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15
Some 3CNF properties are hard to test
 In Proc. 35th ACM Symp. on Theory of Computing
, 2003
"... Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that th ..."
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Cited by 59 (10 self)
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Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires Ω(n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O ( √ n) queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474–483]. Notice that for every negative instance (i.e., an assignment that does not satisfy ϕ) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: 1. In the context of testing for linear properties, adaptive twosided error tests have no more power than nonadaptive onesided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a onesided error. 2. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires Ω(n) queries.
Testing stconnectivity
 11th International Workshop on Randomization and Computation (RANDOM 2007
, 2007
"... We continue the study, started in [9], of property testing of graphs in the orientation model. A major question which was left open in [9] is whether the property of stconnectivity can be tested with a constant number of queries. Here we answer this question on the affirmative. To this end we const ..."
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Cited by 6 (5 self)
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We continue the study, started in [9], of property testing of graphs in the orientation model. A major question which was left open in [9] is whether the property of stconnectivity can be tested with a constant number of queries. Here we answer this question on the affirmative. To this end we construct a nontrivial reduction of the stconnectivity problem to the problem of testing languages that are decidable by branching programs, which was solved in [11]. The reduction combines combinatorial arguments with a concentration type lemma that is proven for this purpose. Unlike many other property testing results, here the resulting testing algorithm is highly nontrivial itself, and not only its analysis.
Testing Properties of ConstraintGraphs
 Proceedings of the 22 nd IEEE Annual Conference on Computational Complexity (CCC 2007
"... We study a model of graph related formulae that we call the ConstraintGraph model. A constraintgraph is a labeled multigraph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over ..."
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Cited by 6 (4 self)
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We study a model of graph related formulae that we call the ConstraintGraph model. A constraintgraph is a labeled multigraph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over the variables that label its adjacent edges. A Boolean assignment to the variables satisfies the constraint graph if it satisfies every vertex function. We associate with a constraintgraph G the property of all assignments satisfying G, denoted SAT (G). We show that the above model is quite general. That is, for every property of strings P there exists a property of constraintgraphs PG such that P is testable using q queries iff PG is thus testable. In addition, we present a large family of constraintgraphs for which SAT (G) is testable with constant number of queries. As an implication of this, we infer the testability of some edge coloring problems (e.g. the property of two coloring of the edges in which every node is adjacent to at least one vertex of each color). Another implication is that every property of Boolean strings that can be represented by a Readtwice CNF formula is testable. We note that this is the best possible in terms of the number of occurrences of every variable in a formula. 1
Approximate Satisfiability and Equivalence
 IN LOGIC IN COMPUTER SCIENCE
, 2006
"... Inspired by Property Testing, we relax the classical satisfiability U  = F between a finite structure U of a class K and a formula F, to a notion of εsatisfiability U =ε F, and the classical equivalence F1 ≡ F2 between two formulas F1 and F2, toεequivalence F1 ≡ε F2 for ε>0. We consider the clas ..."
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Cited by 4 (2 self)
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Inspired by Property Testing, we relax the classical satisfiability U  = F between a finite structure U of a class K and a formula F, to a notion of εsatisfiability U =ε F, and the classical equivalence F1 ≡ F2 between two formulas F1 and F2, toεequivalence F1 ≡ε F2 for ε>0. We consider the class of strings and trees with the edit distance with moves, and show that these approximate notions can be efficiently decided. We use a statistical embedding of words (resp. trees) into ℓ1, which generalizes the original Parikh mapping, obtained by sampling O(f(ε)) finite samples of the words (resp. trees). We give a tester for equality and membership in any regular language, in time independent of the size of the structure. Using our geometrical embedding, we can also test the equivalence between two regular properties on words, defined by Monadic Second Order formulas. Our equivalence tester has polynomial time complexity in the size of the automaton (or regular expression), for a fixed ε, whereas the exact version of the equivalence problem is PSPACEcomplete. Last, we extend the geometric embedding, and hence the tester algorithms, to infinite regular languages and to contextfree languages. For contextfree languages, the equivalence tester has an exponential time complexity, whereas the exact version is undecidable.
Property Testing: Theory and Applications
, 2003
"... Property testers are algorithms that distinguish inputs with a given property from those that are far from satisfying the property. Far means that many characters of the input must be changed before the property arises in it. Property testing was introduced by Rubinfeld and Sudan in the context of l ..."
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Cited by 2 (0 self)
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Property testers are algorithms that distinguish inputs with a given property from those that are far from satisfying the property. Far means that many characters of the input must be changed before the property arises in it. Property testing was introduced by Rubinfeld and Sudan in the context of linearity testing and first studied in a variety of other contexts by Goldreich, Goldwasser and Ron. The query complexity of a property tester is the number of input characters it reads. This thesis is a detailed investigation of properties that are and are not testable with sublinear query complexity. We begin by characterizing properties of strings over the binary alphabet in terms of their formula complexity. Every such property can be represented by a CNF formula. We show that properties of nbit strings defined by 2CNF formulas are testable with O ( √ n) queries, whereas there are 3CNF formulas for which the corresponding properties require Ω(n) queries, even for adaptive tests. We show that testing properties defined by 2CNF formulas is equivalent, with respect to the number of required queries, to several other function and graph testing problems.
Rougemont. Property and equivalence testing on strings
, 2004
"... We investigate property testing and related questions, where instead of the usual Hamming and edit distances between input strings, we consider the more relaxed edit distance with moves. Using a statistical embedding of words which has similarities with the Parikh mapping, we first construct a toler ..."
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Cited by 1 (0 self)
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We investigate property testing and related questions, where instead of the usual Hamming and edit distances between input strings, we consider the more relaxed edit distance with moves. Using a statistical embedding of words which has similarities with the Parikh mapping, we first construct a tolerant tester for the equality of two words, whose complexity is independent of the string size, and we derive an approximation algorithm for the normalized edit distance with moves. We then consider the question of testing if a string is a member of a given language. We develop a method to compute, in polynomial time in the representation, a geometric approximate description of a regular language by a finite union of polytopes. As an application, we have a new tester for regular languages given by their nondeterministic finite automaton (or regular expressions), whose complexity does not depend on the automaton, except for a polynomial time preprocessing step. Furthermore, this method allows us to compare languages and validates the new notion of equivalent testing that we introduce. Using the geometrical embedding we can distinguish between a pair of automata that compute the same language, and a pair of automata whose languages are not εequivalent in an appropriate sense. Our equivalence tester is deterministic and has polynomial time complexity, whereas the nonapproximated version is PSPACEcomplete. Last, we extend the geometric embedding, and hence the tester algorithms, to infinite regular languages and to contextfree grammars as well. For contextfree grammars the equivalence test has now exponential time complexity, but in comparison, the nonapproximated version is not even recursively decidable. 1
Rougemont. Property and equivalence testing on strings
, 2004
"... Michel de Rougemont £ We investigate property testing and related questions, where instead of the usual Hamming and edit distances between input strings, we consider the more relaxed edit distance with moves. Using a statistical embedding of words which has similarities with the Parikh mapping, we f ..."
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Cited by 1 (0 self)
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Michel de Rougemont £ We investigate property testing and related questions, where instead of the usual Hamming and edit distances between input strings, we consider the more relaxed edit distance with moves. Using a statistical embedding of words which has similarities with the Parikh mapping, we first construct a tolerant tester for the equality of two words, whose complexity is independent of the string size, and we derive an approximation algorithm for the normalized edit distance with moves. We then consider the question of testing if a string is a member of a given language. We develop a method to compute, in polynomial time in the representation, a geometric approximate description of a regular language by a finite union of polytopes. As an application, we have a new tester for regular languages given by their nondeterministic finite automaton (or regular expressions), whose complexity does not depend on the automaton, except for a polynomial time preprocessing step. Furthermore, this method allows us to compare languages and validates the new notion of equivalent testing that we introduce. Using the geometrical embedding we can distinguish between a pair of automata that compute the same language, and a pair of automata whose languages are not ¤equivalent in an appropriate sense. Our equivalence tester is deterministic and has polynomial time complexity, whereas the nonapproximated version is PSPACEcomplete. Last, we extend the geometric embedding, and hence the tester algorithms, to infinite regular languages and to contextfree grammars as well. For contextfree grammars the equivalence test has now exponential time complexity, but in comparison, the nonapproximated version is not even recursively decidable. 1
Testing Computability by Width Two OBDDs
"... Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by ..."
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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a readonce width2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is known. Width2 OBDDs generalize two classes of functions that have been studied in the context of property testing linear functions (over GF (2)) and monomials. In both these cases membership can be tested in time that is linear in 1/ɛ. Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number, n, of variables in the tested function, we show that (onesided error) testing for computability by a width2 OBDD requires Ω(log(n)) queries, and give an algorithm (with onesided error) that tests for this property and performs Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property [RS96,
Testing Computability by WidthTwo OBDDs Where the Variable Order is Unknown
"... Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for a prespecified distance measure) from every object with that property. In this work we design and analyze an algorithm for testing functions for the property of being c ..."
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Cited by 1 (1 self)
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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for a prespecified distance measure) from every object with that property. In this work we design and analyze an algorithm for testing functions for the property of being computable by a readonce width2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is not known to us. That is, we must accept a function f if there exists an order of the variables according to which a width2 OBDD can compute f. The query complexity of our algorithm is Õ(log n)poly(1/ɛ). In previous work (in Proceedings of RANDOM, 2009) we designed an algorithm for testing computability by an OBDD with a fixed order, which is known to the algorithm. Thus, we extend our knowledge concerning testing of functions that are characterized by their computability using simple computation devices and in the process gain some insight concerning these devices. 1