial (n j<n j. We y 1,2 FIG. 1. A numerical integration of Equation (7), where f is a L+ k function such that 2. We obtain an approximation of an extension to the reals of the factorial function. In this example, where we chose a small < 4, the approximation is just su-cient to remove the error with and obtain exactly (5)). numerically integrated Equation (7) using a standard package (Mathemat- ica). An interesting question is whether L+ k is closed under bounded product for functions with real, rather than integer, values. Our conjecture is that it is not, but we have no proof of this. From the previous two lemmas it follows that Proposition 4.8. If f is an elementary function, then L+ k contains an extension of f to the reals. Taken together, Propositions 4.4 and 4.8 show that the analog class L+ k corresponds to the elementary functions in a natural way. 5. Gn In this section we show that we can extend the results of the previous section to the higher levels of the Grzegorczyk hierarchy, E n for n 3. Let us dene a hierarchy of recursive functions on the reals. Each level is denoted by Gn 3. The rst level is G 3 and each following level is dened either by adding a new basic function, or by allowing the system to solve a certain number of non-linear dierential equations. Definition 5.1 (The hierarchy Gn k be the smallest class containing the constants 0, 1, 1, and , the projections, and which is closed under composition and linear integration, and dened up to n 3 applications of the following operator: Non-linear integration (NLI): if a unary scalar funtion f belongs to Gn then the solution of the initial value problem with initial condition y 1 belongs to Gn We will now show that Gn xed k > 2, corresponds to the nth level of the Grzegorczyk hierarchy in the same way that corresponds to the elementary functions. First, we will show the operator NLI carries us up the levels of the Gn just as iteration does for the Proposition 5.2. For any function f 2 Gn there is an extension of the iteration F (x; Proof. F (x; t) is given by y 1 in the Equation (15) with the initial conditions x. Note that jxj k can dened in L it can be proved that, for all t 2 N, y 1 The function y 1 belongs to Gn+1 since it is dened from f , which is in only one application of NLI. The dynamics for Equation (15) is similar to Equation (7) for iterated multiplication, in which y 1 and y 2 are held constant for alternating intervals. The main dierence is that the terms jcos tj k+1 and jsin tj k+1 on the left ensure that y 1 converges exactly to f(y 2 ), and y 2 exactly to y 1 , by the end of the interval [n; n+1]. The term k (t) on the right ensures that the solution is constant y 1 for t < 0. The proof is similar to the one given in [CMC00, Proposition 9]. Now dene a series of functions exp for x 2 N. Since exp 2 2 L+ k , Proposition 5.2 shows that Gn contains extensions to the reals of exp n 1 for all n 3. Since tary, an extension of it to the reals belongs to L by Proposition 4.8, and since En+1 is dened from En by iteration, Gn contains extensions to the reals of En 1 for all n 3. For simplicity, we will also use the notation exp n (x) and En (x) for monotone extensions of exp n and En to x 2 R. In fact, nite compositions of exp n 1 and En 1 put an upper bound on the growth of functions in Gn so in analogy to Proposition 4.3 we have the following: Proposition 5.3. Let f be a function of arity m in Gn 3. Then there is a constant d and constants A; B; C; D such that, for all x kf(~x)k A exp [d] j. Moreover, the same is true (with dierent con- stants) if exp n 1 is replaced by En 1 . Now, as before, to compare an analog class to a digital one we will say that a real function is computable in E n if it can be approximated by a series of rationals with a functional in E n , and we will say that an integer function is in an analog class if some extension of it to the reals is. We will now show that Gn xed k > 2, corresponds to the nth level of the Grzegorczyk hierarchy in the same way that corresponds to the elementary functions. Proposition 5.4. The following correspondences exist between Gn and the levels of the Grzegorczyk hierarchy, E n for all n 3: 1.Any function in Gn is computable in E n . is an extension to the reals of some ~ f on N, then ~ 3.Conversely, if f 2 E n then some extension of it to the reals is in Gn+ k . Proof. To prove that functions in Gn can be computed with functionals in E n , we follow the proof of Proposition 4.4 for composition and linear integration. However, when we use Euler's method for numerical integration, we now apply the bound of Proposition 5.3 and set m, n and N to grow as AE [d] for a certain d. Since this is in E n , so are the functionals and . Numerical integration of functions in Gn dened by Equation (15) can also be done in E n , although the numerical techniques involved are slightly dierent since ~y 0 is dened implicitly. Now, as in Corollary 4.5, if f takes integer values on the integers we just have to approximate it to within an error less than 1=2, so the restriction of f to the integers is in E n . Conversely, the remarks above show that Gn contains an extension to the reals of En 1 , and Lemma 4.6 shows that it contains the other initial functions of E n as well. Furthermore, it is closed under bounded sum and bounded product for integer-valued functions. The proof of Lemma 4.7 proceeds as before, except that using Proposition 5.3 again, () is now an extension of this to the reals can be dened in Gn+ k , so can the linear dierential equation (7). We showed that Gn contains extensions to the reals of all initial functions of E n and is closed under composition and bounded sums and products for integer valued functions. Therefore, contains extensions to the reals of all functions in E n . A few remarks are in order. First, we stress that the analog model we dene contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve up to n 3 non-linear dierential equations of the form of Equation (15) and no other non-linear dierential equations. Secondly, notice that since [nE implies that Corollary 5.5. Any function in [n (Gn computable in PR and all primitive recursive functions are contained in [n (Gn Finally, instead of allowing our model to solve Equation (15), we can keep everything linear and dene Gn by adding a new basic function which is an extension to the reals of En 1 . While this produces a smaller set of functions on R, it produces extensions to R of the same set of functions on N as the class dened here. 6. CONCLUSION We have dened a new version of Shannon's General Purpose Analog Computer in which the integration operator is restricted in a natural way | to solving linear dierential equations. When we add the ability to measure inequalities in a dierentiable way, the resulting system L+ k corresponds exactly to the elementary functions E . Furthermore, we have dened a hierarchy of analog classes Gn by allowing n 3 non-linear equations of a certain form, and we have shown that this hierarchy corresponds, level by level, to the Grzegorczyk hierarchy E n for n 3. When combined with the earlier result [CMC00] that G contains the primitive recursive functions, this suggests that subclasses of the primitive recursive functions correspond nicely to natural subclasses of analog computers. Several open questions suggest themselves: 1. We used a very specic kind of non-linear operator to dene the classes in Section 5. Is there a more natural family of non-linear dierential equations, whose solutions are total functions, which yield non elementary 2. Can we do without in the denition of L+ k ? Note that we do not need to include it in the denition of Gn since we can dene limited integration as linear integration as and nally set However, we have been unable to nd a way to dene from linear integration alone. 3. Is L closed under bounded product for real-valued functions, and not just integer-valued ones? We think this is unlikely, since it would require some form of iteration like that in Equation 15 where y 1 and y 2 converge to the desired values exactly. We see no way to do this without highly non-linear terms. If L+ k is not closed under real-valued bounded products then we could ask what class would result from that additional operation. While the set of integer functions which have real extensions in the class would remain the same, the set of functions on the reals would be larger. 4. By adding to our basis a function that grows faster than any primitive recursive function, such as the Ackermann function, we can obtain transnite levels of the extended Grzegorczyk hierarchy [Ros84]. It would be interesting to nd natural analog operators that can generate such functions 5. How robust are these systems in the presence of noise? Since it is based on linear dierential equations, L may exhibit a fair amount of robustness to perturbations. We hope to quantify this, and explore whether this makes these models more robust than other continuous-time analog models, which are highly non-linear. 6. Our results on the Grzegorczyk hierarchy seem to be somehow related to [Gak99], which framework is the BSS model of computation. In [Gak99] the recursive characterization of the BSS-computable functions [BSS89] is restricted to match the recursive denition of the classes E n . This might suggest that our continuous-time operations on real functions, namely the various forms of integration we consider, are related to certain restricted types of BSS-machines. It is interesting that linear integration alone, in the presence of k , gives extensions to the reals of all elementary functions, since these are all the functions that can be computed by any practically conceivable digital de- vice. In terms of dynamical systems, L corresponds to cascades of nite depth, each level of which depends linearly on its own variables and the output of the level before it. We nd it surprising that such systems, as opposed to highly non-linear ones, have so much computational power. Finally, we note that while including k as an oracle makes these functions non-analytic, by increasing k they can be made as smooth as we like. Therefore, we claim that these are acceptable models of real physical phe- 3nomena, and may be more realistic in certain cases than either discrete or hybrid systems. ACKNOWLEDGMENTS We thank Jean-Sylvestre Gakwaya, Norman Danner, Robert Israel, Kathleen Mer- rill, Spootie Moore, Bernard Moret, and Molly Rose for helpful discussions, and the anonymous referees for important suggestions for improvement. This work was partially supported by FCT PRAXIS XXI/BD/18304/98 and FLAD 754/98. M.L.C. and J.F.C. also thank the Santa Fe Institute, for hosting visits that made this work possible, and LabMAC (Laboratorio Modelos e Arquitecturas Computacionais da FCUL). --R Equations Di Achilles and the tortoise climbing up the hyper-arithmetical hierar- chy Universal computation and other capabilities of hybrid and continuous dynamical systems. On a theory of computation and complexity over the real numbers: NP-completness Computational models and function algebras. An Introduction to Recursive Function Theory. Cambridge University Press Extensions de la Hi Some classes of recursive functions. Computable functionals. On the de Ordinary Di Matrix Analysis. Complexity of primitive recursion. Complexity Theory of Real Functions. Extension de la notion de fonction r Real number models under various sets of operations. Unpredictability and undecidability in dynamical systems. Recursion theory on the reals and continuous-time computation Dynamical recognizers: real-time language recognition by analog com- puters Classical Recursion Theory II. On the computational power of continuous time neural networks. A survey of continuous-time computation theory Abtract computability and its relation to the general purpose analog computer. Computability in Analysis and Physics. Functions and Hierarchies. Analog computation with dynamical systems. Mathematical theory of the di Neural Netwoks and Analog Computation: Beyond the Turing Limit. The complexity of analog computation. Mathematics and Computers in Simulation Computable Analysis. Subclasses of computable real functions. --TR The complexity of analog computation Complexity theory of real functions Universal computation and other capabilities of hybrid and continuous dynamical systems Recursion theory on the reals and continuous-time computation Dynamical recognizers Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy Analog computation with dynamical systems Neural networks and analog computation Computable analysis Iteration, inequalities, and differentiability in analog computers Ordinary Differential Equations U.S. Technological Enthusiasm and British Technological Skepticism in the Age of the Analog Brain Subclasses of Coputable Real Valued Functions The Computational Power of Continuous Time Neural Networks --CTR Giuseppe Trautteur , Guglielmo Tamburrini, A note on discreteness and virtuality in analog computing, Theoretical Computer Science, v.371 n.1-2, p.106-114, February, 2007 Manuel L. Campagnolo , Kerry Ojakian, The Methods of Approximation and Lifting in Real Computation, Electronic Notes in Theoretical Computer Science (ENTCS), 167, p.387-423, January, 2007 Jerzy Mycka , Jos Flix Costa, Real recursive functions and their hierarchy, Journal of Complexity, v.20 n.6, p.835-857, December 2004 Daniel Silva Graa , Jos Flix Costa, Analog computers and recursive functions over the reals, Journal of Complexity, v.19 n.5, p.644-664, October Jerzy Mycka , Jos Flix Costa, The P NP conjecture in the context of real and complex analysis, Journal of Complexity, v.22 n.2, p.287-303, April 2006 John V. Tucker , Jeffery I. Zucker, Computability of analog networks, Theoretical Computer Science, v.371 n.1-2, p.115-146, February, 2007 Manuel Lameiras Campagnolo, Continuous-time computation with restricted integration capabilities, Theoretical Computer Science, v.317 n.1-3, p.147-165, June 4, 2004 Jerzy Mycka , Jos Flix Costa, A new conceptual framework for analog computation, Theoretical Computer Science, v.374 n.1-3, p.277-290, April, 2007 Olivier Bournez , Emmanuel Hainry, Recursive Analysis Characterized as a Class of Real Recursive Functions, Fundamenta Informaticae, v.74 n.4, p.409-433, December 2006 Olivier Bournez , Emmanuel Hainry, Elementarily computable functions over the real numbers and R-sub-recursive functions, Theoretical Computer Science, v.348 n.2, p.130-147, 8 December 2005

elementary functions;dynamical systems;subrecursive functions;recursion theory;analog computation;primitive recursive functions;grzegorczyk hierarchy;differential equations

636581

A polytopal generalization of Sperner''s lemma.

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar. 32 (1996), 71-74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,...., vn. Label the vertices of T by 1,2,..., n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n - d full dimensional simplices of T, each labelled with d different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in Alekseyevskaya (Discrete Math. 157 (1996), 15-37) and Billera et al. (J. Combin. Theory Ser. B 57 (1993), 258-268).

Introduction Sperner's Lemma is a combinatorial statement about labellings of triangulated simplices whose claim to fame is its equivalence with the topological xed-point theorem of Brouwer [8, 16]. In this paper we prove a generalization of Sperner's Lemma that settles a conjecture proposed by K.T. Atanassov [2]. Consider a convex polytope P in R d dened by n vertices d . For brevity, we will call such polytope an (n; d)-polytope. Throughout the paper we will follow the terminology of the book [20]. By a triangulation T of the polytope P we mean a nite collection of distinct simplices such that: (i) the union of the simplices of T is P , (ii) every face of a simplex in T is in T , and (iii) any two simplices in T intersect in a face common to both. The points v are called vertices of P to distinguish them from vertices of T , the triangulation. Similarly, a simplex spanned by vertices of P will be called a simplex of P to distinguish it from simplices involving other vertices of T . If S is a subset of P , then the carrier of S, denoted carr(S), is the smallest face F of P that contains S. In that case we say S is carried by F . A cover C of a convex polytope P is a collection of full dimensional simplices in P such that [ 2C . The size of a cover is the number of simplices in the cover. 2000 Mathematics Subject Classication. Primary 52B11, Secondary 55M20. Key words and phrases. Sperner's lemma, polytope, path-following, simplicial algorithms. [ Department of Mathematics, University of California Davis, Davis, CA 95616, [email protected]. Magdalen College, Oxford University, Oxford OX1 4AU, U.K., [email protected]. \ Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, U.S.A., [email protected]. J. A. DE LOERA, E. PETERSON, AND F. E. SU Let T be a triangulation of P , and suppose that the vertices of T have a labelling satisfying these conditions: each vertex of P is assigned a unique label from the set and each other vertex v of T is assigned a label of one of the vertices of P in carr(v). Such a labelling is called a Sperner labelling of T . A d-simplex in the triangulation is called a fully-labeled simplex or simply a full cell if all its labels are distinct. The following result was proved by Sperner [16] in 1928: Sperner's Lemma. Any Sperner labelling of a triangulation of a d-simplex must contain an odd number of full cells; in particular, there is at least one. Constructive proofs of Sperner's lemma [4, 9, 12] emerged in the 1960's, and these were used to develop constructive methods for locating xed points [18, 19]. Sperner's lemma and its variants continue to be useful in applications. For example, they have recently been used to solve fair division problems in game theory [14, 17]. The main purpose of this paper is to present a solution of the following conjecture. Conjecture (Atanassov). Any Sperner labelling of a triangulation of an (n; d)- polytope must contain at least (n d) full cells. In 1996, K.T. Atanassov [2] stated the conjecture and gave a proof for the case 2. Note that Sperner's lemma is exactly the case 1. In this paper we prove this conjecture for all (n; d)-polytopes. Here is the central result of our paper: Theorem 1. Any Sperner labelling of a triangulation T of an (n; d)-polytope P must contain at least (n d) full cells. Moreover, the collection of full cells in T corresponds to a cover of P under the piecewise linear map that sends each vertex of T to the vertex of P that shares the same label. We provide a non-constructive and a constructive proof of Theorem 1. The non-constructive proof in Section 2 is obtained via a degree argument and the notion of a pebble set. Section 3 develops background on path-following arguments in polytopes that is closely related to classical path-following arguments for Sperner's lemma [4, 9, 18]. This is applied to give a constructive proof for simplicial poly- topes. In Section 4 we extend the construction to prove the conjecture for arbitrary polytopes. The nal section of the paper is devoted to two interesting consequences of Theorem 1 and its proofs. From our rst proof we derive the following corollary: Corollary 2. Let c(P ) denote the covering number of an (n; d)-polytope P , which is the size of the smallest cover of P . Then, c(P ) n d. This result is best possible as the equality is attained for stacked polytopes. We also obtain a slight strengthening of Theorem 10 of [10]. We need to recall the notion of chamber complex of a polytope P (see [1]): let be the set of all d-simplices of P . Denote by bdry() the boundary of simplex . Consider the set of open polyhedra P [ 2 bdry(). A chamber is the closure of one of these components. The chamber complex of P is the polyhedral complex given by all chambers and their faces. Corollary 3. Let P be an (n; d)-polytope with vertices ng be a collection of closed sets covering the (n; d)-polytope P , such that each face F is covered by [fC h j v h 2 Fg. Then, for each p 2 P , there exists a subset J p ng such that (1) p lies in the convex hull of the vertices v j with j 2 J p , (2) J p has cardinality d A POLYTOPAL SPERNER'S LEMMA 3 (3) \ j2Jp C j 6= ;, and (4) if p and q are interior points of the same chamber of P , then . There are at least c(P ), the covering number, dierent such subsets, and the simplices of P indicated by the labels in these subsets form a cover of P . Figure 1 illustrates with an example the content of the above corollary. Figure 1. Part (A) shows several closed sets covering a hexagon and their four intersection points. The points of intersection correspond to a cover of the hexagon, in this case a triangulation, illustrated in part (B). 2. A Non-Constructive Proof using Pebble Sets The non-constructive proof of Theorem 1 that we give in this section is an extension of \degree" arguments for proving the usual Sperner's lemma. We rst establish a proposition that yields the covering property, then show how the construction a pebble set will yield a lower bound for the number of full cells. Let P be an (n; d)-polytope with Sperner-labelled triangulation T . Consider the piecewise linear (PL) maps each vertex of T to the vertex of P that shares the same label, and is linear on each d-simplex of T . Proposition 4. The dened as above is surjective, and thus the collection of full cells in T forms a cover of P under f . Before proving this result, we recall a few facts about the degree of a map f between manifolds. If y and the Jacobian determinant of f exists and is non-zero at x, then x is called a regular point of f , and the sign of x is the sign (1) of this determinant. The point y is called a regular value of f if every pre-image of y is a regular point. Any regular value y has nitely many pre-images, and the sum of the signs of its pre-images is, in fact, independent of the choice of y and is known as the degree of the map f . The degree is a homotopy invariant of mappings between manifolds (relative to their boundaries) and may also be computed as the multiplicative factor induced by the map f on the corresponding top homology groups (relative to their boundaries). See [5, Ch.1] or [13, Sec.38] for expositions of the topological degree of simplicial maps, or [7] for the general theory. 4 J. A. DE LOERA, E. PETERSON, AND F. E. SU Thus for the above, the interior points of simplices of T are regular points; interior points of chambers of P are regular values. Observe that the sign of a regular point x depends essentially on the orientation of the labels of the simplex of T that contains x. Proof. We shall show, by induction on the dimension d of P , that f has degree 1. If then P is a point and the statement is clearly true. So assume that the above statement holds for all polytopes of dimension less than d. Given a d-dimensional polytope P , let @P denote the boundary of P (i.e., the union of the facets). If consists of two points on which the map is the identity (due to the Sperner labelling), and therefore @f : is the identity map on (reduced) homology groups. For d > 1, we use the Sperner labelling on the facets and the inductive hypothesis to show that the map @f is the identity map. Specically, if F is any facet of P , . Then the map @f induces the following commutative diagram of (reduced) homology groups: where the rows are exact. Since H i 2g, the maps are isomorphisms. By excision, H d 1 (@P; f to be induced by f on the facet F (a polytope of one lower dimension) which by the inductive hypothesis has degree 1 and must therefore be the identity. Since the maps are isomorphisms and f is the identity map, @f also the identity map. The map @f appears in another commutative diagram of (reduced) homology groups induced by f : where the rows are exact. Since H 1g, the maps @ are isomorphisms, which implies that f is the identity map. Hence f has degree 1. Therefore the number of pre-images of any regular point y in the image of f is (when counted with sign). So the map f is surjective, and full cells in T gives a cover of P under f . Thus if we can nd a set of points in P such that any d-simplex spanned by vertices of P contains at most one such point in its interior, then the pre-image of each such point will correspond to a full cell in P (in fact, an odd number, because the number of pre-images, counted with sign, is 1). Thus nding full cells in Theorem 1 corresponds precisely to looking for the following kind of nite point Denition. A pebble set of a (n; d)-polytope P is a nite set of points (pebbles) such that each d-simplex of P contains at most one pebble in its interior. It is worth remarking now two facts about pebble sets. First, the larger the pebble set, the more full cells we can identify, i.e., the number of full cells is at A POLYTOPAL SPERNER'S LEMMA 5 least the cardinality of the largest size pebble set in P . Second, by the denition of chamber, only one pebble can exist within a chamber and when choosing a pebble p we have the freedom to replace it by any point p 0 in the interior of the same chamber because p and p 0 are contained by the same set of d-simplices. We now show that a pebble set of size (n d) exists for any (n; d)-polytope P by a \facet-pivoting" construction. Figure 2. A pebble set with pebbles In the simplest situation, if one of the facets of P is a simplex, call this simplex the base facet. Choose any point q 0 (the basepoint) in the interior of this base facet. Now for each vertex v i not in the base facet, choose a point p i along a line between very close to v i . Exactly how close will be specied in the proof. The collection of all such points fp i g forms a pebble set; it is size (n d) because the simplicial base facet has d vertices. See, for example, Figure 2 for the case of a pentagon; it is a (5; 2)-polytope with pebble set fp 1 g. If none of the facets are simplicial, then one must choose a non-simplicial facet as base. In this case, choose a pebble set fq i g for the base facet (an inductive hypothesis is used here) and then use any one of them for a basepoint q 0 to construct above. The remaining pebbles are obtained from the other q i by perturbing them so they are interior to P . See Figure 3. Theorem 5. Any (n; d)-polytope contains a pebble set of size (n d). Proof. We induct on the dimension d. For dimension a polytope is just a line segment spanned by two vertices. Hence n clearly any point in the interior of the line segment forms a pebble set. For any other dimension d, let V =fv 1 ; :::; vn g 2 R d denote the vertices of the given (n; d)-polytope P . Choose any facet F of P as a \base facet", and suppose without loss of generality that it is the convex hull of the last k vertices d. Then F is a (d 1)-dimensional polytope with k vertices, and by the inductive hypothesis, F has a pebble set QF with pebbles . (If F is a simplex, then consists of one point which can be taken to be any point on the interior of F .) Let diam(P ) denote the diameter of the polytope P , i.e., the maximum pairwise distance between any two points in P . Let H be the minimum distance between 6 J. A. DE LOERA, E. PETERSON, AND F. E. SU any vertex v 2 V and the convex hull of the vertices in V n fvg. Since there are nitely many such distances and the vertices are in convex position, H exists and is positive. Set (1) Using denote the collection of (n points dened by (2) positive constant given by (1). Thus points in along straight lines extending from q 0 and very close to the vertices of P not in F . Figure 3. A pebble set with pebbles . The pebbles lie just above (not shown) on the base of the polytope. Note how q arise from the pebble set construction in Figure 2. Because q i is in F , it lies on the boundary of P and borders exactly one chamber of P (since by induction it is interior to a single chamber in the facet F ). Ignoring momentarily, for 1 i denote a point obtained by \pushing" q i into the interior of the unique chamber that it borders. Let Q g. We shall show that is a pebble set for the polytope P . Note that if P has a simplicial facet F , then with this facet as base, the set Q suces; for then Q F is empty and the construction (2) yields the required number of pebbles by choosing any q 0 in the interior of a simplicial facet F . First we prove some important facts about the p i and q Lemma 6. Let S be a d-simplex spanned by vertices of P . If S contains p i , it must also contain v i as one of its vertices. A POLYTOPAL SPERNER'S LEMMA 7 Proof. By construction, each p i has the property that p i is not in the convex hull of g. This follows because hence implying that the distance of p i from the convex hull of is greater than or equal to H=2. Since the convex hull of V n fv i g does not contain p i , if S is to contain p i it must contain v i as one of its vertices. Lemma 7. Let S be a non-degenerate d-simplex spanned by vertices of P . Then i is in S if and only if q i is in S \ F . Proof. Since q i is in the unique chamber of P that q i borders, any non-degenerate simplex containing q i must contain q . Conversely, any simplex S containing q must contain its chamber and therefore contains q i . Since q i is in F , then q i is in The next three lemmas will show that Q [ Q F is a pebble set for P . Lemma 8. Any d-simplex S spanned by vertices of P contains no more than one pebble of Q. Proof. If S is degenerate (i.e., the convex hull of those vertices is not full dimen- sional), then it clearly contains no pebbles of Q because the p i are by construction in the interior of a chamber. So we may assume that S is non-degenerate. denote the vertices of S. Suppose by way of contradiction that S contained more than one point of Q. Then are contained in S for distinct implies that v i must both be vertices of S. Without loss of generality, let s A be a matrix whose columns consist of q 0 and the vertices of S, adjoined with a row of 1's: This is a (d+1)(d+2) matrix that has rank (d+1) because the s i are anely independent (by the non-degeneracy of S). So the kernel of A, ker(A) is 1-dimensional. Note that p i 0 2 S implies that it is a convex combination of the rst (d+1) columns of A. On the other hand, by construction, it is also a convex combination of s 1 and . Thus there exist constants 0 x x d+1 8 J. A. DE LOERA, E. PETERSON, AND F. E. SU where the rst equality follows from (2). Similarly, y d+13 for some constants 0 y 1. The above equations show that are both in ker(A). But since ker(A) is 1-dimensional, and the last coordinates of these vectors are equal, all entries of these vectors are identical. In particular, x We now claim that x would show that p j 0 could not have been in S after all, a contradiction. To establish the claim, use equations (2) and (3) to express q 0 as an ane combination of the vertices of S: is not in the interior of S, it must be the case that then by (4), q 0 is on a facet of S. This means that it is spanned by d vertices on the facet F of P . Thus the vertices of S must include those d vertices but by Lemma were not on the facet F (because S is non-degenerate), we obtain a contradiction since S cannot contain more than d vertices. Thus the equality x as desired. Lemma 9. Any d-simplex S spanned by vertices of P contains no more than one pebble in Q F . Proof. Since S \ F is a simplex in F that contains at most one point of QF , then by Lemma 7, S can contain at most one point of Q F . Lemma 10. Any d-simplex S spanned by vertices of P cannot contain pebbles of F simultaneously. Proof. Suppose S contained a point q i of Q F . Then by Lemma 7, S \ F contains q i of QF . Since QF was a pebble set for the facet F , S \F cannot also contain q 0 . If S also contained a pebble p i 0 of Q, then by Lemma 6, S contains v i 0 as a vertex. Since S \ F contains q i which is interior to a chamber of F , S must also contain d vertices of F . Since q 0 is in F (but not in S \ F ), q 0 is expressible as a linear combination (but not convex combination) of those d vertices. This linear combination, when substituted for q 0 in (2), would show that the pebble p i 0 is not a convex combination of v i 0 and those d vertices. This contradicts the fact that p i 0 was in S to begin with. Together, the three lemmas above show that S cannot contain more than one point of Q [ Q F , which concludes the proof of Theorem 5. Together, Proposition 4 and Theorem 5 prove Theorem 1. A POLYTOPAL SPERNER'S LEMMA 9 3. Graphs for Path-following and Simplicial Polytopes. Sperner's lemma has a number of constructive proofs which rely on \path- arguments (see, for example, the survey of Todd [18]). Path-following arguments work by using a labelling to determine a path through simplices in a triangulation, in which one endpoint is known and the other endpoint is a full cell. In this section we adapt these ideas for Sperner-labelled polytopes, which are used in the next section to give a constructive \path-following" proof of Theorem 1. Let P be an (n; d)-polytope with triangulation T and a Sperner labelling using the label set ng. We dene some further terminology and notation that we will use from now on. Let L(), the label set of , denote the set of distinct labels of vertices of . Let L(F ) denote the label set of a face F of P . As dened earlier, a d-simplex in T is a full cell if the vertex labels of are all distinct. Similarly, a (d 1)-simplex in T is a full facet if the vertex labels of are all distinct. Note that a full facet on the boundary of P can be regarded as a full cell in that facet. Denition. Given a Sperner-labelled triangulation T of a polytope P , we dene three useful graphs: 1. The nerve graph G is a graph with nodes that are simplices of T whose label set is of size at least d. Formally, is a node of G if jL()j d. Two nodes in G are adjacent if (as simplices) one is a face of the other. 2. If K is a subset of the label set ng of size (d 1), the derived graph GK is the subgraph of the nerve graph G consisting of nodes in G whose label sets contain K. 3. Let G 0 denote the full cell graph, whose nodes are full cells in the nerve graph G. Two full cells ; are adjacent in G 0 if there exists a path from to in G that does not intersect any other full cell. If G is a connected component of G, construct the full cell graph similarly. Thus the nodes of G and GK are either full cells, full facets, or d-simplices with exactly one repeated label. The full cell graph G 0 only has full cells as nodes. Example. The pentagon in Figure 4 has dimension 2. Let of cardinality (d 1). Then the derived graph GK consists of 1-simplices and 2-simplices that are darkly shaded, and it is a subgraph of the nerve graph G consisting of the dark and light-shaded 1-simplices and 2-simplices in Figure 4. In Figure 4, G 0 is a 3-node graph with nodes A; B; C, the full cells. In G 0 , A is adjacent to B, and B is adjacent to C, but A is not adjacent to C. As the example shows, the nerve graph G branches in (d directions at full cells, while the derived graph GK is the subgraph consisting of paths or loops that \follow" the labels of K along the boundary of the simplices in G. We prove these assertions. Lemma 11. The nodes of the derived graph GK are either of degree 1 or 2 for any K of size d 1. A node is of degree 1 if and only if is a full facet on the boundary of P . Hence GK is a graph whose connected components are either loops or paths that connect pairs of full facets on the boundary of P . Proof. Recall that each node of GK has a label set containing K and is either a full cell, full facet, or d-simplex with exactly one repeated label. J. A. DE LOERA, E. PETERSON, AND F. E. SU134453521 Figure 4. A triangulated (5; 2)-polytope (a pentagon) with Sperner labelling. If nodes of GK consist of the dark- shaded simplices, nodes of G consist of dark and light-shaded sim- plices, and nodes of G 0 consist of the three full cells marked by If is a full cell, since we see that L() consists of labels in K and two other labels l 1 ; l 2 . There are exactly two facets of whose label sets contain K; these are the full facets with label sets K [ l 1 and K [ l 1 , respectively. Thus has degree 2. If is a full facet with label set containing K, then it is the face of exactly two d-simplices, unless is on the boundary of P , in which case it is the face of exactly one d-simplex. Thus is degree 1 or 2 in GK , and degree 1 when is a full facet on the boundary of P . If is a d-simplex with exactly one repeated label, then it must possess exactly two full facets. Since K L(), these full facets must also have label sets that contain K. Hence these two full facets are the neighbors of in GK , so has degree 2. Lemma 12. The nodes of the nerve graph G are of degree 1, 2, or d + 1. A node is of degree 1 if and only if is a full facet on the boundary of P . A node is of degree d only if is a full cell. Proof. As noted before, each node of G is either a full cell, full facet, or d-simplex with exactly one repeated label. The arguments for the latter two cases are identical to those in the proof of Lemma 11 by letting K be the empty set. If is a full cell, then every facet of is a full facet, hence the degree of is G. The nerve graph G may have several components, as in Figure 4. In Theorem 16, we will establish an interesting relation between the labels carried by a component G and the number of full cells it carries. First we show that all the labels in a component are carried by the full cells. A POLYTOPAL SPERNER'S LEMMA 11 Lemma 13. If is adjacent to in G, then L() L(), unless is a full cell, in which case L() L(). Thus adjacent nodes in G carry exactly the same labels unless one of them is a full cell. Proof. Suppose is a d-simplex with exactly one repeated label. Then it is adjacent to two full facets with exactly the same label set, so the conclusion holds. Otherwise, if is a full facet, then it is adjacent to two d-simplices that contain it as a facet. Hence L() L() for adjacent to . Finally, if is a full cell, any simplex adjacent to in G is contained in as a facet, so L() L() in that case. Lemma 14. Suppose G is connected component of G. If G contains at least one full cell as a node, then all the labels in G are carried by its full cells. For example, in Figure 4, G has two components. One of them has no full cells. In the other component, all of its labels f1; 2; 3; 4; 5g are carried by its full cells Proof. Since G is connected and contains at least one full cell, each simplex that is not a full cell is connected to a full cell via a path in G that does not intersect any other full cell in G. Call this path g. By Lemma 13, Therefore labels carried by the full cells contain all labels carried by any other node of the graph. Since the label information in a nerve graph is found in its full cells, it suces to understand how the full cells connect to each other. Lemma 15. Any two adjacent nodes in G 0 are full cells in T whose label sets contain at least d labels in common. Proof. Let 1 and 2 be adjacent nodes in G 0 . By construction they must simplices connected by a path in G; let be any such node along this path. Repeated application of Lemma 13 yields L() contains at least the d labels in L( ). We will say the full cell graph G 0 is a fully d-labelled graph because it clearly satises four properties: (a) all nodes in the graph are assigned (d (simply assign to a node of G 0 the label set L()), (b) all edges are assigned d labels (assign an edge the d labels specied in Lemma 15), (c) the label set of an edge (; ) (denoted by L(; )) is contained in L() \ L( ), and are nodes each adjacent to , then L(; ) Proposition 16. Suppose G 0 is a connected fully d-labeled graph. Let L(G 0 ) denote the set of all labels carried by simplices in G 0 and jG 0 j the number of nodes in G 0 . Then We shall use this theorem for graphs G 0 arising as a full cell graph of one connected component of a nerve graph G. In Figure 4, the full cell graph G 0 has just one connected component, and L(G 0 indeed 3 5 2. J. A. DE LOERA, E. PETERSON, AND F. E. SU Proof. We induct on jG 0 j. If jG the one full cell in G 0 has d labels. Hence so the statement holds. We now assume the statement holds for fully d-labeled graphs with less than j nodes, and show it holds for fully d-labeled graphs G 0 with jG has j full cells. We claim that it is possible to remove a vertex v from G 0 and leave connected. This is true because G 0 contains a maximal spanning tree, and the removal of any leaf from this tree will leave the rest of the nodes in G 0 connected by a path in this tree. Now G 0 with v and all its incident edges removed is a new graph (denoted by nodes. Note that this new graph is still fully d-labeled, so by the inductive hypothesis, jG 0 vj jL(G 0 v)j d. v has at least d labels in common with some vertex in G 0 v, by Lemma 15. Hence jG Adding 1 to both sides gives the desired conclusion. This will prove the following useful result. Theorem 17. Let T be a Sperner-labelled triangulation of an (n; d)-polytope P . If the nerve graph G has a component G that carries all the labels of G, then T contains at least (n d) full cells. Proof. Use G to construct the full cell graph G 0 as above, which is a fully d-labelled graph. Note that if G is connected then G 0 is also connected. By Lemma 14, Using Proposition 16, we have which shows there are at least (n d) full cells in G, and hence in G itself. Thus to prove Atanassov's conjecture for a given (n; d)-polytope it suces to nd some component G of the nerve graph G for which L( This is the central idea of the proofs in the next sections. We now use path-following ideas to outline a proof of Atanassov's conjecture in the special case where the polytope is simplicial. This will motivate the proof of Theorem 1 for arbitrary (n; d)-polytopes in the subsequent section. Theorem 18. If P is a simplicial polytope, there is some component G of the nerve graph G which meets every facet of P , and hence carries all labels of G. Proof. Let F be a simplicial facet of the polytope P . Let (G; F ) count the number of nodes of G that are simplices in F . This may be thought of as the number of endpoints of paths in G that terminate on the facet F . Consider two \adjacent" whose intersection is a ridge of the polytope P spanned by (d 1) vertices of P . These vertices have distinct labels; let K be their label set. The derived graph GK consists of loops or paths whose endpoints in GK must be full facets in F 1 or F 2 , since the Sperner labelling guarantees that no other facet of P has a label set containing K. Since every facet of P is simplicial, all the full facets in F 1 and F 2 contain K in their label set. Thus all the nodes of G that are full facets in F 1 and F 2 must also be nodes in the graph GK . Since GK is a subgraph of G and consists of paths with endpoints that pair up full facets in F 1 and F 2 , we see that (G; F 1 mod 2. In fact, since paths in GK are connected, this argument shows that G; G; A POLYTOPAL SPERNER'S LEMMA 13 for any connected component G of G. were arbitrary, the same argument holds for any two adjacent facets. This yields the somewhat surprising conclusion that the parity of ( G; F ) is independent of the facet F . We denote this parity by ( G). Since (G; F ) is also independent of facet, we can dene (G) similarly. Since (G; F ) is the sum of ( G; F ) over all connected components G of G, it follows that (G) connected components G of G. Moreover, because the usual Sperner's lemma applied to (any) simplicial shows that there are an odd number of full facets of T in the Hence there must be some G such that ( G meets every facet of P . Because the facets of P are simplicial, G carries every label, i.e., Theorem 19. Any Sperner-labelled triangulation of a simplicial (n; d)-polytope must contain at least (n d) full cells. Proof. This follows immediately from Theorem 18 and Theorem 17. To extend this proof for non-simplicial polytopes requires some new ideas but follows the basic pattern: (1) nd a function that counts the number of times a component G of G meets a certain facet in a certain way, and show that this function only depends on G, and (2) appeal to the usual Sperner's lemma for simplices in a lower dimension to constructively show that the parity of summed over all components G must be odd. For the non-simplicial case, we cannot guarantee that any faces of P except those in dimension 1 are simplicial. How to connect dimension 1 to dimension d is tackled in the next section, and the ag graph introduced there gives a constructive procedure for nding certain full cells. Then we construct a counting function to show that there are at least (n d) full cells for an (n; d)- polytope. 4. The Flag Graph and Arbitrary Polytopes. Throughout this section, let the symbol denote equivalence mod 2. Recall that L(F ) denotes the label set of a face F . Let F denote a ag of the polytope a choice of faces F 1 F 2 ::: F d where F i is an i-face of P . When the choice of F i is not understood by context, we refer to the i-face of a particular ag F by writing F i (F). Given a ag F , it will be extremely useful to construct \super-paths" containing simplices of P of various dimensions whose endpoints are either on a 1-dimensional edge or a d-dimensional full cell. Denition. Let P be an (n; d)-polytope with a Sperner-labelled triangulation T . Let F be a ag of P . We dene the ag graph GF in the following way. For d, a k-simplex 2 T is a node in the graph GF if and only if is one of four types: (I). the k-simplex is carried by the k-face F k and (II). the k-simplex is carried by the 1)-face F k+1 and 14 J. A. DE LOERA, E. PETERSON, AND F. E. SU (III). the k-simplex is carried by the k-face F k and (IV). the k-simplex is carried by the k-face F k and there is an I such that Two nodes are adjacent in GF if (as simplices) one is a facet of the other and at least one of the pair is of type (I) or (II). Note that if is a type (I) simplex in GF , then it is a \non-degenerate" full cell of the k-face that it is carried in, i.e., the vertices of P corresponding to the labels in L() span a k-dimensional simplex. A type (II) simplex is a non-degenerate full facet in the 1)-face that it is carried in. A type (III) simplex has just one repeated label and satises a certain kind of non-degeneracy (that ensures its two full facets are non-degenerate). A type (IV) simplex is one kind of degenerate full cell in the k-face that it is carried in (but such that it has exactly two facets which are non-degenerate).422 admits labels 1,2,3,4,5,6,7 F 2admits labels 1,2,3,4,5 F Figure 5. A path in the ag-graph of a (7; 3)-polytope. The gure at left shows simplices along a path in the triangulation. Simplices carried by F 2 are shaded. The gure at right shows the label sets of the simplices along this path. Simplices 1 ; :::; 7 occur in counterclockwise order along this path. Example. Let P be a (7; 3)-polytope P , i.e., a 3-dimensional polytope with 7 vertices, and suppose T is a Sperner-labelled triangulation of P . Let F 1 F 2 F 3 be a ag F of P with label sets f1; 2g f1; 2; 3; 4; 5g f1; 2; :::; 7g, respectively. Consider the following collection of simplices shown in Figure 5. Let be simplices with label sets: L( 1 (with repeated label 2), 6g such that in each pair f i ; i+1 g, one is a facet of the other. The simplices are carried by the face F 2 and others are carried by the face F 3 . Each of these simplices is a node in the graph are of type (II), 2 is of type (III), and 5 is of type (IV). Furthermore, each pair A POLYTOPAL SPERNER'S LEMMA 15 i and i+1 are adjacent in GF . Except for 1 and 7 , each of these simplices has exactly 2 neighbors in GF so the above sequence traces out a path. The following result shows that GF does, in fact, consist of a collection of loops or paths whose endpoints are either 1-dimensional and d-dimensional. Lemma 20. Every node of GF has degree 1 or 2, and has degree 1 only when is a 1-simplex or a d-simplex in GF . Proof. Consider a k-simplex of type (I). If k 2, then has a facet determined by the k labels in L() \L(F k 1 ), and this facet is a (k 1)-simplex of type (I) or so it is adjacent to . No other facets of are types (I)-(IV). If k d 1, then is a facet of exactly one (carried by F k+1 ) that must be of type (I) or (III) or (IV). Thus a type (I) simplex has degree 2 unless d, in which case it has degree 1. In case (II), the k-simplex is the facet of exactly two 1)-simplices in F k+1 ; these are either of type (I) or (III) or (IV) and are thus neighbors of in GF . Facets of are not types (I)-(IV) because they are co-dimension 2 in the face F k+1 . Thus type (II) vertices have degree 2. In case (III), the k-simplex has exactly two facets determined by the k labels in L(); each of these is a (k 1)-simplex adjacent to in GF because it is either of type (I) in F k 1 or of type (II) in F k . No other facets of are types (I)-(IV). Thus type (III) vertices have degree 2. In case (IV), the labelling rules show that the set L() \ (L(F I ) n L(F I 1 )) is of size two. Call these labels a and b. There is exactly one facet of that omits the label a and one facet which omits the label b; each of these is a (k 1)-simplex of type (I) in F k 1 or of type (II) in F k , so is adjacent to in GF . No other facets of are non-degenerate; therefore cannot be types (I) or (II). Thus type (IV) vertices have degree 2. We remark that in the denition of the ag graph, we require at least one of an adjacent pair to be of type (I) or (II) because without this restriction, some type vertices could have degree greater than 2. For instance, in a Sperner-labelled, triangulated (9; 4)-polytope, suppose F 1 F 2 F 3 F 4 is a ag F of P with label sets f1; 2g f1; 2; 3; 4; 5g f1; 2; :::; 7g f1; 2; :::; 9g, respectively. If is a 4-simplex in F 4 with label set f1; 2; 3; 4; 6g such that its face with labels f1; 2; 3; 4g is carried in F 3 , then both and are of type (IV). They each already have two facets in GF of type (I) or (II), so we would not want to dene them to be adjacent to each other. Theorem 21. A Sperner-labelled triangulation of an (n; d)-polytope contains, for each edge F 1 of P , a non-degenerate full cell whose labels contain L(F 1 ). Proof. For any ag F containing the edge F 1 , Lemma 20 shows that GF consist of loops or paths whose endpoints are non-degenerate full cells in F 1 or F d ; thus the total number of such end points full cells in F 1 and in F d must be of the same parity. On the other hand, the 1-dimensional Sperner's lemma shows that the number of full cells in F 1 is odd. So the number of non-degenerate full cells in F d in GF must be odd. In particular there is at least one non-degenerate full cell in F d whose label set contains L(F 1 ). J. A. DE LOERA, E. PETERSON, AND F. E. SU Notice that the above proof is constructive; the graph GF yields a method for locating a non-degenerate full cell for any choice of ag F , by starting at one of the full cells on the edge F 1 (an odd number of them are available). At most an even number of them are matched by paths in GF , so at least one of them is matched by a path to a non-degenerate full cell in F d . However, as we show now, more can be said about the location of full cells. Rather than locating all of them at the endpoints of paths in a ag-graph, we can show that there is some component G of the nerve graph G that contains at least (n d) full cells. One can trace paths through this component to nd them. We nd a component G of G that carries all labels. Theorem 16 will imply that the component must have (n d) full cells. As in the case for simplicial polytopes, the key rests on dening a function that counts the number of times that G meets a facet in a certain way, and then showing that the parity of exhibits a certain kind of invariance| it really only depends on G. Any component with non-zero parity will be the desired component. Denition. Suppose F is a facet of P and R is a ridge of P that is a facet of F . Let (G; F; R) denote the number of nodes of the nerve graph G in the facet F such that jL() \ Similarly if K is any (d 1)-subset of L(F ), let (G; F; K) denote the number of nodes of the graph G in the facet F such that jL() \ If G is a connected component of G, dene ( G; F; R) and ( G; F; K) similarly using G instead of G. Thus (G; F; R) (resp. ( G; F; R)) counts non-degenerate full cells of type (I) from GF (resp. GF ) in the facet F , for all ags F of P such that It is easy to show that the parity of (G; F; R) is independent of both F and R: Theorem 22. Given any ag F of P , suppose Then (G; F; R) 1: Proof. Consider the subgraph contains simplices of dimension (d 1) or lower. This subgraph must be a collection of loops or paths (since GF is) whose endpoints (even number of them) are non-degenerate full cells in either F 1 or F d 1 . But Sperner's lemma in 1-dimension (or simple inspection) shows that the number of full cells in must be odd. Hence the number of non-degenerate full cells of GF that meet must be odd as well. The next two theorems show that for connected components G of G, the parity of ( G; F; R) is also independent of F and R. This fact does not follow directly from Theorem 22 since we do not know that endpoints of GF for dierent ags are connected in G. To establish this we need to trace connected paths in the nerve graph G rather than the ag graph GF . Lemma 23. Let G be a connected component of G. Suppose that R; R 0 are ridges of P and F a facet of P such that R; R 0 are both facets of F . Then G; F; R) ( G; F; R 0 A POLYTOPAL SPERNER'S LEMMA 17 Proof. First assume that R and R 0 are \adjacent" ridges sharing a common facet C (it has dimension d 3, if you are keeping track). We claim that A;x G; where x runs over all labels in L(F ) that are not in L(C) and A runs over all (d 2)- subsets of L(C). (Here we write A[ x instead of A[ fxg to reduce notation.) The above sum holds because it only counts fully-labelled simplices from G in F that contain a (d 2)-subset A of L(C), and every such appears exactly twice in this counted once each in ( G; F; A [ a) and in G; F; A [ b). On the other hand, if x is not the label set of any ridge of P , then any fully-labelled simplex on the boundary of P that contains K must be contained in the facet F . Since K is of size d 1, by Lemma 11, we see that ( G; F; K) 0 because there is an even number of endpoints of paths in GK , and such paths are connected subgraphs of the connected graph G. Thus the only terms surviving the above sum correspond to label sets of the two ridges R; R 0 that are facets of F and share a common face C, i.e., G; G; which yields the desired conclusion for neighboring ridges R, R 0 . Since any two ridges of a facet F are connected by a chain of adjacent ridges, the general conclusion holds. Lemma 24. Let G be a connected component of G. Let F; F 0 be adjacent facets of the polytope P bordering on a common ridge R. Then G; F; R) ( G; Proof. Let R denote the ridge common to both F and F 0 . Let K be any non-degenerate subset of L(R) of size (d 1), i.e., K is not a subset of F i for i < (d 2). Consider the derived graph GK . By Lemma 11, this graph consists of paths connecting full cells from G on the boundary of P that contain the label set K. Since these paths are connected subgraphs of G, there is an even number of endpoints of these paths in G. On the other hand, because of the Sperner labelling, all such endpoints must lie in facets of P that contain R. There are exactly two such facets, F and F 0 . Hence G; G; which produces the desired conclusion. Theorem 25. Let G be a connected component of G. The parity of ( G; F; R) is independent of F and R. Proof. Since all facet-ridge pairs (F; R) are connected by a sequence of adjacent facets and ridges, the statement follows from Lemmas 23 and 24. Hence we may dene the parity of G to be ( G; F; R) for any facet-ridge Similarly, dene the parity of G to be (G) (G; F; R) for any facet- ridge pair (F; R), which is well-dened and equal to 1 in light of Theorem 22. Now we may prove J. A. DE LOERA, E. PETERSON, AND F. E. SU Theorem 26. If P is an (n; d)-polytope, there is some component G of G which carries all labels of P . Proof. Fix some ag F of P , and let (G; F; R) is the sum of ( G; F; R) over all connected components G of G, it follows that (G) connected components G of G. Moreover, Theorem 22 shows that (G) 1. Hence there must be some G such that ( G carries the labels in L(R). Since the ag F was arbitrary, G must carry all labels of P . This concludes our alternate\path-following" proof of Theorem 1, because the (n d) count follows immediately from Theorems 26 and 17, while the covering property follows (as before) from Proposition 4. 5. conclusion As applications of Theorem 1, we prove the corollaries mentioned in the intro- duction, and make some further remarks. Corollary 2 follows directly from the degree and covering arguments in the non-constructive proof of Section 2 and the fact that stacked polytopes have triangulations of size (n d) [11]. Corollary 3, proved below, is stronger than the version stated in [10], which does not mention the covering property of the full cells, nor their cardinality. That weaker version follows also from the combinatorial results of Freund in [6] (in particular his Theorem 4). In fact we should note the similar avor of Freund's results to our theorem although he considers triangulations of polytopes with the number of labels equal to the number of facets (not vertices). His results seem to imply a \dual" version of our polytopal Sperner but without an estimate of how many full cells exist. Proof of Corollary 3. Let C j , be the closed sets in the statement. Consider an innite sequence of triangulations T k of the polytope P with the property that the maximal diameter of their simplices tends to zero as k goes to innity. For each triangulation, we label a vertex y of T k with ng j y 2 g. This is clearly a Sperner labelling. By Theorem 1, each triangulation T k targets a collection of simplices of P corresponding to full cells in T k . There are only nitely many possible collections (since they are subsets of the set of all simplices of P ), and because there are innitely many T k , some collection C of simplices must be targeted innitely many times by a subsequence T k i of T k . By Theorem 1, this collection C is a cover of P and therefore has at least c(P ) elements. For each simplex in C, choose one full cell i in T k i that shares the same label set. The i form a sequence of triangles decreasing in size. By the compactness of subsequence of these triangles converges to a point, which (by the labelling rule) must be in the intersection of the closed sets C j with j 2 L(). Thus given a point choose any simplex of C that contains P (since C is a cover of P ), and let J L(). Then the above remarks show that J p satises the conditions in the conclusion of the theorem. Moreover, there are at least c(P ) dierent such subsets, one for each in C. We remark that in the statement of Theorem 1, the full cells not only correspond to a cover of the polytope P but, in fact, to a face-to-face cover. While this is not A POLYTOPAL SPERNER'S LEMMA 19 apparent from the rst proof of Theorem 1, it may be seen from the second proof; in particular, the adjacencies in full cell graph G 0 indicate how the simplices of the cover meet face-to-face. Thus the subsets in Corollary 1 also satisfy this property. We close with a couple of questions. For a specic polytope P , dene the pebble number p(P ) to be the size of its largest pebble set. The (n d) lower bound of Theorem 1 is tight, achieved by stacked polytopes whose vertices are assigned dierent labels. But for a specic polytope P , the arguments of Section 2 show that the lower bound (n d) can be improved to p(P ). What can be said about the value of p(P )? We can provide at least two upper bounds for this number. On one hand p(P ) c(P ), because for a maximal pebble set, at most one pebble lies in each simplex of a minimal size cover. On the other hand, consider the simplex-chamber incidence introduced in [1]. As the columns correspond to chambers a pebble selection is essentially a selection of a \row-echelon" submatrix; therefore the rank of M is an upper bound on the size of pebble sets. Our pebble construction gives an algorithm for selecting an explicit independent set of columns of M (although this may not always be a basis). A related question is: for a specic polytope P , how can one determine the minimal cover size c(P )? Although Corollary 2 gives a general sharp lower bound for all polytopes, we know that sometimes minimal covers are much larger for specic polytopes, such as for cubes (as the volume arguments in [15] show). Also note that the minimal cover may be strictly smaller than the minimal triangulation (an example is contained in [3]). Finding other explicit constructions of pebble sets (besides our \facet-pivoting" construction of Section 2) that work for specic polytopes may shed some light on these questions. --R Combinatorial bases in systems of simplices and chambers Hungarica Minimal simplicial dissections and triangulations of convex 3-polytopes On the Sperner lemma Combinatorial Analogs of Brouwer's Fixed-Point Theorem on a Bounded Polyhedron Ein Beweis des Fixpunktsatzes f Simplicial Approximation of Fixed Points Intersection theorems on polytopes On triangulations of the convex hull of n points The Approximation of Fixed Points of a Continuous Mapping Seifert and Threlfall: A Textbook of Topology A lower bound for the simplexity of the n-cube via hyperbolic volumes Neuer Beweis f Sperner's lemma in fair division --TR Combinatorial analogs of Brouwer''s fixed-point theorem on a bounded polyhedron Duality and minors of secondary polyhedra Combinatorial bases in systems of simplices and chambers A lower bound for the simplexity of the <italic>n</italic>-cube via hyperbolic volumes Extremal Properties for Dissections of Convex 3-Polytopes --CTR Frdric Meunier, Sperner labellings: a combinatorial approach, Journal of Combinatorial Theory Series A, v.113 n.7, p.1462-1475, October 2006 Timothy Prescott , Francis Edward Su, A constructive proof of Ky Fan's generalization of Tucker's lemma, Journal of Combinatorial Theory Series A, v.111 n.2, p.257-265, August 2005

sperner's lemma;path-following;polytopes;simplicial algorithms

636696

An Efficient Protocol for Authenticated Key Agreement.

This paper proposes an efficient two-pass protocol for authenticated key agreement in the asymmetric (public-key) setting. The protocol is based on Diffie-Hellman key agreement and can be modified to work in an arbitrary finite group and, in particular, elliptic curve groups. Two modifications of this protocol are also presented: a one-pass authenticated key agreement protocol suitable for environments where only one entity is on-line, and a three-pass protocol in which key confirmation is additionally provided. Variants of these protocols have been standardized in IEEE P1363 [17], ANSI X9.42 [2], ANSI X9.63 [4] and ISO 15496-3 [18], and are currently under consideration for standardization and by the U.S. government's National Institute for Standards and Technology [30].

Introduction Key establishment is the process by which two (or more) entities establish a shared secret key. The key is subsequently used to achieve some cryptographic goal such as confidentiality or data integrity. Broadly speaking, there are two kinds of key establishment protocols: key transport protocols in which a key is created by one entity and securely transmitted to the second entity, and key agreement protocols in which both parties contribute information which jointly establish the shared secret key. In this paper, we shall only consider key agreement protocols for the asymmetric (public-key) two-entity setting. Let A and B be two honest entities, i.e., legitimate entities who execute the steps of a protocol correctly. Informally speaking, a key agreement protocol is said to provide implicit key authentication (of B to A) if entity A is assured that no other entity aside from a specifically identified second entity B can possibly learn the value of a particular secret key. Note that the property of implicit key authentication does not necessarily mean that A is assured of B actually possessing the key. A key agreement protocol which provides implicit key authentication to both participating entities is called an authenticated key agreement (AK) protocol. Informally speaking, a key agreement protocol is said to provide key confirmation (of B to is assured that the second entity B actually has possession of a particular secret key. If both implicit key authentication and key confirmation (of B to are provided, then the key establishment protocol is said to provide explicit key authentication (of B to A). A key agreement protocol which provides explicit key authentication to both participating entities is called an authenticated key agreement with key confirmation (AKC) protocol. For an extensive survey on key establishment, see Chapter 12 of Menezes, van Oorschot and Vanstone [27]. Extreme care must be exercised when separating key confirmation from implicit key authen- tication. If an AK protocol which does not offer key confirmation is used, then, as pointed out in [10], it is desirable that the agreed key be confirmed prior to cryptographic use. This can be done in a variety of ways. For example, if the key is to be subsequently used to achieve confiden- tiality, then encryption with the key can begin on some (carefully chosen) known data. Other systems may provide key confirmation during a 'real-time' telephone conversation. Separating key confirmation from implicit key authentication is sometimes desirable because it permits flexibility in how a particular implementation chooses to achieve key confirmation, and thus moves the burden of key confirmation from the establishment mechanism to the application. In this paper, we propose a new and efficient two-pass AK protocol. The protocol is based on Diffie-Hellman key agreement [14], and has many of the desirable security and performance attributes discussed in [10] (see x2). Two modifications of this protocol are also presented: a one-pass AK protocol suitable for environments where only one entity is on-line, and a three-pass protocol in which key confirmation is additionally provided. The protocols described in this paper establish a shared secret K between two entities. A derivation function should then be used to derive a secret key from the shared secret. This is necessary because K may have some weak bits - bits of information about K that can be predicted correctly with non-negligible advantage. One way to derive a secret key from K is to apply a one-way hash function such as SHA-1 [28] to K. With the exception of Protocol 3 in x6.2, this paper does not include key derivation functions in protocol descriptions. All protocols described in this paper have been described in the setting of the group of points on an elliptic curve defined over a finite field. However, they can all be easily modified to work in any finite group in which the discrete logarithm problem appears intractable. Suitable choices include the multiplicative group of a finite field, subgroups of Z n where n is a composite integer, and subgroups of Z p of prime order q. Elliptic curve groups are advantageous because they offer equivalent security as the other groups but with smaller key sizes and faster computation times. The remainder of the paper is organized as follows. x2 discusses the desirable attributes of AK and AKC protocols. x3 describes the elliptic curve parameters that are common to both entities involved in the protocols, public keys, and methods for validating them. x4 reviews the MTI protocols, and describes some active attacks on them. These attacks influenced the design of the new two-pass AK protocol, which is presented in x5. x6 presents the one-pass variant of the protocol, and the three-pass variant which provides explicit key authentication. Finally, x7 makes concluding remarks. Desirable attributes of AK and AKC protocols Numerous Diffie-Hellman-based AK and AKC protocols have been proposed over the years; how- ever, many have subsequently been found to have security flaws. The main problems were that appropriate threat models and the goals of secure AK and AKC protocols lacked formal defini- tion. Blake-Wilson, Johnson and Menezes [10], adapting earlier work of Bellare and Rogaway [9] in the symmetric setting, provided a formal model of distributed computing and rigorous definitions of the goals of secure AK and AKC protocols within this model. Concrete AK and AKC protocols were proposed, and proven secure within this framework in the random oracle model [8]. This paper follows their definitions of goals of secure AK and AKC protocols (described informally in x1) and their classification of threats. A secure protocol should be able to withstand both passive attacks (where an adversary attempts to prevent a protocol from achieving its goals by merely observing honest entities carrying out the protocol) and active attacks (where an adversary additionally subverts the communications by injecting, deleting, altering or replaying messages). In addition to implicit authentication and key confirmation, a number of desirable security attributes of AK and AKC protocols have been identified (see [10] for a further discussion of these): 1. known-key security. Each run of a key agreement protocol between two entities A and B should produce a unique secret key; such keys are called session keys. A protocol should still achieve its goal in the face of an adversary who has learned some other session keys. 2. (perfect) forward secrecy. If long-term private keys of one or more entities are compromised, the secrecy of previous session keys established by honest entities is not affected. 3. key-compromise impersonation. Suppose A's long-term private key is disclosed. Clearly an adversary that knows this value can now impersonate A, since it is precisely this value that identifies A. However, it may be desirable that this loss does not enable an adversary to impersonate other entities to A. 4. unknown key-share. Entity A cannot be coerced into sharing a key with entity B without A's knowledge, i.e., when A believes the key is shared with some entity C 6= B, and B (correctly) believes the key is shared with A. 5. key control. Neither entity should be able to force the session key to a preselected value. Desirable performance attributes of AK and AKC protocols include a minimal number of (the number of messages exchanged in a run of the protocol), low communication overhead (total number of bits transmitted), and low computation overhead. Other attributes that may be desirable in some circumstances include role-symmetry (the messages transmitted between entities have the same structure), non-interactiveness (the messages transmitted between the two entities are independent of each other), and the non-reliance on encryption (to meet export requirements), hash functions (since these are notoriously hard to design), and timestamping (since it is difficult to implement securely in practice). 3 Domain parameters and key pair generation This section describes the elliptic curve parameters that are common to both entities involved in the protocols (i.e., the domain parameters), and the key pairs of each entity. 3.1 Domain parameters The domain parameters for the protocols described in this paper consist of a suitably chosen elliptic curve E defined over a finite field F q of characteristic p, and a base point In the remainder of this subsection, we elaborate on what "suitable" parameters are, and outline a procedure for verifying that a given set of parameters meet these requirements. In order to avoid the Pollard-rho [32] and Pohlig-Hellman [31] algorithms for the elliptic curve discrete logarithm problem (ECDLP), it is necessary that the number of F q -rational points on E, denoted #E(F q ), be divisible by a sufficiently large prime n. As of this writing, it is commonly recommended that n ? 2 160 (see [3, 4]). Having fixed an underlying field F q , n should be selected to be as large as possible, i.e., one should have n - q, so #E(F q ) is almost prime. In the remainder of this paper, we shall assume that n ? 2 160 and that n ? 4 p q. By Hasse's Theorem, Hence implies that n 2 does not divide #E(F q ), and thus E(F q ) has a unique subgroup of order n. Also, since there is a unique integer h such that namely To guard against potential small subgroup attacks (see x4.1), the point P should have order n. Some further precautions should be exercised when selecting the elliptic curve. To avoid the reduction algorithms of Menezes, Okamoto and Vanstone [24] and Frey and R-uck [16], the curve should be non-supersingular (i.e., p should not divide (q generally, one should verify that n does not divide q large enough so that it is computationally infeasible to find discrete logarithms in F q C suffices in practice [3]). To avoid the attack of Semaev [34], Smart [35], and Satoh and Araki [33] on F q -anomalous curves, the curve should not be F q -anomalous (i.e., #E(F q ) 6= q). A prudent way to guard against these attacks, and similar attacks against special classes of curves that may be discovered in the future, is to select the elliptic curve E at random subject to the condition that #E(F q ) is divisible by a large prime - the probability that a random curve succumbs to these special-purpose attacks is negligible. A curve can be selected verifiably at random by choosing the coefficients of the defining elliptic curve equation as the outputs of a one-way function such as SHA-1 according to some pre-specified procedure. A procedure for accomplishing this, similar in spirit to the method given in FIPS 186 [29] for selecting DSA primes verifiably at random, is described in ANSI X9.62 [3]. To summarize, domain parameters are comprised of: 1. a field size q, where q is a prime power (in practice, either an odd prime, or 2. an indication FR (field representation) of the representation used for the elements of F q ; 3. two field elements a and b in F q which define the equation of the elliptic curve E over F q (i.e., in the case p ? 3, and y in the case 4. two field elements x P and y P in F q which define a finite point of prime order in E(F q ) (since P is described by two field elements, this implies that P 6= O, where O denotes the point at infinity); 5. the order n of the point P ; and 6. the cofactor Domain parameter validation. A set of domain parameters (q; FR; a; b; can be verified to meet the above requirements as follows. This process is called domain parameter validation. 1. Verify that q is a prime power. 2. Verify that FR is a valid field representation. 3. Verify that a, b, x P and y P are elements of F q (i.e., verify that they are of the proper format for elements of F q ). 4. Verify that a and b define a (non-singular) elliptic curve over F q (i.e., 4a 3 the case p ? 3, and b 6= 0 in the case 5. Verify that P satisfies the defining equation of E (and that P 6= O). 6. Verify that n ? 4 p q, that n is prime, and that n is sufficiently large (e.g., n ? 2 160 ). 7. Verify that 8. Compute verify that 9. To ensure protection against known attacks on special classes of elliptic curves, verify that n does not divide q 20, and that n 6= q. 3.2 Key pair generation Given a valid set of domain parameters (q; FR; a; b; h), an entity A's private key is an integer d selected at random from the interval [1; n \Gamma 1]. A's public key is the elliptic curve point . The key pair is (Q; d). Each run of a key agreement protocol between two entities A and B should produce a unique shared secret key. Hence, for the protocols described in this paper, each entity has two public keys: a static or long-term public key, and an ephemeral or short-term public key. The static public key is bound to the entity for a certain period of time, typically through the use of certificates. A new ephemeral public key is generated for each run of the protocol. A's static key pair is denoted (WA ; wA ), while A's ephemeral key pair is denoted For the remainder of this paper, we will assume that static public keys are exchanged via certificates. Cert A denotes A's public-key certificate, containing a string of information that uniquely identifies A (such as A's name and address), her static public key WA , and a certifying authority CA's signature over this information. To avoid a potential unknown key-share attack (see x4.2), the CA should verify that A possesses the private key wA corresponding to her static public key WA . Other information may be included in the data portion of the certificate, including the domain parameters if these are not known from context. Any other entity B can use his authentic copy of the CA's public key to verify A's certificate, thereby obtaining an authentic copy of A's static public key. Public-key validation. Before using an entity's purported public key Q, it is prudent to verify that it possesses the arithmetic properties it is supposed to - namely that Q be a finite point in the subgroup generated by P . This process is called public-key validation [19]. Given a valid set of domain parameters (q; FR; a; b; purported public key be validated by verifying that: 1. Q is not equal to O; 2. xQ and y Q are elements in the field F q (i.e., they are of the proper format for elements of 3. Q satisfies the defining equation of E; and 4. Embedded public-key validation. The computationally expensive operation in public-key validation is the scalar multiplication in step 4. For static public keys, the validation could be done once and for all by the CA. However, since a new ephemeral key is generated for each run of the protocol, validation of the ephemeral key places a significant burden on the entity performing the validation. To reduce this burden, step 4 can be omitted during key validation. Instead, the protocols proposed in x5 and x6 ensure that the shared secret K generated is a finite point in the subgroup generated by P . To summarize, given a valid set of domain parameters embedded public-key validation of a purported public key accomplished by verifying that: 1. Q is not equal to O; 2. xQ and y Q are elements in the field F q (i.e., they are of the proper format for elements of F q ); and 3. Q satisfies the defining equation of E. 4 The MTI key agreement protocols The MTI/A0 and MTI/C0 key agreement protocols described here are special cases of the three infinite families of key agreement protocols proposed by Matsumoto, Takashima and Imai [23] in 1986. They were designed to provide implicit key authentication, and do not provide key confirmation. Closely related protocols are KEA [30] and those proposed by Goss [17] and Yacobi [37], the latter operating in the ring of integers modulo a composite integer. Yacobi proved that his protocol is secure against certain types of known-key attacks by a passive adversary (provided that the composite modulus Diffie-Hellman problem is intractable). However, Desmedt and Burmester [13] pointed out that the security is only heuristic under known-key attack by an active adversary. This section illustrates that the MTI/A0 and MTI/C0 families of protocols are vulnerable to several active attacks. The small subgroup attack presented in x4.1 illustrates that the MTI/C0 protocol (as originally described) does not provide implicit key authentication. The unknown attack presented in x4.2 illustrates that the MTI/A0 protocol does not possess the unknown key-share attribute in some circumstances. Other active attacks on AK protocols are discussed by Diffie, van Oorschot and Wiener [15], Burmester [11], Just and Vaudenay [20], and Lim and Lee [22]. 4.1 Small subgroup attack The small subgroup attack was first pointed out by Vanstone [26]; see also van Oorschot and Wiener [36], Anderson and Vaudenay [1], and Lim and Lee [22]. The attack illustrates that authenticating and validating the static and ephemeral keys is a prudent, and sometimes essential, measure to take in Diffie-Hellman AK protocols. We illustrate the small subgroup attack on the MTI/C0 protocol. The protocol assumes that A and B a priori possess authentic copies of each other's static public keys. MTI/C0 AK protocol. 1. A generates a random integer r A , computes the point sends to B. 2. B generates a random integer r computes the point sends TB to A. 3. A computes A r 4. B computes 5. The shared secret is the point K. The small subgroup attack can be launched if the order n of the base point P is not prime; say, . The attack forces the shared secret to be one of a small and known subset of points. A small subgroup attack on the MTI/C0 protocol. 1. E intercepts A's message replaces it with 2. E intercepts B's message TB and replaces it with mTB . 3. A computes A r A (mTB 4. B computes K lies in the subgroup of order t of the group generated by P , and hence it takes on one of only possible values. Since the value of K can be correctly guessed by E with high probability, this shows that the MTI/C0 protocol does not provide implicit key authentication. The effects of this can be especially drastic because a subsequent key confirmation phase may fail to detect this attack - E may be able to correctly compute the messages required for key confirmation. For example, E may be able to compute on behalf of A the message authentication code (MAC) under K of the message consisting of the identities of A and B, the ephemeral point purportedly sent by A, and the point sent by B. The small subgroup attack in MTI/C0 can be prevented, for example, by mandating use of a base point P of prime order, and requiring that both A and B perform key validation on the ephemeral public keys they receive. 1 In this case, static private keys w should be selected subject to the condition 4.2 Unknown key-share attack The active attack on the MTI/A0 protocol described in this subsection was first pointed out by Menezes, Qu and Vanstone [25]. The attack, and the many variants of it, illustrate that it is prudent to require an entity A to prove possession of the private key corresponding to its (static) public key to a CA before the CA certifies the public key as belonging to A. This proof of possession can be accomplished by zero-knowledge techniques; for example, see Chaum, Evertse and van de Graaf [12]. MTI/A0 AK protocol. 1. A generates a random integer r A , computes the point sends (RA ; Cert A ) to B. 2. B generates a random integer r computes the point sends (RB ; Cert B ) to A. 3. A computes 4. B computes 5. The shared secret is the point K. In one variant of the unknown key-share attack, the adversary E wishes to have messages sent from A to B identified as having originated from E herself. To accomplish this, E selects an integer e, 1 - e - and gets this certified as her public key. Notice that E does not know the logical private key which corresponds to her public key (assuming, of course, that the discrete logarithm problem in E(F q ) is intractable), although she knows e. An unknown key-share attack on the MTI/A0 protocol. 1. E intercepts A's message (RA ; Cert A ) and replaces it with (RA ; Cert E ). 2. B sends (RB ; Cert B ) to E, who then forwards (eRB ; Cert B ) to A. 3. A computes 4. B computes A and B now share the secret K even though B believes he shares the secret with E. Note that does not learn the value of K. Hence, the attack illustrates that the MTI/A0 protocol does not possess the unknown key-share attribute. A hypothetical scenario where the attack may be launched successfully is the following; this scenario was first described by Diffie, van Oorschot and Wiener [15]. Suppose that B is a bank branch and A is an account holder. Certificates are issued by the bank headquarters and within each certificate is the account information of the holder. Suppose that the protocol for electronic deposit of funds is to exchange a key with a bank branch via a mutually authenticated key agreement. Once B has authenticated the transmitting entity, encrypted funds are deposited to the account number in the certificate. Suppose that no further authentication is done in the encrypted deposit message (which might be the case to save bandwidth). If the attack mentioned above is successfully launched then the deposit will be made to E's account instead of A's account. 5 The new authenticated key agreement protocol In this section, the two-pass AK protocol is described. The following notation is used in x5 and x6. f denotes the bitlength of n, the prime order of the base point If Q is a finite elliptic curve point, then Q is defined as follows. Let x be the x-coordinate of Q, and let x be the integer obtained from the binary representation of x. (The value of x will depend on the representation chosen for the elements of the field F q .) Then Q is defined to be the integer . Observe that (Q mod n) 6= 0. 5.1 Protocol description We now describe the two-pass AK protocol (Protocol 1) which is an optimization and refinement of a protocol first described by Menezes, Qu and Vanstone [25]. It is depicted in Figure 1. In this and subsequent Figures, A (w A;WA ) denotes that A's static private key and static public key are wA and WA , respectively. Domain parameters and static keys are set up and validated as described in x3. If A and B do not a priori possess authentic copies of each other's static public then certificates should be included in the flows. RA Figure 1: Two-pass AK protocol (Protocol 1) Protocol 1. 1. A generates a random integer r A , computes the point sends this to B. 2. B generates a random integer r computes the point sends this to A. 3. A does an embedded key validation of RB (see x3.2). If the validation fails, then A terminates the protocol run with failure. Otherwise, A computes and If terminates the protocol run with failure. 4. B does an embedded key validation of RA . If the validation fails, then B terminates the protocol run with failure. Otherwise, B computes and If terminates the protocol run with failure. 5. The shared secret is the point K. 5.2 Security notes and rationale Although the security of Protocol 1 has not been formally proven in a model of distributed computing, heuristic arguments suggest that Protocol 1 provides mutual implicit key authenti- cation. In addition, Protocol 1 appears to have the security attributes of known-key security, forward secrecy, key-compromise impersonation, and key control that were listed in x2. Another security attribute of Protocol 1 is that compromise of the ephemeral private keys (r A and r B ) reveals neither the static private keys (wA and wB ) nor the shared secret K. Kaliski [21] has recently observed that Protocol 1 does not possess the unknown key-share attribute. This is demonstrated by the following on-line attack. The adversary E intercepts A's ephemeral public key RA intended for B, and computes certified as her static public key (note that E knows the corresponding private key wE ), and transmits RE to B. B responds by sending RB to E, which E forwards to A. Both A and B compute the same secret K, however B mistakenly believes that he shares K with E. We emphasize that lack of the unknown key-share attribute does not contradict the fundamental goal of mutual implicit key authentication - by definition the provision of implicit key authentication is only considered in the case where B engages in the protocol with an honest entity (which E isn't). If an application using Protocol 1 is concerned with the lack of the unknown key-share attribute under such on-line attacks, then appropriate confirmation should be added, for example as specified in Protocol 3. Protocol 1 can be viewed as a direct extension of the ordinary (unauthenticated) Diffie-Hellman agreement protocol. The quantities s A and s B serve as implicit signatures for A's ephemeral public key RA and B's ephemeral public key RB , respectively. The shared secret is rather than r A r BP as would be the case with ordinary Diffie-Hellman. The expression for RA uses only half the bits of the x-coordinate of RA . This was done in order to increase the efficiency of computing K because the scalar multiplication RAWA in (4) can be done in half the time of a full scalar multiplication. The modification does not appear to affect the security of the protocol. The definition of RA implies that RA 6= 0; this ensures that the contribution of the static private key wA is not being cancelled in the formation of s A in (1). Multiplication by h in (2) and (4) ensures that the shared secret K (see equation (5)) is a point in the subgroup of order n in E(F q ). The check ensures that K is a finite point. If Protocol 1 is used to agree upon a k-bit key for subsequent use in a symmetric-key block cipher, then it is recommended that the elliptic curve be chosen so that n ? 2 2k . Protocol 1 has all the desirable performance attributes listed in x2. From A's point of view, the dominant computational steps in a run of Protocol 1 are the scalar multiplications r AP , RBWB , and hsA (RB +RBWB ). Hence the work required by each entity is 2:5 (full) scalar mul- tiplications. Since r AP can be computed off-line by A, the on-line work required by each entity is only 1:5 scalar multiplications. In addition, the protocol has low communication overhead, is role-symmetric, non-interactive, and does not use encryption or timestamping. While a hash function may be used in the key derivation function (to derive a session key from the shared secret K), the security of Protocol 1 appears to be less reliant on the cryptographic strength of the hash function that some other AK protocols (such as Protocol 3 in [10]). In particular, the requirement that the key derivation function be preimage resistant appears unnecessary. Non- reliance on hash functions is advantageous because history has shown that secure hash functions are difficult to design. 6 Related protocols This section presents two related protocols: a one-pass AK protocol (Protocol 2), and a three-pass AKC protocol (Protocol 3). 6.1 One-pass authenticated key agreement protocol The purpose of the one-pass AK protocol (Protocol 2) is for entities A and B to establish a shared secret by only having to transmit one message from A to B. This can be useful in applications where only one entity is on-line, such as secure email and store-and-forward. Protocol 2 is depicted in Figure 2. It assumes that A a priori has an authentic copy of B's static public key. Domain parameters and static keys are set up and validated as described in x3. RA Figure 2: One-pass AK protocol (Protocol 2) Protocol 2. 1. A generates a random integer r A , computes the point sends this to B. 2. A computes s terminates the protocol run with failure. 3. B does an embedded key validation of RA (see x3.2). If the validation fails, then B terminates the protocol run with failure. Otherwise, B computes s n and terminates the protocol run with failure. 4. The shared secret is the point K. Heuristic arguments suggest that Protocol 2 offers mutual implicit key authentication. The main security drawback of Protocol 2 is that there is no known-key security and forward secrecy since entity B does not contribute a random per-message component. Of course, this will be the case with any one-pass key agreement protocol. 6.2 Authenticated key agreement with key confirmation protocol We now describe the AKC variant of Protocol 1. It is depicted in Figure 3. Domain parameters and static keys are set up and validated as described in x3. Here, MAC is a message authentication code algorithm and is used to provide key confirmation. For examples of provably secure and practical MAC algorithms, see Bellare, Canetti and Krawczyk [5], Bellare, Guerin and Rogaway [6], and Bellare, Kilian and Rogaway [7]. H 1 are (independent) key derivation functions. Practical instantiations of H 1 include H 1 Protocol 3. 1. A generates a random integer r A , computes the point sends this to B. 2. 2.1 B does an embedded key validation of RA (see x3.2). If the validation fails, then B terminates the protocol run with failure. 2.2 Otherwise, B generates a random integer r computes the point RA Figure 3: Three-pass AKC protocol (Protocol 2.3 B terminates the protocol run with failure. The shared secret is K. 2.4 B uses the x-coordinate z of the point K to compute two shared (z) and (z). sends this together with RB to A. 3. 3.1 A does an embedded key validation of RB . If the validation fails, then A terminates the protocol run with failure. 3.2 Otherwise, A computes s terminates the protocol run with failure. 3.3 A uses the x-coordinate z of the point K to compute two shared (z) and (z). 3.4 A computes MAC - 0 verifies that this equals what was sent by B. 3.5 A computes MAC - 0 sends this to B. 4. B computes MAC - 0 verifies that this equals what was sent by A. 5. The session key is -. AKC Protocol 3 is derived from AK Protocol 1 by adding key confirmation to the latter. This is done in exactly the same way AKC Protocol 2 of [10] was derived from AK Protocol 3 of [10]. Protocol 2 of [10] was formally proven to be a secure AKC protocol. Heuristic arguments suggest that Protocol 3 of this paper is a secure AKC protocol, and in addition has all the desirable security attributes listed in x2. 7 Concluding remarks The paper has presented new AK and AKC protocols which possess many desirable security attributes, are extremely efficient, and appear to place less burden on the security of the key derivation function than other proposals. Neither of the protocols proposed have been formally proven to possess the claimed levels of security, but heuristic arguments suggest that this is the case. It is hoped that the protocols, or appropriate modifications of them, can, under plausible assumptions, be proven secure in the model of distributed computing introduced in [10]. Acknowledgements The authors would like to thank Simon Blake-Wilson and Don Johnson for their comments on earlier drafts of this paper. Ms. Law and Dr. Solinas would like to acknowledge the contributions of their colleagues at the National Security Agency. --R "Minding your p's and q's" "Keying hash functions for message authenti- cation" "XOR MACs: New methods for message authentication using finite pseudorandom functions" "The security of cipher block chaining" "Random oracles are practical: a paradigm for designing efficient protocols" "Entity authentication and key distribution" "Key agreement protocols and their security analysis" "On the risk of opening distributed keys" "An improved protocol for demonstrating possession of discrete logarithms and some generalizations" "Towards practical 'proven secure' authenticated key distri- bution" "New directions in cryptography" "Authentication and authenticated key ex- changes" "A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves" "Cryptographic method and apparatus for public key exchange with authenti- cation" Contribution to ANSI X9F1 working group "Authenticated multi-party key agreement" Contribution to ANSI X9F1 and IEEE P1363 working groups "A key recovery attack on discrete log-based schemes using a prime order subgroup" "On seeking smart public-key distribution sys- tems" "Reducing elliptic curve logarithms to logarithms in a finite field" "Some new key agreement protocols providing mutual implicit authentication" "Key agreement and the need for authentication" Handbook of Applied Cryptography "Secure Hash Standard (SHS)" "Digital signature standard" "SKIPJACK and KEA algorithm specification" "An improved algorithm for computing logarithms over GF (p) and its cryptographic significance" "Monte Carlo methods for index computation mod p" "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves" "Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p" "The discrete logarithm problem on elliptic curves of trace one" "On Diffie-Hellman key agreement with short expo- nents" "A key distribution paradox" --TR --CTR Johann Groschdl , Alexander Szekely , Stefan Tillich, The energy cost of cryptographic key establishment in wireless sensor networks, Proceedings of the 2nd ACM symposium on Information, computer and communications security, March 20-22, 2007, Singapore Kyung-Ah Shim , Sung Sik Woo, Cryptanalysis of tripartite and multi-party authenticated key agreement protocols, Information Sciences: an International Journal, v.177 n.4, p.1143-1151, February, 2007 Kyung-Ah Shim, Vulnerabilities of generalized MQV key agreement protocol without using one-way hash functions, Computer Standards & Interfaces, v.29 n.4, p.467-470, May, 2007 Maurizio Adriano Strangio, Efficient Diffie-Hellmann two-party key agreement protocols based on elliptic curves, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico Mario Di Raimondo , Rosario Gennaro , Hugo Krawczyk, Secure off-the-record messaging, Proceedings of the 2005 ACM workshop on Privacy in the electronic society, November 07-07, 2005, Alexandria, VA, USA

diffie-hellman;key confirmation;authenticated key agreement;elliptic curves

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Modeling random early detection in a differentiated services network.

An analytical framework for modeling a network of random early detection (RED) queues with mixed traffic types (e.g. TCP and UDP) is developed. Expressions for the steady-state goodput for each flow and the average queuing delay at each queue are derived. The framework is extended to include a class of RED queues that provides differentiated services for flows with multiple classes. Finally, the analysis is validated against ns simulations for a variety of RED network configurations where it is shown that the analytical results match with those of the simulations within a mean error of 5%. Several new analytical results are obtained; TCP throughput formula for a RED queue; TCP timeout formula for a RED queue and the fairness index for RED and tail drop.

INTRODUCTION The diverse and changing nature of service requirements among Internet applications mandates a network architecture that is both flexible and capable of differentiating between the needs of different applications. The traditional Internet architecture, however, offers best-effort service to all traffic. In an attempt to enrich this service model, the Internet Engineering Task Force (IETF) is considering a number of architectural extensions that permit service discrimi- nation. While the resource reservation setup protocol (RSVP) [1], [2] and its associated service classes [3], [4] may provide a solid foundation for providing different service guarantees, previous efforts in this direction [5] show that it requires complex changes in the Internet architecture. That has led the IETF to consider simpler alternatives to service differentiation. A promising approach, known as Differentiated Services (DiffServ) [6], [7], [8] allows the packets to be classified (marked) by an appropriate class or type of service (ToS) [9] at the edges of the network, while the queues at the core simply support priority handling of packets based on their ToS. Another Internet-related quality of service (QoS) issue is the queue's packet drop mechanism. Active queue management (AQM) has been recently proposed as a means to alleviate some congestion control problems as well as provide a notion of quality of service. A promising class is based on randomized packet drop or marking. In view of the IETF informational RFC Random Early Detection (RED) [11] is expected to be widely deployed in the Internet. Two variants of RED, RIO [12] and WRED [13] have been proposed as a means to combine the congestion control features of RED with the notion of DiffServ. This work aims to make a contribution towards analytical modeling of the interaction between multiple TCP flows (with different round-trip times), non-responsive flows (e.g. UDP) and active queue management (AQM) approaches (e.g. RED) in the context of a differentiated services architecture. A new framework for modeling AQM queues is introduced and applied to a network of RED queues with DiffServ capabilities termed DiffRED which is similar to the RIO and WRED algorithms. Our work complements previous research efforts in the area of Manuscript first submitted December 7, 2000. A. A. Abouzeid is with the Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, 8th St., Troy NY 12180, USA (e-mail: [email protected]). S. Roy is with the Department of Electrical Engineering, University of Washington, Box 352500, Seattle WA 98195-2500, USA (e-mail:[email protected]). Part of the material in this paper (Section III.A,B) was presented at IEEE Global Communications Conference, San Francisco, November 2000, and is included in a revised form in this paper. TCP modeling under AQM regimes. Specifically, this work has the following unique contributions C1. TCP timeout in AQM Networks: We derive an expression for the probability of a timeout that reduces the inaccuracy by as much as an order of magnitude vis-a-vis previous state-of-art [14], [15] C2. Poisson Process Splitting (PPS) Based Analysis for TCP flows: We show by analysis and simulations that the PPS approximation holds for a population of heterogeneous (in terms of delay) TCP flows sharing a bottleneck RED queue. C3. Fairness Results: We derive expressions for the fairness index of a population of heterogeneous TCP flows for the case of RED and compare it to Tail Drop. C4. Network of AQM (RED) Routers: We extend the result to a network of AQM queues with TCP (and UDP) flows. C5. Network of DiffRED Routers: Finally, the results are extended to a network of RED and DiffRED queues. While the details of the above contributions are presented in the following sections, we first comment on the relation between the above contributions and previous/concurrent related work. Literature Review The steady-state throughput of a single TCP-Reno (or NewReno) flow with Tail Drop (TD) queueing policy where packet drops can be modelled as an i.i.d loss with probability p has been shown to have the well-known inverse- # p dependance. This result was derived by several authors [16], [17], [18] using different analytic approaches; for example, [16] used a non-homogeneous Poisson process approximation of the TCP flow while [17], [18] use a fixed point approximation method, both yielding the same dependance to within a multiplicative constant. This result which we call throughput formula1, does not incorporate the effect of TCP timeout mechanism. In [14], the above was extended to include timeouts for a Tail-Drop queue for scenarios where it is reasonable to assume that conditioned on a packet loss, all remaining packets in the same window of data are also lost (i.e. burst loss within the same window round). Based on this assumption, the authors derive a more accurate formula which we term throughput formula2, for the steady-state TCP throughput. Specifically, this derivation includes (as an intermediate step) an expression for the probability, given a packet loss, that this loss will be detected by a timeout - we term this the the timeout formula. An alternative timeout probability was derived in [15] with independent packet losses (i.e. without the assumption of burst loss within a loss window). However, the result is not available in closed form and requires the knowledge of the window size probability distribution function. Building on the above two formulae for the throughput of an individual TCP flow, Bu et al. [19] extend the analysis to a network of RED queues by solving numerically coupled equations representing the throughput for each individual flow. A similar approach is adopted in [20], [21]. Our contribution C4 listed above is similar to [19], [20], [21] insofar that we also seek to model a network of AQM queues with TCP flows. However, unlike [19], [20], [21], we do not use any of the formulae derived earlier. Specifically, we show that the timeout formula derived in [14] largely over-estimates the probability of a timeout detection for an AQM queue due to the assumption of burst loss within a loss window, via analysis that is supported by ns2 simulation. Thus in C1, we derive a new formula for the timeout probability and compare the accuracy of the new formula with those derived earlier. In [11], the authors predict that the well known bias of TCP against connections with larger delay sharing a bottleneck Tail Drop queue will only be slightly affected by the introduction of RED without any definitive analytical support. Our results in C2, C3 precisely quantify the amount of unfairness experienced by TCP flows over a bottleneck RED vis-a-vis a Tail Drop Packet Drop ProbabilityAverage Queue (packets) minth maxth Fig. 1. RED packet drop probability as a function of the average queue size (1) (2)Average Queue Size (1) (2) Drop Probability minth maxth Fig. 2. Schematic of the DiffRED packet drop probability for two classes of service. queue by using a PPS approximation (that is shown to hold within an acceptable error margin) and hence deriving a fairness index for a population of heterogeneous TCP flows for the two cases of RED and Tail Drop. Paper outline The paper is organized as follows. In Section II, we present the network model considered in this paper, describe the RED and DiffRED algorithms and outline our modeling approach. Section III considers a single congested RED queue, the PPS property is shown to hold for a population of heterogeneous flows, the analytical results for a single RED queue are validated against ns simulations and the fairness results of RED versus Tail-Drop are presented. Section IV presents a derivation of our timeout formula and a comparison between our results and previous results from [14] and [15] against ns simulations. Section V presents the analysis of a network of DiffRED 1 queues with TCP and UDP flows. The network model validation is presented in Section VI. Section VII concludes the paper outlining future extensions. II. MODELING AND NOTATION A. The RED Algorithm We briefly review the original RED algorithm (see [11] for further details). A RED router calculates a time-averaged average queue size using a low-pass filter (exponentially weighted moving average) or smoother over the sample queue lengths. The average queue size is compared to two thresholds: a minimum threshold (minth) and a maximum threshold (maxth). When the average queue size is less than minth no packets are dropped; when the average queue size exceeds maxth, every arriving packet is dropped. Packets are dropped probabilistically when the time-averaged queue size exceeds minth with a probability p that increases linearly until it reaches maxp at average queue size maxth as shown in Fig. 1. RED also has an option for marking packets instead of dropping them. B. The DiffRED algorithm We propose to investigate a DiffRED algorithm that is identical to the original RED algorithm except for the following important change - DiffRED differentiates between different packets by appropriately labelling each with the class it belongs to. If a packet is not labelled, it will be assigned the default (lowest) class. The RED queue associates a priority coefficient, denoted by # (c) , for each class 1 # c # C where C is the total number of classes. The lowest priority class corresponds to # and the c-th (higher) priority class packets have coefficient # (c) , It will be shown in Section II that RED is a special case of a DiffRED queue with no service differentiation. Packet Drop Probability(packets) minth Average Queue Fig. 3. Packet drop probability for the "Gentle" variant of RED. Packet Drop Probability(packets) minth Average Queue (2) (1) Fig. 4. Packet drop probability for the "Gentle" variant of DiffRED. # (2) # (c) # (C) . When a DiffRED queue receives a packet, it updates the average queue size and drops the packet with probability equal to p # (c) for a class c packet. Thus, for the same congestion level (i.e. average queue size), higher priority packets are dropped with a lower probability. A schematic diagram of the DiffRED dropping algorithm for the case of depicted in Figure 2. The plot shows that the packet drop probability function (as a function of the average queue size) has a lower slope for the higher priority class (for while the drop function for the lower priority class has a higher slope. C. The Gentle Variant of RED In the recently recommended "gentle" 2 variant of RED [22], the packet-dropping probability varies from maxp to 1 as the average queue size varies from maxth to 2maxth . Figures 3-4 depict the packet drop probability for RED and DiffRED with this option enabled. Our model applies equally well to both cases, but the model with the "gentle" option requires less restrictive assumptions and hence is applicable to a wider range of parameters as will be discussed in Section II-E. D. Network Model and Notations Assume a network of V queues that support M TCP flows and N UDP flows. Each TCP connection is a one way data flow between a source S j and destination D j , with reverse traffic consisting only of packet acknowledgments (ACKs) for each successfully received packet at the destination (i.e. no delayed-ACKs). Throughout the paper, we will assume that the TCP flows are indexed by 1 while UDP flows are indexed by M < shows a network with denote the total round-trip propagation and transmission delays for the j th , flow (not including queuing delay) and let - v denote the v th queue link capacity (packets/sec. Denote a zero-one (M +N)-V dimensional order matrix O with elements that specifies the order in which the j th flow traverses the v th queue. A 0 entry indicates that the corresponding flow does not pass through that queue. Since each flow passes through at least one queue, each row in O must have at least one entry equal to 1. In this work, we consider TCP-NewReno and assume the reader is familiar with the key aspects of the window adaptation mechanism [23], [24] - i.e., the two modes of window increase (slow start and congestion avoidance) and the two modes of packet loss detection (reception of multiple ACKs with the same next expected packet number or timer expiry). Our analysis This option makes RED much more robust to the setting of the parameters maxth and maxp. S5 Fig. 5. Experimental network topology with two RED queues. applies best to TCP-NewReno since we assume that multiple packets within the same loss window will trigger at most one duplicate ACK event (rather than one for Modeling Approach The primary challenge facing any analytical approach is to capture the complex interplay of TCP congestion control algorithm and active queue management policies as RED. We adopt a mean value analysis (using fixed point approximation) to estimate (long-term) averages of various parameters as done in [17], [18], [19], [25], [26], while recognizing that such analysis does not allow for accurate prediction of higher order statistical information (e.g. variances). We next state and discuss the following modeling assumptions: A1: A congested queue is fully utilized (except possibly at some isolated instants). A2: Each TCP flow passes through at least one congested queue. A3: The probability of a forced packet loss is negligible. 3 A1 is a well known result that has been observed in the literature through Internet measurements as well as simulations and was also confirmed in all the simulations we performed. A2 is reasonable if the TCP flows window size are not limited by the receiver's advertised window sizes. A3 implies that we model only 'unforced' (or probabilistic) RED packet losses and ignore forced losses which should be avoided when configuring RED queue parameters 4 . Note that it is much easier to configure a RED (or DiffRED) queue to avoid forced packet drops when the "gentle" option is used, as noted in [22] due to its avoidance of the abrupt increase in the drop probability from maxp to 1. Nevertheless, our analysis results hold even if the "gentle" option is not used, as long as A3 holds. III. SINGLE QUEUE The aim of this section is two-fold: (i) we outline an analysis using the PPS approximation 5 and (ii) derive a result for the probability that a loss is detected by a timeout (TO) and not Duplicate Acknowledgment (DA). Both of these results will be used in the following section in a network (rather than a single queue) set-up. A. PPS Based Analysis for TCP flows Consider M Poisson processes, each with an arrival rate # j at a single queue that randomly drops packets with a probability p. Let P i,j denote the probability that the i th packet loss event belongs to the j th flow. From the PPS (Poisson Process Splitting) property, (1) independent of i. 3 Forced packet losses are those due to either buffer overflow or due to packet drop with probability one. 4 This assumption has been adopted in many previous AQM-related research efforts (e.g. [27]). 5 Part of this Section was presented in a preliminary form in [28]. Consider now M TCP flows - let X i denote the i-th inter-loss duration at the queue as shown in Fig. 6 and Q i,j the queue size for session j packets at the end of epoch i. Then the net round-trip propagation and queuing delay for connection j at the end of the i th epoch is # flows share the same buffer, Q the steady-state average queue size. Let W i,j denote the window size of the j th flow just before the end of the i th epoch and W av j denote flow j time-averaged window size (as opposed to E[W i,j that represents the mean window size just prior to the end of the epoch). We postulate that, for M TCP flows sharing a queue, (2) Equation 2 is a postulate that the PPS approximation (1) holds for a population of TCP flows sharing a random drop queue. To justify the above we first consider the case of TCP flows that are perpetually in the congestion avoidance phase (additive increase - multiplicative decrease (AIMD) mode of window size) and use an analytical description of AIMD to derive expressions for the steady state average window and throughput. These expressions are then validated against ns simulations in support of (2). A modified PPS approximation for the case of timeouts is postponed to Sec. V. The window evolution of the TCP-Reno sessions is governed by the following system of equations (refer to Figure 6), w.p. P i,j Taking the expectation of both sides of (3) where we used the independence approximation E[W i,j P fixed point approximation Similarly, denoting # , then, by fixed point approximation to (2), . (5) Substituting by (5) in (4), Equation (6) is a system of M quadratic equations in the M unknowns W j (or # it can be readily verified by direct substitution that time Window 4,1 4,2 Fig. 6. A schematic showing the congestion window size for two (i.e. sessions with different round-trip delays sharing a bottleneck link with RED queue. Packet loss events are indicated by an X. is an explicit solution for (6) and hence, Remarkably, the above result implies that the average steady-state window size is the same for all flows; also, W j implicitly depends on Q and X that need to be determined. To relate the above result for W j to W av j , let W # i,j denote flow j window size just after the start of epoch i. It is straightforward to verify that and from (3) hence where Finally, let r j denote flow j steady-state throughput (in packets/sec). Then, In case of full bottleneck link utilization, which, by substituting from (14) and Finally, let p denote the average packet drop probability of the RED queue. Then, (RED or TD)G Sink Bottleneck Linkt l nt Fig. 7. Schematic of multiple TCP connections sharing a bottleneck link. I th max th 1/max p Q (an.) Q Since the RED average packet drop probability also satisfies Q- min th th - min th a relationship between Q and X follows by equating (17) and (18). B. Single Queue Validation The analytic expressions for various parameters of interest for RED queues derived in the preceding section will be compared against results from ns simulations. Figure 7 depicts the ns simulation model used for validating the above results. M TCP flows share a common bottleneck link with capacity - packets/sec and (one-way) propagation delay # . Forward TCP traffic is a one way flow between a source S j and the sink, with reverse traffic consisting only of packet acknowledgments (ACKs) for each successfully received packet at the sink. We consider the long run (steady-state) behavior and assume that the sources always have packets to transmit (e.g. FTP of large files). The j-th flow encounters a one-way propagation delay # j , and is connected via a link with capacity # j packets/sec. to the bottleneck queue. The total round-trip time for propagation and transmission (but excluding queuing) for the j-th flow is thus Table I lists the parameters used for two sets of simulation experiments with RED policy using the ns simulator [29] (more validation experiments are available in [30]); the link capacities - and # j (packets/second), propagation delays # and # j (seconds) for the topology in Fig. 7 as well as the number of competing flows used for each simulation. In each simulation, a value for the link delay # 1 of the first flow and a factor d > 1 is chosen such that the remaining # j are specified according to # i.e. the access link (from source to queue) propagation delay profile increases exponentially. The values of # 1 and d for each simulation are listed in Table I. For each simulation we compare the analysis results to the simulations measurements of Q in Table I. All simulations use equal packet size of 1 KBytes. Figures 8-9 show the measured and the computed r j for each competing flow, where the x-axis indicates the computed We choose to show the un-normalized r j (packets/second) rather than the normalized Average round-trip time E[g (seconds) Average transmission rate (packets/second) Analysis Simulation Fig. 8. Results for Exp.1. Average transmission rate (packets/second) Analysis Simulation Fig. 9. Results for Exp.2. throughput since the difference between the simulations measurement and the analysis results are indistinguishable if shown normalized to the link capacity. C. Fairness Properties of RED versus Tail-Drop TCP's inherent bias towards flows with shorter round-trip time as a result of its congestion window control mechanism is well-known - in [31], it has been shown that when multiple TCP flows with different round-trip times share a Tail Drop queue, the steady-state throughput of a flow is inversely proportional to the square of the average round-trip delay (see Appendix A for a simple derivation of this result). Our analysis/simulations results (specifically, (9) and (14)) of a single RED queue with multiple TCP flows and without timeouts (see comment at the end of this section) reveal a dependence of the throughput that differs from that of Tail Drop. Consider parameter K j defined by (13). For a population of heterogeneous flows, P j is directly proportional to the TCP flow throughput and hence inversely proportional to the round-trip delay - K j increases with round-trip delay. Since mild variation. Hence, for a large number of flows, the dependence of flow throughput on K j can be neglected 6 . Thus, the conclusions in the remainder of this section will be a lower bound to the improvement in fairness compared to Tail Drop. With the above assumptions (i.e. neglecting the effect of K j ), the results of (9) and (14) show that the average transmission rates of competing connections at a congested RED queue is inversely proportional to the average round-trip times and not to the square of the average round-trip times as in Tail Drop. A fairness coefficient # may be defined as the ratio of the lowest to the highest average throughput among the competing flows sharing a queue (i.e. ideally, denote the fairness coefficients of the equivalent 7 Tail Drop and RED queues. Then, implying that the RED queue achieves a higher fairness coefficient. The previous definition of fairness only considers the flows with the maximum and minimum transmission rates; in the pioneering work of [32], the following index #(r) was introduced to quantify fairness based on the rates of all flows: ideally fair allocation (r results in #(.) = 1; and a worst case allocation ( r , which approaches zero for large n. 6 In the limit as the number of flows tends to infinity, K flows. Note that if the impact of K j were to be included, it would help decrease TCP bias against larger propagation delay since it increases with the round-trip time. 7 An equivalent queue is one that has identical TCP flows and the same average queue size ([11]) With this definition, the fairness index for a queue with n TCP flows yields As an illustrative example, consider TCP flows with an exponentially decaying (or increasing) delay profile with factor d. Then, assuming large n (and hence neglecting the effect of K j for the case of RED), it follows after some simple algebraic manipulations that #RED (d) =n and i.e. # TD An intuitive explanation for this decrease in bias (i.e. fairness coefficient closer to one) for RED is in order. TCP increases its window size by one every round-trip time - thus, lower delay links see a faster increase in their window size. In case of synchronized packet loss (Tail Drop queues), all windows are (typically) reduced at the same time, and hence the average window size is inversely proportional to the average round-trip time. But since the throughput of a flow is proportional to the window size divided by the round-trip time, the average rate of each TCP session is inversely proportional to the square of the average round-trip time. In case of RED, when a packet is dropped, the chances that the packet will belong to a certain connection is (on the average) proportional to that connection transmission rate- thus TCP sessions with lower delays are more likely to have their windows reduced. The analytical results show that this causes the average window size to be equal for all connections (6) and thus the throughput of a connection is only inversely proportional to the round-trip-delay. Thus while the basic RED algorithm does not achieve throughput fairness among the competing flows, it substantially reduces the bias as compared to Tail Drop. Finally, the above fairness results do not take into account the effect of TCP timeouts. For reasons previously shown via simulations in the literature, the occurrence of timeouts will, in general, increase the unfairness in TCP against longer delay links. IV. TIMEOUT FORMULA FOR A TCP FLOW WITH RANDOM PACKET LOSS In this section, we first derive an expression for the probability that a packet loss event will be detected by a timeout (TO) (i.e. not a duplicate ACK detection (DA) event). Next, we compare our result and those in the literature against ns simulations. A. Analysis Consider an instant during a TCP-Reno session at which the window size is in the congestion avoidance phase. Let w denote the current congestion window size, in packets, during which at least one packet loss takes place; each packet is assumed lost independently with probability p. Let the current round be labelled as round i for reference (see Figure 10). Let P (w, the probability, conditioned on at least one packet loss during the current window round, that packet k will be the first packet lost during the round. Then, The k-1 packets transmitted before the first loss in the current round will cause the reception of k - 1 ACKs in the next round (i hence the release of an equal number of packets. Packet sequence number w-k round i round i+1 round i+2 new packets Time Fig. 10. Analysis of the probability of a TO vs. TD. Successful packets (i.e. not dropped), dropped packets, and ACK packets for packet k are represented by white-filled, completely shaded and partly shaded rectangles respectively. In this example, denote the number of duplicate ACKs received in round due to the successful transmission of N 2 out of the k-1 packets in round i+1. Similarly, consider the w-k packets transmitted after the first loss during round i (the current round), and let N denote the number of packets successfully transmitted out of the w - k packets. Notice that, unlike the N 2 packets, the N 1 packets are transmitted after the first loss packet in the current round. Hence, N 1 duplicate ACKs will be generated immediately in round i + 1. Let D(w, denote the probability that a DA detection takes place in round 2. Then, Notice that the above expression is independent of k. This is expected, and can be stated simply as the probability of at least 3 out of Let Q(w) denote the probability of a TO detection in rounds 2. Then For retaining only the most significant terms yields (w - 1)(w - 2)p (w-3) (27) Following a DA detection, the window is halved and TCP attempts to return to its normal window increase mode in congestion avoidance. However, one or more losses may take place in the immediate following round (i.e. before the window is increased). In this paper, consecutive congestion window rounds in which packet loss is detected by DA without any separating additive window increase will be treated as one loss window, that ultimately terminates via either a DA or a TO. Let Z(w) denote the probability that a TO will take place (either immediately or following a sequence of consecutive DA's). Then, log w where the logarithm is base 2. Finally, recalling that W i denotes the window size at the end of the i th epoch, applying a fixed point approximation to (28) yields where Z(W ) is given by (28). validation and comparison with previous results In this section, previous results for Z(w) are listed and compared with the results of a representative simulation using ns consisting of multiple TCP flows sharing a bottleneck RED queue. In [14], the following expression for Z(W ) was derived assuming that, once a packet is lost, all remaining packets in the same congestion window are also lost (which is probably reasonable for a Tail Drop queue). In [19], the authors argue that this is still a good result when applied to an AQM like RED which drops packets independently. We show here that this is not true. In [15], the following expression for Z(w) was derived which is the same as our expression for Q(w) in the previous section (26). However, the authors do not analyze the case where multiple consecutive DA's ultimately cause a TO. Also, the expression for E[Z(w)] was not derived in [15], since the analysis relies on the knowledge of the congestion window probability distribution function. Nevertheless, we show here that applying a fixed point approximation to (31), yielding would result in an under-estimation of the timeout probability. In order to compare between the accuracy of our result (29) and the previous results (30) and (32) when applied to a RED queue, we construct the following simulation using ns. A single congested RED queue is considered with a number of identical TCP flows ranging from 5 to 40. The RED parameters are set as follows: and ECN disabled. The link capacity is 1 Mbps and two-way propagation delay (no including queuing) is 20 ms. The packet size is 1 KB. The buffer size is selected large enough to avoid buffer overflow. The trace functionality within ns was enhanced so as to record, for one of the flows, the time instant, congestion window size and the reason (TO or DA) for each window decrease. Post-processing scripts were used to compute the average window size just before a packet loss detection (i.e. W ), the average packet drop through the queue (p), the total number of packet loss detection events that are detected by TO and those detected by DA, and hence the 0.40.81.2 Number of TCP flows Measured E[W]/10 Measured E[p] Measured TO prob. E[Z(W)] Z(E[W]) from our analysis Z(E[W]) from Padhye et al. Fig. 11. Comparison between different timeout formulae against ns simulations for TCP-NewReno. measured E[Z(W )]. Finally, the measured p and W from the simulations were substituted in (29), (30) and (32) to compute Z(W ). The results are shown in Figure 11. Figure 11 does not show the results from (32) since the values produced are practically 0,i.e., excessively under-estimates the probability of a timeout. Figure 11 shows that our result is much more accurate than that of [14] (as expected). The improvement accuracy is up to an order of magnitude for small number of competing flows (and hence small p). The reason is that, for small p, the assumption of conditional burst loss used in [14] becomes very unrealistic. As p increases, the RED queue operation approaches that of Tail Drop, since p approaches maxp (and hence all arriving packets are dropped with higher probability, resulting in synchronous operation of the flows and thus burst losses). However, even for high p, our formula is more accurate than that of [14]. V. NETWORK ANALYSIS Since a RED queue is a special case of a DiffRED queue with # we present an analysis for the case of DiffRED queues. In the first subsection, we consider a network of TCP and UDP flows and we model the additive increase/multiplicative decrease dynamics of TCP only. Next, we extend the model results to capture the TCP timeout behavior. A. Modeling without timeouts Let minth v , maxth v and maxp v (c) denote the DiffRED parameters of the v th queue. Let (c) denote the priority coefficient of class c, where # v denote the average (steady state) packet drop probability in the v th queue. Let p v (c) denote the average packet drop probability for class c at the v th queue. Then for the case of the "gentle" variant, qv -minthv th th (c) while for the case without the "gentle" variant, qv -minthv According to the description of the DiffRED dropping algorithm (Figure 2) where (35) assumes that all the DiffRED queues in a network use the same value of the class priority coefficients (i.e. # (c) is independent of v) 8 . Let #(j) and p v (j) denote the priority coefficient and average drop probability at the v th queue for the class that flow j belongs to. For the case of the "gentle" variant, it follows from (33) and (35) that -maxp (1) while for the case without the "gentle" variant Let r j denote the steady state packet transmission rate of flow j. Let the generator matrix G denote the fraction of the j th flow transmission rate at the input of the v th queue. Then the elements of G are h(o(j, v), o(j, k)) (38) where h(o(j, v), o(j, denote the steady state probability that the dropped packet belongs to the v th queue. Applying A4, Let P j|v denote the probability that, conditioned on a packet loss from the v th queue, that packet loss belongs to the j th flow, then, using A4, Let P j denote the probability that a packet loss belongs to the j th flow. Then, which by substituting from (40) and (41) yields, 8 It is possible to extend the analysis by assuming that each queue has a different set of priority coefficients by defining #v (c) for each queue. This may be specifically valuable when modeling the interaction between DiffServ enabled and non-DiffServ queues. However, for initial model simplicity, we postpone this to future extensions. where the above is simply the ratio of the sum of the packet drop rates of flow j at all the queues to the sum of the packet drop rates of all flows in the network. Now, we can consider one of the TCP flows in the network and consider its window evolution. Let Y i denote the time at which the i th random packet loss (caused by any of the V queues and causing a packet loss from one of the M event takes place, and X the i-th inter-loss duration (epoch). Let the j th TCP session window size (in packets) at the beginning and end of the i th epoch be denoted by W i,j and W i+1,j respectively. Let q i,v denote the queue size of queue v at the end of epoch i; then the net round-trip propagation and queuing delay for connection j at the end of the i th epoch is # Following the same steps as before, the window evolution of the TCP sessions is governed by the system of equations (4), where E[# This system of equations captures only the TCP additive increase/multiplicative decrease dynamics but not the impact of timeout - this is done in the next section. From Sec. III-A, and where K j is given by (12). Substituting by X from (44), P j from (43) and r j from (45) in (4), Finally, for congested links (A1), M+N while for uncongested (underutilized) links, M+N The set of M equations (46)-(48) in the M are then solved numerically as explained in Section IV. B. Modeling TCP timeout mechanism In this section, we show how the model can be extended to incorporate the effect of timeouts. Since it is well known that the slow start behavior has a negligible effect on the TCP steady-state throughput, we neglect slow start and assume that the window size after a timeout is set to 1 and that TCP continues in the congestion avoidance phase after some idle duration (discussed below). For simplicity, only single timeouts are considered since the difference in performance between a single timeout and multiple timeouts is insignificant (both result in very time window Fig. 12. Approximate timeout behavior. After timeout, slow-start is neglected, and TCP is assumed to resume in congestion avoidance mode with an initial window size value of one packet. low throughput). Thus, our objective is to model the approximate timeout behavior depicted in Fig.12 , where a single timeout causes the window size to be reduced to one, and TCP refrains from transmission (i.e. is idle) for a period of one round-trip time (# i,j ). Let Z(W i,j ) denote the probability that, given a packet loss belonging to flow j, that loss will result in a timeout. Then, following the same steps as before (for w.p. w.p. where Taking the expectations of both sides and simplifying, denote the steady-state proportion of time that TCP flow j is busy (b the non-responsive flows). Following the same derivation steps as before, and An expression for derived (and validated) in Sec. IV (equation 29). An expression for b j is derived as follows. Let X i,j denote the inter-loss times between packet losses of TCP flow j (as opposed to X i which denotes the inter-loss times in the network) and consider the window size evolution of flow j in between two consecutive Duplicate ACK (DA) events (see Fig. 12) Then, and hence Let NDA (W j ) denote the average (from a fixed point approximation argument) expected number of consecutive DA's. Then NDA distribution with parameter and finally, Thus, a set of M equations in the M+V unknowns is obtained by substituting by b j from (57) and P j from (52) in (51). The remaining V equations correspond to (47)-(48) with modification to include the timeout behavior yielding, M+N for fully utilized links and M+N for underutilized links. A solution for the M +V unknowns is obtained by solving the M +V equations numerically as explained in the following section. VI. NETWORK MODEL RESULTS In order to validate the analysis presented in the previous sections, we present a representative number of comparisons between the numerical results obtained from the solution of the M +V non-linear equations and the simulation results from the ns-2 simulations. The simulations reported here use the "gentle " option of RED in the ns-2 simulator set to "true" (i.e. it uses the gentle variant) and the "setbit " option set to false (i.e. packet drops instead of marking). Other experiments without the "gentle" option provide results with similar accuracy (not reported here due to space constraints). We first describe the technique used for solving the non-linear equations. We then summarize the modifications to the RED algorithm in ns-2 to simulate the DiffRED behavior. And finally, we present the validation experiments. A. The network solver In the analysis section, the V +M unknowns (W j , are related by a set of non-linear equations. In general, a set of non-linear equations can be solved using a suitable numerical technique. Two main problems encountered in solving non-linear equations are; (i) the uniqueness of the solution and (ii) the choice of an appropriate initial condition that guarantees the convergence towards a true solution. In this section, we describe the algorithm used to achieve both objectives. Our algorithm is based on the observation that there exists a unique solution for (46)-(48) that satisfies 0 # q v # maxth v and W j > 0. Thus, the network solver is composed of the following two steps (or modules): Step I The set of equations (46)-(48) is solved using any iterative numerical technique. We used a modified globally convergent Newton-Raphson technique [33] that operates as follows: (1) It chooses initial conditions randomly within the solution space; (2) Performs Newton-Raphson technique while checking for convergence; (3) If the algorithm converges to a valid solution, the program terminates; else, the program repeats from step (1) again. Note that the resulting solution does not take into account timeouts or non-responsive flows - hence, step II. Step II This step of our algorithm uses the solution provided in step I as initial conditions - it also uses a globally convergent Newton-Raphson technique applied this time to the extended modeling equations describing the timeout behavior in (51)-(52), (58)-(59) (and, if applicable, the UDP behavior). Thus, the network solver is composed of two modules, each of which uses an iterative numerical method; the first module solves the simplified model and thus provides an approximate solution for the network which is then fed into the next module that refines the network solution providing a more accurate one. This technique has proved to converge to the correct solution in all experiments we tried, a representative of which (a total of 160 experiments) are reported in the following sections. B. Modifying ns-2 to model DiffRED The following modifications to ns-2 were incorporated. In the packet common header, we added a priority field that contains an integer specifying the priority class of the packet. Sim- ilarly, an additional parameter for the TCP object specifying its class of service was added; when a TCP session is established, this parameter is set to the selected class of service for that session. The TCP mechanism is also modified to label each packet by the TCP session priority (by copying the TCP session class of service to the packet header of the packet being transmit- ted). Finally, the RED algorithm is modified so as to first check for the packet class (from the packet header) and compute the packet drop probability according to the mechanism outlined in Sections II.B (for the case without the "gentle" option) or II.C (for the "gentle" variant). C. Experimental Topology We use the topology of Figure 5. It consists of two RED/DiffRED queues Q 1 and Q 2 . There are a total of five sets of flows going through the queues and each flow set is composed of four identical flows. In the figure, S denotes the source (origin) while D denotes the destination for those flows. Three sets of flows (S1, S2 and S3) arrive at Q 1 , out of which only one traverses through Q 2 . Two other sets of flows arrive at Q 2 . The only two bottleneck links in the topology are those of Q 1 and Q 2 , where the link speeds are - 1 and - 2 (packets/sec) and In the experiments, we vary the priority class of S2, allowing it to have a higher priority in some of the experiments to compensate it for the higher loss rate it suffers since it has to traverse both queues. Each experiment is composed of five simulation runs, at the end of each we compute the parameters of interest; average queue size for each queue, average window size for each TCP flow and the average goodput (number of packets received at the destination per second). The duration of each run is long enough to guarantee the operation of the network in the steady state for a long duration. We then compute the average, among the simulation runs, of each of these parameters and compare with the network solver results. In the figures, we plot one point (x, y) for each experiment, where x represents the simulation result and y represents the corresponding result obtained from applying our analytical network solver. Hence, ideally, all points should lie on a line with a slope of 45 degrees. II PARAMETERS SETTINGS FOR THE EXPERIMENTAL NETWORK TOPOLOGY OF FIG.5 AND FIG.16. LINK size obtained from simulations (packets) Average queue size obtained from analysis (packets) Fig. 13. Average queue size obtained from simulations and analysis for the experiments with TCP traffic only. D. Experiments with TCP only A total of eighty experiments have been performed with various parameters. The parameters of five of these experiments are shown in Table I. Three additional similar sets are performed but by reducing the bottleneck link speeds by half for each set (i.e. the first experiment in the last set would configure the bottleneck links at 4Mbps) thus forming twenty experiments. These twenty experiments are repeated four times, with the priority coefficient of the higher priority flow S2 varying from 0.1 to 1.0 (in steps of 0.3). The results are shown in Figures 13-15. Note that this also validates the analysis for the RED case, since setting the priority coefficient to 1.0 corresponds to RED operation. In Figure 13, each experiment results in two points, one for each queue. Hence, the figure contains a total of 160 simulation/analysis comparison points. On the other hand, in Figure 14 and Figure 15, each experiment results in three comparison points. The reason is that TCP flows and S3 (on one hand) and S4 and S5 have identical goodput and congestion window size values. Thus, each experiment renders three window sizes (and three goodput values), one for and S3, one for S2 and a third for S4 and S5 for a total of 240 comparison points for each of these figures. E. Experiments with TCP and UDP To validate the model results for mixed TCP and UDP traffic, we modify the topology of Figure 5 to Figure 16 by adding two sets of UDP (constant bit rate) flows, one that originates at Average window size for TCP flows obtained from simulations (packets) Average window size for flows obtained from analysis (packets) Fig. 14. average congestion window size from simulations and analysis for the experiments with TCP traffic only. Average goodput obtained from simulations (kbps) Average goodput obtained from analysis Fig. 15. The average goodput obtained from simulations and analysis for the experiments with TCP traffic only. S6 and passes through both queues before exiting the network while the other originates at S7 and passes only through Q2. The transmission rates of the UDP flows are set such that the total transmission rates of S1 and S2 equals 10% of the link capacities - 1 and - 2 . The results are shown in Figures 17-19. Just like Figures 13-14, Figures 17-18 contains a total of 160 and 240 simulation/analysis comparison points (respectively). However, unlike Figure 15 which contains 240 comparison points Figure 19 contains an additional 160 comparison points accounting for the 2 UDP flows in each of the 80 experiments. Finally, in Figure 20, we show the results of varying the priority coefficient for one of the experiments (the first experiment in Table I) separately. The figure shows how the priority coefficient is effective in boosting S2 goodput from its low value when to a much higher value (almost equal to the other TCP flows rate) when S5 Fig. 16. Experimental network topology with two RED queues. size obtained from simulations (packets) Average queue size obtained from analysis (packets) Fig. 17. Average queue size obtained from simulations and analysis for mixed TCP and UDP traffic. 3051525Average window size for TCP flows obtained from simulations (packets) Average window size for flows obtained from analysis (packets) Fig. 18. Average congestion window size obtained from simulations and analysis for mixed TCP and UDP traffic. Average goodput obtained from simulations (kbps) Average goodput obtained from analysis Fig. 19. Average goodput obtained from simulations and analysis for mixed TCP and UDP traffic. Coefficient for the higher priority flows (Set 2) Goodput flows S1 and S3 flows S4 and S5 UDP flows S6 UDP flows S7 Fig. 20. Effect of varying the priority coefficient of S2 on its goodput. The experiment parameters are shown in the first entry of Table I. VII. CONCLUSION In this paper, an analytical model for a network of RED/DiffRED queues with multiple competing TCP and UDP flows was presented. Unlike previous work, our analysis is specifically targeted towards AQM schemes which are characterized by random packet drop (unlike Tail Drop queues which drop packets in bursts). Our main contributions are (i) An accurate timeout formula that is orders of magnitude more accurate than the best-known analytical formula, (ii) Closed form expressions for the relative fairness of RED and Tail-Drop towards heterogeneous flows (iii) Analysis of RED queues in a traditional as well as differentiated services net- work. Our analysis has relied on a set of approximations to the timeout dynamics as well as to the loss rate process of each flow in the network. The analytical results were validated against ns simulations. These show that the results are accurate within a mean margin of error of 2% for the average TCP throughput, 5% for the average queue size and 4% for the average window size attributable to the approximations introduced in Section II. The model proposed should be applicable to a variety of AQM schemes that rely on randomized (instead of deterministic) packet drops. A number of avenues for future research remain. First, the model can be extended to the analysis of short-lived flows and the effects of limiting advertised congestion window. Also model extensions to other versions of TCP (e.g. Tahoe, SACK,.etc.) and to the case of ECN [34] may be considered. Finally, as noted in (8), especially with the expected coexistence of DiffServ networks with best effort networks, an extension of this model that captures the interaction between traditional RED queues and DiffRED would be valuable. In Sec. III-C, we have stated that the average window size of a TCP flow passing through a Tail Drop queue is inversely proportional to its round-trip delay. Proof: We show below that, for a Tail-Drop queue, the steady state window size for any flow is inverse proportional to the round-trip time. In Tail Drop queues, buffer overflow causes (typically) at least one packet loss from each flow passing through the queue ([35], [36]). Using the same notations as Sections II and III, Taking expectation of both sides of (60), and denoting proving the claim. --R Resource ReSerVation Protocol (RSVP)-Version 1 functional specification RSVP: A new resource ReSerVation protocol. Specification of controlled-load network element service Specification of guaranteed quality of service. The Use of RSVP with IETF Integrated Services. An architecture for differentiated services. An approach to service allocation in the Internet. Definition of the differentiated services field (DS filed) in the IPv4 and IPv6 headers. Recommendations on queue management and congestion avoidance in the Internet. Random early detection gateways for congestion avoidance. Explicit allocation of best-effort packet delivery service Technical Specification from Cisco. Modeling TCP Reno performance: A simple model and its empirical validation. Comparative performance analysis of versions of TCP in a local network with a lossy link. The stationary behavior of ideal TCP congestion avoidance. The macroscopic behavior of the TCP congestion avoidance algorithm. Promoting the use of end-to-end congestion control in the internet Fixed point approximations for TCP behavior in an AQM network. A framework for practical performance evaluation and traffic engineering in ip networks. Recommendation on using the gentle variant of RED. TCP/IP Illustrated The NewReno Modification to TCP's Fast Recovery Algorithm. Blocking probabilities in large circuit-switched networks The performance of TCP/IP for networks with high bandwidth-delay products and random loss Analytic understanding of RED gateways with multiple competing TCP flows. Stochastic Models of Congestion Control in Heterogeneous Next Generation Packet Networks. Flow Control in High-Speed Networks Analysis of the increase and decrease algorithms for congestion avoidance in computer networks. Numerical Recipes in C: The Art of Scientific Computing. A proposal to add explicit congestion notification (ECN) to IP. Oscillating behavior of network traffic: a case study simulation. Congestion avoidance and control. --TR Congestion avoidance and control Analysis of the increase and decrease algorithms for congestion avoidance in computer networks Numerical recipes in C (2nd ed.) TCP/IP illustrated (vol. 1) Random early detection gateways for congestion avoidance The performance of TCP/IP for networks with high bandwidth-delay products and random loss The macroscopic behavior of the TCP congestion avoidance algorithm Explicit allocation of best-effort packet delivery service Comparative performance analysis of versions of TCP in a local network with a lossy link Promoting the use of end-to-end congestion control in the Internet Modeling TCP Reno performance Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED Stochastic models of congestion control in heterogeneous next generation packet networks

differentiated services;performance analysis;network modeling;random early detection;UDP;TCP

637151

Massively parallel fault tolerant computations on syntactical patterns.

The general capabilities of reliable computations in linear cellular arrays are investigated in terms of syntactical pattern recognition. We consider defects of the processing elements themselves and defects of their communication links. In particular, a processing element (cell) is assumed to behave as follows. Dependent on the result of a self-diagnosis it stores its working state locally such that it becomes visible to the neighbors. A defective cell cannot modify information but is able to transmit it unchanged with unit speed. Cells with link failures are not able to receive information via at most one of their both links to adjacent cells. Moreover, static and dynamic defects are distinguished.It is shown that fault tolerant real-time recognition capabilities of two-way arrays with static defects are characterizable by intact one-way arrays and that one-way arrays are fault tolerant per se. For arrays with dynamic defects it is proved that all failures can be compensated as long as the number of adjacent defective cells is bounded.In case of arrays with link failures it is shown that the sets of patterns that are reliably recognizable are strictly in between the sets of (intact) one-way and (intact) two-way arrays.

Introduction Nowadays it becomes possible to build massively parallel computing systems that consist of hundred thousands of processing elements. Each single component is subject to failure such that the probability of misoperations and loss of function of the whole system increases with the number of its elements. It was von Neumann [19] who rst stated the problem of building reliable systems out of unreliable components. Biological systems may serve as good examples. Due to the necessity to function normally even in case of certain failures of their components nature developed mechanisms which invalids the errors, with other words, they are working in some sense fault tolerant. Error detecting and correcting components should not be global to the whole system because they themselves are subject to failure. Therefore, the fault tolerance has to be a design feature of the single elements. A model for massively parallel, homogenously structured computers are the cellular arrays. Such devices of interconnected parallel acting nite state machines have been studied from various points of view. In [4, 5] reliable arrays are constructed under the assumption that a cell (and not its links) at each time step fails with a constant probability. Moreover, such a failure does not incapacitate the cell permanently, but only violates its rule of operation in the step when it occurs. Under the same constraint that cells themselves (and not their links) fail (i.e. they cannot process information but are still able to transmit it unchanged with unit speed) fault tolerant computations have been investigated, e.g. in [6, 14] where encodings are established that allow the correction of so-called K-separated misoperations, in [9, 10, 17, 20] where the famous ring squad synchronization problem is considered in defective cellular arrays, and in terms of interacting automata with nonuniform delay in [7, 11] where the synchronization of the networks is the main object either. Here we are interested in more general computations. In terms of pattern recognition the general capabilities of reliable computations are considered. Since cellular arrays have intensively been investigated from a language theoretic point of view, pattern recognition (or language acceptance) establishes the connection to the known results and, thus, inheres the possibility to compare the fault tolerant capabilities to the non fault tolerant ones. In the sequel we distinguish three dierent types of defects. Static defects are the main object of Section 3. It is assumed that each cell has a self-diagnosis circuit which is run once before the actual computation. The results are stored locally in the cells and subsequently no new defects may occur. Otherwise the whole computation would become invalid. A defective cell cannot modify information but is able to transmit it with unit speed. Otherwise the parallel computation would be broken into two non interacting parts and, therefore, would become impossible at all. In Section 4 the defects are generalized. In cellular arrays with dynamic defects it may happen that a cell becomes defective at any time. The formalization of the corresponding arrays includes also the possibility to repair a cell dynamically The remaining sections concern another natural type of defects. Not the cells themselves cause the misoperation but their communication links. It is assumed that a defective cell is not able to receive information via at most one of its both links to adjacent cells. The corresponding model is introduced in Section 5 in more detail. In Section 6 it is shown that the real-time arrays with link failures are able to reliably recognize a wider range of sets of patterns than intact one-way arrays. In order to prove this result some auxiliary algorithmic subroutines are given. Section 7 concludes the investigations by showing that the devices with link failures are strictly weaker than two-way arrays. Hence, link failures cannot be compensated in general but, on the other hand, do not decrease the computing power to that one of one-way arrays. In the following section we dene the basic notions and recall the underlying intact cellular arrays and their mode of pattern recognition. Preliminaries We denote the integers by Z, the positive integers f1; 2; g by N and the set N [ f0g by N 0 . X 1 X d denotes the Cartesian product of the sets we use the notion X d alternatively. We use for inclusions and if the inclusion is strict. Let M be some set and be a function, then we denote the i-fold composition of f by f [i] , A two-way resp. one-way cellular array is a linear array of identical nite state machines, sometimes called cells, which are connected to their both nearest neighbors resp. to their nearest neighbor to the right. The array is bounded by cells in a distinguished so-called boundary state. For convenience we identify the cells by positive integers. The state transition depends on the current state of each cell and the current state(s) of its neighbor(s). The transition function is applied to all cells synchronously at discrete time steps. Formally: Denition 1 A two-way cellular array (CA) is a system hS; -; #; Ai, where 1. S is the nite, nonempty set of cell states, 2. S is the boundary state, 3. A S is the set of input symbols, is the local transition function. If the ow of information is restricted to one-way (i.e. from right to left) the resulting device is a one-way cellular array (OCA) and the local transition function maps from (S [ f#g) 2 to S. A conguration of a cellular array at some time t 0 is a description of its global state, which is actually a mapping c t The data on which the cellular arrays operate are patterns built from input symbols. Since here we are studying one-dimensional arrays only the input data are nite strings (or words). The set of strings of length n built from symbols from a set A is denoted by A n , the set of all such nite strings by A . We denote the empty string by " and the reversal of a string w by w R . For its length we write jwj. The set A + is dened to be A n f"g. In the sequel we are interested in the subsets of strings that are recognizable by cellular arrays. In order to establish the connection to formal language theory we call such a subset a formal language. Moreover, sets L and L 0 are considered to be equal if they dier at most by the empty word, i.e. L n Now we are prepared to describe the computations of (O)CAs. The operation starts in the so-called initial conguration c 0;w at time 0 where each symbol of the input string is fed to one cell: c 0;w During a computation the (O)CA steps through a sequence of congurations whereby successor congurations are computed according to the global transition function be a conguration, then its successor conguration is as follows: c t+1 c t+1 for CAs and c t+1 for OCAs. Thus, is induced by -. An input string w is recognized by an (O)CA if at some time i during its course of computation the leftmost cell enters a nal state from the set of nal states F S. Denition be an (O)CA and F S be a set of nal states. 1. An input w 2 A + is recognized by M at time step the conguration c 2. recognizes w at some time stepg is the set of strings (language) recognized by M. 3. mapping. If M can recognize all w 2 L(M) within at most t(jwj) time steps, then M is said to be of time complexity t. The family of all sets which are recognizable by some CA (OCA) with time complexity t is denoted by L t (CA) (L t (OCA)). If t equals the identity function recognition is said to be in real-time, and if t is equal to k id for an arbitrary rational number k 1, then recognition is carried out in linear-time. Correspondingly, we write L rt ((O)CA) and L lt ((O)CA). In the sequel we will use corresponding notations for other types of recognizers. Now we are going to explore some general recognition capabilities of CAs that contain some defective cells. The defects are in some sense static [17]: It is assumed that each cell has a self-diagnosis circuit which is run once before the actual computation. The result of that diagnosis is stored in a special register of each cell such that intact cells can detect defective neighbors. Moreover (and this is the static part), it is assumed that during the actual computation no new defects may occur. Otherwise the whole computation would become invalid. What is the eect of a defective cell? It is reasonable to require that a defective cell cannot modify information. On the other hand, it must be able to transmit information in order to avoid the parallel computation being broken into two not interacting lines and, thus, being impossible at all. The speed of information transmission is one cell per time step. Another point of view on such devices is to dene a transmission delay between every two adjacent cells and to allow nonuniform delays [7, 11]. Now the number of defective cells between two intact ones determine the corresponding delay. Since the self-diagnosis is run before the actual computation we may assume that defective cells do not fetch an input symbol. Nevertheless, real-time is the minimal possible time needed for non-trivial computations and, consequently, is dened to be the number of all cells in the array. In order to obtain a computation result here we require the leftmost cell not to be defective. Later on we can omit this assumption. Formally we denote CAs with static defects by SD-CA and the corresponding language families by L t (SD-CA). Considering the general real-time recognition capabilities of SD-CAs the best case is trivial. It occurs when all the cells are intact: The capabilities are those of CAs. On the other hand, fault tolerant computations are concerned with the worst case (with respect to our assumptions on the model). The next two results show that in such cases the capabilities can be characterized by intact OCAs from what follows that the bidirectionality of the information ow gets lost. Theorem 3 If a set is fault tolerant real-time recognizable by a SD-CA, then it is real-time recognizable by an OCA. Proof. Let D be a SD-CA and let k 2 N be an arbitrary positive integer. Set the number of cells of D to 1. For the mapping f ng. Now assume the cells at the positions f(i), 1 i k, are intact ones and all the other cells are defective (cf. Figure 1). In between the cells f(i) and f(i there are f(i) f(i defective ones. During a real-time computation the states of a cell f(i) at time t 2 i cannot in uence the overall computation result. The states would reach the leftmost a a 1 a a 2 a 1 a a 3 a 2 a 1 a 0 a 4 a 3 a 2 a 1 a a 5 a 4 a 3 a 2 a 1 a 0 b 5 b 4 b 3 a 6 a 5 a 4 a 3 a 2 a 1 a 0 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 b 7 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 9 a 8 a 7 a 6 a 5 a 4 a 3 a a 9 a 8 a 7 a 6 a 5 a a 9 a 8 a 7 a 12 a 11 a 10 a 9 a 13 a 12 a 11 a 14 a 13 a 15 Figure 1: One-way information ow in SD-CAs. cell after another f(i) steps. This gives the arrival time which is greater than real-time. Conversely, the cell f(i) computes all its states up to time t independently of the states of its intact neighbors to the left: The nearest intact neighbor to the left is cell f(i+1) and there are 2 i 1 defective cells in between Up to now we have shown that the information ow in D is one-way. But compared to OCAs the cells in D are performing more state changes. It remains to show that this does not lead to stronger capabilities. Let i be some intact cell of D. As long as it operates independently on its intact neighbors it runs through state cycles provided that the adjacent defective regions are long enough. Let s be such a cycle. Now one can always enlarge the lengths of the defective regions such that they correspond to j Therefore, during their isolated computations the cells run through complete cycles. Obviously, such a behavior can be simulated by the cells of an OCA since the cycle lengths are bounded by the number of states of D. 2 In order to obtain the characterization of real-time SD-CAs by real-time OCAs we need the converse of Theorem 3. Theorem 4 If a set is real-time recognizable by an OCA, then it is fault tolerant real-time recognizable by a SD-CA. Proof. The idea of the simulation is depicted in Figure 2. Each cell of a SD-CA that simulates a given OCA waits for the rst information from its right intact neighbor. The waiting period is signaled to its left intact neighbor by signals labeled . This information leads to a waiting period of the left intact neighbor. Each intact cell performs a simulation step when it receives a non-waiting signal. It follows that a cell sends exactly as many waiting signals to the left as are defective cells located to its right. Therefore, the leftmost cell needs exactly one simulation step for each intact cell and one waiting step for each defective cell and, thus, computes the result in real-time. 2 a 0 a 0 a 0 a 0 a 0 a 0 a 0 a 0 a 1 a 1 a 1 a 2 a 3 a 4 Figure 2: OCA simulation by SD-CAs. The following corollary formalizes the characterization: Corollary 5 L rt From the previous results follows the interesting fact that OCAs are per se fault tolerant. Additional defective cells do not decrease the recognition capabilities. Corollary 6 L rt It is often useful to have examples for string sets not recognizable by a certain device. Example 7 Neither the set of duplicated strings fww nor the set of strings whose lengths are a power of 2 fw are fault tolerant real-time recognizable by SD-CAs. (It has been shown in [13] resp. [15] that they do not belong to the family The previous results imply another natural question. Is it possible to regain the recognition power of two-way CAs in fault tolerant SD-CA computations by increasing the computation time? How much additional time would be necessary? In [3, 18] it has been shown that an OCA can simulate a real-time computation in twice real-time if the input word is reversed. It is an open problem whether or not L rt (CA) is closed under reversal. But nevertheless, a piece of good news is that only one additional time step for each intact cell is necessary in order to regain the computation power in a fault tolerant manner. Theorem 8 A set is recognizable by an OCA in twice real-time (and, thus, its reversal by an CA in real-time) if and only if it is fault tolerant recognizable by a SD-CA in real-time+m, where m denotes the number of intact cells. Proof. Let L be a set that is recognizable by an OCA in twice real-time and denote the length of an input word by n. Let us consider the situation in the proof of Theorem 4 at real-time: The corresponding SD-CA has simulated n steps of the corresponding OCA. But due to the previous delay, from now on no further delay of the intact cells is necessary. Thus, the next n steps of the OCA can be simulated in n steps by the SD-CA either. Since the number of intact cells is exactly n the only if part follows. Now let L be a set that is recognizable by a SD-CA in time real-time+m. We follow the same idea as in the proof of Theorem 3. The distance between two intact cells has to be enlarged. If we number the intact cells from right to left by then the distance between cells i and has to be enlarged at least by 2 i 1 (m 1) in order to prevent additional ow of information. 2 One assumption on our model has been an intact leftmost cell. Due to Corollary 5 we can omit this requirement. Now the overall computation result is indicated by the leftmost intact cell of the one-way array which operates per se independently on its defective left neighbors. In the following cellular arrays with dynamic defects (DD-CA) are introduced. Dynamic defects can be seen as generalization of static defects. Now it becomes possible that cells fail at any time during the computation. Afterwards they behave as in the case of static defects. In order to dene DD-CAs more formally it is helpful to suppose that the state of a defective cell is a pair of states of an intact one. One component represents the information that is transmitted to the left and the other one the information that is transmitted to the right. By this formalization we obtain the type indication of the cells (defective or not) for free: Defective cells are always in states from S 2 and intact ones in states from S. A possible failure implies a weak kind of nondeterminism for the local transition function. Denition 9 A two-way cellular array with dynamic defects (DD-CA) is a system hS; -; #; Ai, where 1. S is the nite, nonempty set of cell states which satises S \ 2. S is the boundary state, 3. A S is the set of input symbols, Sg is the local transition function which satises -(s If a cell works ne the local transition function maps to a state from S. Otherwise it maps to a pair from S 2 indicating that the cell is now defective. The denition includes the possibility to repair a cell during the computation. In this case - would map from a pair to a state from S. Note that the nondeterminism in a real computation is a determinism since the failure or repair of a cell is in some sense under the control of the outside world. We assume that initially all cells are intact and as in the static case that the leftmost cell remains intact. In the sequel we call an adjacent subarray of defective cells a defective region. The next results show that dynamic defects can be compensated as long as the lengths of defective regions are bounded. Theorem 10 If a set is real-time recognizable by a CA, then it its real-time recognizable by a DD-CA if the lengths of its defective regions are bounded by some k 2 N 0 . Proof. Assume for a moment that the lengths of the defective regions are exactly k. A DD-CA D that simulates a given CA hS; -; #; Ai has the state set The general idea of the proof is depicted in Figure 3. As long as a cell does not detect a defective neighbor it stores the states of its neighbors and its own state in some of its additional registers as shown in the gure. time t the state of cell i might be as follows: center Assume now that the right neighbor of cell i becomes defective. Due to our assumption we know that there must exist a defective region of length k at the right of cell i. During the next k time steps cell i stores the received states and computes missing states from its register contents as shown in Figure 3. Subsequently its state might be as follows | {z } center a a 0 a 1 a 0 a 0 a a t a c a d e c d e a t a t+1 a t d c a t c t+1 e c d e e t+1 d t+1 d t+2 c t+2 c t+3 a t+3 e t+2 d t+3 c t+4 Figure 3: Compensation of From now on cell i receives the states that the intact cell in at time t; t is able to compute the necessary intermediate states from its register contents. A crucial point is that the lengths of defective regions are xed to k. Due to that assumption a cell i knows when it receives the valid states from its next intact neighbor i+k+1 or i k 1. We can relax the assumption as required to lengths of at most k cells by the following extension of the simulation. Each cell is equipped with a modulo k counter. Since the current value of the counter is part of the cell state it is also part of the transmitted information. A cell that stores received information in its additional registers stores also the received counter value. Now it can decide whether it receives the valid state from its next intact neighbor by comparing the received counter value to the latest stored counter value. If they are equal then the received information is from a defective cell, otherwise it is valid and the cell uses no more additional registers. New failures in subsequent time steps can be detected by the same method. If the received counter value is equal to the latest stored counter value then additional cells have become defective. In such cases the cell uses correspondingly more additional registers in order to compensate the new defects. It remains to explain what happens if two defective regions are joint by failure of a single connecting intact cell. Up to now we have used the transmitted contents of the main registers only. But actually the whole state, i.e. all register contents, are transmitted. In the case in question the next intact cells to the left and right of the joint defective region can ll additional registers as desired. 2 Corollary 11 If a set is real-time recognizable by an OCA, then it is real-time recognizable by a DD-OCA if the lengths of its defective regions are bounded by some k 2 N 0 . In order to provide evidence for general fault tolerant DD-CA computations we have to relax the assumption of bounded defective region lengths. We are again concerned with the worst case. The hardest scenario is as follows. Initially all cells are intact and thus fetching an input symbol. During the rst time step all but the leftmost cell fail. (Needless to say, if the leftmost cell becomes also defective then nobody would expect a reasonable computation result.) It is easy to see that in such cases the recognition capabilities of DD-CAs are those of a single cell, a nite state machine (see Figure 4). Lemma 12 If a set is fault tolerant recognizable by a DD-CA, then it is recognizable by a nite state machine and thus regular. Corollary 13 If a set is fault tolerant recognizable by a DD-OCA, then it is recognizable by a nite state machine and thus regular. 5 Devices with Link Failures In Section 3 it has been shown for CAs with defective cells that in case of large adjacent defective regions the bidirectional information ow gets lost. This means that the fault tolerant computation capabilities of two-way arrays are those of one-way arrays. The observation gives rise to investigate cellular arrays with defective links for their own. In order to explore the corresponding general a a 1 a a a a 5 g 0 a 6 a 7 # a 8 Figure 4: Worst case DD-CA computation. reliable recognition capabilities we have to take a closer look on the device in question. Again we assume that each cell has a self-diagnosis circuit for its links which is run once before the actual computation, and again the result of that diagnosis is indicated by the states of the cells such that intact cells can detect defective neighbors. What is the eect of a defective link? Suppose that each two adjacent cells are interconnected by two unidirectional links. On one link information is transmitted from right to left and on the other one from left to right. both links are failing, then the parallel computation would be broken into two not interacting lines and, thus, would be impossible at all. Therefore, it is reasonable to require that at least one of the links between two cells does not fail. Suppose for a moment that there exists a cell that, due to a link failure, cannot receive information from its right neighbor. This would imply that the overall computation result (indicated by the leftmost cell) is obtained with no regard to the input data to the right of that defective cell. So all reliable computations would be trivial. In order to avoid this problem we extend the hardware such that if a cell detects a right to left link failure it is able to reverse the direction of the other (intact) link. Thereby we are always concerned with defective links that cannot transmit information from left to right. Another point of view on such devices is that some of the cells of a two-way array behave like cells of a one-way array. Sometimes in the sequel we will call them OCA-cells. The result of the self-diagnosis is indicated by the states of the cells. Therefore we have a partitioned state set. Denition 14 A cellular array with defective links (mO-CA) is a system is the partitioned, nite, nonempty set of cell states satisfying 2. S is the boundary state, 3. A S i is the set of input symbols, 4. m 2 N 0 is an upper bound for the number of link failures, 5. is the local transition function for intact cells, is the local transition function for defective cells. A reliable recognition process has to compute the correct result for all distributions of the at most m defective links. In advance it is, of course, not known which of the links will fail. Therefore, for mO-CAs we have a set of admissible start congurations as follows. For an input string the conguration c 0;w is an admissible start conguration of an mO-CA if there exists a set D ng of defective cells, jDj m, such that c 0;w ng n D and c 0;w For a clear understanding we dene the global transition function of mO-CAs as follows: Let c t , t 0, be a conguration of an mO-CA with defective cells D, then its successor conguration is as follows: c t+1 c t+1 c t+1 c t+1 c t+1 c t+1 Due to our denition of - i and - d once the computation has started the set D remains xed, what meets the requirements of our model. In the following we are going to answer the following questions. Do some link failures reduce the recognition power of intact CAs or is it possible to compensate the defects by modications of the transition function as shown for CAs with defective cells? Can mO-CAs recognize a wider range of string sets than intact OCAs? 6 mO-CAs are Better than OCAs The inclusions L rt (OCA) L rt (mO-CA) L rt (CA) are following immediately from the denitions. Our aim is to prove that both inclusions are strict. 6.1 Subroutines In order to prove that real-time mO-CAs are more powerful than real-time OCAs we need some results concerning CAs and OCAs which will later on serve as subroutines of the general construction. 6.1.1 Time Constructors A strictly increasing mapping f : N ! N is said to be time constructible if there exists a CA such that for an arbitrary initial conguration the leftmost cell enters a nal state at and only at time steps f(j), 1 j n. A corresponding CA is called a time constructor for f . It is therefore able to distinguish the time steps f(j). The following lemma has been shown in [1]. Lemma 15 The mapping Proof. The idea of the proof is depicted in Figure 5. At initial time the leftmost cell of a CA sends a signal with speed 1=3 to the right. At the next time step a second signal is established that runs with speed 1 and bounces between the slow signal and the left border cell. The leftmost cell enters a nal state at every time step it receives the fast signal. The correctness of the construction is easily seen by induction. 2 Figure 5: A time constructor for 2 n . A general investigation of time constructible functions can be found, e.g. in [12, 2]. Especially, there exists a time constructor for 2 n that works with an area of at most n cells. Actually, we will need a time constructor for the mapping 2 2 n . Fortunately, in [12] the closure of these functions under composition has been shown. Corollary 16 The mapping There exists a corresponding time constructor that works with an area of at most 2 n cells. 6.1.2 Binary OCA-Counters Here we need to set up some adjacent cells of an OCA as a binary counter. Actually, we are not interested in the value of the counter but in the time step at which it over ows. Due to the information ow the rightmost cell of the counter has to contain the least signicant bit. Assume that this cell can identify itself. In order to realize such a simple counter every cell has three registers (cf. Figure 6). The third ones are working modulo 2. The second ones are signaling a carry-over to the left neighbor and the rst ones are indicating whether the corresponding cell has generated no carry-over (0), one carry-over (1) or more than one carry-over (2) before. Now the whole counter can be tested by a leftmoving signal. If on its travel through the counter all the rst registers are containing 0 and additionally both carry-over registers of the leftmost cell are containing 1, then it recognizes the desired time step. Observe that we need the second carry-over register in order to check that the counter produces an over ow for the rst time. Figure binary OCA-counter. Figure 7: A binary CA-shift-right- counter. 6.1.3 Binary CA-Shift-Right Counters For this type of counter we need two-way information ow. It is set up in a single (the leftmost) cell of a CA. Since we require the least signicant bit to be again the rightmost bit in the counter we have to extend the counter every time it produces an over ow. The principle is depicted in Figure 7. Each cell has two registers. One for the corresponding digit and the other one for the indication of a carry-over. Due to the two-way information ow the leftmost cell can identify itself. Every time it generates a carry-over the counter has to be extended. For this purpose the leftmost cell simulates an additional cell to its left appropriately. This fact signals its right neighbor the need to extend the counter by one cell. The right neighbor reacts by simulating in addition the old process of the leftmost cell which now computes the new most signicant bit. After the arrival of this extension signal at the rightmost cell of the counter the extension is physically performed by the rst cell at the right of the counting cells which now computes the least signicant bit. Obviously, it can be checked again by a leftmoving signal whether the counter represents a power of 2 or not. 6.2 Proof of the Strictness of the Inclusion Now we are prepared to prove the main result of this section. Let a set of strings L be dened as follows: The easy part is to show that L does not belong to L rt (OCA). Lemma Proof. In [8] it has been shown that for a mapping f with the property lim the set of strings fb n a f(n) j n 2 Ng does not belong to L rt (OCA). Applying the result to L we obtain lim and therefore It remains to show that L is real-time recognizable by some mO-CA. Theorem Proof. In the following a real-time 1O-CA M that recognizes L is constructed. On input data b n a m we are concerned with three possible positions of the unique defective cell: 1. The position is within the b-cells. 2. The position is within the leftmost 2 n a-cells. 3. The position is at the right hand side of the 2 n th a-cell. At the beginning of the computation M starts the following tasks in parallel on some tracks: The unique defective cell establishes a time constructor M 1 for if it is an a-cell. The leftmost a-cell establishes another time constructor and, additionally, a binary shift-right counter C 1 that counts the number of a's. The rightmost b-cell starts a binary OCA-counter C 2 and, nally, the rightmost a-cell sends a stop signal with speed 1 to the left. According to the three positions the following three processes are superimposed. Case 3. (cf. Figure 8) The shift-right counter is increased by 1 at every time step until the stop signal arrives. Each over ow causes an incrementation (by 1) of the counter C 2 . Let i be the time step at which the stop signal arrives at the shift-right counter C 1 and let l be the number of digits of C 1 . During the next l time steps the signal travels through the counter and tests whether its value is a power of 2, from which Subsequently, the signal tests during another n time steps whether the value of the binary counter C 2 is exactly 2 n , from which l = 2 n follows. If the tests are successful the input is accepted because the input string is of the form b n a l a a 2 l and, thus, belongs to L. OCA b b a a a a a a a a a a a a a a a a a a a a overflow Figure 8: Example for case 3, Case 2. (cf. Figure 9) In this case the space between the b's and the defective cell is too small for setting up an appropriate counter as shown for Case 3. Here a second binary counter C 3 within the b-cells is used. It is increased by 1 at every time step until it receives a signal from the defective cell and, thus, contains the number of cells between the b-cells and the defective cell. Its value x is conserved on an additional track. Moreover, at every time step at which the time constructor M 1 marks the defective cell to be nal, a signal is sent to the b-cells that causes them to reset the counter to the value x by copying the conserved value back to the counter track. After the reset the counter is increased by 1 at every time step. Each reset signal also marks an unmarked b-cell. The input is accepted if exactly at the arrival of the stop signal at the leftmost OCA b b a a a a a a a a a a a a a a a a a a a a overflow f f f f f f f overflow Figure 9: Example for case 2, cell the counter over ows for the rst time and all b-cells are marked: Let the last marking of M 1 happen at time for some r 2 N. The corresponding leftmoving signal arrives at time 2 2 r x at the b-cells and resets the counter C 3 to x. The stop signal arrives at time 2 2 r N, at the counter that has now the value x+ s. Since the counter produces an over ow it holds . Moreover, since M 1 has sent exactly r marking signals and all b-cells are marked it follows Therefore, the stop signal arrives at the rightmost b-cell at time 2 2 r and the input belongs to L. Case 1. Since the binary counter M 3 within the b-cells is an OCA-counter it works ne even if the defective cell is located within the b-cells. Case 1 is a straightforward adaption of case 2 (here M 2 is used instead of M 1 ). 2 Corollary 19 L rt (OCA) L rt (1O-CA) This result can be generalized to devices with a bounded number of defective cells as follows. Theorem 20 Let m 2 N be some constant then L 2 L rt (mO-CA). Proof. If all defective cells are located within the b-cells, or within the rst or within the last 2 2 n a-cells, then the proof follows from the three cases in the proof of Theorem 18. Otherwise we are concerned with two more cases: Case 4: The leftmost defective a-cell is not within the rst 1 a-cells. By standard compression techniques we can simulate m cells by one. Now the construction of Case 3 in the proof of Theorem solves this case. Case 5: The leftmost defective a-cell is within the rst 1 a-cells. In order to adapt Case 2 of the previous proof due to Corollary 16 we may assume the time constructor for 2 n works in at most n cells. By standard compression techniques we obtain at most 1 cells. According to our assumption not all defective cells are within the rst 2 n a- cells. So we have two defective cells with a distance of more than 1 cells. This allows us to use the compressed time constructor in that area. The remaining construction has been shown in Case 2. At the very beginning of the computation it is not know which area of the possible m areas is the related one. In order to determine it we start time constructors at each defective cell. Now we are able to choose a correct track of a successful computation and the set is recognized if one of the m time constructors recognizes the input. 2 Corollary be some constant then L rt (OCA) L rt (mO-CA). 7 CAs are Better than mO-CAs In order to complete the comparisons we have to prove that the computational power of real-time mO-CAs is strictly weaker than those of CAs. For this purpose we can adapt a method developed in [16] for proving that certain string sets do not belong to L rt (OCA). The basic idea in [16] is to dene an equivalence relation on string sets and bound the number of distinguishable equivalence classes of real-time OCA computations. be an OCA and X;Y A . Two strings w; w are dened to be (M;X;Y )-equivalent i for all x 2 X and y 2 Y the leftmost states of the congurations [jwj] are equal (cf. Figure 10). OCA Figure 10: Principle of bounding real-time equivalence classes. The observation is that the essential point of the upper bound on equivalence classes is due to the fact that the input sequences x and y are computational unrelated. Therefore, we can assume that the cell obtaining the rst symbol of w resp. of w 0 as input is defective and so adapt the results in [16] to 1O-CAs immediately: Corollary be some constant, then Corollary Finally, it follows for a constant m 2 N: --R Some relations between massively parallel arrays Fault tolerant cellular automata The synchronization of nonuniform networks of Pushdown cellular automata Minimal time synchronization in restricted defective cellular automata Synchronization of a line of Signals in one dimensional cellular automata Fault tolerant cellular spaces Language recognition and the synchronization of cellular au- tomata Language not recognizable in real time by one-way cellular automata A fault-tolerant scheme for optimum-time ring squad synchro- nization Deterministic one-way simulation of two-way real-time cellular automata and its related problems Probabilistic logics and the synthesis of reliable organisms from unreliable components --TR On real-time cellular automata and trellis automata Reliable computation with cellular automata A simple three-dimensional real-time reliable cellular array Minimal time synchronization in restricted defective cellular automata The synchronization of nonuniform networks of finite automata Language not recognizable in real time by one-way cellular automata Some relations between massively parallel arrays Pushdown cellular automata Signals in one-dimensional cellular automata Real-Time Language Recognition by One-Way and Two-Way Cellular Automata

static and dynamic defects;syntactical pattern recognition;fault-tolerance;link failures;cellular arrays

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A case study of OSPF behavior in a large enterprise network.

Open Shortest Path First (OSPF) is widely deployed in IP networks to manage intra-domain routing. OSPF is a link-state protocol, in which routers reliably flood "Link State Advertisements" (LSAs), enabling each to build a consistent, global view of the routing topology. Reliable performance hinges on routing stability, yet the behavior of large operational OSPF networks is not well understood. In this paper, we provide a case study on the eharacteristics and dynamics of LSA traffic for a large enterprise network. This network consists of several hundred routers, distributed in tens of OSPF areas, and connected by LANs and private lines. For this network, we focus on LSA traffic and analyze: (a) the class of LSAs triggered by OSPF's soft-state refresh, (b) the class of LSAs triggered by events that change the status of the network, and (c) a class of "duplicate" LSAs received due to redundancy in OSPF's reliable LSA flooding mechanism. We derive the baseline rate of refresh-triggered LSAs automatically from network configuration information. We also investigate finer time scale statistical properties of this traffic, including burstiness, periodicity, and synchronization. We discuss root causes of event-triggered and duplicate LSA traffic, as well as steps identified to reduce this traffic (e.g., localizing a failing router or changing the OSPF configuration).

INTRODUCTION Operational network performance assurances hinge on the stability and performance of the routing system. Understanding behavior of routing protocols is crucial for better operation and management of IP networks. In this pa- per, we focus on Open Shortest Path First (OSPF) [1], a widely deployed Interior Gateway Protocol (IGP) in IP Aman Shaikh is at the University of California, Santa Cruz, CA 95064. E-mail: [email protected] Chris Isett is with Siemens Medical Solutions, Malvern, PA 19355. E-mail: [email protected] Albert Greenberg, Matthew Roughan and Joel Gottlieb are with falbert,roughan,[email protected] networks today to control intradomain routing. Despite wide-spread use, behavior of OSPF in large and commercial IP networks is not well understood. In this paper, we provide a case study of the dynamic behavior of OSPF in a large enterprise IP network, using data gathered from the deployment of a novel and passive OSPF monitoring sys- tem. To our knowledge, this case study represents the first detailed report on OSPF dynamics in any large operational IP network. OSPF is a link-state protocol, where each router generates "Link State Advertisements" (LSAs) to create and maintain a local, consistent view of the topology of the entire routing domain. Tasks related to generating and processing LSA traffic form a major chunk of OSPF process- ing. In fact, OSPF LSA storms that cripple the network are not unheard of [2]. Therefore, understanding the dynamics of LSA traffic are vital to manage OSPF networks. Such an understanding can also lead to realistic workload models which can be used for a variety of purposes like realistic simulations and scalability studies. Therefore, we focus on the LSA traffic in this case study. Specifically, we introduce a general methodology and associated predictive model to investigate what the LSA traffic reveals about network topology dynamics and failure modes. The enterprise network under investigation provides highly available and reliable connectivity from customer's facilities to applications and databases residing in a data center. Salient features of the network are: ffl OSPF is used for routing in the data center. The OSPF domain consists of about 15 areas and 500 routers. This paper presents dynamics of OSPF for 8 areas (including the backbone area) covering about 250 routers over a one month period of April, 2002. ffl The OSPF domain has a hierarchical structure with application and database servers at the root and customers at the leaves. The domain uses Ethernet LANs extensively for connectivity. This is in contrast to ISP networks which rely on point to point link technologies. ffl Customers are connected over leased lines to the OSPF network in the data center. EIGRP [1] is run over the leased lines. Customer reachability information learnt via EIGRP is subsequently imported into the OSPF domain. This is in contrast to many ISP networks which propagate external reachability information using an internal instance of BGP (I-BGP [1]). We believe the salient characteristics of the enterprise net-work are common to a wide class of networks. To understand characteristic of LSA traffic of the enterprise network, we classify the traffic into three classes: Refresh-LSAs - the class of LSAs triggered by OSPF's soft-state refresh mechanism, ffl Change-LSAs - the class of LSAs triggered by events that change the status of the network, and ffl Duplicate-LSAs - the class of extra copies of LSAs received as a result of the redundancy in OSPF's reliable LSA flooding mechanism. In Section V, we provide a simple formula to predict the rate of refresh-LSA traffic, with parameters that can be determined using information available in the router configuration files. Our measurements confirm that the prediction is accurate. To understand finer grained refresh traffic characteristics, we propose and carry out simple time-series analysis. In the case study, this analysis revealed that the routers fall into two classes with different periodic refresh behavior. As it turned out, the two classes ran two versions of the router operating systems (Cisco IOS). Our measurements showed that refresh traffic is not synchronized across routers. In contrast, Basu and Riecke [3] reported evidence of synchronization from their OSPF model simulations. We believe that day to day variations in the operational context tends to break synchronization arising from initial conditions. We saw no evidence of forcing functions (however that push the network towards synchronization. Having baselined the refresh-LSA traffic, we move on to analysis of traffic triggered by topology changes in Section VI. We isolate change-LSAs and attribute them to either internal or external topology changes. Internal changes are changes to the topology of the OSPF domain, whereas external changes are changes in the reachability information imported from EIGRP. We found that the bulk of change-LSAs were due to external changes. In addition, the overwhelming majority of change-LSA traffic came from persistent yet partial failure modes. Internal change-LSAs arose from failure modes within a single router. Bulk of External change-LSAs arose from a single EIGRP session which was flapping due to congestion on the link. Interestingly, in one critical internal router failure case, an impending failure eluded the SNMP based fault and performance management system, but showed up prominently in spikes in change-LSA traffic. As a result of these LSA measurements, proactive maintenance was carried out, moving the network away from an operating point where an additional router failure would have had catas- trophic, network-wide impact. Because OSPF uses reliable flooding to disseminate LSAs, a certain level of duplicate-LSA traffic is to be ex- pected. However, in the case study we observed certain asymmetries in duplicate-LSA traffic that were initially surprising, given the complete symmetry of the physical network design (Section VII). However, a closer look revealed asymmetries in the logical OSPF control plane topology. This analysis then led to a method for reducing duplicate-LSA traffic by altering the routers' logical OSPF configurations, without changing physical structure of the network. A. Related Work For the most part, previous studies of OSPF have been model or simulation-based [3] [4], or have concentrated on measuring OSPF implementation behavior on a single router or in a small testbed [5]. The only exception is a paper by Labovitz et al. [6] in which the authors analyzed OSPF instability for a regional ISP network. However, our work is a first comprehensive analysis of OSPF LSA traffic and can lead to development of realistic network-wide modeling parameters and simulation scenarios of greatest interest. Very interesting work related to IS-IS [1] convergence in ISP networks (and the potential for much faster convergence) has appeared in talks and Internet drafts from Packet Design [7] [8]. In the realm of interdomain routing, numerous studies have been published about the behavior of BGP in the Internet; some examples of which are [6] [9] [10]. These studies have yielded many interesting and important insights. IGPs, such as OSPF, need similar at- tention, and we believe that this paper is a first step in that direction. II. OSPF FUNDAMENTALS AND LSAS OSPF is a link state routing protocol, meaning that each router within the domain discovers and builds an entire view of the network topology. This topology view is conceptually a directed graph. Each router represents a node in this topology graph, and each link between neighboring routers represents a unidirectional edge. Each link also has an associated weight that is administratively assigned in the configuration file of the router. Using the weighted topology graph, each router computes a shortest path tree with itself as the root, and applies the results to build its forwarding table. This assures that packets are forwarded along the shortest paths in terms of link weights to their destinations [11]. We will refer to the computation of the shortest path tree as an SPF computation, and the resultant tree as an SPF tree. For scalability, an OSPF domain may be divided into Y I F A A CBorder router AS border router112 Area 2115 I F OSPF domain G I F A Fig. 1. From left to right, the figure depicts an example OSPF topology, the view of that topology from router G, and the shortest path tree calculated at G. (Though we show the OSPF topology as an undirected graph here for simplicity, in reality the graph is directed.) areas determining a two level hierarchy as shown in Figure 1. Area 0, known as the backbone area, resides at the top level of the hierarchy and provides connectivity to the non-backbone areas (numbered 1, 2, . OSPF assigns each link to exactly one area. The routers that have links to multiple areas are called border routers. For example, routers C , D and G are border routers in Figure 1. Every router maintains a separate copy of the topology graph for each area it is connected to. The router performs the SPF computation on each such topology graph and thereby learns how to reach nodes in all the areas it is connected to. In general, a router does not learn the entire topology of remote areas (i.e., the areas in which the router does not have links), but instead learns the weight of the shortest paths from one or more border routers to each node in remote areas. Thus, after computing the SPF tree for each area, the router learns which border router to use as an intermediate node for reaching each remote node. In ad- dition, the reachability of external IP prefixes (associated with nodes outside the OSPF domain) can be injected into OSPF (X and Y in Figure 1). Roughly, reachability to an external prefix is determined as if the prefix were a node linked to the router that injects the prefix into OSPF. A. Link State Advertisements (LSAs) Routers running OSPF describe their local connectivity in Link State Advertisements (LSAs). These LSAs are flooded reliably to other routers in the network, which the routers use to build the consistent view of the topology described earlier. Flooding is made reliable by mandating that a router acknowledge the receipt of every LSA it receives from every neighbor. The flooding is hop-by-hop and hence does not itself depend on routing. The set of LSAs in a router's memory is called the link state database and conceptually forms the topology graph for the router. It is worth noting that the term LSA is commonly used to describe both OSPF messages and entries in the link state database. An LSA has essentially two parts: (a) an identifier - three parameters that uniquely define a topological element (e.g. a link or a network), and (b) the rest of the contents, describing the status of this topological element OSPF uses several types of LSAs for describing different parts of topology. Every router describes links to all neighboring routers in a given area in a Router LSA. Router LSAs are flooded only within an area and thus are said to have an area-level flooding scope. Thus, a border router has to originate a separate router LSA for every area it is connected to. For example, router G in Figure 1 describes its links to E and F in its area 0 router LSA, and its links to H and I in area 2 router LSA. OSPF uses a Network LSA for describing routers attached to a broadcast network (e.g., Ethernet LANs). These LSAs also have an area-level flooding scope. Section II-B describes OSPF operation in broadcast networks in more detail. Border routers summarize information about one area into another by originating Summary LSAs. It is through summary LSAs that other routers learn about nodes in the remote areas. For example, Information Flooding Scope Router The router's OSPF links belonging to the area Area Network The routers attached to the broadcast network Area Summary The nodes in remote areas reachable from the border router Area External The external prefixes reachable from the ASBR Domain I LSA TAXONOMY router G in Figure 1 learns about A and B through summary LSAs originated by C and D. Summary LSAs have area-level flooding scope. As mentioned earlier, OSPF allows routing information to be imported from other routing protocols, e.g., RIP, EIGRP or BGP. The router that imports routing information from other protocols into OSPF is called an AS Border Router (ASBR). An ASBR originates external LSAs to describe external routing infor- mation. In Figure 1 all the routers learn about X and Y through external LSAs originated by ASBR A. External LSAs are flooded in the entire domain irrespective of area boundaries, and hence have domain-level flooding scope. Table I summarizes this taxonomy of OSPF's LSAs. A change in the network topology requires affected routers to originate and flood appropriate LSAs. For in- stance, when a link between two routers comes up, the two ends have to originate and flood their router LSAs with the new link included in it. Moreover, OSPF employs periodic refresh of LSAs. So, even in the absence of any topological changes every router has to periodically flood self-originated LSAs. The default value of the refresh- period is minutes. The refresh mechanism is jittered and driven by timer expiration. Due to reliable flooding of LSAs, a router can receive multiple copies of a change or refresh triggered LSA. We term the first copy received at a router as new and copies subsequently received as du- plicates. Note that LSA types introduced in Table I are orthogonal to refresh or change triggered LSA, and new versus duplicate instances of an LSA. B. OSPF Operation over a Broadcast Network As noted in the introduction, the enterprise network makes extensive use of Ethernet LANs which provide broadcast capability. OSPF represents such broadcast networks via a hub-and-spoke topology. One router is elected as the Designated Router (DR). The DR originates a net-work LSA representing the hub, describing links (repre- senting the spokes) to the other routers attached to the broadcast network. To provide additional resilience, the routers also elect a Backup Designated Router (BDR), which becomes the new DR if the DR fails. OSPF flooding over a broadcast network is a two step process: 1. A router attached to the network sends an LSA only to the DR by sending it to a special multicast group DR-Rtrs. Only the DR and the BDR listen to this group. 2. The DR in turn floods the LSA back to other routers on the network by sending it to another special multicast All-Rtrs. All the routers on the network listen to this group. The BDR participates in the DR-Rtrs group so that it can remain in sync with DR. However, the BDR does not flood an LSA to All-Rtrs unless the DR fails to do so. III. ENTERPRISE NETWORK AND ITS INSTRUMENTATION In this Section, we first describe the OSPF topology of the enterprise network used for our case study. We then describe the OSPF monitoring system we deployed in that network, for collecting LSAs and providing real-time monitoring of the OSPF network. A. Enterprise Network Topology The enterprise network provides highly available and reliable ("always on") connectivity from customer's facilities to applications and databases residing in a data center (see Figure 2). The network has been designed to provide a high degree of reliability and fault-tolerance. Customer- premise routers are connected to the data center routers via leased lines. An instance of EIGRP runs between the endpoints of each leased line. The routers in the data center form an OSPF domain which is the focus of this pa- per. Customer reachability information learnt via EIGRP is imported as external LSAs into the OSPF domain. The domain consists of Cisco routers and switches. For scal- ability, the OSPF domain is divided into about 15 areas forming a hub-and-spoke topology. Servers hosting applications and databases are connected to area 0 (the back-bone area) whereas customers are connected to routers in non-backbone areas. Certain details of the topology of non-backbone areas are relevant to our analysis. Figure 3 shows the topology of a non-backbone area. Two routers - termed B1 and B2 are connected to all areas (the backbone area and every non-backbone area), and serve as OSPF border routers. Each non-backbone area has up to 50 routers. As shown in the figure, each area consists of two Ethernet LANs. All the routers of the area are connected to these LANs. Routers B1 and B2 have connections to both LANs and provide the interconnection between the two LANs. Other routers of the area are connected to exactly one of the two LANs. OSPF Domain Data Center Customer-site Applications Customer-site Servers Customer-site Customer-site Customer-site Fig. 2. Enterprise network topology. Other routers Area A Customer-premise router Other routers External (EIGRP) Border rtr Border rtr R R' Fig. 3. Structure of a non-backbone OSPF area. All the areas are connected via two border routers B1 and B2. Since customer-premise routers (e.g., R 0 in Figure 3) are not part of the OSPF domain, and all data center routers are not part of EIGRP domain, routes from one protocol are injected into the other for ensuring connectivity. Thus, router R of area A in Figure 3 which is connected to a customer router R 0 , injects EIGRP routes into OSPF as external LSAs. Route injection into OSPF is carefully controlled through configuration. B. OSPF Monitoring The architecture of the OSPF monitor consists of two basic components: LSARs (LSA Reflectors) and LSAGs (LSA aGgregators) [4]. By design, LSARs are extremely simple devices that connect directly to the network and capture OSPF LSAs, and "reflect" them to LSAG for further processing. In the case study here, the LSARs connect to LANs and join the appropriate multicast groups to receive LSAs. At least one LSAR was connected to each area under study. In a point to point deployment, the LSARs form "partial adjacencies". These adjacencies fall short of full OSPF adjacencies but are sufficient to receive OSPF traffic. LSARs speak enough OSPF to capture OSPF LSA traffic. However, the design rules out the possibility of the LSAR itself getting advertised for potential use for routing regular traffic. All code complexity is concentrated in the LSAGs. In the case study, we deployed a single LSAG in the network. The LSARs reliably feed the LSAs to the LSAG, which aggregates and analyzes the LSA stream to provide real-time monitoring and fault management capability. For lack of space in this paper, we do not go in further details of the monitoring system architecture. We deployed three LSARs and one LSAG, running on four Linux servers. Each LSAR has a number of interfaces connected to different areas. LSARs currently monitor area zero and seven non-backbone areas, covering a total of about 250 routers. The LSARs are connected to LANs and configured to monitor LSAs sent to the multicast group All-Rtrs. One advantage of this approach is that LSAR does not have to establish adjacencies with any routers, and remains completely passive and invisible to the OSPF domain. Since LSAR listens to group All-Rtrs, LSA traffic seen by it is essentially identical to that seen by a regular (i.e., non-DR, non-BDR) router on the LAN. IV. RESULTS We carried out the following steps to analyze the LSA traffic: ffl Baseline. We analyze the refresh-LSA traffic to base-line the protocol dynamics, arising from soft state refresh. Specifically, we predict the rate of refresh-LSA traffic from information obtained from the router configuration files, and then carry out a time-series analysis of finer time scale characteristics. ffl Analyze and fix anomalies. We take a closer look at the change-LSA traffic, and identify root causes. In the operational setting, the heavy-hitter root causes correspond to failure modes. Identifying these failure modes at incipient stages enables proactive maintenance. ffl Analyze and fix protocol overheads. We take a closer look at duplicate-LSA traffic, identify root causes, and identify configuration changes for reducing the traffic. To get a general sense for the nature of observed LSA traffic, consider Figure 4. The Figure shows the number of refresh, change and duplicate LSAs received per day, in April, 2002, for four OSPF areas. The other OSPF areas monitored exhibited similar patterns of behavior. First, note that refresh-LSA traffic is roughly constant throughout the month for all areas. (The small dip in the refresh traffic on April 7 is a statistical artifact due to rolling the clocks forward by one hour during the switch Number of LSAs Day Number of LSAs per day: April, 2002 Refresh LSAs Change LSAs Duplicate LSAs2000600010000 Number of LSAs Day Number of LSAs per day: April, 2002 Refresh LSAs Change LSAs Duplicate LSAs (a) Area 0 (b) Area 210003000500070009000 Number of LSAs Day Number of LSAs per day: April, 2002 Refresh LSAs Change LSAs Number of LSAs Day Number of LSAs per day: April, 2002 Refresh LSAs Change LSAs Duplicate LSAs (c) Area 3 (d) Area 4 Fig. 4. Number of refresh, change and duplicate LSAs received at LSAR during each day in April. to daylight savings time.) Second, all four areas show differences in change and duplicate-LSA traffic. In the backbone area (area refresh-LSA traffic is about two orders of magnitude greater than change and duplicate- LSA traffic. Non-backbone areas have very similar physical topologies, but show markedly different change and duplicate-LSA traffic. In area 2, change-LSA traffic is significant, though duplicate-LSA traffic is negligible. In area 3, we note significant duplicate-LSA traffic, and negligible traffic. Finally, area 4 saw negligible traffic for both change and duplicate LSAs. The reasons for these variations in LSA traffic patterns will become apparent in sections VI and VII. V. REFRESH-LSA TRAFFIC A. Predicting Refresh-LSA Traffic First, let us consider how to determine the average rate NR of refresh-LSAs received at a given router R. For the purposes of the calculation, we assume that the set LR of unique LSA-identifiers in router R's link-state database is constant. That is, network elements are not being introduced or withdrawn. We will use the term LSA interchangeably with LSA-identifier. Let F l denote the average rate of refreshes for a given LSA l in the link-state database. Then, l2LR F l (1) Let D denote the set of LSAs originated by all routers in the OSPF domain, and S l the set of routers that receive a given LSA l. Then, the set LR can be expressed as which together with Eq. 1 determines NR . Thus, we see that estimating the refresh-LSA traffic at a router requires determining three parameters: ffl D; the set of LSAs originated by all the routers in the OSPF domain. ffl For each LSA l in D, S l ; the set of routers that can receive l. ffl For each LSA l in D, the associated refresh-rate F l of l. We next describe how to estimate these three parameters from the configuration files of routers. Number of LSAs Day Number of LSAs per day: April, 2002 Refresh-LSAs (actual) Number of LSAs Day Number of LSAs per day: April, 2002 Refresh-LSAs (actual) Refresh-LSAs (expected:config) (a) Area 2 (b) Area 3 Fig. 5. Expected refresh-LSA traffic versus actual refresh-LSA traffic for two OSPF areas. A.1 Parameter Determination To determine D, it is possible to use information available in router configuration files. In particular, it is not hard to count the exact number of internal LSAs using configuration files. For example, a router configuration file specifies the OSPF area associated with each interface of the router. We can derive the number of router LSAs a given router originates by counting the number of unique areas associated with the router's interfaces. On the other hand, it is impossible to estimate the exact number of external LSAs from configuration files. In general, the number of external LSAs depend on which prefixes are dynamically injected into OSPF domain. However, one can use heuristics to determine external LSAs using the filtering clauses in configuration files that control external route injection Calculating the parameter S l for LSA l is equivalent to counting the routers in the flooding scope of l. The count can be easily determined by constructing the OSPF topology and area structure from the configuration files. To estimate the refresh-rate F l of LSA l, a crude option is to use the recommended value of 30 minutes from the OSPF specification [12]. In practice, better estimates can be obtained by combining configuration information with published information on the router vendor's refresh algorithm We determined all three parameters from the network's router configuration files using an automated router configuration analysis tool, NetDB [13]. Specifically, we computed the set D for router, network, summary and external LSAs. We estimated external LSAs using the heuristic that every external prefix explicitly permitted via configuration is in fact injected as an external LSA. As it turned out, this heuristic underestimated the number of external LSAs by about 10%, owing to the injection of more-specific prefixes than those present in filters within the configuration files. For refresh-rates, the tool first determined operating system version of each router from the configuration files. It then consulted a table of refresh rates using the operating system version as the index. The table itself was populated from information published on the vendor web-site [14] [15]. Figure 5 shows the expected refresh-LSA traffic per day versus the actual number of LSAs received by LSAR, for two areas. Clearly, the actual refresh-LSA traffic is as predicted B. Time-series Analysis In this section, we report on a time-series analysis of refresh-LSA traffic. The analysis revealed that the traffic is periodic, as expected. Recently, a paper by Basu and Riecke [3] suggested that LSA refreshes from different routers could become synchronized. We tested the hypotheses and found refresh traffic not to be synchronized across different routers. B.1 Periodicity of Refresh-LSA Traffic The time-series analysis revealed that the routers fall into two classes: ffl The first class has a refresh-period of minutes and exhibits very strong periodic behavior. ffl The second class has a refresh-period of about 33 min- utes, with a jittered refresh pattern. As it turned out, the analysis picked up differences in the refresh algorithms, associated with different releases of the router operating system. Specifically, the first class of routers ran IOS 11 (11.1 and 11.2) whereas the second class of routers ran IOS 12 (12.0, 12.1 and 12.2). The OSPF implementation in IOS 11 follows a simple refresh freqency (cycles per hour) LSAs per minute Fig. 6. Refresh-LSA traffic for routers running IOS 11. The upper graph shows the time-series for a few hours of a typical day. The lower graph shows the power-spectrum analysis of the time-series. strategy. The router scans the OSPF database every minutes and refreshes all its LSAs by reflooding them in the network [15]. Figure 6 shows an example from the time-series obtained by binning the LSAs into sampling intervals of size 1 minute (the horizontal line in the upper graph shows the average LSA rate based on bins). It seems obvious from the graph that there is a periodicity in the time-series. To test this, and determine the period we plot the power spectrum of the time-series in the lower graph of the figure (based on a longer 1 week sample of data). The power spectrum shows a distinct peak at a frequency of 2 cycles per hour (a period of The subsequent peaks are the harmonics of this distinct peak, and so we can conclude that the time-series shows strong periodicity. The refresh algorithm underwent a change when IOS 11.3 was introduced [15]. The router running IOS 12 has a timer which expires every refresh-int seconds. Upon expiry of the timer, the router refreshes only those LSAs whose last refresh-time is more than refresh-int is configurable with a default of 4 min- utes. Furthermore, the timer is jittered. Since the routers of the enterprise network use the default, we expect the refresh interval to be about 32 minutes (the smallest multiple of 4 which is greater than minutes). The effect can be seen in the upper graph of Figure 7 which shows the LSA refresh pattern for routers running IOS 12. The power spectrum in the lower graph of the figure shows that the data has a strong component at 1.79 cycles per hour, which is roughly 33 minutes as expected. (We have correspondingly chosen the bin size for this data to be 67.189 seconds to minimize aliasing in the results.) Notice that freqency (cycles per hour) LSAs per minute Fig. 7. Refresh-LSA traffic for routers running IOS 12. The upper graph shows the time-series for a few hours of a typical day. The lower graph shows the power-spectrum analysis of the time-series. there is considerably more noise in the spectrum due to the jitter algorithm. B.2 Synchronization of Refresh-LSA Traffic It has been suggested that LSA refreshes are likely to be synchronized with the undesirable consequence that they are all sent nearly simultaneously creating a burst of LSA traffic at a router [3]. We analyze refresh-LSA traffic to see how bursty the traffic appears. In general, the burstiness of LSA traffic received by a router depends on two things: ffl The burstiness of refresh-LSA traffic originated by a single router. ffl Synchronization between refresh-LSAs originated by different routers. We have observed that LSAs originated by a single router are usually clumped together during refresh. With IOS 11 this is expected since a router refreshes all LSAs on expiry of a single timer. Even with IOS 12, we have observed that LSAs originated by a single router are clumped together. Specifically, summary and external LSAs originated by some routers tend to be refreshed in big bursts. This explains the periodic spikes seen in Figures 6 and 7. Next, we consider how refresh-LSAs coming from different routers interact. A recent paper [3] suggested that LSA refreshes from different routers are likely to be syn- chronized. The mechanism that creates this synchronization is related to the startup of the routers. However, in general, in network related phenomena, synchronization is only a real problem when there are forces driving the system toward synchronization, which is not the case here. For example, see [16] [17] where synchronization occurs as a result of the dynamics of the system pushing it towards time (in period=1800 seconds) number of routers (a) time (in periods=2015 seconds) number of routers (b) (a) IOS 11 (b) IOS 12 Fig. 8. Number of routers whose LSAs were received within one second intervals at LSAR during one refresh cycles. The routers belong to area 8. synchronization in a similar manner to the Huygens' clock synchronization problem [18]. To understand why synchronization is only a real problem if the system is pushed towards it, consider that in a real network it would be very rare for all the routers in a network to be rebooted simultaneously. Over time though, individual links and routers are added, dropped, and restarted. Each time the topology is changed in this way, a little part of the synchronization is broken. The larger the network, the more often topology changes will occur, and so the synchronization is broken more quickly in the cases where it might cause problems. Furthermore, there is always a small drift in any periodic signal, and this drift breaks the synchronization over time. Moreover, there is no "weak coupling" [16] in OSPF LSA refresh process, i.e., the LSAs generation at a router is not driven by that at other routers. Finally, the addition of jitter in IOS 12 onwards quickly removes any synchronization between these routers. If there is no force driving the system towards synchronization, then it is unlikely to be seen outside of simulations. For the enterprise network, we have observed that refresh traffic from different routers is not strongly synchronized Figure 8(a) and (b) show the number of routers (from area 8) whose LSAs were received at LSAR during a one second interval for the duration of a typical refresh- cycle. Neither graphs display evidence of strong synchronization between routers. We have also performed stastical tests which show that at least at time scales below a minute the LSA traffic from different routers is not at all synchro- nized, and appears to be uniformly distributed over the minute refresh period. On larger time scales, there is some apparent weak correlation (see the clustering of routers at and 0.75 in Figure 8(a)), but the degree of correlation seen should not have practical importance even if it is not a statistical anomaly. Area 8 was chosen because it contained a good mix of routers with IOS 11 and 12. Other areas show similar characteristics VI. CHANGE-LSA TRAFFIC Figure 4 shows that some areas receive significant change-LSA traffic. In this section, we first classify these LSAs by whether they indicate internal or external changes. Then, we look at the underlying causes. Internal changes are conveyed by router and network LSAs within the area in which change occurs and by summary LSAs outside the area. External changes are conveyed by external LSAs. Figure 9 shows the number of change LSAs for the month of April. The figure provides curves for selected areas, accounting for more than 99% of the corresponding LSA traffic in April. Figure 9 shows that external changes constitute the largest component of change-LSAs generated in the net- work. External changes from area 2 dominate those seen in other areas (Figure 9(c)). Among internal changes, most occurred in area 0 (Figure 9(a)). Internal change-LSAs in area 0 were not propagated to other areas, since the net-work was configured to allow only summary LSAs representing default route (0.0.0.0/0) into non-backbone areas. The spike in Figure 9(b) is due to a border router withdrawing and re-announcing summary LSAs. A. Root Cause Analysis We saw that area 0 accounted for most of the internal changes seen in April. It turns out that almost all these changes were due to an internal error in a crucial router in area 0. This router was the DR on all the LANs of area 0. Because of the error, there would be episodes lasting a few minutes during which the problematic router would drop and re-establish adjacencies with other routers on the LAN. Accordingly, a flurry of change-LSAs were gener- fchange LSAs Day Number of change LSAs (Router Area 5 Number of change LSAs Day Number of change LSAs (Summary) per day: April, 2002 Number of change LSAs Day Number of change LSAs (External) per day: April, 2002 Area 6 (a) Router network LSAs (b) Summary LSAs (c) External LSAs Fig. 9. Change-LSA traffic for each day in April. Each graph shows those areas which together account for more than 99% of change-LSA in April. For example, areas 0, 1, 2 and 6 account for 99% external Number of LSAs Day Number of LSAs per day: April, 2002 Router network LSAs in area 0 Router network change LSAs in area 0 Change LSAs due to problematic router2060100140 Number of LSAs Hour Number of LSAs per hour: April 16, 2002 Router network LSAs in area 0 Router network change LSAs in area 0 Change LSAs due to problematic router (a) Entire April (b) April 16. Fig. 10. Effect of a problematic router on the number of router network LSAs in area 0. ated during each such episode. Each episode lasted only for a few minutes and there were only a few episodes each day. The data suggests that during the episodes the net-work was at risk of partitioning or was in fact partitioned. In April, these episodes account for more than 99% of total internal change-LSAs observed in area 0. Figure 10(a) shows the number of router and network LSAs for each day of April, and Figure 10(b) shows the same statistic for each hour of one of these days when area 0 witnessed a few episodes. On April 19th, acting on the data gathered by the OSPF monitor, the operator changed the configuration of the problematic router to prevent it from becoming the DR, and rebooted it. As a result, the network stabi- lized, and changes in the area 0 topology vanished. Inter- estingly, this illustrates the potential of OSPF monitoring for localizing failure modes, and proactively fixing the net-work before more serious failures occur. Figure 9(c) shows that among all areas, area 2 witnessed the maximum number of external changes in April. A large percentage of these changes were caused by a flapping external link. One of the routers (call it A) in area 2 maintains a link to a customer premise router (call it B) over which it runs EIGRP, as mentioned in Section III-A. Router A imports 4 EIGRP routes into OSPF as 4 external LSAs. Closer inspection of network conditions revealed that the EIGRP session between A and B started flapping when the link between A and B became overloaded. This leads to router A repeatedly announcing and withdrawing EIGRP prefixes via external LSAs. The flapping of the link between A and B happened nearly every day in April between 9 pm and 3 am. These link flaps accounted for about 82% of the total external change-LSAs and 99% of the total external change-LSAs witnessed by area 2. At the time of writing of this paper, the network operator is still looking into ways of minimizing the impact of these external EIGRP flaps without impacting customer's connectivity or performance. VII. DUPLICATE-LSA TRAFFIC In Section IV, we remarked that area 3 received significant duplicate-LSA traffic (almost 33% of the total LSA traffic in that area). On the other hand, area 2 saw negligible duplicate-LSA traffic. Since processing duplicate- LSAs wastes CPU resources, it is important to under- fLSAs Day Number of LSAs per day: April, 2002 Total LSAs in area 2 Total change LSAs in area 2 Total change LSAs due to flappers20060010004 8 12 Number of LSAs Hour Number of LSAs per hour: April 11, 2002 Total LSAs in area 2 Total change LSAs in area 2 Total change LSAs due to flappers (a) Entire April (b) April 11. Fig. 11. Effect of a flapping external link on area 2 external LSAs. stand the circumstances that lead to duplicate-LSA traffic in some areas and not others. As we will see, a detailed analysis of the OSPF control plane connectivity explains the variation in duplicate LSA traffic seen in areas 2 and 3, and leads to a configuration change that would reduce duplicate LSA traffic in area 3. In general, we believe such analysis can provide operational guidelines for lowering the level of duplicate LSA traffic, at the cost of small trade-offs in network responsiveness. A. Causes of Duplicate-LSA Traffic In the enterprise network under study, all areas have identical physical connectivity. Thus, it initially came as a surprise that one area saw significant duplicate-LSA traffic and another area did not. As it turns out, though all areas have identical physical structure, the difference in how LSAs propagate through the areas gives rise to the differences observed in duplicate-LSA traffic. Recall that the areas are LAN-based, and that the DR and BDR behave differently than other routers on the LAN, as described in Section II-B: The DR and BDR send LSAs to all the routers (and the monitoring system's LSAR) on the LAN, whereas other routers send LSAs only to the DR and BDR. Thus, the LSA propagation behavior on a LAN depends strongly on which routers play the role of the DR or BDR, and how these routers are connected to the rest of the net-work The analysis is rather intricate. Recall that every area has two LANs, and that the LSAR is attached to one of the LANs. Let us denote the LAN on which the LSAR resides as LAN 1, and the other LAN as LAN 2. Recall that B1 and B2 are connected to both LANs; other routers are connected to only one of the LANs. We denote B1 and B2 as the B-pair, and rest of the routers as LAN1- router or LAN2-router, based on which LAN the routers reside on. Since the B-pair routers are connected to both LANs, the role they play on LAN 1 (DR, BDR or regu- lar) is very important in determining whether the LSAR receives duplicate-LSAs or not. Indeed, it is the B-pair routers' difference in role in areas 2 and 3 that gives rise to different duplicate-LSA traffic in these two areas. We arrive at four cases based on the roles B-pair routers play on LAN 1: Case 1: fDR, BDRg Case 2: fDR, regularg Case 3: fBDR, regularg Case 4: fregular, regularg To understand which of these cases leads to duplicate-LSA traffic on LAN 1 of a given area, we model LSA propagation on LAN 1 with a "control-plane" diagram in Figure 12. This diagram shows links between those routers that can send LSAs to each other. In addition, the figure shows how one or more copies of LSA L may propagate to the LSAR via the B-pair routers. Suppose LSA L is originated by a LAN2-router. The B-pair routers receive copies of L on their LAN 2 interfaces and further propagate the LSA to the LSAR on LAN 1. We denote the copies of L propagated via B1 and B2 as L1 and L2 respectively in Figure 12. The figure makes it clear that cases 1 and 3 lead to duplicate-LSAs whereas cases 2 and 4 do not. Table II shows the cases we encounter in different areas. Note that area 3 encounters case 3 whereas area 2 encounters case 2. This explains why area 3 receives duplicate- LSA traffic and area 2 does not. Note that under cases 1 and 3, whether the LSAR actually receives multiple copies of LSA L depends on the LSA arrival times at various routers. For example, consider case 1. Whether the B-pair routers send LSA L to the LSAR or not depends on the order in which the LSA arrives at these two routers. At least one of B-pair routers (a) Case 1 (b) Case 2 L1 or L2 (d) Case 4 (c) Case 3 LSAR LSAR LSAR LSAR Fig. 12. Control-plane diagram for LAN 1 under different roles played by the B-pair routers. The figure also shows how different copies of LSA L can arrive at the LSAR via the B-pair routers. L1 and L2 are copies of LSA L. Area DR on LAN 1 BDR on LAN 1 Case above Area 4 B2 LAN 1 rtr case 2 Area 5 B2 B1 case 1 Area 6 B2 B1 case 1 Area 8 LAN 1 rtr B2 case 3 II DR AND BDR ON LAN 1 OF VARIOUS AREAS. must send the LSA to the LSAR on LAN 1. However, whether the other router also sends the LSA to LSAR depends on the order of LSA arrival at this router. If the router receives the LSA on LAN 2 first, it sends the LSA to LSAR resulting in a duplicate being seen at LSAR. On the other hand, if the router receives the LSA on LAN 1 first, it does not send the LSA to LSAR. In this case, the LSAR does not receive a duplicate-LSA. A similar argument can be made regarding case 3. To summarize, an LSA originated by a LAN2-router may get duplicated on LAN 1 under cases 1 and 3, if it arrives in a particular order at different routers. Figure 13 shows the number of duplicate-LSAs originated by various routers for two representative areas. All the duplicate- LSAs seen by the LSAR are originated by LAN2-routers and the B-pair routers. The LSAR does not see duplicate- LSAs for LSAs originated by a LAN1-router. This is because irrespective of which router is DR and BDR on LAN 1, LSAR receives a single copy of an LSA originated by a LAN1-router from the DR. Figure 14 shows the fraction of total LSAs originated by LAN2-routers that are duplicated. As the figure indi- cates, even under cases 1 and 3, not all the "duplicate- susceptible" LSAs are actually duplicated. We observed that within a given area, the percentage of LSAs originated by LAN2-routers that become duplicated remains roughly constant for days. However, this percentage varies widely across areas. Understanding this behavior requires understanding the finer time scale behavior of the routers in- volved, and is ongoing work. Number of duplicate LSAs Day Number of duplicate LSAs per day: April, 2002 Total duplicate LSAs (LAN2-routers) Total duplicate LSAs (B1) Total duplicate LSAs (B2) Total duplicate LSAs Number of duplicate LSAs Day Number of duplicate LSAs per day: April, 2002 Total duplicate LSAs (LAN2-routers) Total duplicate LSAs (B1) Total duplicate LSAs (B2) Total duplicate LSAs (LAN1-routers) (a) Area 3 (case Fig. 13. Duplicate-LSA traffic from various routers. B. Avoiding Duplicate-LSAs Having uncovered the causes of duplicate-LSAs, we explore ways to reduce their volume. The enterprise network operator can avoid duplicate-LSAs if he can force case 2 or 4, by controlling which router becomes the DR and/or the BDR on LAN 1. This depends on a complex election algorithm executed by all routers on the LAN. The input to this algorithm is priority parameter, configurable on each interface of a router. The higher the priority, the greater the chance of winning the election, though these priorities provide only partial control. As a result, the operator cannot force case 2 to apply. Even if the network operator assigns highest priority to one of the B-pair and zero priority to the other routers on LAN 1, there is no guarantee that the high priority router will become DR. Fortunately, the operator can force case 4 to apply by ensuring that neither of the B-pair routers become DR or BDR on LAN 1. This is accomplished by setting the priority of these two routers to 0, so that they become ineligible to become DR or BDR [12]. Whether forcing case 4 is sensible depends on at least two factors. First, the DR and BDR play a very important role on a LAN, and bear greater OSPF processing load than the regular routers on the LAN. Therefore, the operator has to ensure that the most suitable routers (taking into account load and hardware capabilities) can become DR and BDR. The second factor is more subtle. Typically, reducing duplicate-LSAs requires reducing the number of alternate paths that the LSAs take during reliable flood- ing. This can increase the LSA propagation time, which in turn can increase convergence time. With case the LSAs originated on LAN 2 have to undergo an extra hop before the other routers on LAN 1 receives them. This means that the LSA propagation time may increase Fraction Day Fraction of LAN 2 LSAs that get duplicated: April, 2002 area 3 area 6 area 8 Fig. 14. Fraction of LSAs originated by LAN2-routers that get duplicated. case 4 is forced. VIII. CONCLUSIONS In this paper, we provided a case study of OSPF behavior in a large operational network. Specifically, we introduced a methodology for OSPF traffic analysis, treating LSA traffic generated by soft-state refresh, topology change, and redundancies in reliable flooding, in turn. We provided a general method to predict the rate of refresh LSAs from router configuration information. We found that measured refresh-LSA traffic rates matched predicted rates. We also looked at finer time scale behavior of refresh traffic. The refresh period of different routers was in conformance with the expected behavior of their IOS versions. Though LSAs originated by a single router tend to come in bursts, we found no evidence of synchronization across routers. This may reduce scalability concerns, which would arise if refresh synchronization were present, leading to spikes in CPU and bandwidth usage. We found that LSAs indicating topology change were mainly due to external changes. This is not unexpected since the network imports customer reachability information into OSPF domain which is prone to change as customers are added, dropped or their connectivity is changed. Moreover, since customers are connected over leased lines, their reachability information is likely to be more volatile. For both internal and external topology changes, persistent but partial failure modes produced the vast majority of change LSAs, associated with flapping links. Interest- ingly, the internal change-LSA traffic pointed to an intermittently failing router, leading to a preventative action to protect the network. It is fair to say that any time a new way to view networks is introduced (route monitoring in this case), new phenomena are observed, leading to better network visibility and control. Though further and wider studies are needed, we suspect that persistent and partial failure modes are typical, and the development of strategies for stabilizing OSPF would benefit from focusing on such modes. During the study, we did not observe any instance of network-wide meltdown or network-wide in- stability. We also investigated the nature of the duplicate- LSA traffic seen in the network. The analysis led to a simple configuration change that reduces duplicate traffic, without impacting the physical structure of the network. The findings of this case study are specific to the enterprise network and the duration of the study. Similar studies for other OSPF networks (enterprise and ISP) and studies over longer durations are needed to further enhance understanding of OSPF dynamics. This forms a part of our future work. Furthermore, In the ISP setting, we intend to join the OSPF and BGP monitoring data to analyze the interactions of these protocols. Another direction of future work is to develop realistic workload models for OSPF emulation, test and simulations. Our methodology for predicting refresh-LSA traffic is a first step in that direction. The workload models can also be used in conjunction with work on OSPF processing delays on a single router [5] to investigate network scalability. ACKNOWLEDGMENTS We are grateful to Jennifer Rexford and Matt Grossglauser for their comments on the paper. We thank Russ Miller for his encouragement and guidance in the operational deployment. Finally, we thank the anonymous reviewers for their comments. --R Routing in the Internet "Can One Rogue Switch Buckle AT&T's Net- work?," "Stability Issues in OSPF Rout- ing," "An OPSF Topology Server: Design and Evaluation," "Experience in Black-box OSPF Measurement," "Experimen- tal Study of Internet Stability and Wide-Area Network Failures," "Toward Milli-Second IGP Convergence," "ISIS Routing on the Qwest Backbone: a Recipe for Subsecond ISIS Convergence," "Internet Routing Stability," "Origins of Pathological Internet Routing Instability," "OSPF Version 2," "IP Network Configuration for Intra-domain Traffic Engineering," "Cisco Systems," "OSPF LSA Group Pacing," "The Synchronization of Periodic Routing Messages," "Oscillations and Chaos in a Flow Model of a Switching System," --TR The synchronization of periodic routing messages Internet routing instability Routing in the Internet (2nd ed.) Stability issues in OSPF routing Experience in black-box OSPF measurement OSPF --CTR Renata Teixeira , Aman Shaikh , Tim Griffin , Jennifer Rexford, Dynamics of hot-potato routing in IP networks, ACM SIGMETRICS Performance Evaluation Review, v.32 n.1, June 2004 Aman Shaikh , Albert Greenberg, OSPF monitoring: architecture, design and deployment experience, Proceedings of the 1st conference on Symposium on Networked Systems Design and Implementation, p.5-5, March 29-31, 2004, San Francisco, California Aman Shaikh , Rohit Dube , Anujan Varma, Avoiding instability during graceful shutdown of multiple OSPF routers, IEEE/ACM Transactions on Networking (TON), v.14 n.3, p.532-542, June 2006 Yin Zhang , Matthew Roughan , Carsten Lund , David L. Donoho, Estimating point-to-point and point-to-multipoint traffic matrices: an information-theoretic approach, IEEE/ACM Transactions on Networking (TON), v.13 n.5, p.947-960, October 2005 Yin Zhang , Matthew Roughan , Carsten Lund , David Donoho, An information-theoretic approach to traffic matrix estimation, Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communications, August 25-29, 2003, Karlsruhe, Germany Ruoming Pang , Mark Allman , Mike Bennett , Jason Lee , Vern Paxson , Brian Tierney, A first look at modern enterprise traffic, Proceedings of the Internet Measurement Conference 2005 on Internet Measurement Conference, p.2-2, October 19-21, 2005, Berkeley, CA Yin Zhang , Zihui Ge , Albert Greenberg , Matthew Roughan, Network anomography, Proceedings of the Internet Measurement Conference 2005 on Internet Measurement Conference, p.30-30, October 19-21, 2005, Berkeley, CA Haining Wang , Cheng Jin , Kang G. Shin, Defense against spoofed IP traffic using hop-count filtering, IEEE/ACM Transactions on Networking (TON), v.15 n.1, p.40-53, February 2007

routing;LSA traffic;OSPF;enterprise networks

637247

Active probing using packet quartets.

A significant proportion of link bandwidth measurement methods are based on IP's ability to control the number of hops a packet can traverse along a route via the time-to-live (TTL) field of the IP header. A new delay variation based path model is introduced and used to analyse the fundamental networking effects underlying these methods. Insight from the model allows new link estimation methods to be derived and analysed. A new method family based on packet quartets: a combination of two packet pairs each comprising a probe following a pacesetter packet, where the TTL of the pacesetter is limited and the end-to-end delay variation of the probes is measured, is introduced. The methods provide 'pathchar-like' rate estimates over multiple links without relying on the delivery of ICMP messages, with reduced invasiveness and other advantages. The methods are demonstrated using simulations, and measurements on two different network routes are used for illustration and comparison against available tools (pathchar and clink). A comprehensive analysis of practical issues affecting the accuracy of the methods, such as link layer headers, is provided. Particular attention is paid to the consequences of 'invisible' hops: nodes where the TTL is not decreased.

INTRODUCTION Recent papers of Lai and Baker [1] and Dovrolis [2] identify two primary network effects at the foundation of existing probing techniques. The one-packet methods are based on the assumption that the transmission delay is linear in the packet size. The idea was first introduced by V. Jacobson in his pathchar tool: to estimate link bandwidths from round trip delays of different sized packets from successive routers along the path [3], [1]. The packet pair family is based on the 'spacing' effect of the bottleneck link [4], the fact that the minimum inter-departure time of consecutive packets from a link is attained when they are back-to-back. This effect can be utilised to estimate bottleneck bandwidth (the smallest link rate) [5], [6], [7], [8], [1], [2]. Methods estimating available bandwidth are also based on the observation of the inter-departure time of consecutive probe packets, taking into consideration the cases when cross traffic packets are inserted between them. This possibility is discussed in [6], [7], [9], [1], [2], [10]. In a recent paper [11], we enhanced the packet-pair family through thoroughly investigating its dependence on probe size and the effects of link layer headers. In this paper we contribute to the development of hybrid methods exploiting both of Research Center for Ultra Broadband Information Networks, Dept. of Electrical and Electronic Engineering, The University of Melbourne. Hungary R&D, fa.pasztor, [email protected] The authors gratefully acknowledge the support of Ericsson. the above effects. In doing so we make extensive use of the IP header time-to-live (TTL) field as a means of accessing different hops along the path. Pathchar was the first link bandwidth estimation method published based on this idea. It used the TTL field to induce reverse path ICMP messages, allowing round-trip delay measurements (figure (1)). Downey suggested a number of improvements to the filtering method used and implemented them in his clink tool [3]. Forward path Backward path ICMP returned UDP sent RECEIVER FW Hop #3 FW Hop #2 FW Hop #1 FW Hop #4 Expired Fig. 1. IP time to live (TTL) expiration explained, the TTL of the UDP packet is set to 2. In real networks the backward route can be different for each TTL. In this paper we expand and generalise on these ideas in several ways. First, we describe the advantages of moving from an approach based on measuring delays, and minimum filter- ing, to one based on delay variation, and peak detection in his- tograms. We give a thorough description of the nature of the noise encountered by probes as they traverse the route, and introduce new methods exploiting the characteristics of this noise (for more details see [12]). We also point out the crucial importance for all one-packet methods of probes remaining in different busy periods at each hop. This distinction is central to the performance of probing methods, but has not been clearly appreciated before (see however [11], [12]). When this condition is satisfied, the noise typically takes on symmetric characteristics which allows highly efficient median based filtering. Using these ideas, we propose ACCSIG as a more efficient pathchar-like method. The resulting discussion also prepares the ground for the second and main part of the paper, where we follow in the steps of Lai and Baker [1] in using TTL, but without using the returning ICMP messages, with their associated disadvantages. We introduce a flexible packet quartet (PQ) probe class, where probes are replaced by a probe and pacesetter pair, and in this framework define and analyse 5 new estimation methods based on delay variation and peak detection. We determine their key characteristics including their susceptability to different kinds of error. Simulation based comparisons are performed amongst the new methods, and real network measurements on two different network routes are used to illustrate the methods and compare them to pathchar and clink. A full network based validation however is beyond the scope of the pa- per, whose main aim is to introduce the packet quartet methods and to derive their central properties. Finally, several important Segment Local route International route 1. 100Mbps LAN 100Mbps LAN 2x100Mbps switched 2x100Mbps switched 2. Bottleneck 10Mbps LAN 10Mbps LAN 2x10Mbps switched 3. 100Mbps FDDI 4. 155Mbps ATM 5. 155Mbps ATM (2x155Mbps ATM) 6. 22Mbps 7. Unknown est. 100Mbps single hop 8. Unknown est. 10Mbps single hop 9. Suspected Bottleneck estimated 2x2Mbps 10. Unknown estimated 22Mbps 11. Unknown estimated >100Mbps 12. 100Mbps LAN (only last link known) I IP-SEGMENT COMPOSITION OF THE TWO TEST ROUTES. practical issues are discussed in detail, in particular the common problem of invisible hops, where TTL cannot be decremented, which leads to serious estimation errors and is one of the key drawbacks of TTL based approaches. The behaviour of the proposed and existing methods with respect to this problem is given a thorough treatment. It is shown how it can be detected and in some cases corrected by combining different packet quartet methods. We model a route as a chain of ordered hops, H , where a hop is a FIFO queue with deterministic service rate h followed by a transmission link of rate h and latency D h . If a packet is to be dropped at hop h due to an expired TTL field, then it does so at the instant it arrives there, without actually entering the queue at hop h. Physically, this hop model is consistent with store and forward routers where packet arrivals and departures are counted from the end of the last bit: the input queue is seen as part of the previous transmission link, and only when the packet has fully arrived is it either dropped, or transferred (in negligible time) to the output queue, which is the queue modelled. The network measurements in this paper were taken over two one-way routes. One, in the laboratory, was only two IP- hops (we call this a segment) long and completely known. The other exited the country, was 12 segments long (as measured by traceroute), and was only partially known, as detailed in table I. A high accuracy sender, similar to that described in [13], was used capable of sending arbitrary streams of UDP probes. The raw experimental data is the arrival and departure timestamps of the probes. These were collected with a modified tcpdump using a modified software clock, described in [14], with a skew of less than 0.1 parts per million. Software measurement noise was measured to be significant in just a few measurements per 10000. II. PATHCHAR REVISITED We begin with some generic definitions relating to packets traversing a single hop. Although the equations hold true for arbitrary packets, in this paper the index i refers to probe packets only. Probe i arrives to the hop by appearing in the queue at the time instant i . It begins service after a waiting time of w i 0, completes it after a service time of x i > 0, and after a constant propagation delay of D > 0, exits the hop at time . The 1-hop delay is therefore and comparing two successive probes we have the 1-hop inter-departure time: t delay in which D plays no role. Pathchar and its variants are based on measuring the round-trip delay between the release of TTL limited UDP probes and the receipt of their corresponding ICMP error messages. If we label the forward route hop after which the TTL expires by h ttl , the backward route hop at which the ICMP error message enters by k ttl , and let K be the last hop on the backward route, then using equation (1) the delay can be expressed as: h=k ttl icmp Since the service time of probe i on hop h can be expressed as is the probe size, and as the size of ICMP messages is fixed, we see that there exists a deterministic linear relationship between probe size and delay. This can be exploited by varying the probe size, and filtering to eliminate the contribution of the waiting time terms, which can be thought of as a kind of noise. In pathchar, after measuring this round-trip delay for many probes, a filtering based on minima is applied to the delay series, which aims to select probes for which the waiting times were in fact zero. For such probes the delay can be expressed as where C represents the packet size independent remainder of equation (6). This operation is performed for a number of different probe sizes (default values: pathchar 45, clink 93) to obtain samples of a function d(p) describing the minima as a function of probe size p. Finally, a slope is fitted through the estimated d(p) function values. From equation (7) it is clear that the slope estimate corresponds to . The link rates h are then determined by perfoming the procedure for all TTL values from 1 up to the number of IP hops, and estimating the h from the increase in the slope at each stage. In the ACCSIG method described below, we change the above methodology in one key respect, resulting in a significant improvement in efficiency. The change is based on a novel approach of observing and analysing a network route, using the delay variation of probes, rather than the delay. We now introduce this approach, which is also the basis of the packet quartet methods described in section III. A. A new delay variation based route description Probing analysis has typically centered on the fd i g series, as in pathchar-like methods, or on the time series of receiver g, as in packet-pair type methods. For either purpose, we claim that it is advantageous to focus on the delay variation f- i g, for two reasons. First, as - and the are part of probe design and therefore known, f- i g includes but allows richer structure through the choice of ft i g. Sec- ond, up to a constant f- i g is equivalent to fd i g, but like ft only requires accurate clock rates for its measurement, not end- to-end clock synchronisation, an important practical advantage. In fact,through its reliance on slope estimates, pathchar is also based on differences of delay. Here we show some of the advantages of moving to an explicit delay variation framework, the development of which was motivated by repeated observations in network measurements of very characteristic features in delay variation histograms. As the route delay is simply the sum of the individual hop delays, the delay variation over an H-hop route can be written using equation (5) as: (w h Consider each of the terms in the above equation. The first expresses the contribution of the probe service times. It is a deterministic effect independent of cross traffic which accumulates over hops where adjacent probes have unequal service times. The second term is due to cross traffic. It can be thought of a kind of random noise, however the nature of that noise depends critically upon the interaction between the cross traffic and the probe stream. During network experiments we observed that probe streams with probe separations in the range of 100s of ms and above, produce characteristic symmetric delay variation histograms. This can be be understood in terms of the nature of the noise in such a case, as follows. Recall that the busy periods for a queue are the time intervals where the server is continuously active, which separate the idle periods when the queue is empty. If, for each hop, each probe arrives in a different busy period, which can be achieved with high probability by sending them well separated, then each waiting term w h i comes from a different busy period, and contains no explicit probe component. Provided the cross traffic does not change systematically over the measurement interval, it is then reasonable to regard these waiting times as an approximately IID sequence across probes, and hence (w h as a symmetric, near white noise at each hop. This does not mean that a strong assumption of low correlations between hops is being made. Indeed, only if a queue is unstable will we not be able to ensure that probes arrive to it in different busy periods for sufficiently large separation. In figure (2a) the delay variation histogram is given from a simulation with stationary cross traffic, where the fixed sized probes do in fact arrive to the hop in different busy periods. The service time term cancels, leaving only the waiting time terms, and we observe, as argued above, that they behave as a symmetric noise. When however consecutive probes arrive during the same busy period the nature of the noise is substantially differ- ent. We briefly consider this case in section II-D.3. We now describe the accumulation signature, the - i based observable which serves as the basis of our new estimation methods. With constant probe size (assuming x h accumulation term (p i 1= h vanishes, corresponding to a peak at - in the histogram, which the noise spreads out. If instead the probe sizes are made to alternate (the alternation is essential, other orderings would produce different distri- butions), the peak splits into two, and the distance between the twin peaks is proportional to the quantity 1= h , and allows it to be measured. In figure (2b) it is seen in the context of a real measurement how these symmetric peaks combine with the symmetric noise to produce a symmetric bimodal distribution - the signature. d [sec] (a) d [sec] (b) Fig. 2. Accumulation signature in delay variation histograms. (a) 1 hop simula- tion. Constant probe size: the service time cancels, the noise is symmetric, (b) 12 hop measurement. Alternating probe size: the peak splits and the signature emerges: a symmetric bimodal histogram. To reduce measurement error it is desirable to increase the signature, that is to increase the distance between the peaks. This suggests constructing a probe stream with sizes which alternate between the minimum and maximum possible values, rather then using a large number of different packet sizes as in pathchar-like methods. A second order effect, the fact that the efficiency of peak detection may depend on packet size, will be discussed below. The delay variation framework is inherently robust in that the key assumption of probes arriving in different busy periods can be verified by checking for the symmetry of the his- togram. Although non-stationarity in the cross traffic will distort the distribution, in practice it will remain approximately symmetric unless a systematic change occurs over the course of the measurement. A level shift for example in one of the cross traffic streams, results in the received delay variation histogram being the sum of two symmetric components, together with a non-symmetric component consisting of just a single - i value. B. ACCSIG - A new link bandwidth estimation method In this section we apply the accumulation signature idea to the TTL scenario of pathchar. Hence, as above, we assume that well spaced probes are dropped after hop h ttl and that ICMP messages are generated and sent back to the sender along some backward path. Formally, we can consider these ICMP packets as the original probes which have undergone a size change after hop h ttl . One can then use equation (8) to write the round-trip (sender-to-sender) delay variation as h=k ttl (w h h=k ttl (w h where as before K is the number of hops on the backward route, and k ttl is the hop where the ICMP packet enters the backward route. As the ICMP packets share a common size, independently of the size of the probe which triggered their generation, the second component of the above equation becomes zero, leaving an accumulation term generated by the alternating initial probe sizes up to hop h ttl only, but noise terms corresponding to both the forward and backward paths. By the same argument as in section II-A, these noises should be symmetric, resulting in a bimodal, symmetric delay variation histogram. The resulting delay variation analogue of equation (7) is where N i is the symmetric noise term. Here we are assuming that the time to generate the ICMP packet is independent of the probe size. The estimation of link bandwidths is performed recursively, similarly to pathchar and clink. A group of probes with a fixed limited TTL is sent to measure and the procedure is repeated for steadily increasing TTL until the receiver is reached. The h are determined from the difference in the estimates at consecutive steps. To complete our description of ACCSIG the question of peak detection in the - i histogram must be addressed. A number of different methods could be applied to perform this task, including kernel density estimation as suggested in [8], or using conditional sample means or quantiles. Both here and throughout this paper we used the following median related method which we found to be more accurate than kernel estimation, and which avoids the outlier sensitivity of moments. As the probe sizes alternate, the (p i has two possible values. Assume that p 1 < p 2 . Two subsets can be defined depending on the parity of i: Each of these is symmetric in its own right and corresponds roughly to the parts of the histogram to either side of the origin. The peaks are detected by taking the medians of these subsets. The absolute values + and of the peak locations can also Relative error upper estimate lower Pathchar Clink Accsig Accsig* PQ1 PQ2 PQ3 PT1 Measurement method Relative error upper estimate lower (b) 22 Mbps Fig. 3. Relative error of different methods over 1-hop segments. (a) Last (2 nd ) segment (10Mbps) of local route, (b) 6 th segment (22Mbps) of international route. be used to provide an informal interval estimate, in a similar way to clink [3]. This is defined by the rate estimates corresponding to the closest of the two over ttl 1 and ttl, and the furthest. The key advantage of the above approach is that, unlike methods such as pathchar and variants which attempt to find the minimum delay and in so doing discard the majority of the delay values, the accumulation signature utilises the information from all the probes that are received. This greatly reduces the number of probes that need to be sent, especially in the case of longer routes and higher link utilisations, as minimum detection filtering requires the probes to arrive to empty queues at each hop along the route. C. Comparison with pathchar and clink A full comparison of ACCSIG against pathchar and clink is beyond the scope of this paper, as it would involve covering many different route and traffic scenarios, as well as tackling the issues of how to conduct a truly fair comparison and how to measure accuracy appropriately. Instead we offer a simple network based illustration of their comparative performance for lightly loaded routes of short and medium length (be- tween midnight and 6 a.m in both the the sender and receiver timezones). Under these conditions we expect each method to produce similar results. It is well known that under heavy traffic conditions minimum delay detection is problematic and estimates from pathchar and its variants typically suffer significant errors [3], [1]. ACCSIG will perform better under these circumstances due to the symmetry of the independence signa- ture, which allows the detection of the delay variation peak positions even if no probes traverse the path with minimum delay. In effect, the nature of the noise is used to advantage. The measurements were performed over a two and a twelve segment route. The short route was on the local LAN and was fully known, whereas the other was international (between the University of Melbourne and the WAND group at the University of Waikato in New Zealand) and only partially known. As be- fore, by segment we mean a hop at the IP level, many of which consist of multiple link layer hops, as indicated in table I. As a rough measure of fairness we selected the number of probes transmitted in the ACCSIG probe stream to be equal to the number used by pathchar, which is roughly twice that of clink under default settings. In addition, we produced ACCSIG estimates with only the first quarter of the - i values used (marked by Accsig ), which corresponds to half the number used by clink. Figure 3 shows the relative errors of the link rate estimates produced by different methods on two selected 1-hop segments (the other methods will be introduced in section III). Results over other 1-hop segments with low rate are similar to those of figure 3b, the difference in estimates between the three methods being usually within 10%. ACCSIG performs better on the last hop of a route, such as in figure 3a, as it does not rely on the generation the ICMP 'UDP port un- reachable' packet, which incurs an additional latency, but uses receiver timestamps instead. This however is not intrinsic to ACCSIG, but a function of the current implementation. -50%+50% Bw. Estimates upper estimate lower -50%+50% Bw. Estimates upper estimate lower -50%+50% Number of probe pairs Bw. Estimates upper estimate lower segment #2 segment #6 segment #9 Fig. 4. Efficiency of ACCSIG. Link rate estimates as a function of the number of probe pairs in the international route. (a) Segment #2 (2-hops), (b) Segment #6 (1-hop) and (c) (suspected bottleneck), Segment #9 (suspected 2-hops). The expected benefit of ACCSIG is its more efficient use of the probes sent. To quantity this further, figure 4 plots a sequence of estimates based on truncating the received delay variation series from the international route to emulate shorter probe streams. We see that the 'time' to convergence to the long term estimates varies from as little as 10 probes, up to 200, depending on the distance of the segment from the sender, and its link rate. Larger distances increase the accumulated cross traffic noise, resulting in slower convergence (compare charts 4a and 4b). How- ever, segment #9 is further away yet convergence is faster than at segment #6. This is due to the fact that the delay variation increment induced by a hop is inversely proportional to its rate, and segment #9 is the (suspected) bottleneck. Because the assumption of probes arriving in different busy periods is fundamental for the ACCSIG method, it is not possible to obtain estimates more quickly by sending probes closely spaced. It should be noted however that this assumption is of key importance for pathchar and its variants as well: reducing the time of the measurement while keeping the number of probes constant has a negative impact on the accuracy of all one-packet based methods. In the case of minimum filtering the reason for this is particularly clear. Probes which closely follow each other Relative error uncorrected est. upper estimate lower Relative error uncorrected est. upper estimate lower Pathchar Clink Accsig Accsig* PQ1 PQ2 PQ3 PT1 Measurement methods Relative error uncorrected est. upper estimate lower (a) 100 Mbps (c) 1.94 Mbps Correction for invisible hops Correction for invisible hops Correction for invisible hops Fig. 5. Relative error over multiple hop segments. (a) 1 st segment of local route (2x100Mbps LAN), (b) 2 nd segment of international route (2x10Mbps LAN), (c) 9 th segment of international route, measured as the bottleneck (2x2Mbps). will be likely to share busy periods with other probes in at least one router in the route, and therefore will not be traversing the system with minimal delay as required. In figure 5, three further examples from the same experiments are given for segments with multiple hops. Once again pathchar, clink and ACCSIG give very similar estimates, however they are half of the true link rate! One of the main contributions of this paper is to point out that this is due to invisible hops. We describe this problem in more detail in the next section ( II-D.1), and contribute to its resolution in section III. D. Practical Issues The measurement results reveal that all three methods sometimes suffer significant errors even in the case of relatively low bandwidth segments. This section describes a number of factors which can significantly effect the accuracy of estimates. Some are generic and shared by all three methods and even more widely, others hold only for pathchar and clink. D.1 Invisible hops Unfortunately, in real networks not all switching elements test and decrease the TTL value in the IP header, and therefore be- come, for practical purposes, invisible to IP-TTL based probing [15]. For example Ethernet and ATM switches, although store and forward switching elements, are below the IP layer and do not touch IP's TTL, as illustrated on figure (6). Some IP routers are even configured to not decrease TTL. As a result of this, TTL based measurement methods sometimes give estimates corresponding to TTL 'segments', that is sequences of hops between which the TTL does not change, rather than individual hops. For example the bandwidth estimate IP;1 from the first stage of clink, pathchar and ACC- SIG for the route shown on figure (6) (which corresponds to the Link level ("real") path representation path representation IP,1 IP,2 level SENDER RECEIVER LAN Switch Hop #3 Hop #2 Hop #1 IP Router Packets with IP-TTL=1 dropped here Fig. 6. Invisible hops: the lower view of the route shows all hops, the upper only those that test and decrease IP's TTL. three-hop two-segment local test route), will be: IP;1 =X which, using ttl 1 to index segments, generalises to where h ttl is the last hop before the pacesetter is dropped, and by convention h This effect is very likely to be responsible for the errors in the estimates of figure (5). The original estimates are marked as uncorrected, and show significant estimation errors. However using our knowledge about the presence of invisible hops and their effects, as detailed for example in equation (13) for ACC- SIG, the estimates can be corrected. Unfortunately this correction is impossible if the presence of invisible hops is unknown. Packet quartet based methods, introduced in section III, provide an efficient means of detection. D.2 High bandwidth links The estimates for high bandwidth links unfortunately are too unreliable to support meaningful comparison. To understand why this is the case for the delay variation and peak detection underlying ACCSIG, consider that with probes of 1500 and 40 bytes, a 12sec error in the estimate of the increase in inter- peak distance for two consecutive ttl values, corresponding to a 6sec error for a single peak, produces a 10% error in the estimate of a 100Mbps link. The accuracy of the peak detection is mainly a function of the number of probes used and the amount of cross traffic noise entering via the waiting time components of equation (9), which increase with ttl. Note that pathchar and clink estimates are subject to time-stamping errors, ranging from 10s of sec to 100s of ms, which can lead to similar estimation errors. D.3 Invasiveness Pathchar and clink determine the sending times of probes, by default, based on the round-trip times. This, as noted in [8], can result in loads as high as 60% on a 10 Mbps LAN with a round-trip delay of 1ms. Especially on highly utilised network routes, this approach can lead to probes joining the queues in the same busy period as the previous probe. As such a probe by definition won't experience minimum delay, the minimum filtering step of Pathchar and clink either filters out these probes, in which case the methods do not gather any information form these probes, or they will lead to erroneous minimum delay estimates leading to link rate estimation errors. In the case of ACCSIG, by recursively applying Lindley's equation (16), and using equation (8), it can be shown that the delay variation for such a probe is (w h where s(i) is the last hop along the route where probe i shared a busy period with probe i 1, and c s(i) i is the aggregated service time of the cross traffic entering that queue between probes and i. By comparing equations (8) and (14) it is clear that the accumulation signature before hop s(i) is destroyed by this new effect, so that if a significant number of probes experience it, an erroneous link bandwidth estimate will result. This condition can be detected however, as the definition of the peaks and the symmetry of the histogram is lost, and estimates become er- ratic. Our suggestion in this case it to send probes with increased spacing. This is less invasive, and probably much more efficient, than the traditional wisdom of simply transmitting more probes. To limit invasiveness, in all the measurements in this paper the probe streams (for ACCSIG and for the PQ methods) were dimensioned to remain below 64kbps on average. Increasing the spacing of probes will naturally decrease the probability of them sharing busy periods, but, if the number of probes remains the same the measurement time will increase proportionally. D.4 Lower layer packet headers Packet pair like methods are based on estimates from the service time of a single probe, whereas those underlying pathchar,clink and ACCSIG are based on differences of service times. Because of this the additional headers of link layers can affect packet pair based estimates, as investigated in detail in [11], but they leave unaffected the results of the pathchar variant methods, where the packet size change cancels out. Thus, different methods see different values, an important point to understand regardless of whether one prefers to measure the bandwidth as seen by IP or the link layer. Non-linearities Another practical problem can arise when the delay is non-linear in packet size, as shown in [15]. Such non-linearities are typically the result of lower layer effects, such as layer 2 frag- mentation. If the network layer payload is fragmented, the resulting additional lower layer header(s) decreases the effective rate seen by the network layer. Another example could be a minimum frame size at layer 2, as for example in the case of Ethernet. In such cases the spread of different packet sizes used by pathchar and clink can be an advantage. ACCSIG can also use a large(er) number of different sized probes, at the possible expense of an increase in peak detection errors. D.6 Forwarding time If the time needed to generate the ICMP message in response to the UDP packet is not constant, for example if it has a component depending on packet size, it can lead to erroneous band-width estimates. This will not only effect the estimate of a given hop, but will propagate as TTL is increased, as the link rates are extracted using a sum over previous TTL values. This error is large on the last hop for both pathchar and clink as discussed above. III. PACKET QUARTETS The methods of the previous section rely on the accurate timing of returning ICMP packets. The resulting reliance on appropriate router behaviour, plus the additional noise of the return path(s), are key drawbacks. One would prefer to exploit TTL's ability to access hops individually along the route in some other way. The tailgater method described by Lai and Baker in [1] is one way of doing this. It is derived from a deterministic model of packet delay and consists of two phases. First, by sending isolated probes and using linear regression on the minimum delay for each size, 1= h and are estimated. In the second phase pairs of packets are sent, each consisting of two packets sent back-to-back: a packet of the smallest possible size, the tailgater, follows the largest possible (non-fragmented) packet with a limited TTL. The goal is to cause the tailgater to queue directly behind the large packet at each hop until the TTL expires, enabling the accumulated service time to be measured from that hop on. The final bandwidth estimates are determined using minima filtering for each TTL, combined with the components measured in the first phase. Pacesetter packet, with a limited TTL Probe packet, delivered to the receiver Sender Receiver Fig. 7. Packet quartets: pacesetter packets have limited TTL, while the probes reach the receiver. Drawbacks of the tailgater method include those related to minima filtering, and the need to perform two phases. To address these and other issues we introduce packet quartets (PQ) (see figures (7) and (8)) as a flexible probe pattern underlying a new family of link bandwidth estimation methods. A packet quartet is formed by grouping any two of a sequence of packet pairs, each consisting of a probe packet closely following a pacesetter packet, which has a limited TTL. The pairs, similarly to the probes of the ACCSIG method, are sent sufficiently separated to avoid joining queues during the same busy period. Within pairs the opposite is true, the probe and the pace- setter are designed to be in the same busy period at each hop. Practical limitations to achieving this, and their consequences, are discussed in the next paragraph. Apart from the pair spac- ing, packet quartets have six parameters: the two probe sizes the two pacesetter sizes p s;i 1 and p s;i , and the last hops that the pacesetters traverse before dropping out: hops A and B. In what follows we discuss a four packet pattern as in figure 7. It should not be forgotten however that when i is incremented the order of A and B reverse, negating the accumulation terms to produce the other half of the symmetric histograms, as before. SENDER RECEIVER Packet #3 Packet #2 Packet #4 Packet #1 Hop A Hop B Fig. 8. Packet quartets: pacesetter packets #1 and #3 have limited TTL, while probes, packets #2 and #4, reach the receiver. Within a pair, the following effect is important to understand. Assume that the probe and its pacesetter are back-to-back and that the latter has just arrived at an empty queue on hop h. Because of the store and forward router behaviour, the probe will not enter the queue until after an interval x h 1 controlled by the previous hop, whereas the pacesetter will leave the queue after a time x h controlled by hop h. Thus, as pointed out in [1], the ability of the pacesetter to guarantee that the probe queues behind it is bounded by the ratio of the probe and pacesetter service times, which depends on the ratio of the link rates. Thus, the ratio of the packet sizes is both a significant design issue and a practical limit of the applicability of these methods to high speed links. However, separation of the pair members does not invalidate packet quartets for three reasons. First, as the delay variation based model will make clear, it is not essential that the probes and pacesetters be back-to-back, the fundamental requirement is that they share the same busy period. Second, the separation may only persist over a few hops. If the probe catches up to its pacesetter's busy period then the pair will resume its role from that point. Finally, it is not neccessary that all pairs stay together, but only that enough do so to make the target signature large enough to be detected. A. A delay variation based model We adopt the convention that probes are those packets we observe and measure, as they arrive at the receiver. As in section II, we focus on the delay variation of the probes, which are indexed by i. As pacesetter packets are (typically) dropped before reaching the receiver, they are not classified as probes. Instead, they manifest as special components of the waiting times that probes experience, characterised by a known duration and persistence across hops. From equation (8) it is obvious that - (j) i , the delay variation up to hop j, can be expressed as: Lindley's equation connects the waiting times of successive packets. With an infinite buffer, it states If we assume that probe i will enter hop h to join the same busy period as its pacesetter, the probe's waiting time becomes: where c h s;i is the aggregate service time of cross traffic packets which arrived at hop h between the arrival of the pacesetter at the instant h s;i , and of the probe at h . Since by definition the arrival time to hop h is the departure time from hop h 1, the last component can be rewritten as: for all h > 1. Here we are assuming that the probe and its pacesetter also share the same busy period at hop i 1. Recursively applying equations (15), (17) and (18), the delay variation of the probes of a general packet quartet becomes: A A (w h (w h (w h s;i is the interarrival time of the pacesetters at the first hop. The properties of the packet quartet delay variation are qualitatively different depending on which of the following scenarios the TTL values lie: where H is the number of hops. The case with corresponds to probes only being sent from the sender, which was discussed in section II. In each of these regions of the quartet design space, new methods will be defined. Their properties and limitations will be compared against each other as well as the methods of section II. IV. PACKET QUARTET LINK ESTIMATION METHODS In this section 5 canonical estimation approaches, PQ1, PQ2, PQ3, PT1 and PT2, arising out of packet quartets, will be described and their main properties derived. Simulations are used to illustrate these properties, but are not meant to test performance under all conditions. Similarly, the network comparisons provide a basis for discussion of the primary features of the methods, and of important issues such as invisible hops, rather than a full scale validation. A comparison against the tail- gater method of [1] is not provided as an implementation was not available (we expected it to be included in the nettimer package, however its documentation states that it is not [16]). A. PQ1 and SENDER RECEIVER Packet #3 Packet #2 Packet #4 Packet #1 Acc. Component Sender -> A Acc. Component A->Receiver Hop A Fig. 9. Packet quartets with 0 < This TTL scenario is illustrated in figure (9). For it equation (19) reduces to: A A (w h (w h There are two service time, or more specifically accumula- tion, components in this equation. The first is an accumulation signature due to the difference in the size of the pacesetters, and as such it is created between hop #1 and A, after which the pace- setters are dropped. The second is due to the difference in the size of the probes, and is created from hop A to the receiver. If both of these were active it would be difficult to determine which term had contributed in what proportion to the resulting - i value. Fortunately, they can be selectively eliminated by choosing probes or pacesetters to be of equal size. This leads to two more specialised probe patterns which form the basis of two new bandwidth estimation methods. We first define the methods, then compare them and investigate their properties. For the remainder of this subsection we collect the non service time components of equation (20) into A.1 Method PQ1 Setting a constant, for each probe cancels the second accumulation component, leaving the defining equation for method PQ1: A Equations (21) and (10) differ only in their noise terms. As a result, link bandwidth estimates of individual hops can be made in exactly the same way as for ACCSIG. A.2 Method PQ2 Setting constant for each pacesetter cancels the first accumulation component, leaving the defining equation for method PQ2: Equations (21) and (22) are formally very similar, but the methods defined by them have some very different properties. A.3 PQ1 and PQ2 compared The essential difference with PQ2 is that, to determine the bandwidth of hop h ttl , the difference of delay variations from stages h ttl and h ttl+1 must be taken, instead of from h ttl and h ttl 1 as for PQ1, ACCSIG and the pathchar variant methods. As one important consequence of this, the link bandwidth estimates of segments including invisible hops will be different for the two methods. While the estimates from PQ1 will be formed as described in equation (13) for ACCSIG, those from PQ2 will obey This effect is shown in figure (10), from which it is easily seen Link level ("real") path representation path representation IP,1 IP,2 level SENDER RECEIVER LAN Switch Hop #3 Hop #2 Hop #1 IP Router Packets with IP-TTL=1 dropped here pathchar,clink,ACCSIG and PQ1 based estimate PQ2 based estimate Fig. 10. Invisible hops - PQ1 and PQ2 produce different hop bandwidth estimates in the presence of an invisible hop that the bandwidth of the last hop, hop 2, of the first segment, can be measured. This is true more generally: the last hop of segments containing invisible hops can be accessed. In effect this exploits a kind of 'edge effect' arising from the differences in the two equations. The second important consequence regards the peak detection accuracy needed. Recall that measuring the distance between the symmetric peaks in the - i histogram is the basis of the accumulation signature based estimate here just as for ACCSIG. In PQ2 the peak detection is based on the difference in the probe sizes. However this is necessarily small, since by the definition of packet quartets, probes are significantly smaller than the pacesetters, and this will also be true of their difference. As a result the distance between the peaks for PQ2 will be significantly smaller than that for PQ1 on the same hop, which makes PQ2 intrinsically less accurate than PQ1. The third consequence again favors PQ1. Comparing equation (20) to equation (9), we find the additional noise component c A describing the cross traffic arriving between probes and their pacesetters. On heavily loaded hops this term can significantly bias the estimates. Now for simplicity assume stationary cross traffic on hop A and let the components of each pair be back-to-back on leaving hop A 1. Since for PQ1 the probe size is constant, the time period x A 1 i in which the cross traffic may infiltrate between the probe and its pacesetter on hop A is the same for all i. The cross traffic components will then be statistically identical and tend to cancel each other. For PQ2 this is not the case, as the probes sizes are different (in fact deliberately as different as possible to minimise sensitivity to peak detection errors). For PQ2 this additional asymmetric cross traffic breaks the symmetry of the distributions of - causing significant problems for median based peak detection. Other peak detection methods should therefore be used under heavy network load. A.4 PQ1, PQ2 simulation results Simulations results of a 6 hop route with both zero and moderate cross traffic are shown in table II. The accuracy of PQ2 drops sharply in the presence of cross traffic, whereas PQ1 performs extremely well in both cases. Similar simulations of the same 6 hop route, but with two hops made invisible, leaving 4 segments, is given in table III. Just as for ACCSIG, the results for segments may not correspond directly to the known link rates, but are predicted by the 'hop mixing' equations, equation (13) for PQ1 (and ACCSIG) and equation (23) for PQ2. Of these, that of PQ1 is easier to interpret. From table III (no CT) we see that for single hop segments we recover the link rate, and for segments with two hops of the same rate, the method recovers half of the rate. Although less obvious, with no cross traffic PQ2 also gives values which obey its respective mixing equation. Method PQ2 however performs less well under cross traffic, as can be seen by comparing the method's no CT and with CT columns of table III. Large differences between PQ1 and PQ2 can be seen as evidence for the presence of invisible hops. A.5 PQ1, PQ2 measurement based comparisons Results for selected segments of the 2 netork routes our new methods are shown in figures 3 and 5. The PQ1 based estimates are in good agreement with pathchar, clink and ACCSIG results. The main difference with PQ1 is that the number of noise components only depends on the number of hops, rather than varying with TTL across the different stages of the measurement. The aggregate noise is therefore lower, roughly speaking, for the second half of the route, due to the lack of backward path. This is a great advantage on long routes. The above feature of PQ1 is shared by PQ2. Another is that neither rely on receiving ICMP error messages, which eliminates forwarding time errors entirely. The cost of these advantages is a higher sensitivity to peak detection errors. For example, the PQ1 measurements were performed using 1500 and 730 byte pacesetter packets followed by probes, and the PQ2 measurements employed 1500 byte pacesetters followed by 40 and 100 byte probes. With these val- ues, a given peak detection error would cause an estimation error for PQ1 approximately twice as large as that for ACCSIG, and this ratio increases to approximatly 20 for PQ2. PQ2's higher sensitivity to peak detection errors is demonstrated by the first hop estimate of the two hop route (fig. 5), which is off by 50% from the 100Mbps expected from equation (23). The PQ2 based estimates of the 1 st and 2 nd , and 8 th and 9 th segments of the international route, by their inconsistency with Nominal PQ1 PQ2 PQ3 PT1 PT2 link bandwidth no CT with CT no CT with CT no CT with CT no CT with CT no CT with CT 100 100 100 20:7 100 143 100 143 inf 0 1. 100Mbps 100 100 99.99 15.8 100 120 100 120 0.4 0 2. 10Mbps 4. 2Mbps 2 5. 2Mbps 2 2.02 2 3.08 2 2.00 2 1.98 2 0.24 II SIMULATION BASED LINK BANDWIDTH ESTIMATES OF A 6 HOP ROUTE. FOR EACH METHOD: COLUMN #1 NO CROSS TRAFFIC, COLUMN #2 60% CROSS TRAFFIC LOAD ON HOP #4, 30%ON THE OTHER HOPS. PQ1 as discussed above, reveal the invisible hops we know (seg- ments 1, 2) or expect (9) are there. B. SENDER RECEIVER Packet #3 Packet #2 Packet #4 Packet #1 Hop A Hop B Acc. Component Sender -> A Edge Component A Edge Component B Acc. Component B->Receiver Acc. Component Fig. 11. Packet quartets with 0 < A < B This TTL scenario is illustrated in figure (11). Unlike the case where A = B, here even when the probes sizes are chosen and the pacesetter sizes also, p an accumulation term remains, and the delay variation equation (19 reduces to: where N A;B denotes all the non service time terms. In the special case of becomes the defining equation of method PQ3: With this method in effect a single hop has been isolated through an edge effect. It is also different in that the histograms have only a single peak if we set t . If we assume that A is known, and the noise term has symmetric char- acteristics, then an estimate of the unique peak ^ - can be easily obtained, leading to: The link bandwidth of the first hop, the access link rate of the sender, is usually known, allowing the remaining bandwidths to be estimated recursively from equation (25) (if it is not known it can be estimated by starting the measurement from A = 0, which results in - . Further discussion of the cases is deferred until section IV-C). Note that the estimate from one stage of the measurement is used to obtain the estimate for the next, which can potentially lead to a magnification of errors. This is not the case with the previously discussed pathchar variants, nor with PQ1 and PQ2, as there the estimates follow directly from the difference of the estimates of only two measurement stages. PQ3's estimates for segments with invisible hops will suffer from estimation errors of a different nature to those seen so far, as can be understood from setting equation (24). From the difference of equations (24) and (25), the bandwidth estimate in such a case, the hop mixing equation, is seen to be: IP;h ttl 1+h ttl which is similar to equation (13), and for sufficiently small p=p s ratio, it would give the same results. For the packet sizes used in our measurements, 1500 byte pacesetters and 40 byte probes, this ratio is 0:026. B.1 PQ3 simulation results Returning to the 6 hop simulation comparison in table II, we find that PQ3 performs similarly to PQ1, but with some positive bias for large link rates. In the case of invisible hops in table III it is again very consistent with PQ1 with no cross traffic, but has more errors with it. Nominal link bandwidth PQ1 PQ2 PQ3 PT1 (Segment details) no CT with CT no CT with CT no CT with CT no CT with CT 100 100 9:09 15:8 100 143 100 143 1. 100Mbps 100 100 9.09 10.5 100 120 100 120 100 100 9:09 7:8 100 103 100 103 5 5:06 1:67 1:83 5:07 5:18 5:07 5:09 2. 10Mbps 5 5.04 1.67 1.74 5.07 5.15 5.07 5.04 3. 2Mbps 1 III SIMULATION BASED LINK BANDWIDTH ESTIMATES OF THE SAME 6 HOP ROUTE, 2 HOPS (2ND AND 4TH) MADE INVISIBLE. FOR EACH METHOD: COLUMN #1 NO CROSS TRAFFIC, COLUMN #2 60% CROSS TRAFFIC LOAD ON HOP #3, ELSE 30%. B.2 PQ3 measurement based comparison Returning to figures 3 and 5, we find that, again, the estimates are in good agreement with pathchar, clink and ACCSIG results for link bandwidths up to approximately 30Mbps, however the differences seem to be slightly larger than in the case of PQ1. The differences could be caused by the somewhat different estimation of invisible segments, as well as by the fact that estimates from equation (25) are affected by lower layer packet headers in a similar way to packet pair based estimates, as discussed in section II-D.4. PQ3's sensitivity to peak detection errors is similar to that of ACCSIG, which is an advantage over PQ1. Another advantage is that due to the fixed and equal probe sizes, and pacesetter sizes, the achievable pacesetter/probe ratio is higher than that of PQ1 and PQ2, allowing the measurement of a wider spectrum of hop bandwidths. However, the possible magnification of estimation errors is an important disadvantage of this method. C. PT1 and The special case of packet quartets with to only one pacesetter packet being sent out from the sender, as illustrated in figure (12), in effect reducing the quartets to triplets. SENDER RECEIVER Packet #3 Packet #2 Packet #4 Packet #1 Acc. Component B+1->Receiver Acc. Component Sender -> B Edge Component B Fig. 12. Packet quartets with As before, packet sizes can be selected to cause the cancellation of the accumulation terms. However it is impossible to cancel both terms simultaneously, unless the probes and their pacesetters have the same size, which is incompatible with the latter's aim of causng the probes to queue, of controlling their pace. Hence, two methods, corresponding to the cancellation of only one accumulation term at a time, will be defined. Setting for each probe cancels the second accumulation component, leaving the defining equation for method PT1: The properties of this method are very similar to those of PQ3. The link bandwidths in this case can be estimated from the estimate of the unique delay variation peak ^ as: where again we have set t . The comment regarding the potential accumulation of errors in the case of method PQ3 is also valid here, and the effect of invisible hops on PT1's bandwidth estimates, equation (27), is exactly the same. Note, that PT1 holds a similar relation to the tailgater method as ACCSIG does to pathchar and clink. As a con- sequence, most of the causes of error in PT1's estimates are shared with tailgater. The most important differences between the two are the result of the delay variation foundation of replacing delay: there is no need for two different probing phases, nor for minimum filtering or linear regression. C.2 Method PT2 Setting constant for each pacesetter results in the cancellation of the first accumulation component, leaving the defining equation for method PT2: Unfortunately, as the result of increased sensitivity to the peak detection errors and the propagation of these errors, this method is useless in practice as we now show. For completeness we include here the hop mixing equation, which is similar but different to equation (23): IP;h ttl 1+h ttl+1 C.3 PT1, PT2 simulation results We return a final time to the 6 hop simulation comparison in table II. The striking result is that even with no cross traf- fic, PT2's sensitivity and error propagation causes its estimates to deteriorate rapidly. The initial peak detection error in this case has its origin in the limited resolution of the simulation timestamps, which was chosen to be equivalent to the 2 resolution of the timestamp representation of our measurement infrastructure. This tiny error becomes exponentially amplified. In constrast, the results of PT1 are generally very good, and very consistent with PQ3 as expected. The same is true for the invisible hop results of table III for PT1. C.4 PT1 measurement results We only present measurement results in the case of PT1, as performs so poorly. Again, from figures 3 and 5 we find that the estimates are very similar to those of PQ3. The comments regarding the effect of the link layer headers, the sensitivity to peak detection errors, and the propagation of errors, also hold. Note that PT1, in conjunction with other methods, has the potential to detect invisible hops. Let us assume the knowledge of the 1= h of the ttl segments, for example using PQ1. In this case, x B remains the only unknown of equation (28), which corresponds to the service time of the probes on the last hop of the route segment containing an invisible hop or hops, ie., the service time of an invisible hop. However, the sensitivity to peak detection errors becomes similar to that of PQ2, although the propagation of the errors won't affect the estimates of B for fixed. However, the fact that PT1's estimate can potentially suffer from the link layer header error, means that PQ2 seems a more efficient tool for detecting the invisible hop anatomy of segments. V. INVESTIGATING THE BOTTLENECK Based on the properties discussed above of PQ1 and PQ2 in estimating segments with invisible hops (see equations (13) and (23)) and the results for segments 8 to 10 of the international route (PQ1: [10:1; 0:98; 22:2] Mbps, and PQ2: [2:13; 1:9; 27:4] Mbps), we concluded that the ninth hop consists of two 2Mbps hops. This conclusion is supported by independent measurements made using the fundamentally different packet-pair based methods [11]. VI. CONCLUSIONS A variety of link bandwidth measurement methods were presented that exploit the ability of the IP-TTL field to target hops within a route to enable measurements over multiple links. The approach is novel in its use of delay variation, and histogram based peak detection using medians, rather than delay series and filtering on minima. The information of each probe packet is used, resulting in greater efficiency, that is using less probes for fixed accuracy. The first new method presented, ACCSIG, uses returning ICMP messages like pathchar and clink, and gives similar results, but is more efficient. ICMP based methods however suffer from greater noise due to a longer round-trip path, and router dependent processing delays. To avoid these, packet quartets were introduced: a 6 parameter probe stream class incorporating pairs consisting of probes trapped behind larger pacesetter packets with limited TTL values. From this class, 5 new measurement methods were defined: PQ1, PQ2, PQ3, PT1 and PT2, their main characteristics analysed, and their performance illustrated and evaluated with simulation and accurate network measurements over two routes. One of these, PT1, is closely related in terms of TTL use to the existing tail- gater method, but with the advantages of being delay variation based. The properties of the new methods are summarized in table VI with respect to several metrics of practical importance, which were described in detail. Rows 1-4 give rankings, where best. We analysed in detail the consequences of invisible hops: the fact that TTL detects IP hops or segments, which often consist of multiple link layer hops. The methods were shown to fall into 2 fundamental categories depending on their behaviour over seg- ments. The first group includes Pathchar (and its variants), ACCSIG, PQ1, PQ3 and PT1, while PQ2 and PT2 belong to the second group. Invisible hops can be detected by comparing the results from a member of each group. The last invisible hop in the segment can be accessed by cancelling terms between the methods, for example by combining results from PQ1 and PQ2. Each group can be divided further into two sub-categories, defined by their hop mixing equations, summarised in table VI. In summary, method PQ1 is the best overall, offering accurate estimates with few probes and low noise. ACCSIG is also efficient and accurate. PQ2 can also be useful but is less ac- curate, whereas PQ3 is considerably less accurate. PT1 is very similar to PQ3 but even less accurate. Finally PT2 is very sensitive and magnifies errors, and is useless in most situations. Performance Metric ACCSIG PQ1 PQ2 PQ3 PT1 PT2 Sensitivity to peak 2 3 detection errors (1-4) Sensitivity to lower layer headers (1-3) propagation Sensitivity to forwarding errors (1-2) Mixing equation IV COMPARISON OF NEW METHODS. Further work is needed to fully characterise and compare the methods' performance under a variety of traffic conditions and parameter settings. In addition, methods to automatically detect invisible hops and to fully resolve them are the subject of ongoing work. --R "Measuring link bandwidths using a deterministic model of packet delay," "What do packet dispersion techniques measure?," "Using pathchar to estimate internet link characteristics," "Congestion avoidance and control," "Characterizing end-to-end packet delay and loss in the inter- net," "Measuring bottleneck link speed in packet-switched networks," "End-to-end internet packet dynamics," "Measuring bandwidth," "Multifractal cross-traffic estimation," "End-to-End Available Band- Measurement Methodology, Dynamics, and Relation with TCP Throughput," "The packet size dependence of packet-pair like methods," "On the scope of end-to-end probing methods," "A precision infrastructure for active probing," "PC based precision timing without GPS," "patchar - a tool to infer characteristics of internet paths," "Nettimer: A tool for measuring bottleneck link bandwidth," --TR Congestion avoidance and control Measuring bottleneck link speed in packet-switched networks End-to-end Internet packet dynamics Using pathchar to estimate Internet link characteristics Measuring link bandwidths using a deterministic model of packet delay PC based precision timing without GPS End-to-end available bandwidth --CTR A. Novak , P. Taylor , D. Veitch, The distribution of the number of arrivals in a subinterval of a busy period of a single server queue, Queueing Systems: Theory and Applications, v.53 n.3, p.105-114, July 2006 Guojun Jin , Brian L. Tierney, System capability effects on algorithms for network bandwidth measurement, Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, October 27-29, 2003, Miami Beach, FL, USA Matthew Luckie , Tony McGregor, Path diagnosis with IPMP, Proceedings of the ACM SIGCOMM workshop on Network troubleshooting: research, theory and operations practice meet malfunctioning reality, September 03-03, 2004, Portland, Oregon, USA Sridhar Machiraju , Darryl Veitch , Franois Baccelli , Jean Bolot, Adding definition to active probing, ACM SIGCOMM Computer Communication Review, v.37 n.2, April 2007 Constantinos Dovrolis , Parameswaran Ramanathan , David Moore, Packet-dispersion techniques and a capacity-estimation methodology, IEEE/ACM Transactions on Networking (TON), v.12 n.6, p.963-977, December 2004 Sridhar Machiraju , Darryl Veitch, A measurement-friendly network (MFN) architecture, Proceedings of the 2006 SIGCOMM workshop on Internet network management, p.53-58, September 11-15, 2006, Pisa, Italy Neil Spring , David Wetherall , Thomas Anderson, Reverse engineering the Internet, ACM SIGCOMM Computer Communication Review, v.34 n.1, January 2004 Mahajan , Neil Spring , David Wetherall , Thomas Anderson, User-level internet path diagnosis, Proceedings of the nineteenth ACM symposium on Operating systems principles, October 19-22, 2003, Bolton Landing, NY, USA

active probing;cross-traffic;delay variation;bottleneck bandwidth;TTL;internet measurement

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The monadic theory of morphic infinite words and generalizations.

We present new examples of infinite words which have a decidable monadic theory. Formally, we consider structures {N, >, P} which expand the ordering {N, >} of the natural numbers by a unary predicate P; the corresponding infinite word is the characteristic 0-1-sequence xp of P. We show that for a morphic predicate P the associated monadic second-order theory MTh{N, >, P) is decidable, thus extending results of Elgot and Rabin (1966) and Maes (1999). The solution is obtained in the framework of semigroup theory, which is then connected to the known automata theoretic approach of Elgot and Rabin. Finally, a large class of predicates P is exhibited such that the monadic theory MTh(N, >, P) is decidable, which unifies and extends the previously known examples.

Introduction In this paper we study the following decision problem about a xed !-word x: Given a Buchi automaton A, does A accept x? If the problem (Acc x ) is decidable, this means intuitively that one can use x as external oracle information in nonterminating nite-state systems and still keep decidability results on their behaviour. We solve this problem for a large class of !-words, the so-called morphic words and some generalizations, complementing and extending results of Elgot and Rabin [11] and Maes [15]. The problem (Acc x ) is motivated by a logical decision problem regarding monadic theories, starting from the fundamental work of Buchi [8] on the equivalence between the monadic second-order theory MThhN;<i of the linear order hN; <i and !-automata (more precisely: Buchi automata). Buchi used this reduction of formulas to automata to show that MThhN;<i is decidable. The decidability proof is based on the fact that a sentence of the monadic second-order language of hN; <i can be converted into an input-free Buchi automaton A such that holds in hN; <i i A admits some successful run; the latter is easily checked. It was soon observed that Buchi's Theorem is applicable also in a more general situation, regarding expansions of hN; <; P i of the structure hN; <i by a xed predicate P N. Here one starts with a formula (X) with one free set variable and considers an equivalent Buchi automaton A over the input alphabet is true in hN; <i with P as interpretation of X i A accepts the characteristic word xP over B associated with P (the i-th letter of xP is 1 In other words: The theory MThhN; <; P i is decidable if one can determine, for any given Buchi automaton A, whether A accepts xP . Elgot and Rabin [11] followed this approach via the decision problem (Acc xP ) and found several interesting predicates P such that MThhN; <; P i is decidable, among them the set of factorial numbers n!, the set of k-th powers n k for any xed k, and the set of k-powers k n for any xed k. Elgot and Rabin (and later also Siefkes [18]) used an automata theoretic anal- ysis, which we call contraction method, for the solution of the decision problem (Acc xP ). The idea is to reduce the decision problem to the case of ultimately periodic !-words (which is easily solvable). Given P as one of the predicates mentioned above, Elgot and Rabin showed that for any Buchi automaton A, the 0-sections in xP can be contracted in such a way that an ultimately periodic !-word is obtained which is accepted by A i xP is accepted by A. For the ultimately periodic word , one can easily decide whether it is accepted by A. More recently, A. Maes [15] studied \morphic predicates", which are obtained by iterative application of a morphism on words (see Section 2 for deni- tions). The morphic predicates include examples (for instance, the predicate of the Fibonacci numbers) which do not seem to be accessible by the Elgot-Rabin method. Maes proved that for any morphic predicate P , the rst-order theory decidable, and he also introduced appropriate (although spe- cial) versions of morphic predicates of higher arity. It remained open whether for each morphic predicate P , the monadic theory MThhN; <; P i is decidable. In the present paper we answer this question positively, based on a new (and quite simple) semigroup approach to the decision problem (Acc xP In Section 2 we show that for morphic predicates P the problem (Acc xP ) is decidable. As a consequence, we nd new examples of predicates P with decidable monadic theory MThhN; <; P i. Prominent ones are the Fibonacci predicate (consisting of all Fibonacci numbers) and the Thue-Morse word predicate (consisting of those numbers whose binary expansion has an even number of 1's). In the second part of the paper, we embed this approach into the framework of the contraction method. This method is shown to be applicable to predicates which we call \residually ultimately periodic". We prove two results: Each mor- phic predicate is residually ultimately periodic, and a certain class of residually ultimately periodic predicates shares strong closure properties, among them under sum, product, and exponentiation. This allows to obtain many example predicates P for which the monadic theory MThhN; <; P i is decidable. It should be noted that for certain concrete applications (such as for the Fibonacci predicate) the semigroup approach is much more convenient than an explicit application of the contraction method, and only an analysis in retrospect reveals the latter possibility. Also morphic predicates like the Thue-Morse word predicate are not approachable directly by the contraction method, because there are no long sections of 0's or 1's. Let us nally comment on example predicates P where the corresponding Buchi acceptance problem (Acc xP ) and hence MThhN; <; P i is undecidable, and give some comments on unsettled cases and on related work. First we recall a simple recursive predicate P such that MThhN; <; P i is undecidable. For this, consider a non-recursive, recursively enumerable set Q of positive numbers, say with recursive enumeration m recursive but even the rst-order theory FThhN; <; P i is undecidable: We have some element x of P , the element x is the next element after x in P , a condition which is expressible by a rst-order sentence k over hN; <; P i. Buchi and Landweber [9] and Thomas [19] determined the recursion theoretic complexity of theories MThhN; <; P i for recursive P ; it turns out, for example, that for recursive P the theory MThhN; <; P i is truth-table-reducible to a complete -set, and that this bound cannot be improved. The situation changes when recursive predicates over countable ordinals are considered (see [22]). In [20] it was shown that for each predicate P , the full monadic theory MThhN; <; P i is decidable i the weak monadic theory WMThhN; <; P i is (where all set quantiers are assumed to range only over nite sets). However, there are examples P such that the rst-order theory FThhN; <; P i is decidable but WMThhN; <; P i is undecidable ([19]). In the present paper we restrict ourselves to expansions of hN; <i by unary predicates. For expansions hN; <; fi by unary functions f , the undecidability of MThhN; <; fi arises already in very simple cases, like for the doubling function 2n. For a survey on these theories see [16]. On the other hand, it seems hard to provide mathematically natural recursive predicates P for which MThhN; <; P i is undecidable. A prominent example of a natural predicate P where the decidability of MThhN; <; P i is unsettled is the prime number predicate. As remarked already by Buchi and Landweber in [9], a decidability proof should be dicult because it would in principle answer unsolved number theoretic problems like the twin prime hypothesis (which states that innitely many pairs primes exist). The only known result is that MThhN; <; P i is decidable under the linear case of Schinzel's hypothesis [4]. Part of the results of the present paper has been presented at the conference MFCS'2000 [10]. automata over morphic predicates A morphism from A to itself is an application such that the image of any word an is the concatenation (a (an ) of the images of its letters. A morphism is then completely dened by the images of the letters. In the sequel, we describe morphisms by just specifying the respective images of the letters as in the following example Let A be a nite alphabet and let be a morphism from A to itself. For any integer n, we denote n the composition of n copies of . Let the sequence of nite words dened by (a) for any integer n. If the rst letter of (a) is a, then each word xn is a prex of xn+1 . If furthermore the sequence of length jx n j is not bounded, the sequence converges to an innite word x which is denoted by ! (a). The word x is a xed point of the morphism since it satises Example 1 Let consider the morphism given by The words are respectively equal to It can be easily proved by induction on n that n+1 Therefore, the xed point ! (a) is equal to the innite word abc 2 bc 4 bc 6 bc An innite word x over B is said to be morphic if there is a morphism from A to itself and a morphism from A to B such that In the sequel, the alphabet B is often the alphabet and the morphism is letter to letter. The characteristic word of a predicate P over the set N of non-negative integers is the innite word over the alphabet B dened by b predicate is said to be morphic i its characteristic word is morphic. These denitions are illustrated by the following two examples. Example 2 Consider the morphism introduced in the preceding example and the morphism given by The morphic word actually the characteristic word of the predicate Ng. This can be easily proved using the equality 1. The square predicate is therefore morphic. It will proved in the sequel that the class of morphic predicates contains all predicates of the form fn k jn 2 Ng and fk n jn 2 Ng for any xed integer k. Example 3 Consider the morphism from B to B dened by The xed point is the characteristic word of the predicate P with the binary expansion of n contains an even number of 1. The xed point ! (1) is the well-known Thue-Morse word. We refer the reader to [5] for an interesting survey on that sequence and related works of A. Thue. Recall that a Buchi automaton is an automaton is a nite set of states, E Q A Q is the set of transitions and I and F are the sets of initial and nal states. A path is successful if it starts in an initial state and goes innitely often through a nal state. An innite word is accepted if it is the label of a successful path. We refer the reader to [21] for a complete introduction. Now we can state the main result and in the corollary, its formulation in the context of monadic theories: Theorem 4 Let (a)) be a xed morphic word where : A ! A and are morphisms. For any Buchi automaton A, it can be decided whether x is accepted by A. As explained in the introduction, the theorem can be transferred to a logical decidability result, invoking Buchi's Theorem [8] on the equivalence between monadic formulas over hN; <i and Buchi automata: Corollary 5 For any unary morphic predicate P , the monadic second-order theory of hN; <; P i is decidable. Proof (of Theorem automaton. Dene the equivalence relation over A by F F means that there is a path from p to q labeled by u and p u F means that there is a path from p to q labeled by u which hits some nal state. This equivalence relation captures that two nite words have the same behavior in the automaton with respect to the Buchi acceptance condition. It was already introduced by Buchi in [8]. Denote by the projection from A to A = which maps each word to its equivalence class. The equivalence relation is a congruence of nite index. Indeed, for any words u, v, u 0 and v 0 , the following implication holds: If This property allows us to dene a product on the classes which endows the set A = with a structure of nite semigroup. The projection is then a morphism from A onto A =. Furthermore, for any xed states p and q, there are at most three possibilities for any word u: Either there is a path from p to q through a nal state, or there is a path from p to q but not through a nal state or there is no path from p to q labeled by u. This proves that the number of classes is bounded by 3 n 2 where n is the number of states of the automaton. The following observation gives the main property of the congruence . Suppose that the two innite words x and x 0 can be factorized and x k for any k 0. Then x is accepted by A accepted by A. Indeed, suppose that x is the label of a successful path. This path can be factorized where q 0 is initial and where the nite path q k a nal state for innitely many k. Since u k u 0 k for any k 0, there is another path where the nite path q k a nal state whenever q k does. This proves that x 0 is also the label of a successful path and that it is accepted by A. In particular, if an innite word x can be factorized that u 1 u 2 , then x is accepted by A i u 1 is also accepted by A. Since a is the rst letter (a), the word (a) is equal to au for some nonempty nite word u. It may be easily veried by induction on n that for any integer n, one has The word can be factorized We claim that there are two positive integers n and p such that for any k n, the relation u k u k+p holds. This relation is equivalent to (u k morphism from A into a semigroup is completely determined by the images of the letters. Therefore, there are nitely many morphism from A into the nite semigroup A =. This implies that there are two positive integers n and p such that This implies that k greater than n, and thus u k u k+p . Note that these two integers n and p can be eectively computed. It suces to check that ( n (b)) ( n+p (b)) for any letter b of the alphabet A. Dene the sequence (v k ) k0 of nite words by v 1. The word x can be factorized and the relations v 1 v 2 v 3 hold. This proves that the word x is accepted by the automaton A i the word v 1 is accepted by A. This can be obviously be decided. 3 A large class of morphic predicates The purpose of this section is to give a uniform representation of a large class of morphic predicates: We will show that each predicate k 0 and a polynomial Q(n) with non-negative integer values is morphic. This supplies a large class of predicates P where the decision problem (Acc xP ) and hence the monadic theory of hN; <; P i is decidable. In particular, the classical examples fk xed k in N of [11] are covered by this. As a preparation, we shall develop a sucient condition on sequences to dene a morphic predicate. It will involve the notion of a N-rational sequence. These considerations will also show that the Fibonacci predicate fFn (where We refer the reader to [17] for a complete introduction and to [3] for a survey although we recall here the denitions. Definition 6 A sequence (un ) n0 of integers is N-rational if there is a graph G allowing multiple edges, sets I and F of vertices such that un is the number of paths of length n from a vertex of I to a vertex of F . The graph G is said to recognize the sequence (un ) n0 . An equivalent denition is obtained by considering non-negative matrices. A sequence (un ) n0 is N-rational i there is a matrix M in N kk and two vectors L in B 1k and C in B k1 such that un = LM n C. It suces indeed to consider the adjacency matrix M of the graph and the two characteristic vectors of the sets I and F of vertices. It also possible to assume that the two vectors L and respectively belong N 1k and N k1 instead of B 1k and B k1 since the class of N-rational sequences is obviously closed under addition. A triplet (L; M;C) such that un = LM n C is called a matrix representation of the sequence and the integer k is called the dimension of the representation. The following example illustrates these notions. Example 7 The number of successful paths of length n in the graph pictured in Figure 1 is the Fibonacci number Fn where F This shows that the sequence (Fn ) n0 is N-rational. This sequence has the following matrix representation of dimension 2 which can be deduced from the graph of Figure 1. Figure 1: A graph for the Fibonacci sequence We now state the main result of this section. Theorem 8 Let un be a sequence of non-negative integers and let un be the sequence of dierences. If there is some integer l such that dn 1 for any n l and such that the sequence (d n ) nl is N-rational, then the predicate As a rst illustration of the Theorem, reconsider the predicate of the Fibonacci Each dierence is equal to Fn 1 , it obviously satises dn 1 for n 1, and the sequence (Fn ) n0 is N-rational as shown in Example 7. So the Fibonacci predicate is morphic. With the following corollary we obtain a large collection of morphic pred- icates, including the predicates of the form fn k considered in [11]. Corollary 9 Let Q be a polynomial such that Q(n) is integer for any integer n and let k be a positive integer. The predicate 0g is morphic. The proof of the corollary is entirely based on the following lemma. a polynomial with a positive leading coecient such that Q(n) is integer for any integer n. Let k be a positive integer and let un be dened by un = Q(n)k n . There is a non-negative integer l such that the sequence Note that such a polynomial may have non-integer coecients as the polynomial Proof Let d be the degree of Q. We claim that there are then a non-negative integer l and positive integers a a d such that for any n, d a i Let Q l (n) be the polynomial Q(n + l). Since Q l (n) is integer for any integer n, the polynomial Q l (n) is equal to a linear combination of the binomials with integer coecients (see [12, p. 189]). There is a unique sequence a of d integers such that for any integer n d a l;i Since the binomials satisfy the well-known relation n+1 , it follows that the coecients a l;i satisfy the following relation. For any integer l, one has a l+1;d = a l;d and a a l;i+1 for i < d. Since the leading coecient of Q l is positive, the coecient a l;d is also positive for any l. Using the relation on the coecients on the a l;i , it can be proved by induction on the dierence d i that for any i d, there is an integer l i such that a l;i is positive for any l l i . The integer l dened the required property. We now dene the matrix M and the vectors L and C as follows. The vector L is the vector of dimension d d. The matrix M is the square matrix of dimension d dened by d. The vector C is the the vector of dimension d It is pure routine to prove by induction on n that LM n is the vector Ln given by L for 0 i d. Therefore un+l is equal to LM n C for any and the sequence (un+l ) n0 is N-rational. The proof of Theorem 8 needs some preliminary result that we now state. This lemma makes easier the proof that certain predicates are morphic. It essentially states that the property of being morphic is preserved by shifting and by changing a nite number of values. In particular, if two predicates P and P 0 coincide for almost every n, then P is morphic i P 0 is morphic. Lemma 11 Let P be a predicate and let R be a nite set of integers. Let k be a non-negative integer and let P 0 be the predicate R [ is morphic i P 0 is morphic. Proof Let P be a morphic predicate. Assume that the characteristic word xP of P is equal to ( ! (a)) where and are respectively morphisms from A into itself and from A into B . Let x be the xed point ! (a) and let u be the nite word such that (a) = au. The innite word x can be then factorized We rst prove that both predicates P are also morphic. Let A 0 be the alphabet A [ where a 0 and a 1 are two new symbols. Dene the morphism 0 from A 0 into itself by 0 (a any b in A. It is clear that the xed point Dene the morphism 0 from A 0 into B by 0 (a 1 any b in A. If 0 (a 0 ) is set to 0, then the word is the characteristic word of P 0 and if 0 (a 0 ) is set to 1, then the word is the characteristic word of P 00 We now prove that there is a positive integer k such that the predicate be the alphabet A [ where a 0 is a new symbol. Dene the morphism 0 from A 0 into itself by any b in A. It is clear that the xed point Let k be the length of u. Dene the morphism 0 from A 0 into B by (a any b in A. The word 0 is the characteristic word of P 0 . The claimed result follows then easily from the two previous results. The following two results will be used in the proof of Theorem 8. The rst result is due to Schutzenberger. We refer the reader to [17, Thm II.8.6] and [6, Thm V.2.1] for complete proofs. Theorem 12 (Sch utzenberger 1970) Let (un ) n0 be a N-rational sequence such that un 1 for any n. The sequence (un 1) n0 is also N-rational. For the next result, we need the notion of a D0L-sequence. A N-rational sequence (un ) n0 of integers is said to be a D0L-sequence if it can be recognized by a graph all of whose states are nal. Equivalently, a N-rational sequence (un ) n0 is a D0L-sequence if it has a matrix representation (L; M;C) such that all components of L are either 0 or 1 and such that all components of C are equal to 1. We have then the following result [17, Lem III.7.4]. Every N-rational sequence (u n ) n0 can be decomposed into D0L- sequences. This means that there are two integers K and p such that each sequence (u k+np ) n0 for k K is a D0L-sequence. We nally come to the proof of Theorem 8. Proof We suppose that un satises dn 1 for n l and that the sequence (d n )n l is N-rational. By Lemma 11, it may be assumed without loss of generality that l = 0. By Theorem 12, the sequence (v n ) n0 dened by that sequence can be decomposed into D0L-sequences. There are two two integers K and p such that each sequence (v k+np ) n0 for k K is a D0L-sequence. By Lemma 11, it may be assumed again without loss of generality that and that u the sequence v Each sequence v k has a matrix representation such that each component of L k are either 0 or 1 and such that all components of C k are equal to 1. Let d k be the dimension of that representation. Dene the alphabet A by Dene the morphism from A into itself by a 7! ac L0;1 c k;i 7! c Mk;i;1 It can be easily proved by induction on n that Y where each w k;i is a word on the alphabet fc k;j j 1 j d k g of length v k;i . Actually the number occurrences of the letter c k;j in the word w k;i is the j-th component of the vector L k M i k . Dene then the morphism from A into B by As a complement to Corollary 9 (that predicates fQ(n)k n j n 2 Ng are morphic), we formulate a necessary condition for a predicate P to be morphic. It turns out that the factorial predicate fn! j n 2 Ng does not fall under this condition, which shows that it is not morphic. In the subsequent section, we shall develop a framework where such predicates can be handled together with the morphic ones, thus unifying the contraction method of Elgot and Rabin with the results above. Proposition 14 Let (un ) n0 be a strictly increasing sequence of integers. If the predicate fun integer k. Proof Suppose that the characteristic word of the predicate fun j n 2 Ng is equal to ( ! (a)) where and are respectively morphisms from A into itself and from A into B . Let x be the xed point ! (a) and let u be the nite word such that (a) = au. The innite word x can be then factorized Let k be the integer dened by (b)j. It can be easily shown by induction on n that j n (b)j k n for any letter b. Let B the set of letters b such that (b) contains at least one 1, that is g. We claim that if for a xed letter b, there is an integer n such that n (b) contains a letter of B, the smallest integer satisfying this property is smaller than the cardinality of A. This follows from the fact that a letter c appears in n (b) i there is a sequence of letters such that b appears in (b i ) for 0 i < n. For any integer n, the word n (a) contains therefore a letter of B and the result follows easily. Residually ultimately periodic predicates In this section, we develop a framework which merges the contraction method of Elgot and Rabin with the semigroup approach used for morphic predicates. The common generalization of the two approaches is captured by the notion of residually ultimately periodic innite words. We introduce these words and we show that morphic words belong to this class. The application to the contraction method is developed in the subsequent section. Definition 15 A sequence (un ) n0 of words over an alphabet A is said to be residually ultimately periodic if for any morphism from A into a nite semigroup S, the sequence (un ) is ultimately periodic. This property is said to be eective i for any morphism from A into a nite semigroup, two integers n and p such that for any k n, can be eectively computed. An innite word x is called residually ultimately periodic if it can be factorized where the sequence (u n ) n0 is eectively residually ultimately periodic. The following proposition shows how this property can be used for deciding the problem (Acc x Proposition 16 If the innite word x is residually ultimately periodic, the problem (Acc x ) is decidable. Proof Let A = (Q; E; I ; F ) be a Buchi automaton. Dene the equivalence relation over A by The relation is a congruence of nite index. The application which maps any nite word to its class is therefore a morphism from A into the nite semigroup A =. This congruence has the following main property. Suppose that the two innite words x and x 0 can be factorized such that u k u 0 k for any k 0. Then x is accepted by A i x 0 is accepted by A. Since the sequence (u n ) n0 is residually ultimately periodic, there are two integers n and p such that for any k greater than n, Therefore, x is accepted by A i the word v 1 is accepted by A where v We illustrate this notion by the following example. Example 17 The sequence of words (a n! ) n0 is residually ultimately periodic. This sequence is actually residually ultimately constant. It is indeed well-known that for any element s of a nite semigroup S and any integer n greater than the cardinality of S, s n! is equal to a xed element usually denoted s ! in the literature [2, p. 72]. The sequences of words which are residually ultimately constant have been considerably studied. They are called implicit operations in the literature [2]. A slight variant of the previous example shows that the sequence (u n ) n0 dened by un residually ultimately constant. Since (n equal to nn! 1, the word u 0 u 1 is the characteristic word of the factorial predicate Ng. The monadic theory of hN; <; P i where P is the factorial predicate is therefore decidable by the previous proposition. In the following propositions we connect the residually ultimately periodic sequences to the innite words obtained by iterating morphisms. Proposition be a morphism from A into itself and let u be a word over A. The sequence un = n (u) is residually ultimately periodic, and this property is eective. Proof Let be a morphism from A into a nite semigroup S. Since S is nite, there are nitely many morphisms from A into S. Therefore, there are two integers n and p such that This implies that for any k greater that n, one has . Note that the two integers n and p can be eectively computed. It suces to nd n and p such that ( n any letter a in A. It follows from the proposition that a morphic word has a factorization whose factors form a residually ultimately periodic sequence. Let x be a morphic word are morphisms and let u be the nite word such that au. The word x can be factorized and un = ( n (u)) for n 1. By the proposition, the sequence ( n (u)) n0 is residually ultimately periodic. The sequence (u n ) n0 is therefore also residually ultimately periodic. 5 The predicate class K The decidability of the decision problem (Acc x ) for an innite word x involves a \good" factorization of the word x. In the case morphic words, this factorization is naturally provided by the generation of x via a morphism. Another approach is to consider the canonical factorization of the word x induced by the blocks which form the word x (representing the distances between the successive elements of the predicate). The contraction method as developed by Elgot and Rabin [11] and Siefkes [18] reduces the sequence of these nite words from 0 1 to an ultimately periodic sequence. In this section we embed this method into the framework developed above. If strictly increasing sequence of integers, the characteristic word xP of the predicate can be canonically factorized un is the word 0 kn+1 kn 1 1 over the alphabet B . If this sequence of words is residually ultimately periodic and if furthermore this property is eective, it is decidable whether the word xP is accepted by a Buchi automaton and the monadic theory of hN; <; P i is therefore decidable. We will prove that the class of sequences such that this property holds contains interesting sequences like (n k ) and that it is also closed under several natural operations like sum, product and exponentiation The following lemma essentially states that it suces to consider the sequence un = a kn+1 kn over a one-letter alphabet. Lemma 19 Let (un ) n0 be a sequence of words over A and let a be a letter. The sequence (un ) n0 is residually ultimately periodic i the sequence (u n a) n0 is residually ultimately periodic. Moreover this property is eective for i it is eective for (un a) n0 . Proof It is clear that if the sequence (u n ) n0 is residually ultimately periodic, then the sequence (un a) n0 is also residually ultimately periodic. Indeed, for any morphism from A into nite semigroup, the relation implies that if the sequence (un ) n0 is ultimately periodic, then the sequence (una) is also ultimately periodic. Conversely let be a morphism from A into a nite semigroup S. Let ^ S be the semigroup S 1 S with the product dened by (s; t)(s be the morphism from A into ^ S dened by It may be easily veried by induction on the length on the word w that Therefore, if the sequence (un a) n0 is residually ultimately periodic, the sequences (una) and (un ) are ultimately periodic. The sequence (u n ) n0 is then residually ultimately periodic If A is the one-letter alphabet fag, the semigroup A is isomorphic to the set N of integers by identifying any word a n with the integer n. Therefore, a sequence of integers is said to be residually ultimately periodic i the sequence a kn is residually ultimately periodic. K be the class of increasing sequences (k n ) n0 of integers such that the sequence (k n+1 kn ) n0 is residually ultimately periodic. By Lemma 19 and by Proposition 16, we conclude: Theorem be in K and let xP be the characteristic word of the predicate Ng. Then the decision problem (Acc xP ) is decidable and the monadic theory MThhN; <; P i is also decidable. We give two comments, the rst one on the relation between the class K and residually ultimately periodicic sequences, the second one on a similar class of predicates introduced by Siefkes [18]. As stated in Corollary 28 below, each sequence in K is residually ultimately periodic. The converse is not true as the following example shows. Let be an innite word over the alphabet B , that is b Consider the sequence dened by )!. If the word x has innitely many occurrences of 1, the sequence (k n ) n0 is then residually ultimately periodic (cf. Example 17). However, if the word x is not ultimately periodic, the sequence residually ultimately periodic. In [18], Siefkes studies predicates generated by sequences two conditions: to be \eectively ultimately reducible" (which corresponds to being eectively residually ultimately periodic), and to have an essentially increasing sequence of dierences kn+1 kn , i.e., for each d 0, we have eectively computable from d). The second assumption ensures that the sequence (k n+1 kn ) n0 is residually ultimately periodic in our sense. Although most natural predicates of the class K can also be treated by Siefkes' approach, there are some exceptions. For instance, let dened by is even and by is odd. This sequence (k n ) n0 is in the class K but Siefkes' second assumption does not apply. The following theorem shows that the class K contains interesting sequences and that it is closed under several natural operations. Theorem 22 Any sequence (k n ) n0 such that the sequence (k n+1 kn ) n0 is N-rational belongs to K. If the sequences belong to K, the following sequences also belong to K: (sum and product) kn (exponentiation) k l n for a xed integer k and k l n (generalized sum and product) By Lemma 10, the class K contains any sequence of the form k n Q(n) where k is a positive integer and Q is a polynomial such that Q(n) is integer for any integer n. By applying the generalized product to the sequences the sequence (n!) n0 belongs to K. The closure by dierences shows that K contains any rational sequence of integers such that lim n!1 (k n+1 kn rational sequence of integers is the dierence of two N-rational sequences [17, Cor. II.8.2]. The class K is also closed by other operations. For instance, it can be proved that if both sequences belong to K, then the sequence dened by belongs to K. The class K is closed under sum, dierence product and exponentiation but the following example shows that it not closed under quotient. Example 23 Consider the sequence This sequence is not residually ultimately periodic since the sequence (k n mod 4) is not ultimately periodic. It turns out that (k n mod p) is not ultimately periodic unless [1]. It can be easily seen that the greatest integer l such that 2 l divides kn is the number of 1 in the binary expansion n. Therefore, (k n mod 4) is equal to 2 if n is a power of 2 and to 0 otherwise. For two integers t and p, dene the equivalence relation t;p on N as follows. For any integers k and k 0 , one has The integers t and p are respectively called the threshold and the period of the relation t;p . Note that the relation k t;p k 0 always implies that mod p. The equivalence relation t;p is of nite index and it is compatible with sums and products. Indeed if k t;p k 0 and l t;p l 0 hold then both relations relations t;p for t and p capture the property of being residually ultimately periodic for sequences of integers. Lemma 24 A sequence (k n ) n0 of integers is residually ultimately periodic i for any integers t and p there are two integers t 0 and p 0 such that for any n greater than t 0 , one has kn t;p kn+p 0 . Proof Indeed, this condition is sucient since for any element s of a nite semigroup, there are two integers t and p such that s . Conversely, this condition is also necessary. The set N= t;p equipped with addition is a nite semigroup and the canonical projection from N to N= t;p is a morphism. The following result is almost trivial but it will often be be used. Lemma l be l relations associated with xed integers residually ultimately periodic sequences of integers. There are two integers r and q such that k j;n t i ;p i for any 1 i l, any 1 j m and any n greater than r. Proof Since each sequence (k i;n ) n0 is residually ultimately periodic, there are two integer r i;j and q i;j such that k j;n t i ;p i for any n greater than t i;j . The two integers r and q dened by the required property. The following lemma states that the class of residually ultimately periodic sequences of integers is closed under sum and product. These results follow from a more general result which states the class of residually ultimately periodic sequences of words are closed under substitution. More precisely, if the sequence of morphisms from A into B is such that each sequence ( n (a)) n0 is residually ultimately periodic and if the sequence (u n ) n0 of words over A is also residually ultimately periodic, then the sequence ( n (un residually ultimately periodic. If each word un is for instance equal to the xed word ab and if the morphism n maps a to vn and b to wn , the word n (u) is equal to vn wn . The class of residually ultimately periodic sequences of words are closed under concatenation. If un is equal to a kn and if n (a) is equal to a l n , then n (un ) is equal to a kn l n . We give here a direct proof of these results which relies on the compatibility of any relation t;p with sums and products. Lemma 26 Let (k n ) n0 and (l n ) n0 be two residually ultimately periodic sequences of integers. Both sequences are also residually ultimately periodic. If lim n!1 (k n l n then the sequence (k n l n ) n0 is also residually ultimately periodic. The following example shows that the assumption on the limit of the sequence kn l n is really necessary. Let be an innite word over the alphabet B . Consider the sequences (k n ) n0 and (l n ) n0 dened by These two sequences are obviously residually ultimately periodic (cf. Example 17). However, the dierence kn l n is equal to n!. If the word x is not ultimately periodic, the sequence (k n l n ) n0 is not residually ultimately periodic. Proof By Lemma 25, there are then r and q such that kn t;p kn+q and l n t;p l n+q for any n greater than r. This yields kn and kn l n t;p kn+q l n+q for n greater than r. The sequences are residually ultimately periodic. The relations kn t;p kn+q and l n t;p l n+q imply that and l This yields kn l the dierence kn l n is greater than t for all n greater than some r 0 . Then for any n greater than r and r 0 , one has kn l n t;p kn+q l n+q for n greater than r 0 and the sequence (k n l n ) n0 is residually ultimately periodic. The following lemma states the class of residually ultimately periodic sequences of integers is closed under generalized sum and product when l It will be used to prove the general case. Lemma be a residually ultimately periodic sequence of inte- gers. The sequences (Kn ) n0 and (Ln ) n0 dened by are also residually ultimately periodic. Proof Let t;p be the relation associated with two xed integers t and p. There are then two integers r and q such that kn t;p kn+q for any n grater than r. Let k be the sum . Note that integer l. There are then two integers r 0 and q 0 such that r 0 k t;p (r that , Kn t;p Kn+qq 0 holds for any n greater than r Indeed, one has r t;p r t;p r t;p r The proof for Ln is very similar. It suces to replace each sum by a product. For instance, the constant k is dened by and the two integers r 0 and q 0 are chosen such that k r 0 . The previous lemma has the following corollary. Corollary 28 Any sequence (k n ) n0 in K is residually ultimately periodic. The following lemma is needed to prove that the class K is closed under generalized sum and product. residually ultimately periodic sequences of integers. Both sequences (Kn ) n0 and (Ln ) n0 dened by are then residually ultimately periodic. Proof Let t;p be the relation associated with two xed integers t and p. There are then two integers r and q such that kn t;p kn+q for any n grater than r. Let k be the sum . Note that integer n greater than r. There are then two integers r 0 and q 0 such that r 0 k t;p (r Lemma 25, there are also two integers r 00 and q 00 such that l n r;q l n+q 00 and dn r+r 0 q;qq 0 dn+q 00 for any n greater than r 00 . We claim that Kn t;p Kn+q 00 for any n greater than r 00 . We rst claim that Kn+q 00 t;p the result obviously holds. Otherwise, one has l n r and l n+q 00 r. Since l has k i t;p k i+l where l = l n+q 00 l n for any l n i. This proves the claim. We now prove that i=ln t;p Kn . If the result holds obviously. Otherwise, one has dn r we may assume that dn+q 00 > dn . Since dn+q suppose that dn+q 00 l. One has then t;p l n+dn+lqq 0 t;p l n+dn r 0 q t;p l n+dn r 0 q t;p l n+dn X The proof for Ln is similar. It suces to replace each sum by a product. We nally come to the proof of theorem 22. Proof We prove that any sequence (k n ) n0 such that the sequence (k n+1 kn ) n0 is N-rational belongs to K. This follows from Theorem 8 and from Proposition but we also provide a direct proof. If the sequence (d n ) n0 is dened by is N-rational, there is a matrix representation (L; M;C) such that the relation t;p to matrices by setting M t;p M 0 i the relation M k;l t;p M 0 k;l holds for any (k; l)-entry of the matrices. There are then two integers r and q such that M r t;p M r+q and this implies M n t;p M n+q for n greater than r. Thus, one has dn t;p dn+q for n greater than r. If both sequences belong to to K, Lemma 26 applied to the sequences (k n+1 kn ) n0 and (l n+1 l n ) n0 shows that the sequence belongs then to K. If furthermore, the assumption on the limit is fullled, it also shows that the sequence (k n l n ) n0 belongs then to K. The dierence kn+1 l n+1 kn l n is equal to kn+1 (l n+1 l n )+(kn+1 kn )l n . By Lemma 27, the sequences (k n ) n0 and (l n ) n0 are residually ultimately periodic. By Lemma 26, the sequence of dierences is then residually ultimately periodic and the sequence (k n l n ) n0 belongs then to K. The dierence k l n+1 k l n is equal to k l n (k l n+1 l n 1). By lemma 27, both sequences k l n and k l n+1 l n are residually ultimately periodic. By Lemma 26, the sequence of dierences is then residually ultimately periodic and the sequence belongs then to K. Let Kn be the sum . The dierence Kn+1 Kn is equal to the sum . By Lemma 27, the sequence (l n ) n0 is residually ultimately periodic and by Lemma 29, the sequence of dierences is then residually ultimately periodic. Let Kn be the product . The dierence Kn+1 Kn is equal to l n Y l n+1 Y By Lemma 29, the sequence (Ln ) n0 dened by residually ultimately periodic. By Lemma 27 applied to the sequence (Ln ) n0 , the sequence dened by L 0 residually ultimately periodic. By Lemma 26, the sequence of dierences is then residually ultimately periodic and the sequence (Kn ) n0 belongs then to K. Let dn be the dierence kn+1 kn , let d 0 n be the dierence k l n+1 n . We assume that the sequence (k n ) n0 is strictly increasing and we also assume that l n 2 for n greater than some constant which can be eectively computed. We prove that the sequence (d 0 residually ultimately periodic. Let t;p be the relation associated with two xed integers t and p. We rst claim that greater than t. On has indeed the following inequalities Since the sequence (k n ) n0 is assumed to be strictly increasing, dn is non-zero and kn is greater than t for any n greater than t. By Lemma 25, there are two integers s and m such that k n t;p k n+m for any integer k and any integer n greater than s. Note that if n s and if k t;p l, then k n t;p l n t;p l n+m . By Lemma 25, there are two integers r and q such that kn t;p kn+q , dn t;p dn+q 0 , l n s;m l n+q for any integer n greater than r. We claim that d 0 for any integer n greater than r. n+q t, it suce to prove that d 0 n+q mod p. The relations kn t;p kn+q and l n s;m l n+q 0 imply k l n n+q . The relation kn+1 t;p kn+q+1 and l n+1 r;q l n+q+1 imply k l n+1 n+q+1 . This two relations then imply d 0 n+q mod p. Conclusion We have introduced a large class of unary predicates P over N such that the corresponding Buchi acceptance problem Acc xP (and hence the monadic theory decidable. The class contains all morphic predicates (which solves a problem of Maes [14, 15]). The connection to the work by Elgot and Rabin [11] and Siefkes [18] was established by extending the class of morphic predicates to the class of the residually ultimately periodic predicates. Finally, strong closure properties (under sum, product, and exponentiation) were shown for certain residually ultimately periodic predicates (where the sequence of differences of successive elements is residually ultimately periodic). Altogether we obtain a large collection of concrete examples P such that MThhN; <; P i is de- cidable, containing the k-th powers and the k-powers for each k, the value sets of polynomials over the integers, the factorial predicate, the Fibonacci predicate, as well as the predicates derived from morphic words like the Thue-Morse word and the Fibonacci word. Let us mention some open problems. Our results do not cover expansions of hN; <i by tuples predicates rather than by single predicates. In his dissertation, Hosch [13] has solved the problem for the special case of the predicates P Ng. We do not know whether MThhN; <; decidable if the P i are just known to be residually ultimately periodic. There should be more predicates P for which the Buchi acceptance problem Acc xP and hence the theory MThhN; <; P i is decidable. A possible next step is to consider Sturmian words, a natural generalization of morphic words (see [7]). Finally, we should recall the intriguing question already asked by Buchi and Landweber in [9] (\Problem 1"): Is there an \interesting" recursive predicate P such that MThhN; <; P i is undecidable? How about P being the prime number predicate? Acknowledgement The authors would like to thank Jorge Almeida, Jean-Paul Allouche, Jean Berstel and Jacques Desarmenien for very interesting suggestions and comments. --R Linear cellular automata Finite Semigroups and Universal Algebra. Decidability and undecidibility of theories of with a predicate for the primes. Axel Thue's work on repetitions in words. Rational Series and Their Languages. The monadic theory of morphic in Decidability and undecidibility of extensions of second Concrete Mathe- matics Decision Problems in B Decidability of the An automata theoretic decidability proof for the Open questions around B Decidable extensions of monadic second order successor arith- metic The theory of successor with an extra predicate. On the bounded monadic theory of well-ordered structures Automata on in Ehrenfeucht games --TR Rational series and their languages Automata on infinite objects Linear cellular automata, finite automata and Pascal''s triangle Open questions around BuMYAMPERSANDuml;chi and Presburger arithmetics An automota theoretic decidability proof for first-order theory of <inline-equation> <f> <fen lp="ang"><blkbd>N</blkbd>,MYAMPERSANDlt;,P<rp post="ang"></fen></f> </inline-equation> with morphic predicate P Concrete Mathematics Automata The Monadic Theory of Morphic Infinite Words and Generalizations Ehrenfeucht Games, the Composition Method, and the Monadic Theory of Ordinal Words Decision problems in buechi''s sequential calculus --CTR Alexander Rabinovich, On decidability of monadic logic of order over the naturals extended by monadic predicates, Information and Computation, v.205 n.6, p.870-889, June, 2007 Jean Berstel , Luc Boasson , Olivier Carton , Bruno Petazzoni , Jean-Eric Pin, Operations preserving regular languages, Theoretical Computer Science, v.354 n.3, p.405-420, 4 April 2006 Erich Grdel , Wolfgang Thomas , Thomas Wilke, Literature, Automata logics, and infinite games: a guide to current research, Springer-Verlag New York, Inc., New York, NY, 2002

morphic predicates;second-order

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New Interval Analysis Support Functions Using Gradient Information in a Global Minimization Algorithm.

The performance of interval analysis branch-and-bound global optimization algorithms strongly depends on the efficiency of selection, bounding, elimination, division, and termination rules used in their implementation. All the information obtained during the search process has to be taken into account in order to increase algorithm efficiency, mainly when this information can be obtained and elaborated without additional cost (in comparison with traditional approaches). In this paper a new way to calculate interval analysis support functions for multiextremal univariate functions is presented. The new support functions are based on obtaining the same kind of information used in interval analysis global optimization algorithms. The new support functions enable us to develop more powerful bounding, selection, and rejection criteria and, as a consequence, to significantly accelerate the search. Numerical comparisons made on a wide set of multiextremal test functions have shown that on average the new algorithm works almost two times faster than a traditional interval analysis global optimization method.

Introduction . In this paper the problem of nding the global minimum f of a real valued one-dimensional continuously dierentiable function f and the corresponding set S of global minimizers is considered, i.e.: In contrast to one-dimensional local optimization problems which were very well studied in the past, the univariate global optimization problems are in the area of interest of many researchers now (see, for example, [1, 6, 10, 9, 11, 14, 16, 15, 17, 21, 28, 30, 32]). Such an interest is explained by existence of a large number of applications where it is necessary to solve this kind of problems (see [2, 10, 12, 29, 30, 31]). On the other hand, numerous approaches (see, for example, [5, 11, 13, 18, 19, 22, 23, 26, 31]) enable to generalize to the multidimensional case methods developed to solve univariate problems. In those cases where the objective function f(x) is given by a formula, it is possible to use an Interval Analysis Branch-and-Bound approach to solve the problem (1.1) (see [13, 19, 20, 23]). A general global optimization algorithm based on this approach is shown in Algorithm 1. The algorithm selects the next interval to be processed (selection rule, line 3), which can be totally or partially rejected when it is guaranteed that it does not contain any global minimizer, (elimination rule, line 4). This elimination process is carried out by using information obtained from the inclusion functions which return y E-mail: [email protected] z E-mail: [email protected] x E-mail: [email protected] { Department of Computer Architecture and Electronics, University of Almera, Spain. This work was supported by the Ministerio de Educacion y Cultura of Spain (CICYT TIC99-0361). k E-mail: [email protected]. ISI-CNR c/o DEIS, Universita della Calabria, 87036 Rende (CS) Italy, and University of Nizhni Novgorod, Nizhni Novgorod, Russia. L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV general interval Branch-and-Bound global optimization algorithm. Funct IGO(S; f) 1. Set the working list L := fSg and the nal list Q := fg 2. while ( L 6= fg ) 3. Select an interval X from L 4. if X cannot be eliminated 5. Divide X generating the termination criterion 7. Store X i in Q 8. else 9. Store X i in L 10. return Q an enclosure of the real range of f(x) (and in some cases of f 0 (x) and f 00 (x)) on X (bounding rule). If the interval cannot be rejected, it is subdivided (division rule, line 5). When the generated subintervals are informative enough, they are stored in the nal list (termination rule, line 7). Otherwise, they are stored in the working list for further processing (line 9). The algorithm nishes when there are no intervals to be processed (line 2) and returns the set of intervals with valuable information (line 10). An overview on theory and history of these rules can be found, for example, in [13]. Of course, every concrete realization of Algorithm 1 depends on the available information about the objective function f(x). In this paper it is supposed that inclusion functions can be evaluated for f(x) and its rst derivative f 0 (x) on X . Thus, the information about the objective function which can be obtained during the search is the following: where dened by its lower and upper bound. inclusion function of f(x) at X obtained by Interval Arithmetic. For the real range of f(x) over X the following inclusion inclusion of the function f(x) at a point x 2 X . inclusion function of f 0 (x) over X . When information (1.2) is available, the rules of a traditional realization of Algorithm 1 can be written more precise. Below we present a Traditional Interval analysis global minimization Algorithm with Monotonicity test (TIAM) which is used frequently for solving the problem (1.1) using the information (1.2) (see [13]). Selection rule: Among all the intervals X i stored in the working list L select an interval X such that Bounding rule: The fundamental Theorem of Interval Arithmetic provides a natural and rigorous way to compute an inclusion function. In the present study the inclusion function F of the objective function f is available by Extended Interval Arithmetic (see [7, 13]). Elimination rule: The common elimination rules are the following: Midpoint test: An interval X is rejected when F (X) > f~, where f~ is the best known upper bound of f . The value of f~ is updated by evaluating F (m(X)) in selected intervals, where is the midpoint of the interval X . Cut-o test: When f~ is improved, all intervals X stored in the working and nal lists satisfying the condition F (X) > f~ are rejected. Monotonicity test: If for an interval X the condition then this means that the interval X does not contain any minimum and, therefore, can be rejected. Division rule: Usually two subintervals are generated using m(X) as the subdivision point (bisection). Termination rule: A parameter determines the desired accuracy of the problem solution. Therefore, intervals X that have a width w(X) less than , i.e., are moved to the nal list Q. Other termination criteria can be found in [23]. As can be seen from the above description, the algorithm evaluates lower bounds for f(x) over every interval separately, without taking into consideration information which can be obtained from other intervals. F 0 (X) is used only during the Monotonicity test and is not connected with the information F (m(X)) and F (X). Only F (X) is used in order to obtain a lower bound for f(x) over X , all the rest of the search information is not used for this goal. The only exchange of information between the intervals is done through f~. In Lipschitz global optimization there exist algorithms for solving problem (1.1) evaluating lower bounds by constructing support functions (in fact, F (X) can be viewed as a special support function { constant { for f(x) over X) for the objective function too (see, for example, [6, 10, 11, 17, 18, 21, 22, 27, 28, 30, 31]). They work in a way similar to the TIAM support functions are built and successively improved in order to obtain a better lower bound for the global minimum. Of course, these support functions are completely dierent and are built on the basis of dierent ideas. An interesting aspect of the support function concept in the context of this paper is the use of the search information. When a support function is built for an interval [a; b], the information regarding neighbors [c; a] and [b; d] is also used in order to construct a better support function and to obtain a better lower bound for f . In this paper, a new Interval analysis global minimization Algorithm using Gradient information (IAG) is proposed for solving problem (1.1). It uses the same information (1.2) as TIAM but, due to a more e-cient usage of the search information, constructs support functions which are closer to the objective function and enables to obtain better lower bounds. As it will be shown hereinafter, the new method IAG has a quite promising performance in comparison with the traditional TIAM. The rest of the paper is structured as follows. In Section 2 some theoretical results explaining construction of the support functions and lower bounds are presented. The algorithm IAG is described in Section 3. Numerical experiments comparing performance of TIAM and IAG are presented in Section 4. Finally, Section 5 concludes the paper. 2. New support functions based on interval evaluations of the objective function and its rst derivative. In order to proceed with the description of the new algorithm, theoretical results are presented to explain how the new support functions and the corresponding lower bounds are constructed in IAG. We start with the following lemma illustrated in Figure 2.1, in a similar way as was done for non- 4 L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV __ __ __ __ F(c) f ~ Fig. 2.1. Graphical example of Lemma 2.1 and Theorem 2.2 dierentiable functions in both, Lipschitz optimization (see [9, 11, 18, 21, 22, 28, 30, 31]) and approaches based on evaluation of slopes as presented in [24]. Lemma 2.1. Given a continuously dierentiable function f a closed interval in R, an interval X S, an enclosure F (c) of f(c), c 2 X, and an enclosure then the following bounds hold for f(x); x 2 X: Proof. It follows from Mean Value Theorem that there exists a point 2 [x; c] such that By extending the equation (2.2) to intervals the following inclusion is obtained. Let us take a generic point x 2 X . In dependence on the mutual disposition of the points c and x in X , three results can be deduced from (2.3): Lemma is proved. This Lemma gives us a possibility to construct a new interval analysis support function for f(x). It can be seen from Figure 2.1 that it is similar to that ones built in Lipschitz global optimization (see, for example, [10, 21, 22, 27, 31]). The Lipschitz support functions are piece-wise linear. The slope of each linear piece is L or L, where L is the Lipschitz constant. In our approach, for every interval X the slopes of support functions are equal to F 0 (X) for all x c and to F 0 (X) for all x c (see Figure 2.1). The following results are the basics for the new support functions and explain how the new lower bounds for f are evaluated. Theorem 2.2. Given closed intervals X;S such that X S R and a continuously dierentiable function f R. Let for a point c 2 X a lower bound lb(c) of f(c) be determined and an enclosure F 0 (X) of f 0 (X) be obtained. For a given current upper bound f~ of f , there exists a set V X where all the global minimizers of X, if any, are included. Proof. For a minimizer point x 2 S it applies that f(x ) f~. Combining with Lemma 2.1 a minimizer x 2 X \ S has to fulll: and therefore can only be located in set: f~ From Theorem 2.2 it can be derived that if f~ < lb(c) then can be constructed as: As an example, depicted in Figure 2.1 for the case F 0 (X) > 0 Notice that V can be obtained in a similar way by applying the interval Newton operator to nd the roots of the function f(x) f~ on X , under the Theorem conditions [8, 19, 20]. Theorem 2.3. Let us consider a continuously dierentiable function f where S is a closed interval in R and intervals X; Y such that X Y S. If: 6 L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV f ~ __ G __ G __ __ Fig. 2.2. Interval V is the region which can contain global minimizers. The set XnV does not contain any global minimizer. 1. lower bounds lb(x) and lb(x) of, respectively, f(x) and f(x), have been evaluated 2. a current upper bound f~ of f is such that f~ minflb(x); lb(x)g; 3. bounds Then only the interval G can contain global minimizers and a lower bound z(X; lb(x); lb(x); G) of f(x) over the interval X can be calculated as follows: (see Figure 2.2). Proof. By applying Theorem 2.2 with x the interval V 2 from (2.7) is obtained. The same operation with the point us the interval V 1 from (2.6). Then, the interval V from (2.8) is obtained as Let us prove now the formula (2.9). Since X Y , F applying the Mean Value Theorem, we have and From these two inequalities follows so and which proves the theorem. Corollary 2.4. If for an interval X the inequality z(X; lb(x); lb(x); G) > f~ is fullled then it can be derived that X does not contain any global minimizer. Proof. Proof is evident and so it is omitted. Let us return now to the problem (1.1). We can use the information (1.2) during the global search. Thus, by using F (X) together with the function d(x) from (2.10) we can build a new support function D(x) for f(x) over each interval The corresponding new lower bound F z(X) for f(x) over the interval X is calculated in the following way Essential in the use of the algorithm is that for obtained from set X according to (2.8) the current value of f~ is a lower bound of f at v and v; i.e. f~ f(v) and f~ f(v), so are easily available bounds. 3. Description of the new algorithm. On the basis of theoretical results presented in the previous section we can determine new rules for Algorithm 1 in order to introduce the new Interval analysis global minimization Algorithm using Gradient information (IAG) described in Algorithm 2: Selection rule: Select an interval X such that where L is the working list ordered by non-decreasing values of F z(X i ) as the rst ordering criterion and non-increasing order with respect to the age of the intervals as the second ordering criterion. Therefore, the selected interval will always be at the head of the working list. Bounding rule: The lower bound F z(X) from (2.12) is used. Elimination rule: Four elimination rules are used: Monotonicity test: If for an interval X the condition then this means that the interval X does not contain any minimum and, therefore, can be rejected. RangeUp test: An interval X is rejected if f~ < F z(X). Gradient test: The subregion fXnV g, where V is dened by (2.8), is rejected. Of course, when the whole interval X can be eliminated. Cut-o test: When f~ is improved, all intervals X stored in the working and nal lists for which the condition F z(X) > f~ is fullled are rejected. Note, that this Cut-o test is dierent from the Cut-o test of TIAM where the condition F (X) > f~ was used. Division rule: Suppose that an interval X has been obtained as a result of application of RangeUp and Gradient tests to an interval Y and then stored in the working list L. If the interval X is chosen for subdivision, the point m(X) is used as the subdivision point. Note, that in general m(X) 6= m(Y ) and therefore this division rule does not coincide with the division rule of the TIAM algorithm. 8 L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV Algorithm 2 Interval analysis global minimization Algorithm using Gradient information Funct 1. 2. if ( F 3. x~ := s; f~ := F 4. else 5. x~ := s; f~ := F Test 7. return (f 8. 9. X := GradTest ( 10. lb(x) := lb(x) := f^(X) := f~ 11. F z(X) := maxfF (X); z(X; lb(x); lb(x); F 0 (S))g Lower bound of f(X) 12. if ( w(X) ) 13. Save fX; f^(X); F z(X)g in Q 14. else 15. Save fX; f^(X); F z(X)g in L 16. while ( L 6= fg ) 17. fX; f^(X); F z(X)g := Head(L) 18. comment X := 19. if (0 2 F 0 (X)) Monotonicity Test 20. if 21. f~ := F (m(X)) 22. CutOTest ( f~; 25. 26. for i := 1,2 GradTest 28. 29. if ( w(X fully rejected test 33. Save fX 34. else 35. Save fX 36. return Q Every element X i in the working and nal lists, L and Q, respectively, is a structure with the following data: Bounds x i and x i of the interval X i . The value being the value of f~ at the moment of creation of the The lower bound F z(X i ). Let us comment Algorithm 2. The IAG algorithm starts by evaluating F (s) and F (line 2) and initializing x~ and 5). If the monotonicity SUPPORT FUNCTIONS USING GRADIENT INFORMATION 9 Algorithm 3 Gradient test Funct GradTest(X; lb(x); lb(x); f~; G) 1. if ( lb(x) > f~ ) 2. 3. 4. if ( w(X) > 0 and lb(x) > f~ ) 5. 7. return X __ _ _ __ __ __ __ f ~ _ f ~ __ __ __ __ Fig. 3.1. The upper graph is an example of the initial phase of IAG. Bottom graphs show an example of how the intervals X1 and X2 are built from X. The bottom left hand graph shows the case F (m(X)) > f~ and the bottom right hand f~ < minflb(x); lb(x)g. Only shaded areas can contain f . test is satised (line 6), the algorithm nishes and the solution is given by ff~; x~g. Otherwise, in order to apply the Gradient test, lb(s) and lb(s) are initialized in line 8. The Gradient test is implemented in the GradTest procedure presented in Algorithm 3 which is applied to S on line 9. The GradTest procedure applies Theorems 2.2 and 2.3 to S, using returns an interval X S, such that the set of global minimizers of S are also in X . Lower bounds of f(x) and f(x) (lb(x) and lb(x), respectively) and f^(X) are set to f~ and F z(X) is computed (lines 10 and 11). The interval X is stored in the nal list Q (line 13) or in the working list L (line 15), depending on the value of w(X). Graphically, this initialization stage is shown in the top graph of Figure 3.1. L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV After this initialization stage and while the list L is not empty (line 16), IAG will select the interval at the head of L (line 17) for further processing. If F 0 (X) does not satisfy the Monotonicity test (line 19), F (m(X)) is evaluated (line 20). If F (m(X)) is a better upper bound of f than f~, f~ is updated to F (m(X)) (line 21) and the Cut-o test is applied to the intervals in the working and nal lists (line 22). Then, the interval X is subdivided into two subintervals X 1 and X 2 (line 23). These subintervals inherit from X the lower bound of f(x) in one of their bounds (line 24) and the other shared bound is set to F (m(X)) (line 25). For each subinterval the Gradient test is carried out using the derivative information of X (line 27) instead of the value of F 0 (X i ) which has not been evaluated 1 . If an interval 2g is not rejected (line 29), F z(X i ) is evaluated using also the value of Only when the RangeUp test is not satised (line 31), the interval will be saved in the nal (line 33) or working lists (line 35). Bottom graphs of Figure 3.1 show how the intervals X 1 and X 2 are built from X . In case of F (m(X)) > f~ (see lower left hand graph), interval X 1 and X 2 can be shortened by applying the GradTest procedure. In addition, if f~ < lb(x) and/or f~ < lb(x) (see lower right hand graph of Figure 3.1), X 1 and/or X 2 can be shortened again by the GradTest procedure. Notice that in the IAG algorithm, if f~ < lb(x) then f~ < lb(x), too, and vice versa. 4. Numerical results. The new algorithm IAG has been numerically compared with the method TIAM on a set of 40 test functions. This set of test functions is described in Table 4.1 and has been taken from [3, 4, 24]. The search region and the number of local and global minimizers are shown for all the functions. For both algorithms the stopping criterion was Table 4.2 shows numerical comparison between TIAM and IAG. Column NFE presents the number of Interval Function Evaluations, i.e., the number of F (X) evaluations plus the number of interval point evaluations F (x). Column NDE shows the number of Interval Function Evaluations of the derivative F 0 (X). Columns TM and present NFE +NDE for algorithms TIAM and IAG, respectively. Finally, column TM=TG provides information of the relative speedup of the IAG algorithm compared to TIAM algorithm. The last row of Table 4.2 shows the average values of data at columns TG, TM , and TM=TG. It can be seen from Table 4.2 that the ratio TM=TG is always greater than one, so IAG outperforms TIAM for all the functions. For this set of functions the speedup TM=TG ranges between [1:22; 6:98] and in average is 1:78. It can also be seen from Table 4.2 that for those functions where TIAM needs a lot of function evaluations the largest values of TM=TG were obtained (see functions which obtained speed up of 4:56 and 6:98, respectively). Figures 4.1 and 4.2 graphically show how algorithms TIAM and IAG work. The function presented in Figure 4.1 has only one global minimizer while the function has two global minimizers. In both gures the left hand graph refers to algorithm TIAM while right hand graphs depict the performance of algorithm IAG. For all the graphs the termination criterion was w(X) 0:05. 1 Evaluation of these values will become crucial only for that interval which is chosen later for subdivision. By using the derivative information of X we avoid additional computations of which can be useless if the interval will never be chosen for subdivision. Table Description of the test functions. Column N shows the number of the function, S the initial search interval, column M presents the number of local minimizers, and G shows the number of global minimizers. Function 26 e sin(3x) [0:2; 7:0] 5 3 28 sin(x) [0; 20] 4 3 29 2(x Horizontal arrows represent the values of f~ during the execution. Boxes represent the margins of all the evaluated intervals X and the lower and upper bounds of F (X). For the IAG algorithm, the new support function d(x) from (2.10) is also shown. At the top of the graphs, colored boxes represent the set of rejected intervals as well as intervals which contain a global minimizer. The color of a box species the criterion responsible for the rejection of that interval (Blue = GradTest procedure, midpoint and cut-o tests for TIAM and RangeUp and cut-o tests for IAG, Yellow = Boxes in the nal list Q). From these graphs is easy to realize how e-cient every rejection criterion is. L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV Table Results of numerical comparison between TIAM and IAG. 9 156 115 41 101 76 25 1.54 28 1.42 28 1.51 14 22 284 209 75 178 132 46 1.6 26 336 268 68 271 214 57 1.24 28 366 292 74 261 206 55 1.40 34 636 460 176 350 259 91 1.82 36 982 711 271 445 331 114 2.21 772.1 282.33 1.78 Figures 4.1 and 4.2 show that for these examples more than 50% of the initial interval S was rejected due to the GradTest procedure. It is also clearly shown that TIAM had to evaluate more intervals than IAG. Figure 4.2 shows some intervals where the best lower bound of f(X) was that obtained by the computation of z(X) instead of F (X), i.e., that F It should be noticed also that the TIAM algorithm is unable to take advantage of the information provided by the evaluation of F (m(X)) when F (m(X)) > f~. In contrast, IAG is able to reduce the interval even in this case (clearly shown in Figures 4.1 and 4.2). 5. A brief conclusion. In this paper a new way to calculate support functions for multiextremal univariate functions has been presented. The new support functions are based on the usual information used in global optimization working with Interval Analysis: interval evaluations of the objective function at a point, at an interval, and interval evaluation of the rst derivative of the objective function at an interval, i.e., F (x); F (X), and F 0 (X). Traditional interval analysis global optimization algorithms use this information Fig. 4.1. Graphical representation for the execution of TIAM (left hand graph) and IAG (right hand graph) algorithms for function Fig. 4.2. Graphical representation for the execution of TIAM (left hand graph) and IAG (right hand graph) algorithms for function separately: F (x) is used to obtain an upper bound for the global minimum, F (X) is used to determine a support function { being a constant { for the objective function nally, F 0 (X) is used in the Monotonicity test for rejecting intervals which do not contain global minimizers. In contrast, the new method uses the whole information jointly in order to construct a support function which is closer to the objective function. The new support function enables to develop more powerful rejection and bounding criteria and to accelerate the search signicantly. In fact, the new algorithm works almost two times faster in comparison with a traditional interval analysis method on a wide set of multiextremal test functions. The new approach has several possibilities for generalization. First, interval analysis bounds for F 0 (X) can be substituted by other estimates (for example, slope tools developed in [25] for non-smooth problems) in order to obtain new support functions. Second, the new method can be generalized to the multi-dimensional case by diago- 14 L.G. CASADO, I. GARCIA, J.A. MARTINEZ AND YA.D. SERGEYEV nal approach proposed in [22] or by using adaptively constructed space-lling curves proposed in [26]. Acknowledgement . The authors would like to thank E.M.T. Hendrix for his useful remarks and suggestions. --R An algorithm for State of the Art in Global Optimization. A global search algorithm using derivatives Global Optimization Using Interval Analysis Global optimization of univariate lipschitz functions: 1. Handbook of Global Optimization Guaranteed ray intersections with implicit surfaces Continuous Problems A method for converting a class of univariate functions into d. A bridging method for global optimization An adaptive stochastic global optimization algorithm for one-dimensional functions Equivalent methods for global optimization Convergence rates of a global optimization algorithm Interval analysis Interval Methods for Systems of Equations An algorithm for New computer methods for global optimization Automatic Slope Computation and its Application in Nonsmooth Global Optimiza- tion Two methods for solving optimization problems arising in electronic measurement and electrical engineering Numerical Methods on Multiextremal Problems Global optimization with non-convex constraints: Sequential and parallel algorithms An improved univariate global optimization algorithm with improved linear bounding functions --TR --CTR Tams Vink , Jean-Louis Lagouanelle , Tibor Csendes, A New Inclusion Function for Optimization: Kite&mdashlThe One Dimensional Case, Journal of Global Optimization, v.30 n.4, p.435-456, December 2004

interval arithmetic;global optimization;branch-and-bound

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Steps toward accurate reconstructions of phylogenies from gene-order data.

We report on our progress in reconstructing phylogenies from gene-order data. We have developed polynomial-time methods for estimating genomic distances that greatly improve the accuracy of trees obtained using the popular neighbor-joining method; we have also further improved the running time of our GRAPPA software suite through a combination of tighter bounding and better use of the bounds. We present new experimental results (that extend those we presented at ISMB'01 and WABI'01) that demonstrate the accuracy and robustness of our distance estimators under a wide range of model conditions. Moreover, using the best of our distance estimators (EDE) in our GRAPPA software suite, along with more sophisticated bounding techniques, produced spectacular improvements in the already huge speedup: whereas our earlier experiments showed a one-million-fold speedup (when run on a 512-processor cluster), our latest experiments demonstrate a speedup of one hundred million. The combination of these various advances enabled us to conduct new phylogenetic analyses of a subset of the Campanulaceae family, confirming various conjectures about the relationships among members of the subset and confirming that inversion can be viewed as the principal mechanism of evolution for their chloroplast genome. We give representative results of the extensive experimentation we conducted on both real and simulated datasets in order to validate and characterize our approaches.

INTRODUCTION Genome rearrangements. Modern laboratory techniques can yield the ordering and strandedness of genes on a chromosome, allowing us to represent each chromosome by an ordering of signed genes (where the sign indicates the strand). Evolutionary events can alter these orderings through rearrangements such as inversions and transpositions, collectively called genome rearrangements. Because these events are rare, they give us information about ancient events in the evolutionary history of a group of organisms. In consequence, many biologists have embraced this new source of data in their phylogenetic work [16, 26, 27, 29]. Appropriate tools for analyzing such data remain primitive when compared to those developed for DNA sequence data; thus developing such tools is becoming an important area of research, as attested by recent meetings on this topic [14, 15]. Optimization problems. A natural optimization problem for phylogeny reconstruction from gene-order data is to reconstruct an evolutionary scenario with a minimum number of the permitted evolutionary events on the tree-what is known as a most parsimonious tree. Unfortunately, this problem is NP-hard for most criteria-even the very simple problem of computing the median of three genomes under such models is NP-hard [9, 28]. However, because suboptimal solutions can yield very different evolutionary reconstructions, exact solutions are strongly preferred over approximate solutions (see [33]). Moreover, the relative probabilities of each of the rearrangement events (inversions, transpositions, and inverted are difficult to estimate. To overcome the latter problem, Blanchette et al. [5] have proposed using the breakpoint phylogeny, the tree that minimizes the total number of breakpoints, where a breakpoint is an adjacency of two genes that is present in one genome but not in its neighbor in the tree. Note that constructing the breakpoint phylogeny remains NP-hard [7]. Methods for reconstructing phylogenies. Blanchette et al. developed the BPAnalysis [31] software, which implements various heuristics for the breakpoint phylogeny. We reimplemented and extended their approach in our GRAPPA [17] software, which runs several orders of magnitude faster thanks to algorithm engineering techniques [24]. Other heuristics for solving the breakpoint phylogeny problem have been proposed [8, 12, 13]. Rather than attempting to derive the most parsimonious trees (or an approximation thereof), we can use existing distance-based methods, such as neighbor-joining (NJ) [30] (perhaps the most popular phylogenetic method), in conjunction with methods for defining leaf-to-leaf distances in the phylogenetic tree. Leaf-to-leaf distances that can be computed in linear time currently include breakpoint distances and inversion distances (the latter thanks to our new algorithm [3]). We can also estimate the "true" evolutionary distance (or, rather, the expected true evolutionary distance under a specific model of evolution) by working backwards from the breakpoint distance or the minimum inversion distance, an approach suggested by Sankoff [32] and Caprara [10] and developed by us in a series of papers [23, 34, 35], in which we showed that these estimators significantly improve the accuracy of trees obtained using the neighbor-joining method. Results in this paper. This paper reports new experimental results on the use of our distance estimators in reconstructing phylogenies from gene-order data, using both simulated and real data. We present several new results (the first two are extensions of the results we presented at ISMB'01 [23] and the third an extension of the results we presented at 510 MORET, TANG, WANG, AND WARNOW . Simulation studies examining the relationship between the true evolutionary distance and our distance estimators. We find that our distance estimators give very good predictions of the actual number of events under a variety of model conditions (including those that did not match the assumptions). . Simulation studies examining the relationship between the topological accuracy of neighbor-joining and the specific distance measure used: breakpoint, inversion, or one of our three distance estimators. We find that neighbor-joining does significantly better with our distance estimators than with the breakpoint or inversion distances. . A detailed investigation of the robustness of neighbor-joining using our distance estimators when the assumed relative probabilities of the three rearrangement events are very different from the true relative probabilities. We find that neighbor-joining using our estimators is remarkably robust, hardly showing any worsening even under the most erroneous assumptions. . A detailed study of the efficacy of using our best distance estimator, significantly improved lower bounds (still computable in low polynomial time) on the inversion length of a candidate phylogeny, and a novel way of structuring the search so as to maximize the use of these bounds. We find that this combination yields much stronger bounding in the naturally occurring range of evolutionary rates, yielding an additional speedup by one to two orders of magnitude for our GRAPPA code. . A successful analysis of a dataset of Campanulaceae (bluebell flower) using a combination of these techniques, resulting in a one-hundred-million-fold speedup over the original approach-in particular, we were able to analyze the dataset on a single workstation in a few hours, whereas our previous analysis required the use of a 512-node supercluster. Our research combines the development of mathematical techniques with extensive experimental performance studies. We present a cross-section of the results of the experimental study we conducted to characterize and validate our approaches. We used a large variety of simulated datasets as well as several real datasets (chloroplast and mitochondrial genomes) and tested speed (in both sequential and parallel implementations), robustness (in particular against mismatched models), efficacy (for our new bounding technique), and accuracy (for reconstruction and distance estimation). 2. BACKGROUND 2.1. The Nadeau-Taylor model of evolution When each genome has the same set of genes and each gene appears exactly once, a genome can be described by an ordering (circular or linear) of these genes, each gene given with an orientation that is either positive (g i ) or negative (-g i ). Let G be the genome with signed ordering g 1 , g 2 , . , g k . An inversion between indices a and b, for a # b, produces the genome with linear ordering A transposition on the (linear or circular) ordering G acts on three indices, a, b, c, with a # b and c / # [a, b], picking up the interval g a , g a+1 , . , g b and inserting it immediately after g c . Thus the genome G above (with the assumption of c > b) is replaced by An inverted transposition is a transposition followed by an inversion of the transposed subsequence. The (generalized) Nadeau-Taylor model [25] of genome evolution uses only genome rearrangement events, so that all genomes retain equal content. The model assumes that the number of each of the three types of events obeys a Poisson distribution on each edge, that the relative probabilities of each type of event are fixed across the tree, and that events of a given type are equiprobable. Thus we can represent a Nadeau-Taylor model tree as a triplet (T , {# e }, (# I , # T , # IT )), where the the triplet (# I , # T , # IT ) defines the relative probabilities of the three types of events (inversions, transpositions, and inverted transpositions). For instance, the triplet ( 1, 1, 1) indicates that the three event classes are equiprobable, while the triplet (1, 0, 0) indicates that only inversions happen. 2.2. Distance-based estimation of phylogenies Given a tree T on a set S of genomes and given any two leaves i, j in T , we denote by the path in T between i and j. We let # e denote the number of events (inversions, transpositions, or inverted transpositions) on the edge e during the evolution of the genomes in S within the tree T . This is the actual number of events on the edge. We can then define the of actual distances, which is additive. When given an additive matrix, many distance-based methods are guaranteed to reconstruct the tree T and the edge weights (but not the root). Atteson [1] showed that NJ is guaranteed to reconstruct the true tree T when given an estimate of the additive matrix [# ij ], as long as the estimate has bounded error: Theorem 2.1. (From [1]) Let T be a binary tree and let # e and # ij be defined as described above. Let Let D be any n - n dissimilarity matrix (i.e. D is symmetric and zero on the diagonal). If x then the NJ tree, NJ(D), computed for D is identical to T . That is, NJ is guaranteed to reconstruct the true tree topology if the input distance matrix is sufficiently close to an additive matrix defining the same tree topology. Consequently, techniques that yield a good estimate of the matrix [# ij ] are of significant interest. Distance measures. The edit distance between two gene orders is the minimum number of inversions, transpositions, and inverted transpositions needed to transform one gene order into the other. The inversion distance is the edit distance when only inversions are permitted. The inversion distance can be computed in linear time [3, 18]; the transposition distance is of unknown computational complexity [4]. Given two genomes G and G # on the same set of genes, a breakpoint in G is an ordered pair of genes (g a , g b ) such that g a and g b appear consecutively in that order in G, but neither (g a , g b ) nor (-g b , -g a ) appear consecutively in that order in G # . The number of breakpoints in G relative to G # is the breakpoint distance between G and G # . The breakpoint distance is easily calculated by inspection in linear time. See Figure 1 for an example of these distances. Estimations of true evolutionary distances. Estimating the true evolutionary distance requires assumption about the model; in the case of gene-order evolution, the assumption 512 MORET, TANG, WANG, AND WARNOW FIG. 1. Example of transposition, inversion, and breakpoint distances. We obtain G 3 from G 0 after 3 inversions (the genes in the inversion interval are highlighted at each step). G 3 can also be obtained from G 0 with one transposition: move the gene segment (6, 7, 8) to the position between genes 1 and 3. d T , d I , and d B are the transposition, inversion, and breakpoint distances, respectively. is that the genomes have evolved from a common ancestor under the Nadeau-Taylor model of evolution. Sankoff's technique [32], applicable only to inversions, calculates this value exactly, while IEBP [35] and EDE [23], applicable to very general models of evolution, obtain approximations of these values, and Exact-IEBP [34] calculates the value exactly for any combination of inversions, transpositions, and inverted transpositions. These estimates can all be computed in low polynomial time. 2.3. Performance criteria Let T be a tree leaf-labelled by the set S. Deleting some edge e from T produces a bipartition # e of S into two sets. Let T be the true tree and let T # be an estimate of T , as illustrated in Figure 2. The false negatives of T # with respect to T , denoted FN(T , T # ), are those bipartitions that appear in T that do not appear in T # . The false negative rate is the number of false negatives divided by the number of non-trivial bipartitions of T . Similarly, the false positives of T # with respect to T are defined as those bipartitions that appear in T # but not in T , and the false positive rate is the ratio of false positives to the number of nontrivial edges. For example, in Figure 2, the edge corresponding to the bipartition {1, 2, 3 | 4, 5, 6, 7, 8} is present in the true tree, but not in the estimate, and is thus a false negative. Note that, if both trees are binary, then the number of false negatives equals the number of false positives. In reporting our results, we will use the false negative rate.254713 57 4 (a) True tree T (b) Estimate T # FIG. 2. False positive and false negative edges. T is the true tree, T # is a reconstructed tree, and bold edges are false negative edges in (a) and false positive edges in (b). 3. TRUE DISTANCE ESTIMATORS 3.1. Definitions Given two signed permutations, we can compute their breakpoint distance or one of the edit (minimum) distances (for now, the inversion distance), but the actual number of evolutionary events is not directly recoverable. All that can be done is to estimate that number under some assumptions about the model of evolution. Thus our true distance estimators return the most likely number of evolutionary events for the given breakpoint or inversion distance. We developed three such estimators: the IEBP estimator [35] approximates the most likely number of evolutionary events working from the breakpoint distance, using a simplified analytical derivation; the Exact-IEBP estimator [34] refines the analytical derivation and returns the exact value for that quantity; and the EDE estimator [23] uses curve fitting to approximate the most likely number of evolutionary events working from the inversion distance. All three estimators provide considerably more accurate estimates of true evolutionary distances than the breakpoint or inversion distances (at least for large moreover, trees obtained by applying the neighbor-joining method to these estimators are more accurate than those obtained also using neighbor-joining, but based upon breakpoint distances or inversion distances. 3.2. Comparison of distance estimates We simulated the Nadeau-Taylor model of evolution under different weight settings to study the behavior of different distance estimators. The numbers of genes in the datasets are 37 (animal mitochondria [6]), and 120 (chloroplast genome in many plants [20]). For each dataset in the experiment, we chose a number between 1 and some upper bound B as the number of rearrangement events. B is chosen to be 2.5 times the number of genes, which (according to our experimental results) is enough to make the distance between two genomes similar to the distance between two random genomes. We then computed the BP (breakpoint) and INV (inversion) distances and corrected them to get IEBP, Exact-IEBP, and EDE distances. In Figures 3, 4, and 5, we plot the (unnormalized) computed distances against the actual number of events-using an inversion-only scenario for the case of 37 genes and a scenario with equally likely events as well as an inversion-only scenario for the case of 120 genes. These figures indicate that, as expected, BP and INV distances underestimate the actual number of events-although, when the number of events is low, they are highly accurate and have small variance. The linear region-the range of the x-coordinate values where the curve is a straight line-is longer for INV distances than for BP distances so that INV distances produce unbiased estimates in a larger range than do BP distances. In contrast, the three estimators provide good estimates on the average, although their variances increase sharply with the edit distance values. Exact-IEBP produces good estimates over all ranges (clearly improving on IEBP), while EDE tends to underestimate the distance unless the scenario uses only inversions. We also plotted the difference between the actual (true) evolutionary distance and the minimum or estimated distance, under various models of evolution (mixtures of inver- sions, transpositions, and inverted transpositions), for two different genome sizes (37 and 120), and for various number of events (rates of evolution). Figure 6 shows the results for three different models of evolution on 37 and 120 genes, respectively; the values are plotted in a cumulative fashion: at position x along the horizontal axis, we plotted the mean absolute difference for all generated pairs with a true evolutionary distance of at most x. These figures show that Exact-IEBP is usually the best choice, but that EDE distances are 514 MORET, TANG, WANG, AND WARNOW Breakpoint Distance Actual number of events Inversion Distance Actual number of events (a) BP distance (b) INV distance IEBP Distance Actual number of events Exact-IEBP Distance Actual number of events Distance Actual number of events (c) IEBP distance (d) Exact-IEBP distance (e) EDE distance FIG. 3. Mean and standard deviation plots for the two distances and three distance estimators, for 37 genes under an inversion-only scenario. The datasets are divided into bins according to their x-coordinate values (the BP or INV distance). Actual number of events Actual number of events (a) BP distance (b) INV distance Actual number of events Actual number of events Actual number of events (c) IEBP distance (d) Exact-IEBP distance (e) EDE distance FIG. 4. Mean and standard deviation plots for the two distances and three distance estimators, for 120 genes under a scenario in which all three types of events are equally likely. The datasets are divided into bins according to their x-coordinate values (the BP or INV distance). Actual number of events Actual number of events (a) BP distance (b) INV distance Actual number of events Actual number of events Actual number of events (c) IEBP distance (d) Exact-IEBP distance (e) EDE distance FIG. 5. Mean and standard deviation plots for the two distances and three distance estimators, for 120 genes under an inversion-only scenario. The datasets are divided into bins according to their x-coordinate values (the BP or INV distance). Actual number of events Absolute difference Actual number of events Absolute difference Actual number of events Absolute difference (a) inversions only (b) transpositions only (c) equally likely events Actual number of events Absolute difference Actual number of events Absolute difference Actual number of events Absolute difference (a) inversions only (b) transpositions only (c) equally likely events FIG. 6. The mean difference between the true evolutionary distance and our five distances estimates, under three models of evolution, plotted as a function of the tree diameter, for 37 genes (top) and 120 genes (bottom). 516 MORET, TANG, WANG, AND WARNOW nearly as good (and occasionally better) when the model uses only inversions-although the variance for larger distances is high, making it difficult to draw firm conclusions. IEBP and EDE clearly improve on BP and INV. 3.3. Neighbor-joining performance We conducted a simulation study to compare the performance of NJ using the same five distances. In Figure 7, we plot the false negative rate against the normalized pairwise inversion distances, under three different model weights settings: (1, 0, (0, 1, (transpositions only), and ( 1, 1, 1) (all three events equally likely). In each plot we pool the results for the same model weight but different numbers of genomes: 10, 20, 40, 80, and 160. Note that NJ(EDE) is remarkably robust: even though EDE was engineered for an inversion-only scenario, it can handle datasets with a significant number of transpositions and inverted transpositions almost as well. NJ(EDE) recovers 90% of the edges even for the nearly saturated datasets where the maximum pairwise inversion distance is close to 90% of the maximum value. That NJ(EDE) improves on NJ(IEBP), in spite of the fact that IEBP is a comparable estimator, may be attributed to the greater precision (smaller variance) of EDE for smaller distances-most of the choices made in neighbor-joining are made among small distances, where EDE is more likely to return an approximation within a small factor of the true distance. Exact-IEBP, the most expensive of our three estimators to compute, yields the second best performance, although the difference between the error rates of NJ(EDE) and NJ(Exact-IEBP) is too small to be statistically significant. Normalized Maximum Pairwise Inversion Distance False Negative Rate Normalized Maximum Pairwise Inversion Distance False Negative Rate Normalized Maximum Pairwise Inversion Distance False Negative Rate (a) inversions only (b) transpositions only (c) equally likely events FIG. 7. False negative rates of NJ methods under various distance estimators as a function of the maximum genomes. (Results for different numbers of genomes are pooled into a single figure when the model weights are identical.) 3.4. Robustness of distance estimators As discussed earlier, estimating the true evolutionary distance requires assumptions about the model parameters. In the case of EDE, we assume that evolution proceeded through inversions only-so how well does NJ(EDE) perform when faced with a dataset produced through a combination of transpositions and inverted transpositions? In the case of the two IEBP methods, the computation requires values for the respective rates of inver- sion, transposition, and inverted transposition, respectively, which obviously leaves a lot of room for mistaken assumptions. We ran a series of experiments under conditions similar to those shown earlier, but where we deliberately mismatched the evolutionary parameters used in the production of the dataset and those used in the computation of the distance estimates used in NJ. Figure 8 shows the results for the Exact-IEPB estimator (results for 1515Normalized Maximum Pairwise Inversion Distance False Negative Rate 1515Normalized Maximum Pairwise Inversion Distance False Negative Rate 1515Normalized Maximum Pairwise Inversion Distance False Negative Rate (a) inversions only (b) transpositions only (c) equally likely events FIG. 8. Robustness of the Exact-IEBP method with respect to model parameters. Triples in the legend indicate the model values used in the Exact-IEBP method. the other estimators are similar), indicating that our estimators, when used in conjunction with NJ, are remarkably robust in the face of erroneous model assumptions. 4. MAXIMUM PARSIMONY AND TOPOLOGICAL ACCURACY The main goal of phylogeny reconstruction is to produce the correct tree topology. Two basic approaches are currently used for phylogeny reconstruction from whole genomes: distance-based methods such as NJ applied to techniques for estimating distances and "maximum parsimony" (MP) approaches, which attempt to minimize the "length" of the tree, for a suitably defined measure of the length. We examine two specific MP problems in this section: the breakpoint phylogeny prob- lem, where we seek to minimize the total number of breakpoints over all tree edges, and the phylogeny problem, where we seek to minimize the total number of inversions. We want to determine, using a simulation study, whether topological accuracy is improved by reducing the number of inversions or the number of breakpoints. If possible, we also want to determine whether the breakpoint phylogeny problem or the inversion phylogeny problem are topologically more accurate under certain evolutionary conditions, and if so, under which conditions. We ran a large series of tests on model trees to investigate the hypothesis that minimizing the total breakpoint distance or inversion length of trees would yield more topologically accurate trees. We ran NJ on a total of 209 datasets with both inversion and breakpoint distances. Each test consists of at least 12 data points, on sets of up to 40 genomes. We used two genome sizes (37 and 120 genes, representative of mitochondrial and chloroplast genomes, respectively) and various ratios of inversions to transpositions and inverted trans- positions, as well as various rates of evolution. For each dataset, we computed the total inversion and breakpoint distances and compared their values with the percentage of errors (measured as false negatives). We used the nonparametric Cox-Stuart test [11] for detecting trends-i.e., for testing whether reducing breakpoint or inversion distance consistently reduces topological errors. Using a 95% confidence level, we found that over 97% of the datasets with inversion distance and over 96% of those with breakpoint distance exhibited such a trend. Indeed, even at the 99.9% confidence level, over 82% of the datasets still exhibited such a trend. Figures show the results of scoring the different NJ trees under the two optimization criteria: breakpoint score and inversion length of the tree. In general, the relative ordering and trend of the curves agree with the curves of Figure 7, suggesting that de- 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Breakpoint Score (a) breakpoint score, 37 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Inversion Length (c) inversion length, 37 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Breakpoint Score (b) breakpoint score, 120 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Inversion Length (d) inversion length, 120 genes FIG. 9. Scoring NJ methods under various distance estimators as a function of the maximum pairwise inversion distance for 10, 20, and 40 genomes. Plotted is the ratio of the NJ tree score to the model tree score (breakpoint or inversion) on an inversion-only model tree. 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Breakpoint Score (a) breakpoint score, 37 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Inversion Length (c) inversion length, 37 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Breakpoint Score (b) breakpoint score, 120 genes 11.021.061.1Normalized Maximum Pairwise Inversion Distance Relative Inversion Length (d) inversion length, 120 genes FIG. 10. Scoring NJ methods under various distance estimators as a function of the maximum pairwise inversion distance for 10, 20, and 40 genomes. Plotted is the ratio of the NJ tree score to the model tree score (breakpoint or inversion) on a model tree where the three classes of events are equiprobable. creasing the number of inversions or breakpoints leads to an improvement in topological accuracy. The correlation is strongest for the 120-gene case; this may be because, for the same number of events but a larger number of genes, the rate of evolution effectively goes down and overlap of events becomes less likely. Finally, this trend still holds under the other evolutionary models (such as when only transpositions occur). 5. SEARCHING FOR MAXIMUM PARSIMONY TREES 5.1. The lower bound and its use The following theorem is well known: Theorem 5.1. Let d be a n - n matrix of pairwise distances between the taxa in a set let T be a tree leaf-labelled by the taxa in S; and let w be an edge-weighting on T , so that we have a circular ordering of the leaves of T , under some planar embedding of T , then we have This corollary immediately follows: Corollary 5.1. Let d be the matrix of minimum distances between every pair of genomes in a set S, let T be a fixed tree on S, and let 1, 2, . , n be a circular ordering of leaves in T under some planar embedding of T . Then the length of T is at least 1(d 1,2 This corollary forms the basis of the old "twice around the tree" heuristic for the TSP based on minimum spanning trees [19]. Note that the theorem and its corollary hold for any distance measure that obeys the triangle inequality. In earlier work [22, 24], we used these bounds in a simple manner to reduce the cost of searching tree space. . We obtain an initial upper bound on the minimum achievable inversion length by using NJ with inversion distances. This upper bound is updated every time the search finds a better tree. . Each tree we examine is presented in the standard nested-parentheses format (the nexus format [21]); this format defines a particular circular ordering of the leaves. We use that ordering to compute the lower bound of Corollary 5.1, again using inversion distances. If the lower bound exceeds the upper bound, the tree can be discarded. This bounding may reduce the running time substantially, because the bound can be computed very efficiently, whereas scoring a tree with a tool like GRAPPA [17] involves solving numerous TSP instances. However, when the rate of evolution is high for the size of the genome, the bound often proves loose-the bound is exact when distances are additive, but high rates of evolution produce distances that are much smaller than the additive value. We thus set about improving the bound as well as how it is used in the context of GRAPPA to prune trees before scoring them. Our new results come from three separate ideas: (i) tighten the lower bound; (ii) return more accurate scores for trees with large edge lengths; and (iii) process the trees so as to take better advantage of the bounds. 5.2. Tightening the bound The particular circular order defined by the tree description is only one of a very large number of circular orderings compatible with that tree: since the tree has no internal ordering (i.e., no notion of left vs. right child), swapping two subtrees does not alter the 520 MORET, TANG, WANG, AND WARNOW phylogeny, but does yield a different circular ordering. Any of these orderings defines a valid lower bound, so that we could search through all orderings and retain the largest bound produced in order to tighten the bound. Unfortunately, the number of compatible circular orderings is exponential in the number of leaves, so that a full search is too ex- pensive. We developed and tested a fast greedy heuristic, swap-as-you-go, that provides a high-quality approximation of the optimal bound. Our heuristic starts with the given tree and its implied circular ordering. It then traverses the tree in preorder from an arbitrarily chosen initial leaf, deciding locally at each node whether or not to swap the children by computing the score of the resulting circular ordering and moving towards larger values, in standard greedy fashion. With incremental computations, such a search takes linear time, because each swap only alters a couple of adjacencies, so that the differential cost of a swap can be computed in constant time. In our experiments, the resulting bound is always very close, or even equal, to the optimal bound and much better than the original value in almost all cases. We tested a related bounding technique proposed by Bryant [8], but found it to be very slow (in order to yield reasonably tight bounds, it requires the introduction of Lagrangian variables and the solution of a system of linear equations to determine their values) and always (in our experiments) dominated by our algorithm. Since we use bounding strictly to reduce the total amount of work, it is essential that any lower bound be computable with very little effort. 5.3. Accurate scoring of long edges Long edges in the tree suffer from the same problem as large leaf-to-leaf distances: they are seriously underestimated by an edit distance computation. Thus we decided to use the same remedy presented in the first part of this paper, by correcting edit distances with one of our true distance estimators. We chose the EDE estimator, because it offers the best tradeoff between accuracy and computational cost of our three estimators and used it within GRAPPA wherever distances are computed: in computing the distance matrix for NJ, in computing circular lower bounds from that matrix, and, more importantly, in computing the distance along each edge and in computing the median-of-three that lies at the heart of the GRAPPA approach [24]. Because a distance estimator effectively stretches the range of possible values and because that stretch is most pronounced when the edit distances are already large (the worst case for our circular bounds), using a distance estimator yields significant benefits-a number of our more challenging instances suddenly became very tractable with the combination of EDE distances and our improved circular bound. 5.4. Layered search We added a third significant improvement to the bounding scheme. Since the bound itself can no longer be significantly improved, obtaining better pruning requires better use of the bounds we have. Our original approach to pruning [24] was simply to enumerate all trees, keeping the score of the best tree to date as an upper bound and computing a lower bound for each new tree to decide whether to prune it or score it. In this approach, each tree is generated once, bounded once, and scored at most once, but the upper bound in use through the computation may be quite poor until close to the end if optimal and near-optimal trees appear only toward the end of the enumeration-in which case the program must score nearly every tree. To remedy this problem, we devised a novel and radically different approach, which is motivated by the fact that generating and bounding a tree is very inexpensive, whereas scoring one, which involves solving potentially large numbers of instances of the Travelling Salesperson Problem, is very expensive. Our new approach, which we call a layered search, still bounds each tree once and still scores it at most once, but it typically examines the tree more than once; it works as follows. . In a first phase, we compute the NJ tree (using EDE distances) and score it to obtain an initial upper bound, as in the original code. . In a second phase, every tree in turn is generated, its lower bound computed as described above, and that bound compared with the cost of the NJ tree. Trees not pruned away are stored, along with their computed lower bound, in buckets ordered by the value of the lower bound. 3 . We then begin the layered search itself, which proceeds on the principle that the lower bound of a tree is correlated with the actual parsimony score of that tree. The search looks at each successive bucket of trees in turn, scoring trees that cannot be pruned through their lower bound and updating the upper bound whenever a better score is found. This search technique is applicable to any class of optimization problems where the cost of evaluating an object in the solution space is much larger than the cost of generating that object and works well whenever there is a good correlation between the lower bound and the value of the optimal solution. Our experiments indicate that, unless the interleaf distances are all nearly maximal (for the given number of genes), the correlation between our lower bound and the parsimony score is quite strong, so that our layered search strategy very quickly reduces the upper bound to a score that is optimal or nearly so, thereby enabling drastically better pruning- as detailed below, we frequently observed pruning rates of over 99.999%. 5.5. An experimental assessment of bounding We measured the percentage of trees that are pruned through bounding (and thus not scored) as a function of the three model parameters: number of genomes, number of genes, and number of inversions per edge. We used an inversion-only scenario as well as one with approximately half inversions and half transpositions or inverted transpositions. Our data consisted of two collections of 10 datasets each for a combination of parameters. The number of genomes was 10, 20, 40, 80, and 160, the number of genes was 10, 20, 40, 80, and 160, and the rate of evolution varied from 2 to 8 events per tree edge, for a total of 75 parameter combinations and 1,500 datasets. We used EDE distances to score edges and our "swap-as-you-go" bounding computation, but not layered search-because layered search requires enumerating all trees up front, something that is simply not possible for 20 or more genomes. For each data set with 10 genomes, GRAPPA tested all trees, scored and updated the upper bound if necessary, and kept statistics on the pruning rate. For 20 or more genomes, the number of trees is well beyond the realm of enumeration-with 20 genomes, we have For these cases, we began by running GRAPPA for 6 hours to score as many trees as possible, then used the best score obtained in this first phase as an upper bound in a second phase where we ran another 6 hours during which a random selection 3 The required storage can exceed the memory capacity of the machine, in which case we store buckets on disk in suitably-sized blocks. The cost of secondary memory access is easily amortized over the computation. 522 MORET, TANG, WANG, AND WARNOW # genes FIG. 11. Percentage of trees eliminated through bounding for various numbers of genes and genomes and three rates of evolution. of trees are bounded using our method and their bounds compared to the upper bound obtained in the first phase. The result is a (potentially very) pessimistic estimate of the pruning rate. Figure 11 shows the percentage of trees pruned away by the circular lower bound; in the table, the parameter r denotes the expected number of inversions per edge used in the simulated evolution. (We could not run enough tests for the setting of 160 genes and because merely scoring such a tree accurately can easily take days of computation-the instances of the TSP generated under these circumstances are very time-consuming.) In comparison with the similar table for our earlier approach [23], our new approach shows dramatic improvement, especially at high rates of evolution. We found that most circular orderings in datasets of up to 20 genomes were eliminated. However the table also shows that the bounding does not eliminate many trees in datasets with 40 or more genomes- not unless these datasets have large numbers of genes. The reason is clear: the number of genes dictates the range of values for the pairwise distances and thus also for the tree score-in terms of inversion distances, for instance, we have roughly mn possible tree scores, where m is the number of genes and n the number of genomes; yet the number of distinct trees is (2n - 5)!!, a number so large that, even when only a very small fraction of the trees are near optimal, that fraction contains so many trees that they cannot efficiently be distinguished from others with only mn buckets. Of course, GRAPPA in 6 hours can only examine a vanishingly small fraction of the tree space, so that the upper bound it uses is almost certainly much too high. In contrast with these findings, our layered method gave us pruning rates of 90% in the 10-genome case for genes, a huge improvement over the complete failure of pruning by the normal search method. 6. A TEST OF OUR METHODS ON REAL We repeated our analysis of the Campanulaceae dataset, which consists of 13 chloroplast genomes, one of which is the outgroup Tobacco, but this time to reconstruct the EDE (as opposed to the breakpoint or inversion) phylogeny. Each of the 13 genomes has 105 segments and, though highly rearranged, has what we consider to be a low rate of evolution. In our previous analysis, we found that GRAPPA, with our first bounding ap- proach, pruned about 85% of the trees, for a substantial speedup (on the order of 5-10) over a version without pruning. By using EDE distances and our improved bound compu- tation, we increased this percentage to over 95%, for another substantial speedup of 5-10. By adding layered search, however, we managed to prune almost all of the 13.75 billion trees-all but a few hundred thousand, which were quickly scored and dispatched, for a further speedup of close to 20. The pruning rate was over 99.99%, reducing the number of trees that had to be scored by a factor of nearly 800. As a result, the same dataset that required a couple of hours on a 512-processor supercluster when using our first bounding strategy [2, 22, 24] can now be run in a few hours on a single workstation-and in one minute on that same cluster. In terms of our original comparison to the BPAnalysis code, we have now achieved a speedup (on the supercluster) of one hundred million! We also confirmed the results of our previous analysis-that is, the trees returned in our new analysis, which uses EDE distances, match those returned in an analysis that had used minimum inversion distances, a pleasantly robust result. The speedup obtained by bounding depends on two factors: the percentage of trees that can be eliminated by the bounding and the difficulty of the TSP instances avoided by using the bounds. As Table 11 shows, when the rate of evolution is not too high, close to 100% of the trees can be eliminated by using the bounds. However, the TSP instances solved in GRAPPA can be quite small when the evolutionary rate is low, due to how we compress data (see [24]). Consequently, the speedup also depends on the rate of evolution, with lower rates of evolution producing easier TSP instances and thus smaller speedups. The Campanulaceae dataset is a good example of a dataset that is quite easy for GRAPPA, in the sense that it produces easy TSP instances-but even in this case, a significant speedup results. More generally, the speedup increases with larger numbers of genomes and, to a point, with higher rates of evolution. When one is forced to exhaustively search tree space, these speedups represent substantial savings in time. 7. CONCLUSIONS AND FUTURE WORK We have described new theoretical and experimental results that have enabled us to analyze significant datasets in terms of inversion events and that also extend to models incorporating transpositions. This work is part of an ongoing project to develop fast and robust techniques for reconstructing phylogenies from gene-order data. The distance estimators we have developed clearly outperform straight distance measures in terms of both accuracy (when used in conjunction with a distance-based method such as neighbor-joining) and efficiency (when used with a search-based optimization method such as implemented in our GRAPPA software suite). Our current software suffers from several limitations, particularly its exhaustive search of all of (constrained) tree space. However, the bounds we have described can be used in conjunction with branch-and-bound (based on inserting leaves into subtrees or extending circular orderings) as well as in heuristic search techniques. ACKNOWLEDGMENTS We thank R. Jansen for introducing us to this research area, and D. Sankoff and J. Nadeau for inviting us to the DCAF meeting, during which some of the ideas in this paper came to fruition. This work is supported in part by National Science Foundation grants ACI 00-81404 (Moret), DEB 01-20709 (Moret and Warnow), EIA 01-13095 and by the David and Lucile Packard Foundation (Warnow). --R The performance of the neighbor-joining methods of phylogenetic reconstruction GRAPPA runs in record time. A linear-time algorithm for computing inversion distance between signed permutations with an experimental study Sorting permutations by transpositions. Breakpoint phylogenies. Gene order breakpoint evidence in animal mitochondrial phy- logeny The complexity of the breakpoint median problem. A lower bound for the breakpoint phylogeny problem. Formulations and hardness of multiple sorting by reversals. Experimental and statistical analysis of sorting by reversals. Practical Nonparametric Statistics An empirical comparison of phylogenetic methods on chloroplast gene order data in Campanulaceae. A new fast heuristic for computing the breakpoint phylogeny and experimental phylogenetic analyses of real and synthetic data. Workshop on Gene Order Dynamics DIMACS Workshop on Whole Genome Comparison. Use of chloroplast DNA rearrangements in reconstructing plant phylogeny. GRAPPA: Genome Rearrangements Analysis under Parsimony and other Phylogenetic Algorithms. Transforming cabbage into turnip (polynomial algorithm for genomic distance problems). The travelling salesman problem and minimum spanning trees. Personal communication An extensible file format for systematic infor- mation New approaches for reconstructing phylogenies based on gene order. A new implementation and detailed study of breakpoint analysis. Lengths of chromosome segments conserved since divergence of man and mouse. Chloroplast DNA systematics: a review of methods and data analysis. Chloroplast and mitochondrial genome evolution in land plants. The median problems for breakpoints are NP-complete Chloroplast DNA evidence on the ancient evolutionary split in vascular land plants. The neighbor-joining method: A new method for reconstructing phylogenetic trees Multiple genome rearrangement and breakpoint phylogeny. Probability models for genome rearrangements and linear invariants for phylogenetic inference. Phylogenetic inference. Exact-IEBP: a new technique for estimating evolutionary distances between whole genomes. Estimating true evolutionary distances between genomes. --TR Transforming cabbage into turnip Formulations and hardness of multiple sorting by reversals Probability models for genome rearrangement and linear invariants for phylogenetic inference Sorting permutations by tanspositions Estimating true evolutionary distances between genomes High-Performance Algorithm Engineering for Computational Phylogenetics A New Fast Heuristic for Computing the Breakpoint Phylogeny and Experimental Phylogenetic Analyses of Real and Synthetic Data A Lower Bound for the Breakpoint Phylogeny Problem --CTR Bernard M. E. Moret , Li-San Wang , Tandy Warnow, Toward New Software for Computational Phylogenetics, Computer, v.35 n.7, p.55-64, July 2002 Niklas Eriksen, Reversal and transposition medians, Theoretical Computer Science, v.374 n.1-3, p.111-126, April, 2007 Bernard M. E. Moret , Tandy Warnow, Reconstructing optimal phylogenetic trees: a challenge in experimental algorithmics, Experimental algorithmics: from algorithm design to robust and efficient software, Springer-Verlag New York, Inc., New York, NY, 2002

breakpoint analysis;evolutionary distance;genome rearrangement;inversion distance;distance estimator;ancestral genome

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Experiences in modeling and simulation of computer architectures in DEVS.

The use of traditional approaches to teach computer organization usually generates misconceptions in the students. The simulated computer ALFA-1 was designed to fill this gap. DEVS was used to attack this complex design of the chosen architecture, allowing for the definition and integration of individual components. DEVS also provided a formal specification framework, which allowed reduction of testing time and improvement of the development process. Using ALFA-1, the students acquired some practice in the design and implementation of hardware components, which is not usually achievable in computer organization courses.

Figure 1. Organization of the Integer Unit. This RISC processor is provided with 520 integer registers. Eight of them are global (RegGlob, shared by every procedure), and the remaining 512 are divided in windows of 24 registers each (RegBlock). Each window includes input, output and local registers for every procedure that has been executed recently. When a routine begins, new registers are reserved (8 local and 8 output), and the 8 output records of the calling procedure are used as inputs. A specialized 5-bit register, called CWP (Circular Window Pointer) marks the active window. Every time a new procedure starts, CWP is decremented. Figure 2. Organization of the processor's registers. Besides these general purpose registers, the architecture includes: PCs: the processor has two program counters. The PC contains the address of the next instruction. The stores the address of the PC after the execution of the present instruction. Each instruction cycle finishes by copying the nPC to the PC, and adding 4 bytes (one word) to the nPC. If the instruction is a conditional branch, nPC is assigned to PC, and nPC is updated with the jump address (if the jump condition is valid). Y: is used by the product and division operations; BASE and SIZE: The memory is considered flat (that is, neither segmentation nor pagination mechanisms are included). Likewise, multiprogramming is not supported. The BASE register points to the lowest address a program can access. The SIZE stores the maximum size available for the program. PSR (Processor Status Register): stores the current status for the program. It is interpreted as follows. Bits Content Description Reserved Negative 1 when the result of the last operation is negative 22 Z - Zero 1 when the result of the last operation is zero when the result of the last operation is overflow when the result of the last operation carried one bit 19.12 Reserved Lowest interrupt number to be serviced. 6 PS - Previous State Last mode. Enable Trap 1=Traps enabled; 0=Traps disabled. Current Window Pointer Points to the current register window. Table 1. Contents of the Process Status Register WIM (Window Invalid Mask): this 32-bit register (one bit per window) is used to avoid overwriting a window in use by another procedure. When CWP is decremented, these circuits verify if the WIMbit is active for the new window. In that case, an interrupt is raised and the interrupt service routine stores the content of the window in memory. Usually, WIMonly has one bit in 1 marking the oldest window. TBR (Trap Base Register): it points to the memory address storing the position of a trap routine. Bits Content Description base address Base address of the Trap table 11.4 Trap Type Trap to be serviced 3.0 Constant (0000) Table 2. Contents of the Trap Base Register The first 20 bits (Trap Base Address) store the base address of the trap table. When an interrupt request is received, the number of the trap to be serviced is stored in the bits 11.4. Therefore, the TBR points to the table position containing the address of the service routine. The last 4 bits in 0 guarantees at least 16 bytes to store each routine. When the instruction set level of the SPARC architecture is analyzed, we see that each instruction has a fixed size of 32 bits. Memory operands may be 8, 16 or 32 bits. There are basic Load/Store operations, classified according to the size and sign of their operands. Arithmetic and Boolean operations include add, and, or, div, mul, xor, xnor, and shift. These are able to change the PSR, according to the operation code used. Several jump instructions are available, including relative jumps, absolute jumps, traps, calls, and return from traps. Other instructions include the movement of the register window, NOPs, and read/write operations on the PSR. Multiplication uses 32-bit operands, producing 64-bit results. The most significant 32 bits are stored in the Y register, and the remaining in the ALU-RES register. Integer division operations take a 64-bit dividend and a 32-bit divisor, producing a 32-bit result. The Y register stores the most significant bits of the divi- dend. One ALU input register stores the least significant bits of the dividend, and the other, the divisor. The integer result is stored in the ALU-RES register, and the remainder in the Y register. Most instructions are carried out by the ALU, whose structure is depicted in the following figure. It includes two multiplexers connected to the ALU, Multiplier/Divider unit and shifter. Figure 3. Organization of the ALU. There are two execution modes: User and Kernel. Certain instructions can only be executed in Kernel mode. Also, the Base and Size registers are used only when the program is running in User mode. The CPU executes under the supervision of the Control Unit . It receives signals from the rest of the processor using 64 input bits (organized in 5 groups: the Instruction Register, the PSR, BUS_BUSY_IN, BUS_DACK_IN, and BUS_ERR). Its outputs are sent using 70 lines organized in 59 groups. Some of them include reading/writing internal registers, activating lines for the ALU or multiplexers. Also, connections with the PC, nPC, Trap controller and PSR registers are included. Finally, the Data, Address and Control buses can be accessed. The memory is organized using byte addressing and Little-Endian to store words. The processor issues a memory access operation by writing an address (and data, if needed) in the bus. Then, it turns on the AS (Address Strobe) signal, interpreted by the memory as an order to start the operation. The memory uses the address available and analyzes the RD_WR line to see which operation was asked. If a read was issued, one word (4 bytes) is taken from the specified address and sent through the data lines. In a write operation the address stored in the Byte Select register (lines BSEL0.3) defines the byte to be accessed in the word pointed to by the Address register. If an address is wrong, the ERR line is turned on. A Data Acknowledge (DTACK) is sent when the operation finished. The system components are interconnected using a Bus (see Figure 4). The bus Masters use the BGRANT (Bus Grant) and IACK (IRQ Acknowledgment) lines to be connected to the two devices with the following lower and upper priorities. The device with the highest priority is connected to a constant "1" signal in the BGRANT line. The BGRANT signal is sent to the lower priority devices up to the arrival to a device that requested the bus. When the device finishes the transfer, a IACK is transmitted. Input/output operations are memory mapped. Each device have a fixed set of addresses. Data written in those addresses are interpreted as an instructions for a device. Fifteen IRQ lines (IRQ1.IRQ15) are provided, and devices are connected to these lines. Higher priority devices are connected to lower IRQs. Figure 4. Organization of the Bus. Finally, an external cache memory was defined. The generic structure for the cache controller is defined in figure 5. The design and implementation of these modules were not included in the original version of Alfa- 1. They were defined as an assignment done by undergraduate students, following the procedures that will be presented in the following sections. In a first stage, the circuits were tested separately, different algorithms were implemented, and finally the device was integrated into the architecture. Each model was developed as a DEVS model that was integrated into a coupled model. This extension to the original architecture (which not will be explained in detail) shows some of the capabilities for extensibility and modifiability of Alfa-1. Figure 5. Organization of the Cache memory. 3. IMPLEMENTING THE ARCHITECTURE AS DEVS MODELS The architecture presented in the previous section was completely implemented using CD++. First, the behavior of each component was carefully specified, analyzing inputs, outputs and timing for each element. The specification also provided test cases. Then, each component was defined as a DEVS following the specification. After, each model was implemented in CD++, including an experimental framework following the test cases defined in the specification. Finally, the main model was built as a coupled model connecting all the submodels previously defined. This model follows the design presented in the figure 1, and its detailed definition can be found in [49]. Two implementations were considered. First, we reproduced the basic behavior of each circuit, coded as transition functions. Then, some of them were implemented in detail using Boolean logic. The basic building blocks were developed as atomic models, coupling them using digital logic concepts. In this way, two different abstraction levels were provided. Depending on the interest, each of them can be used. Once thoroughly tested, the basic models were integrated into higher level modules up to completing the definition of the architecture. The following sections will be devoted to present some of the components implemented as assignments done by our students. We show how different abstraction levels can be modeled, and present examples of modifiability of Alfa-1. 3.1. Inc/Dec As explained earlier, we use 520 general purpose registers organized as overlapped windows. In a given time, only one window can be active. The Inc/Dec model is the component that chooses the active window using a 5-bit CWP register. The models that are part of the CWP logic are shown in the figure 2. The CWP is incremented or decremented, and its value (stored in a d-latch represented as another DEVS) is received through the lines OP0-OP4. The outputs are transmitted through the lines RES0-RES4. This atomic model can be defined as: The behavior for the transition functions can be informally defined as follows: { When (x is received in the port y) If increment else decrement dint() { passivate; } { If (RES0.4 is different to the last output) send RESP0.4 through the ports OP0.4; Figure 6. Behavior of the transition functions for the INC/DEC model [50]. The FCOD value is used to tell if the value must be incremented or decremented. The ALU model is used to do this operation. Here we can see that, when an external event arrives, the hold_in function is activated. This macro represents the behavior of the DEVS time advance function (D), and it is in charge of manipulating the sigma variable. This is a state variable predefined for every DEVS model, which represents the remaining time up to the next scheduled internal event. The model will remain in the current state during this time, after which an output and internal transition functions are activated. The hold_in macro makes this timing definition easier. Passivate is another macro, which uses an infinite sigma and puts the model in passive phase (hold_in(passive, infinite)). The following figure shows the implementation of these functions using CD++. As we can see, the external transition function (dext) receives five operands as inputs, together with a function code. According to this code, the parameter is incremented or decremented. After, the model keeps the present value during a delay related with the circuit operation. The output function (l) is activated, and if the circuit changed its state the present value is transmitted. Then, the internal transition function (dint) passivates the model (that is, an internal event with infinite delay is scheduled, waiting for the next input). The constructor allows to specify the model's name, input/output ports, and parameters. As we can see, the definition of a DEVS atomic model is simpler than the use of any standard programming language. We have explained some of the advantages of using DEVS in section 1, but in this case, we can see how to apply it to build our models. DEVS provides a interface, consisting of only four functions to be programmed. This modular definition is independent of the simulator, and it is repeated for every model. Therefore, one can focus in the model development. The user only concentrate in the behavior under external events, the outputs that must be sent to other submodels, and the occurrence of internal events. Behavior for every model is encapsulated in these functions, together with the elapsed time definition. Testing patterns can be easily created, as the model can only activate these functions. IncDec::IncDec( const string &name string time( MainSimulator::Instance().getParameter( Model &IncDec::externalFunction( const ExternalMessage &msg ) { // Check the input ports, assigning the input values. if (_FCOD == 1) { for (int i=0; Increment Increment the va value useing the ALU for (int else { // Decrement for (int i=0; i<=4; i++) Decrement the v value useing the ALU for (int this->holdIn(active, preparationTime); // Schedule a delay for the circuit return *this; Model &IncDec::internalFunction( const InternalMessage & ) { When the delay is consumed, activate the output return *this ; Model &IncDec::outputFunction( const InternalMessage &msg ) { if (_RES[0]!=_OLD[0] || _RES[1]!=_OLD[1] || _RES[2]!=_OLD[2] || { sendOutput(msg.time(), RES0, _RES[0] ); sendOutput(msg.time(), RES1, _RES[1] ); sendOutput(msg.time(), RES2, _RES[2] ); sendOutput(msg.time(), RES3, _RES[3] ); sendOutput(msg.time(), RES4, _RES[4] ); return *this ; Figure 7. INC/DEC model definition: transition functions[50]. Once we have defined the atomic model, we can test it by injecting input values and inspecting the outputs. An experimental frame can be built, including pairs of input/output values to test the model automatically. In any case, we have to build a coupled model including the model to be tested. This is defined as follows: preparation Figure 8. INC/DEC coupled model definition [50]. These definitions follow the DEVS specifications. They are defined by its components (in this case, I_D, an instance of the IncDec model) and external parameters. Then, the links define the influencees and translation functions including the input/output ports for the model. In this case, the I_D model is related with the Top model, using the input/output ports defined earlier. 3.2. RegGlob This model defines the behavior of the global registers. It keeps the contents of the 8 global registers, allowing read/write operations on them. Two auxiliary state variables, olda and oldb, store the last outputs, and output signals are transmitted only for the bits that changed. This model is defined by: {0,1} CIN {0,., 232-1 }; A sketch of this model was shown in the figure 2. As we can see, it uses three select lines (asel, bsel, and csel) to choose two output registers and a register to be modified. An array of 32 integers (IN) keeps the present values of the registers. The Boolean line cen (C enable line) is used to allow write operations. The external transition function models the reception of an input. The function stores the desired operation according to the signal received. Also, we store an input value in the number of register to be activated. A new internal event is scheduled with a predefined delay, which models the circuit delay. If an external event arrives before the end of the delay, the operation is cancelled. Model &Regglob::externalFunction( const ExternalMessage &msg ) { switch (msg.port()) { case cen: line turned on case reset: breset = (int)msg.value(); // Reset Store the input lines this->holdIn ( active, delay ); return *this; Model &Regglob::internalFunction( const if (breset) // for (int i=0; i<255; i++) in[i]=0; if (bcen) for (int i=0; i<32; i++) return *this ; The i-eth line of the A input was enabled Store the register number received The i-eth line of the B input was enabled Store the register number received The i-eth line of the C input was enabled Store the register number received A reset signal was issued // The 8 register (32 bit each) are deleted // The write line was enabled // Update the desired register // Wait the next internal event Model &Regglob::outputFunction( const InternalMessage &msg ) { if (olda[i] != in[selecta*32+i]) { this->sendOutput(msg.time(), aout, if (oldb[i] != in[selectb*32+i]) { this->sendOutput(msg.time(), bout, return *this ; // The register has changed // Transmit it through the output line // The register has changed // Transmit it through the output line Figure 9. RegGlob model definition: transition functions [50] . The output function decides if the register has changed, querying olda and oldb which stores the previous status of the A and B lines. When the register changes, its value is sent through the chosen output or B). This model shows a more interesting use of the internal transition function. In this case, we are considering the internal state to decide how the model must react. The internal transition function sees if the reset line is activated. In that case, it clears the contents of every register. Then, if the cen line was activated, the value of the chosen register is updated with the new input. 3.3. Other basic components The architectural description is completed with other several DEVS models. We include generic aspects, making a brief description of their behavior. We do not include the definition of the model's transition func- tions, built as in the previous examples. Details of these models can be found in [49]. This circuit checks if the next window to be used will be overwritten. The component consists of a Window Invalid Mask register. It returns the value of the CWP-eth bit of the WIM register. {0,., 24-1} RD_WR {0,1} RESET {0, 1}; The memory is provided with three basic operations: read, write and reset. When a reset is issued, the memory initial image is loaded. The processor writes an address in the bus, and signals the memory using the AS signal when the address is ready. Then, a read/write signal is issued. The memory reacts according with this signal, using an output after a time related with the memory latency. The adder receives two inputs. Depending on the result, the Carry bit can be turned on. These models are used to align data read/written during the Load/Store operations. This model represents the behavior of the integer Arithmetic-Logic Unit. It is capable of executing the following operations: add, sub, addx, subx (add/sub with carry), and, or, xor, andn, orn, xnor (negated and, or, xor). This group of models were included to provide the behavior of the most used Boolean gates: AND, OR, NOT and XOR. They receive binary inputs, producing a result according to the desired operation. AS, RD/WR, DTACK, ERR, RESET, BUSY {0,1}; The bus interprets each of the input signals, providing outputs related with them. If a device which received a 1 in the BGRANTin port needs to write data in the Memory, it writes a 0 in the BGRANTout port (no smaller priority device is able to use the bus). Then, the device starts a bus cycle turning on the BUSYsig- nal. The device writes the address to be accessed in the ADDRESSlines, and the data to be written in DATA. After, the Byte Select Mask BSELto define which byte in the word is used. Finally, it turns on the RD/WRout and AS lines to tell a Write operation was issued. When the memory receives the AS signal, it executes a memory cycle that finishes when the DTACKout line is turned on. The device that issued the operation receives this signal in its DTACKin line. When the cycle has finished, if BGRANTin is still in 1, the device is able to transfer new data. Otherwise, it turns off the BUSYline, allowing a new bus operation by other device. This model is used in conditional jumps to decide if a branch must be executed. It represents the CPU clock, which period can be configured. This model is used to decide if access to the global registers or the register window is required. It returns the kind of register (Register window/Global) and its number. This model updates the nPC. This model manages the actions that take place when an interrupt is received. The PIL (Processor Interrupt lines mask the interrupts. If one or more IRQs whose numbers are greater than the PIL are received, the interrupt must be serviced. Then, we see which one has the higher priority, and the TF (Trap Found) bit is turned on. The TT (Trap Type) register is loaded according to the highest level interrupt. This model represents a processor register, implemented as a d-latch. The EIN line enable inputs, and the CLEAR line resets the register to zero. This model is in charge of making multiplication and divisions, turning on the condition bits. These models represent 2 or 4 input multiplexers. To choose them, we receive the 4-bit select signal whose bit turned on marks which input will be sent through the output. This model is in charge of managing the Register Window. This model is in charge of implementing a shifter. These models extend the sign of an operand of 13 or 22 bits to 32 bits. This component defines which trap must be serviced, based on a priority system. One of the input lines defines a non-masked trap. Other 7 bits are used to receive the number of a trap that can be masked. The model returns a bit telling if the trap must be serviced, and 8 bits telling the trap type. The following table shows the kind and priorities for each trap available: Line Description Priority Trap Type INST_ACC_EXCEP Instruction access exception 5 0x01 ILLEG_INST Illegal instruction 7 0x02 PRIV_INST Privileged instruction 6 0x03 WIN_OVER Window overflow 9 0x05 WIN_UNDER Window underflow 9 0x06 ADDR_NOT_ALIGN Address not aligned 10 0x07 DATA_ACC_EXCEP Data access exception 13 0x09 INST_ACC_ERR Instruction access error 3 0x21 DATA_ACC_ERR Data access error 12 0x29 DIV_ZERO Division by zero 15 0x2A DATA_ST_ERR Data store error 2 0x2B Table 3. Available traps According with this table, the model analyzes which is the higher priority trap to be serviced. After a delay, it sends the corresponding index through the output ports. 3.4. Control Unit The Control Unit is in charge of driving the execution flow of the processor. As explained earlier, this model uses several input/output lines. According with the input received, it issues different outputs, activating the different circuits that were defined previously. Here we show part of its behavior. The specification of the input/output sets is not included, because of its size (details can be found in [49]). Model &UC::externalFunction( const ExternalMessage &msg ) { else this->passivate(); } else if( msg.port() == DTACK ) { } else if( msg.port() == CCLOGIC ) { else { string portName; int portNum; nameNum( msg.port().name(), portName, portNum ); portName == "ir" else return *this; Model &UC::internalFunction( const InternalMessage & ) { return *this; Model &UC::outputFunction( const InternalMessage &msg ) { . // See if the c_en line must be activated // Read the Instruction Register and decode the instruction } else { else // See branches else // Transmit the outputs return *this; Figure 10. Control Unit: transition functions . As we can see, this model is activated by the occurrence of a clock tick. In this case, we check if the Control Unit is waiting a result coming from the memory (waitfmc). In that case, we have nothing to do and the model passivates. Otherwise, we register that a clock tick has finished. Other external inputs correspond to the signal DTACK coming from the memory or the CCLOGIC (that is, an input arriving from a register). We also recognize inputs for the Instruction Register (to store a new instruction to execute) or to the PSR (to update the condition codes). The internal transition function records that when the Address Strobe is up, we are waiting the end of a memory transfer. The main tasks of the control unit are executed by the output function. As we can see in the description, the present input values are queried. Depending on the number of clock tick in the instruction cycle, different output lines are activated. 4. THE DIGITAL LOGIC LEVEL The abstraction level of several models was further detailed, letting the students to analyze the digital logic level of the circuits. In the previous stage, the behavior of these circuits was defined by using atomic mod- els. In this case, some of these models were built using atomic models representing the basic Boolean gates (AND, OR, NOT, XOR). These models (described in the previous section) were used as components that were integrated using digital logic. A coupled model representing the complete circuit replaced the old atomic ones. These modifications, also done in course assignments, show the extensibility and modifiability of Alfa-1. Two of the models implemented this way will be explained following. 4.1. CMP model The CMP is a part of the Address Unit that detects addresses falling out of the program boundaries. The model receives two inputs (through the lines OPA and OPB that are connected to the BASE and LIMIT registers). As a result it returns the signal EQ if both values are equal, or LWif A is lower than B. Figure 11. Sketch of the Address Unit. The model is composed of several one-bit comparators, and coupling n of them generates n-bit compara- tors. The following figure shows the basic components of this building block: Figure 12. One-bit comparator [50]. This model is formally described by: select > where each is an atomic defining the corresponding building block, presented previously in section 3.3.; I { AND_n_1 }; I I { Self }; I { Self }; I { AND_n_2 }; I { Self, NOT_n_1, AND_n_1, XOR_n}; and Zij is built using I, as described earlier, and The definition of this coupled model using CD++ is presented in the following figure: in : OPAn OPBn Figure 13. CMP coupled model [50]. First, we define the components of the coupled model (corresponding to the D set). Then, the input/output ports are included (which are related with the X/Y sets defined earlier). Finally, the links show the model influencees (which define the translation function). The select function is implicitly defined by the order of definition for the model components. 4.2. Chip Selector The Chip Selector (CS) circuit is devoted to determine if an address is between two others. The model receives a 32-bit address, an Address Strobe (AS), and it returns a Boolean value telling if the address is between the boundaries. The MASK models provide two 32-bit sets (MAX Mask, MIN Mask) containing the boundaries of the address to be compared. These models, defined originally as latches, were redefined using Boolean gates. The input address for the chip selector is checked using two comparators, instances of the model defined in the previous section. Figure 14. Sketch of the Chip Selector [50]. The result obtained is transmitted through the ports LW and EQ for each of the comparators. Both outputs are ORed for the first register (as we are interested to see if CMP A MAX). After, the LW output of the second register is inverted (as we are interested to see if CMP B MIN). If the circuit is enabled, the result obtained is transmitted. components: MASMAX@MAS MASMIN@MAS CMPA@CMP CMPB@CMP and1@AND and2@AND or@OR not@NOT Link: A31@top OPA31@CMPA A31@top OPA31@CMPB Link: A30@top OPA30@CMPA A30@top OPA30@CMPB Link: out31@MASMAX OPB31@CMPA out31@MASMIN OPB31@CMPB Link: out30@MASMAX OPB30@CMPA out30@MASMIN OPB30@CMPB Link: out0@MASMAX OPB0@CMPA out0@MASMIN OPB0@CMPB Link: AS ina@and2 Link: eq@CMPA ina@or lw@CMPA inb@or Link: out@or ina@and1 out@not inb@and1 Link: out@and1 inb@and2 Link: out@and2 CS@top Figure 15. CS coupled model [50]. 5. SIMULATION RESULTS The present section shows the results obtained when some of the models previously presented are simu- lated. In the first case, we show the results of a value of 20 incremented by the INC/DEC model. The figure shows the model inputs with their timestamps and the output values obtained. The first step consists of giving an initial value to the circuit (in zero by default). The first event generates an output only when the model phase changes. As the preparation time for the circuit is 5 time units, this occurs at 00:00:05:000. The second input does not generate changes in the model and no output is issued. In simulated time 10, a new input is inserted through the port OP2. As this value changed, an output is generated at simulated time 15. The following 2 inputs are not registered because the circuit keeps its present state. The last one increments the value in the register by inserting the value through the FCOD port. The incremented value can be seen 5 time units after. INPUT OUTPUT Figure 16. Inputs and Outputs for the INC/DEC model. The following example shows the execution of the RegGlob model under different inputs. At the instant 0, the C enable line is activated, allowing write operations in the register. In this case, the register 4 is selected (csel2=1, csel1=0 and csel0=0), and the number 0xFFFFFFFF is used as input ter, in 00:00:01:00, the register 2 is selected (csel2=0 and csel1=1 ), and the number 0x55555555 is input (cin0=cin2=cin4.=cin30=1, and cin1= The first value is stored in the register 4, and the second in the register 2. INPUT OUTPUT reset 1 Figure 17. Inputs/Outputs of RegGlob [50]. At 00:00:02:00, C Enable is deactivated. Therefore, the following operations are devoted to read registers. We see that the value in the register 4 is sent through the A output (asel=2) and the register 2 is sent through B (bsel=1). As a result, the values previously loaded are transmitted (that is, 0xFFFFFFFF in A, and 0x55555555 in B). After, Reset is activated. Now, we try to read the register 4 at 00:00:05:00, and we obtain the value 0x00000000. The next test corresponds to the TrapLogic model. Here we can see the result obtained after turning on all the trap bits. Due to this, we expect obtaining the index of the highest priority trap that is pending. The result obtained after the delay time corresponds to the highest one of section 3.4: Data Store Error (whose code is the result we obtained). Also, the Trap Found flag is turned on. inst_acc_excep / 1.000 illeg_inst / 1.000 priv_inst / 1.000 win_over / 1.000 win_under / 1.000 addr_not_align / 1.000 data_acc_excep / 1.000 inst_acc_err / 1.000 data_acc_err / 1.000 div_zero / 1.000 data_st_err / 1.000 trap_inst / 1.000 Figure 18. Execution results for the TrapLogic model [50] . Finally, we show two execution examples that are part of a complete program. All the examples were executed in a Pentium processor (133 MHz), using the Linux version of CD++. The average performance for this model was one instruction per second. The source code was translated to binary using the GNU MASM assembler Linker. The executable is used as the initial memory image for the simulator. The first part of the following figure shows part of a program written in assembly language. The second part presents the binary code generated, together with the addresses for each instruction or data (one word each). set 0x12345678, %r1 st %r1, [dest] sth %r1, [dest+4] sth %r1, [dest+10] stb %r1, [dest+12] stb %r1, [dest+17] stb %r1, [dest+22] stb %r1, [dest+27] unimp Initial Image Addr. Memory Image Final image Addr. Memory Image Load the register 1 with 0x12345678 Store it in the "dest" variable Store the high half-word Store the last byte Interpretation 01001000 Store the register 1 in the address 72 01001100 Store the high part of reg. 1 in address 76 01010010 Store the high part of reg. 1 in address 82 01010100 Store the high byte of reg. 1 in address 84 01011001 Store the high byte of reg. 1 in address 89 01011110 Store the high byte of reg. 1 in address 94 01100011 Store the high byte of reg. 1 in address 99 00000000 unimp 00100000 "dest" variable (20: space character)Values Figure 19. Storing a value in memory . As we can see, this piece of code copies parts of the number 0x12345678 to certain memory addresses. We show the translation of the binary codes based on the specification of the instruction set of the SPARC processor. Finally, we show the memory image after the program execution. As we can see, the values stored in memory follow the instructions defined by the executable code. The following example shows the execution of part of another program. As we can see, the goal is to place a 1 in a given address, and then shift this value to the left, storing the result in the following address. The cycle is repeated 12 times. set 1, %r1 cycle: sll %r1, %r2, %r3 stb %r3, [%r2+dest] subcc %r2, 12, %r0 bne cycle inc 1, %r2 !Delay slot unimp Initial Image Addr. Memory Image 036 10000111 00101000 01000000 00000010 Final image Load the register 1 Shift the value the Store the result in Repeat the cycle 12 Interpretation with the value 1 number of times in r2 the variable dest times set <1>, 1 Take the register 1, shift and store in R3 Store in address substract 12 to R0 Relative jump to address -2 words (40) increment 1 unimp Destination variable A value 0x01 shifted 12 times Figure 20. Shifting and storing results in memory . Once the basic behavior of the simulated computer was verified, a thorough integration test was attacked. As explained earlier, each circuit was defined together with a set of input/output values that were encapsulated in an experimental framework. Once each of the models was tested, each operation in the instruction set was checked. The procedure was developed using the verification facilities of DEVS, defining 17100 test cases. The mechanism consisted in creating an experimental framework, which executed an instruction in the instruction set. The execution result was stored in memory, and a memory dump was executed, obtaining the memory state after the execution. This value is checked against the value obtained when the same program is executed in the real architecture, which is included in the testing experimental framework. This procedure allowed us to find some errors derived of the coupled model. For instance, we could see that the division instruction was not working properly. The generated test included the following sentences: set 274543375, %r24 ! stores a value in register 24 set 13908050 , %r22 ! a second value is stored in the register 22 both values are divided and stored in r10 st %r10 , [dest] ! The result is stored in memory unimp value: .ascii "VALUE:" dest: .word FFFFFFFF ! Result of Test 100 Figure 21. Testing routine for the UDIV instruction. When this example was executed, the testing coupled model found an error: Field Type Expected Found DIF In this case, the destination should have stored the value 19 (274543375 divided by 13908050). Instead, we have found the value 1, allowing us to see that one of the instructions had an unexpected behavior. In this way we could find errors in some of the instructions that could be fixed. We also found errors in some addition instructions, and in conditional jumps with prediction. Finally, we show part of the execution of the simulator for the example presented in Figure 20. We show a log file including the messages interchanged between modules. As in other DEVS frameworks, there are four kinds of messages: * (used to signal a state change due to an internal event), X (used when an external event arrives), Y (the model's output) and done (indicating that a model has finished with its task). The I messages initialize the corresponding models. For each message we show its type, timestamp, value, ori- gin/destination, and the port used for the transmission. Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message I / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message D / 00:00:00:000 / Message X / 00:00:00:000 / Message X / 00:00:00:000 / Message X / 00:00:00:000 / Message X / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message Y / 00:00:00:000 / Message D / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message Y / 00:00:00:000 / Message D / 00:00:00:000 / Message X / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message D / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message * / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Root(00) to top(01) // top(01) to mem(02) // top(01) to bus(03) top(01) to csmem(04) top(01) to cpu(05) top(01) to c1(64) top(01) to dpc(65) mem(02) / . to top(01) // bus(03) / . to top(01) // . to top(01) cpu(05) to ir(06) // cpu(05) to pc_add(07) cpu(05) to pc_mux(08) Initialize the higher level components: memory, bus, CS, etc. The models reply the next scheduled event The CPU initializes the components Root(00) to top(01) top(01) to cpu(05) cpu(05) to npc(10) // Take the nPC . to cpu(05) 1.000 to pc_latch(11) // Send to pc-inc to pc_inc(13) // to increment to pc_latch(11) // the value Schedule the activation of the to cpu(05) // pc-inc model Root(00) to top(01) top(01) to cpu(05) cpu(05) to pc(12) to cpu(05) // Initial address pc(12) / . to cpu(05) // Root(00) to top(01) top(01) to cpu(05) cpu(05) to clock(45) // Clock tick clock(45) / clck / 1.000 to cpu(05) clock(45) / 00:01:00:000 to cpu(05) clck / 1.000 to cu(43) Root(00) to top(01) top(01) to cpu(05) // Arrival to the CU cpu(05) to cu(43) // And activation of the components cu(43) / as / 1.000 to cpu(05) Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message Y / 00:00:00:000 / Message D / 00:00:00:000 / Message * / 00:00:10:000 / Message * / 00:00:10:000 / Message * / 00:00:10:000 / Message D / 00:00:10:000 / Message D / 00:00:10:000 / Message D / 00:00:10:000 / Message * / 00:00:10:000 / Message * / 00:00:10:000 / Message * / 00:00:10:000 / Message Y / 00:00:10:000 / Message Y / 00:00:10:000 / Message D / 00:00:10:000 / Message * / 00:00:20:001 / Message * / 00:00:20:001 / Message Y / 00:00:20:001 / Message Y / 00:00:20:001 / Message Y / 00:00:20:001 / Message Y / 00:00:20:001 / Message Y / 00:00:20:001 / Message Y / 00:00:20:001 / Message D / 00:00:20:001 / Message X / 00:00:20:001 / Message X / 00:00:20:001 / Message X / 00:00:20:001 / Message X / 00:00:20:001 / Message X / 00:00:20:001 / Message X / 00:00:20:001 / Message D / 00:00:20:001 / cu(43) / . to cpu(05) Root(00) to top(01) top(01) to cpu(05) cpu(05) to pc_latch(11) . to cpu(05) Root(00) to top(01) top(01) to cpu(05) cpu(05) to pc_inc(13) // Update the nPC . to cpu(05) Root(00) to top(01) top(01) to mem(02) // Memory returns the first instr. mem(02) / dtack / 1.000 to top(01) mem(02) / . to top(01) Figure 22. Log file of a simple routine . The execution cycle starts by initializing the higher level models (memory, CPU, etc. The message arrived to the CPU model is sent to its lower level components: Instruction Register, PC Adder, PC multiplexer, Control Unit, etc. When the initialization cycle finishes, the imminent model is executed. In this case, the nPC model is acti- vated, transmitting the address of the next instruction. As we can see, the 2nd and 5th bits are returned with a value. That means that the nPC value is (as we see in figure 20, the program starts in the address 32). The value is sent to the pc-inc model, in charge of adding 4 to this register. The update is finished in 10:000, as the activation time of this model was scheduled using the circuit delay. At that moment, a 4 value is added to the nPC, and we obtain the 3rd and 5th bits in 1 (res3 and res5), that is, the next PC. After, the PC is activated and the value 010000 (that is, 32) is obtained. This is the initial address of the program. The following event is the arrival of a clock tick, sent to the processor. The CPU schedules the next tick (in 1:00:000 time units) and transmits the signal to the Control Unit, which activates several components: a-mux, ALU, Addr-mux, IR,etc. We finally see, in the simulated time 20:000, that the memory has returned the first instruction (compare the results with the bit configuration in the address 32). The instruction is sent to the CPU to be stored in the Instruction Register and to follow with the execution. The rest of the instruction cycle is completed in the same way. We are able to follow the execution flow of any program by analyzing this log file. To simplify the analysis of results, we built a set of scripts using Tk, that lets the students to choose which components should be considered. In this way, behavior of each of the subcomponents can be followed more easily, and the students can analyze the behavior of the desired subsystem in detail. 6. CONCLUSION We have presented the use of DEVS to simulate a simple computer. The models were based on the architecture of the SPARC processor, which includes features not existing in simpler CPUs. The tools can be used in Computer Organization courses to analyze and understand the basic behavior of the different levels of a computer system. The interaction between levels can be studied, and experimental evaluation of the system can be done. The use of DEVS allowed us to have reusable models (in this case, Boolean gates, comparators, multiplex- ers, latches, etc. DEVS also allowed us to provide reusable code for different configurations. We provided different machines, one running the digital logic level and the other the instruction set, with different performance in each case, depending on the educational needs. The concept of internal transition functions can be used to improve the definition of the timing properties of each component, letting to define complex synchronization mechanisms. Nevertheless, in this case, most timing delays were represented as simple in- put/output relations. We have met all the goals proposed. Alfa-1 is public domain, and has been developed using CD++ (which is also public domain, and it was built using GNU C++). Therefore, the toolkit is available for its use in most existing Computer Organization courses. We described several levels of the architecture (from the Digital Logic level up to the Instruction Set). The assembly language level was also attacked using public domain assemblers that generated executable code that could run in Alfa-1. We have easily extended the components (for instance, including a cache memory that was not included in the first versions). We also have modified existing components (implementing, for instance, Digital Logic versions of some of the cir- cuits). Thorough testing could be done using an approach based on the construction of experimental frameworks associated with testing functions. An experimental framework was also built for the final integrated model. The most important achievements were related with our educational goals. The whole project was designed with detail as an assignment in a 3rd year Discrete Event Simulation course. The models were formally specified, and the specifications were used by students in a Computer Organization course to build the final version of the architecture. These students had only taken previous prerequisite courses in programming. With only this knowledge the students were able to build all the components here presented. Final integration was planned by a group of undergraduate Teaching Assistants (which also developed the Control Unit and a coupled model representing the whole architecture showed in the Figure 1). Individual and integration testing was also done by 2nd year students. Several of the modifications showed here were developed as course assignments. These facts show the feasibility of the approach from a pedagogical point of view. Upper level courses reported higher success rates and detailed knowledge of the subjects after using Alfa-1. The tools can be obtained at "http://www.sce.carleton.ca/wainer/usenix". Different experiences can be attacked using this toolkit. In the assembly language level, the students can use existing assemblers to build executables that run in the simulator. A complete analysis of the execution flow at the instruction level can be achieved by tracing the execution in the log file. The students can study the flow of a program and each instruction with detail, starting from the memory image of an executable. The instruction cycle and signal flow in the datapath can be easily inspected. Going deeper, we can see the behavior of those circuits implemented in the digital logic level. By extending or changing the existing instructions, and implementing the changes in the Control Unit, the students can experience the design of instruction sets. This allows them to have practice in instruction en- coding, and to relate instruction definition with the underlying architecture. The students also can include new components (as it was showed with the cache memory example), change existing ones, or implement them using digital logic. The hierarchical nature of DEVS provides means to go deeper in the hierarchy. For instance, the logical gates could be implemented defining the transistor level (which has not been implemented in this version). We planned to build an Assembler and Linker, but the code generated by those provided by GNU for SPARC plattforms executed straightforward. Nevertheless, the implementation of an assembler and linker are interesting assignments that can be faced to complete the layered view applied in these courses. Also, a debugger for the Alfa-1 architecture could be built, making easier to study the assembly language level. At present, Alfa-1 is being extended by defining components of the input/output subsystem. Several in- put/output devices, interfaces and DMA controllers will be simulated. Different transference techniques (polling, interrupts, DMA) will be considered. Likewise, the implementation of different cache management algorithms is being finished. Other tasks faced at present include the definition of a graphical interface to enhance the use of the toolkit. The set of scripts mentioned in section 5 will be used to gather the results of the simulations, and will be used as inputs to be displayed in graphical way. In this way, the study and analysis of the different subsystems will be improved. 7. ACKNOWLEDGEMENTS I want to thank to the anonymous referees the detailed comments they made to this article. I also thank Prof. Trevor Pearce at SCE, Carleton University, for his help with the final version. Sergio Zlotnik collaborated in the early stage of this project, presented earlier in [51]. The research was partially funded by the Usenix foundation and UBACYT Project JW10. 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simulation in education;computer organization;applications of DEVS methodology;DEVS models