Recently there has been a significant effort to handle quantitative properties in formal verification and synthesis. While weighted automata over finite and infinite words provide a natural and flexible framework to express quantitative properties, perhaps surprisingly, some basic system properties such as average response time cannot be expressed using weighted automata, nor in any other know decidable formalism. In this work, we introduce nested weighted automata as a natural extension of weighted automata which makes it possible to express important quantitative properties such as average response time. In nested weighted automata, a master automaton spins off and collects results from weighted slave automata, each of which computes a quantity along a finite portion of an infinite word. Nested weighted automata can be viewed as the quantitative analogue of monitor automata, which are used in run-time verification. We establish an almost complete decidability picture for the basic decision problems about nested weighted automata, and illustrate their applicability in several domains. In particular, nested weighted automata can be used to decide average response time properties.
Weighted counterpart of Boolean nested automata. Weighted nested automata have been considered in @cite_3 in the context of finite words, where the weights are given over semirings. It is further required that the semirings of all master and slave automata coincide, while in our case, the value functions may differ. Since the semirings of master and slave automata must coincide for @cite_3 , it can be interpreted as defining weighted counterpart of Boolean nested automata over finite words. Adding nesting structure to words and trees have been extensively studied for non-weighted automata in @cite_12 @cite_4 and also applied to software model checking @cite_24 . The work of @cite_3 defines a weighted counterpart of nesting of finite words, whereas we consider nesting of various weighted automata to express properties of infinite behaviors of systems. Properties such as long-run average response time cannot be expressed in the framework of @cite_3 .
@cite_24 @cite_4 @cite_12 @cite_3
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Weighted automata map input words to numerical values. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. In the 90's, Krob studied the decidability of problems on rational series, which strongly relate to weighted automata. In particular, it follows from Krob's results that the universality problem (that is, deciding whether the values of all words are below some threshold) is decidable for weighted automata with weights in N ∪ ∞ , and that the equality problem is undecidable when the weights are in N ∪ ∞ . In this paper we continue the study of the borders of decidability in weighted automata, describe alternative and direct proofs of the above results, and tighten them further. Unlike the proofs of Krob, which are algebraic in their nature, our proofs stay in the terrain of state machines, and the reduction is from the halting problem of a two-counter machine. This enables us to significantly simplify Krob's reasoning and strengthen the results to apply already to a very simple class of automata: all the states are accepting, there are no initial nor final weights, and all the weights are from the set -1, 0, 1 . The fact we work directly with automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata with weights in N ∪ ∞ , and in fact even with weights in Q≥0 ∪ ∞ , is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of equality vs. universality and the front of the N ∪ ∞ vs. the Z ∪ ∞ domains. We introduce new classes of weighted automata on words. Equipped with pebbles and a two-way mechanism, they go beyond the class of recognizable formal power series, but capture a weighted version of first-order logic with bounded transitive closure. In contrast to previous work, this logic allows for unrestricted use of universal quantification. Our main result states that pebble weighted automata, nested weighted automata, and this weighted logic are expressively equivalent. We also give new logical characterizations of the recognizable series. Weighted finite automata are classical nondeterministic finite automata in which the transitions carry weights. These weights may model, for example, the cost involved when executing a transition, the resources or time needed for this, or the probability or reliability of its successful execution. Weights can also be added to classical automata with infinite state sets like pushdown automata, and this extension constitutes the general concept of weighted automata. Since their introduction in the 1960s they have stimulated research in related areas of theoretical computer science, including formal language theory, algebra, logic, and discrete structures. Moreover, weighted automata and weighted context-free grammars have found application in natural-language processing, speech recognition, and digital image compression. This book covers all the main aspects of weighted automata and formal power series methods, ranging from theory to applications. The contributors are the leading experts in their respective areas, and each chapter presents a detailed survey of the state of the art and pointers to future research. The chapters in Part I cover the foundations of the theory of weighted automata, specifically addressing semirings, power series, and fixed point theory. Part II investigates different concepts of weighted recognizability. Part III examines alternative types of weighted automata and various discrete structures other than words. Finally, Part IV deals with applications of weighted automata, including digital image compression, fuzzy languages, model checking, and natural-language processing. Computer scientists and mathematicians will find this book an excellent survey and reference volume, and it will also be a valuable resource for students exploring this exciting research area. Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages. We propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include executions of structured programs, annotated linguistic data, and HTML XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata—finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-*, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over. The linear encodings of nested words give the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics. We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata. We introduce a weighted logic with discounting and we establish the Buchi-Elgot theorem for weighted automata over finite words and arbitrary commutative semirings. Then we investigate Buchi and Muller automata with discounting over the max-plus and the min-plus semiring. We show their expressive equivalence with weighted MSO-sentences with discounting. In this case our logic has a purely syntactic definition. For the finite case, we obtain a purely syntactically defined weighted logic if the underlying semiring is additively locally finite. We study bisimulation and minimization for weighted automata, relying on a geometrical representation of the model, linear weighted automata ( lwa ). In a lwa , the state-space of the automaton is represented by a vector space, and the transitions and weighting maps by linear morphisms over this vector space. Weighted bisimulations are represented by sub-spaces that are invariant under the transition morphisms. We show that the largest bisimulation coincides with weighted language equivalence, can be computed by a geometrical version of partition refinement and that the corresponding quotient gives rise to the minimal weighted-language equivalence automaton. Relations to Larsen and Skou's probabilistic bisimulation and to classical results in Automata Theory are also discussed. We study efficient translations of Regular Linear Temporal Logic ( ) into automata on infinite words. is a temporal logic that fuses Linear Temporal Logic (LTL) with regular expressions, extending its expressive power to all @math -regular languages. The first contribution of this paper is a novel bottom up translation from into alternating parity automata of linear size that requires only colors @math , @math and @math . Moreover, the resulting automata enjoy the stratified internal structure of hesitant automata. Our translation is defined inductively for every operator, and does not require an upfront transformation of the expression into a normal form. Our construction builds at every step two automata: one equivalent to the formula and another to its complement. Inspired by this construction, our second contribution is to extend with new operators, including universal sequential composition, that enrich the logic with duality laws and negation normal forms. The third contribution is a ranking translation of the resulting alternating automata into non-deterministic automata. To provide this efficient translation we introduce the notion of stratified rankings, and show how the translation is optimal for the LTL fragment of the logic. Unsupervised extractive summarization of emails using a deep auto-encoder with excellent performance.Ensemble Noisy Auto-Encoder runs noisy inputs through one trained network, enhancing performance.Summaries are highly informative and semantically similar to human abstracts. We present methods of extractive query-oriented single-document summarization using a deep auto-encoder (AE) to compute a feature space from the term-frequency (tf) input. Our experiments explore both local and global vocabularies. We investigate the effect of adding small random noise to local tf as the input representation of AE, and propose an ensemble of such noisy AEs which we call the Ensemble Noisy Auto-Encoder (ENAE). ENAE is a stochastic version of an AE that adds noise to the input text and selects the top sentences from an ensemble of noisy runs. In each individual experiment of the ensemble, a different randomly generated noise is added to the input representation. This architecture changes the application of the AE from a deterministic feed-forward network to a stochastic runtime model. Experiments show that the AE using local vocabularies clearly provide a more discriminative feature space and improves the recall on average 11.2 . The ENAE can make further improvements, particularly in selecting informative sentences. To cover a wide range of topics and structures, we perform experiments on two different publicly available email corpora that are specifically designed for text summarization. We used ROUGE as a fully automatic metric in text summarization and we presented the average ROUGE-2 recall for all experiments. We characterize the infinite words determined by one-way stack automata. An infinite language L determines an infinite word alpha if every string in L is a prefix of alpha. If L is regular or context-free, it is known that alpha must be ultimately periodic. We extend this result to the class of languages recognized by one-way nondeterministic checking stack automata (1-NCSA). We then consider stronger classes of stack automata and show that they determine a class of infinite words which we call multilinear. We show that every multilinear word can be written in a form which is amenable to parsing. Finally, we consider the class of one-way multihead deterministic finite automata (1:multi-DFA). We show that every multilinear word can be determined by some 1:multi-DFA, but that there exist infinite words determined by 1:multi-DFA which are not multilinear. 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job J has processing length pj, weight wj, and is processed at a rate of lij when scheduled on machine i. The online scheduler knows the values of wj and lij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1+e)-speed O(1 2)-competitive for any constant e > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f-1i(P) when consuming power P, where fi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms. We describe an automata-theoretic approach for the competitive analysis of online algorithms. Our approach is based on weighted automata, which assign to each input word a cost in R≥0. By relating the "unbounded look ahead" of optimal offline algorithms with nondeterminism, and relating the "no look ahead" of online algorithms with determinism, we are able to solve problems about the competitive ratio of online algorithms, and the memory they require, by reducing them to questions about determinization and approximated determinization of weighted automata. Alur, R. and D.L. Dill, A theory of timed automata, Theoretical Computer Science 126 (1994) 183-235. We propose timed (j&e) automata to model the behavior of real-time systems over time. Our definition provides a simple, and yet powerful, way to annotate state-transition graphs with timing constraints using finitely many real-valued clocks. A timed automaton accepts timed words-infinite sequences in which a real-valued time of occurrence is associated with each symbol. We study timed automata from the perspective of formal language theory: we consider closure properties, decision problems, and subclasses. We consider both nondeterministic and deterministic transition structures, and both Biichi and Muller acceptance conditions. We show that nondeterministic timed automata are closed under union and intersection, but not under complementation, whereas deterministic timed Muller automata are closed under all Boolean operations. The main construction of the paper is an (PSPACE) algorithm for checking the emptiness of the language of a (nondeterministic) timed automaton. We also prove that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the nondeterministic case and PSPACE-complete in the deterministic case. Finally, we discuss the application of this theory to automatic verification of real-time requirements of finite-state systems. Abstract We present an approximation technique, that can render real-time model checking of safety and universal path properties more efficient. It is beneficial, when loops lead to repetition of control situations. Basically we augment a timed automata model with carefully selected extra transitions. This increases the size of the state-space, but potentially decreases the number of symbolic states to be explored by orders of magnitude. We give a formal definition of a timed automata formalism, enriched with basic data types, hand-shake synchronization, urgency, and committed locations. We prove by means of a trace semantics, that if a safety property can be established in the augmented model, it also holds for the original model. We extend our technique to a richer set of properties, that can be decided via a set of traces (universal path properties). In order for universal path properties to carry over to the original model, the semantics of the timed automata formalism is formulated relative to the applied augmentation. Our technique is particularly useful in systems, where a scheduler dictates repetition of control over elapsing time. As a typical example we mention translations of LEGO® RCX™ programs to U ppaal models, where the Round-Robin scheduler is a static entity. We allow scheduler and associated tasks to “park”, until some timing or environmental conditions are met. We apply our technique on a brick-sorter model for a safety property and report run-time data. We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate @math over an alphabet of size independent of block length that can correct a fraction of @math errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the Garcia-Stichtenoth tower, this yields codes of rate @math over an alphabet of size @math , that can correct a fraction of @math errors. In the conclusion of his monumental paper on optimal inapproximability results, Hastad [13] suggested that Fourier analysis of Dictator (Long Code) Tests may not be universally applicable in the study of CSPs. His main open question was to determine if the technique could resolve the approximability of satisfiable 3-bit constraint satisfaction problems. In particular, he asked if the "Not Two" (NTW) predicate is non-approximable beyond the random assignment threshold of 5 8 on satisfiable instances. Around the same time, Zwick [30] showed that all satisfiable 3-CSPs are 5 8-approximable and conjectured that the 5 8 is optimal. In this work we show that Fourier analysis techniques can produce a Dictator Test based on NTW with completeness 1 and soundness 5 8. Our test's analysis uses the Bonami-Gross-Beckner hypercontractive inequality. We also show a soundness lower bound of 5 8 for all 3-query Dictator Tests with perfect completeness. This lower bound for Property Testing is proved in part via a semidefinite programming algorithm of Zwick [30]. Our work precisely determines the 3-query "Dictatorship Testing gap". Although this represents progress on Zwick's conjecture, current PCP "outer verifier" technology is insufficient to convert our Dictator Test into an NP-hardness-of-approximation result. This paper offers characteristic formula constructions in the real-time logic L ν for several behavioural relations between (states of) timed automata. The behavioural relations studied in this work are timed (bi)similarity, timed ready simulation, faster-than bisimilarity and timed trace inclusion. The characteristic formulae delivered by our constructions have size which is linear in that of the timed automaton they logically describe. This also applies to the characteristic formula for timed bisimulation equivalence, for which an exponential space construction was previously offered by Laroussinie, Larsen and Weise. We consider the problem of evaluating the quality of solution sets generated by heuristics for multiple-objective combinatorial optimization problems. We extend previous research on the integrated preference functional (IPF), which assigns a scalar value to a given discrete set of nondominated points so that the weighted Tchebycheff function can be used as the underlying implicit value function. This extension is useful because modeling the decision maker's value function with the weighted Tchebycheff function reflects the impact of unsupported points when evaluating sets of nondominated points. We present an exact calculation method for the IPF measure in this case for an arbitrary number of criteria. We show that every nondominated point has its optimal weight interval for the weighted Tchebycheff function. Accordingly, all nondominated points, and not only the supported points in a set, contribute to the value of the IPF measure when using the weighted Tchebycheff function. Two-and three-criteria numerical examples illustrate the desirable properties of the weighted Tchebycheff function, providing a richer measure than the original IPF based on a convex combination of objectives. Over the last years there has been an increasing research effort directed towards the automatic verification of infinite state systems, such as timed automata, hybrid automata, data-independent systems, relational automata, Petri nets, lossy channel systems, context-free and push-down processes. We present a method for deciding reachability properties of networks of timed processes. Such a network consists of an arbitrary set of identical timed automata, each with a single real-valued clock. Using a standard reduction from safety properties to reachability properties, we can use our algorithm to decide general safety properties of timed networks. To our knowledge, this is the first decidability result concerning verification of systems that are infinite-state in "two dimensions": they contain an arbitrary set of (identical) processes, and they use infinite data-structures, viz real-valued clocks. We illustrate our method by showing how it can be used to automatically verify Fischer's protocol, a timer-based protocol for enforcing mutual exclusion among an arbitrary number of processes.Finally, we show undecidability of the recurrent state problem: given a state in a timed network, check whether there is a computation of the network visiting the state infinitely often. This implies undecidability of model checking for any temporal logic which is sufficiently expressive to encode the recurrent state problem, such as PTL, CTL, etc.