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16
Testing Monotonicity
, 1999
"... We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is fflfar from being monotone (i.e., e ..."
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Cited by 79 (16 self)
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We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is fflfar from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
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Cited by 76 (11 self)
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this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Linearity testing in characteristic two
 IEEE Transactions on Information Theory
, 1996
"... The case we are interested in is when the underlying groups are G=GF(2)n and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(f) measures its distance to a Hadamard code (nor ..."
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Cited by 69 (6 self)
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The case we are interested in is when the underlying groups are G=GF(2)n and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(f) measures its distance to a Hadamard code (normalized so as to be a real number between 0 and 1). The quantity Err(f) is a parameter that is "easy to measure " and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation results. In this context, improved analyses translate into better nonapproximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight.
Combinatorial Property Testing (a survey)
 In: Randomization Methods in Algorithm Design
, 1998
"... We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions diffe ..."
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Cited by 53 (2 self)
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We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We consider (randomized) algorithms which may query the function at arguments of their choice, and seek algorithms which query the function at relatively few places. We focus on combinatorial properties, and specifically on graph properties. The two standard representations of graphs  by adjacency matrices and by incidence lists  yield two different models for testing graph properties. In the first model, most appropriate for dense graphs, distance between Nvertex graphs is measured as the fraction of edges on which the graphs disagree over N 2 . In the second model, most appropriate for boundeddegree graphs, distance between Nvertex ddegree graphs is measured as the fraction of edges on ...
Query efficient PCPs with perfect completeness
 In 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probabil ..."
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Cited by 21 (3 self)
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For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR9987077.
Simple analysis of graph tests for linearity and PCP
 IEEE Conference on Computational Complexity
"... We give a simple analysis of the PCP with low amortized query complexity of Samorodnitsky and Trevisan [16]. The analysis also applies to the linearity testing over finite fields, giving a better estimate of the acceptance probability in terms of the distance of the tested function to the closest li ..."
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Cited by 16 (2 self)
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We give a simple analysis of the PCP with low amortized query complexity of Samorodnitsky and Trevisan [16]. The analysis also applies to the linearity testing over finite fields, giving a better estimate of the acceptance probability in terms of the distance of the tested function to the closest linear function.
Bounds on 2Query Codeword Testing
 IN THE PROCEEDINGS OF RANDOM'03, SPRINGER LNCS
, 2003
"... We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2locally testable code with minimal distance ffin over a finite field F cannot have more than jFj codewords. This result holds even for testers with tw ..."
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Cited by 10 (4 self)
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We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2locally testable code with minimal distance ffin over a finite field F cannot have more than jFj codewords. This result holds even for testers with twosided error. For general (nonlinear) codes we obtain the exact same bounds on the code size as a function of the minimal distance, but our bounds apply only for binary alphabets and onesided error testers (i.e. with perfect completeness). Our bounds are obtained by examining the graph induced by the set of possible pairs of queries made by a codeword tester on a given code. We also demonstrate the tightness of our upper bounds and the essential role of certain parameters.
Exact and approximate testing/correcting of algebraic functions: A survey
 Electronic Colloq. on Comp. Compl., Univ. of Trier TR2001014
, 2001
"... In the late 80’s Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self–testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation ..."
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Cited by 10 (2 self)
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In the late 80’s Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self–testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation of hardness of approximation results. Applications in areas like computer vision, machine learning, and self–correcting programs were also established. In the self–testing problem one is interested in determining (maybe probabilistically) whether a function to which one has oracle access satisfies a given property. We consider the problem of testing algebraic functions and survey over a decade of research in the area. Special emphasis is given to illustrate the scenario where the problem takes place and to the main techniques used in the analysis of tests. A novel aspect of this work is the separation it advocates between the mathematical and algorithmic issues that arise in the theory of self–testing.
Testing and Weight Distributions of Dual Codes
 Theoretical Computer Science
, 1997
"... We study the testing problem, that is, the problem of determining (maybe probabilistically) if a function to which we have oracle access satisfies a given property. We propose a framework in which to formulate and carry out the analyzes of several known tests. This framework establishes a connection ..."
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Cited by 6 (0 self)
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We study the testing problem, that is, the problem of determining (maybe probabilistically) if a function to which we have oracle access satisfies a given property. We propose a framework in which to formulate and carry out the analyzes of several known tests. This framework establishes a connection between testing and the theory of weight distributions of dual codes. We illustrate this connection by giving a coding theoretic interpretation of several tests that fall under the label of lowdegree tests. We also show how the coding theoretic connection we establish naturally suggests a new way of testing for linearity over finite fields. There are two important parameters associated to every test. The first one is the test's probability of rejecting the claim that the function to which it has oracle access satisfies a given property. The second one is the distance from the oracle function to any function that satisfies the property of interest. The goal when analyzing tests is to explai...