Arrow's Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider @math voters who vote independently at random, each following the uniform distribution over the 6 rankings of 3 alternatives. Arrow's theorem implies that any constitution which satisfies IIA and Unanimity and is not a dictator has a probability of at least @math for a non-transitive outcome. When @math is large, @math is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every @math , there exists a @math , which depends on @math only, such that for all @math , and all constitutions on 3 alternatives, if the constitution satisfies: The IIA condition. For every pair of alternatives @math , the probability that the constitution ranks @math above @math is at least @math . For every voter @math , the probability that the social choice function agrees with a dictatorship on @math at most @math . Then the probability of a non-transitive outcome is at least @math .
As noted in @cite_5 , there is an interesting connection between quantitative Arrow statements and the concept of testing introduced in @cite_12 @cite_16 which was studied and used extensively since. Roughly speaking a property of functions is testable if it is possible to perform a randomized test for the property such that if the probability that the function passes the test is close to @math , then the function has to be close to a function with the property (say in the hamming distance). In terms of testing, our result states that among all functions satisfying the IIA property, the Transitivity property is testable. Moreover, the natural test "works": i.e., in order to test for transitivity, one can pick a random input and check if the outcome is transitive.
@cite_5 @cite_16 @cite_12
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In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation. In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k -Colorable, or having a p -Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph. We characterize the set of properties of Boolean-valued functions on a finite domain @math that are testable with a constant number of samples. Specifically, we show that a property @math is testable with a constant number of samples if and only if it is (essentially) a @math -part symmetric property for some constant @math , where a property is @math -part symmetric if there is a partition @math of @math such that whether @math satisfies the property is determined solely by the densities of @math on @math . We use this characterization to obtain a number of corollaries, namely: (i) A graph property @math is testable with a constant number of samples if and only if whether a graph @math satisfies @math is (essentially) determined by the edge density of @math . (ii) An affine-invariant property @math of functions @math is testable with a constant number of samples if and only if whether @math satisfies @math is (essentially) determined by the density of @math . (iii) For every constant @math , monotonicity of functions @math on the @math -dimensional hypergrid is testable with a constant number of samples. The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are far' from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, primarily in the context of the monotonicity of a sequence of integers, or the monotonicity of a function over the n-dimensional hypercube 1,l,m n. These works resulted in monotonicity testers whose query complexity is at most polylogarithmic in the size of the domain.We show that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory. These problems include: testing that a Boolean assignment of variables is close to an assignment that satisfies a specific 2-CNF formula, testing that a set of vertices is close to one that is a vertex cover of a specific graph, and testing that a set of vertices is close to a clique.We then investigate the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We give algorithms and lower bounds for the general problem, as well as for some interesting special cases. In proving a general lower bound, we construct graphs with combinatorial properties that may be of independent interest. Combinatorial property testing, initiated by Rubinfeld and Sudan [23] and formally defined by Goldreich, Goldwasser and Ron in [18], deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (e, q)-test for P is a randomized algorithm that queries at most q places of an input f and distinguishes with probability 2 3 between the case that f has the property and the case that at least an e-fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e. as a product order of total orders). The main result here presents an (e, poly(1 e))-test for every property of 0 1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the “monotonicity” property studied in [9,8,13], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. This class of properties is related to properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations. We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n 1 4) (for a small enough fixed e) exists. The above results identify new classes of properties that are defined by means of restricted logics, and that are efficiently testable. They also lay out a platform that bridges some previous results. We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser, and Ron in [J. ACM, 45 (1998), pp. 653--750]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language @math and an integer n there exists a randomized algorithm which always accepts a word w of length n if @math and rejects it with high probability if @math has to be modified in at least @math positions to create a word in L. The algorithm queries @math bits of w. This query complexity is shown to be optimal up to a factor polylogarithmic in @math . We also discuss the testability of more complex languages and show, in particular, that the query complexity required for testing context-free languages cannot be bounded by any function of @math . The problem of testing regular languages can be viewed as a part of a very general approach, seeking to probe testability of properties defined by logical means. The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property @math has an oblivious one-sided error tester if and only if @math is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable. In property testing, we are given oracle access to a function f, and we wish to test if the function satisfies a given property P, or it is e-far from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the domain. This paper relates the complexity of testing the monotonicity of a function over the d-dimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the query complexity of the property tester. The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions. In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges. We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result. Correlation intractable function ensembles were introduced in an attempt to capture the "unpredictability" property of a random oracle: It is assumed that if R is a random oracle then it is infeasible to find an input x such that the input-output pair (x,R(x)) has some desired property. Since this property is often useful to design many cryptographic applications in the random oracle model, it is desirable that a plausible construction of correlation intractable function ensembles will be provided. However, no plausibility result has been proposed. In this paper, we show that proving the implication, "if one-way functions exist then correlation intractable function ensembles exist", is as hard as proving that "3-round auxiliary-input zero-knowledge Arthur-Merlin proofs exist only for trivial languages such as BPP languages." As far as we know, proving the latter claim is a fundamental open problem in the theory of zero-knowledge proofs. Therefore, our result can be viewed as strong evidence that the construction based solely on one-way functions will be impossible, i.e., that any plausibility result will require stronger cryptographic primitives. We introduce the notion of a canonical tester for a class of properties on distributions, that is, a tester strong and general enough that “a distribution property in the class is testable if and only if the canonical tester tests it.” We construct a canonical tester for the class of properties of one or two distributions that are symmetric and satisfy a certain weak continuity condition. Analyzing the performance of the canonical tester on specific properties resolves two open problems, establishing lower bounds that match known upper bounds: we show that distinguishing between entropy @math on distributions over @math requires @math samples, and distinguishing whether a pair of distributions has statistical distance @math requires @math samples. Our techniques also resolve a conjecture about a property that our canonical tester does not apply to: distinguishing identical distributions from those with statistical distance @math requires @math samples. We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area—graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs: A model for bounded-degree graphs, A model for graphs with a bound on the total number of edges (both models were already considered in the context of uniform distribution testing), and A model which is a combination of the two previous testing models; i.e., bounded-degree graphs with a bound on the total number of edges. We prove that connectivity can be tested in each of these testing models, in a distribution-free manner, using a number of queries that is independent of the size of the graph. This is done by providing a new analysis to previously known connectivity testers (from “standard”, uniform distribution property-testing) and by introducing some new testers. Tolerant testing is an emerging topic in the field of property testing, which was defined in [M. Parnas, D. Ron, and R. Rubinfeld, J. Comput. System Sci., 72 (2006), pp. 1012-1042] and has recently become a very active topic of research. In the general setting, there exist properties that are testable but are not tolerantly testable [E. Fischer and L. Fortnow, Proceedings of the @math th IEEE Conference on Computational Complexity, 2005, pp. 135-140]. On the other hand, we show here that in the setting of the dense graph model, all testable properties are not only tolerantly testable (which was already implicitly proved in [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451-476] and [O. Goldreich and L. Trevisan, Random Structures Algorithms, 23 (2003), pp. 23-57]), but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant. In the course of the proof we develop a framework for extending Szemeredi's regularity lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a @math circuit. This, in some ways, strengthens the results of [N. Alon, R. A. Duke, H. Lefmann, V. Rodl, and R. Yuster, J. Algorithms, 16 (1994), pp. 80-109]. We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangle-freeness. Hence, in particular, testing whether a graph is (K_4 )-free, and testing whether a graph is (C_4 )-free can be done in a constant number of rounds (where (K_k ) denotes the k-node clique, and (C_k ) denotes the k-node cycle). On the other hand, we show that testing (K_k )-freeness and (C_k )-freeness for (k 5 ) appear to be much harder. Specifically, we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test (K_k )-freeness and (C_k )-freeness in a constant number of rounds for (k 5 ). We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT-assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT-assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=12, but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear. The paper tackles the power of randomization in the context of locality by analyzing the ability to boost' the success probability of deciding a distributed language. The main outcome of this analysis is that the distributed computing setting contrasts significantly with the sequential one as far as randomization is concerned. Indeed, we prove that in some cases, the ability to increase the success probability for deciding distributed languages is rather limited. Informally, a (p,q)-decider for a language L is a distributed randomized algorithm which accepts instances in L with probability at least p and rejects instances outside of L with probability at least q. It is known that every hereditary language that can be decided in t rounds by a (p,q)-decider, where p^2+q>1, can actually be decided deterministically in O(t) rounds. In one of our results we give evidence supporting the conjecture that the above statement holds for all distributed languages. This is achieved by considering the restricted case of path topologies. We then turn our attention to the range below the aforementioned threshold, namely, the case where p^2+q . We define B_k(t) to be the set of all languages decidable in at most t rounds by a (p,q)-decider, where p^ 1+1 k +q>1. It is easy to see that every language is decidable (in zero rounds) by a (p,q)-decider satisfying p+q=1. Hence, the hierarchy B_k provides a spectrum of complexity classes between determinism and complete randomization. We prove that all these classes are separated: for every integer k 1, there exists a language L satisfying L B_ k+1 (0) but L B_k(t) for any t=o(n). In addition, we show that B_ (t) does not contain all languages, for any t=o(n). Finally, we show that if the inputs can be restricted in certain ways, then the ability to boost the success probability becomes almost null. We consider monotonicity testing of functions f:[n]d→ 0,1 , in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in @math and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 The research in qualitative reasoning and in spatial CSP is always investigated in the backdrop of its temporal counterpart - qualitative temporal reasoning and TCSP. Unlike the case of interval algebra (IA), the composition table of RCC, IA's so-called spatial counterpart, is in general neither complete nor extensional, the compositional consistency can be still a valid reasoning mechanism. Even in such a restricted situation, many of the known properties of IA have not been investigated for validity in the context of RCC. We address, in this paper two such properties-convexity and minimality. The importance of minimality cannot be underestimated as in a minimal network every label is feasible and hence determining all the consistent scenarios can be accomplished very efficiently. It is known that path consistency does not yield a minimal network for tractable classes of RCC-8. We represent RCC-8 relations as a partially ordered set and exploit the properties of partial ordering to derive very interesting theoretical results. We show here that there exists a convex class of relations of RCC-8 for which path consistency yields a minimal network. Our results are very important as it gives a sufficient condition for minimality and useful to generate all consistent scenarios whenever compositional consistency is a valid reasoning mechanism We present a technique that improves random test generation by incorporating feedback obtained from executing test inputs as they are created. Our technique builds inputs incrementally by randomly selecting a method call to apply and finding arguments from among previously-constructed inputs. As soon as an input is built, it is executed and checked against a set of contracts and filters. The result of the execution determines whether the input is redundant, illegal, contract-violating, or useful for generating more inputs. The technique outputs a test suite consisting of unit tests for the classes under test. Passing tests can be used to ensure that code contracts are preserved across program changes; failing tests (that violate one or more contract) point to potential errors that should be corrected. Our experimental results indicate that feedback-directed random test generation can outperform systematic and undirected random test generation, in terms of coverage and error detection. On four small but nontrivial data structures (used previously in the literature), our technique achieves higher or equal block and predicate coverage than model checking (with and without abstraction) and undirected random generation. On 14 large, widely-used libraries (comprising 780KLOC), feedback-directed random test generation finds many previously-unknown errors, not found by either model checking or undirected random generation. This paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines. Often, counting classes, probabilistic classes, and intersection classes lack such enumerations. Concentrating on the counting class UP, we show that there are relativizations for which UPA has no complete languages and other relativizations for which PB ≠ UPB ≠ NPB and UPB has complete languages. Among other results we show that P ≠ UP if and only if there exists a set S in P of Boolean formulas with at most one satisfying assignment such that S ∩ SAT is not in P. P ≠ UP ∩ coUP if and only if there exists a set S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S. If UP has complete languages then there exists a set R in P of Boolean formulas with at most one satisfying assignment so that SAT ∩ R is complete for UP. Finally, we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP. There is a relativized world where BPPA has no complete languages. If BPP has complete languages then it has a complete language of the form B ∩ MAJORITY, where B ∈ P and MAJORITY = f | f is true for at least half of all assignments is the canonical PP-complete set.