Invariance in Property Testing

Sudan, Madhu

doi:10.1007/978-3-642-16367-8_12

Cited by 27 publications

(31 citation statements)

References 63 publications

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“…More [MORS09], codes [KS07,KS09]. These are documented in several surveys [Fis04,Rub06,Ron08,Sud10], and we refer the reader to these surveys for more background and references on property testing.…”

Section: Background On Property Testingmentioning

confidence: 99%

“…Furthermore, to show the testability of low degree polynomials (a.k.a., Reed-Muller codes), we would only need to resolve Conjecture 4 for a finite 7 family of forbidden induced systems of equations. Regarding the properties of Fourier dimensionality and sparsity, they are currently only known to have two-sided testers [GOS + 09], while Corollary 7 will potentially yield one-sided testers, resolving an issue raised in [Sud10].…”

Section: Proposition 6 a Linear-invariant Property P Is Subspace-herementioning

confidence: 99%

“…We finally note that all the testing algorithms for testable properties of Boolean functions in prior works were indeed oblivious, and that furthermore many of them implicitly consider only oblivious testers. In particular, these types of testers were considered in [Sud10].…”

Section: Queries F On All Elements X ∈ Hmentioning

confidence: 99%

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A Unified Framework for Testing Linear-Invariant Properties

Bhattacharyya¹,

Grigorescu²,

Shapira

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory.Our main contributions here are the following:1. We introduce a simple combinatorial condition, which we call subspace-heredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area.2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with one-sided error, thus addressing a challenge posed by Sudan recently.3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture.Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.

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Section: Background On Property Testingmentioning

confidence: 99%

Section: Proposition 6 a Linear-invariant Property P Is Subspace-herementioning

confidence: 99%

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A Unified Framework for Testing Linear-Invariant Properties

Bhattacharyya¹,

Grigorescu²,

Shapira

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

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“…The affine-invariance of linear codes has been intensely investigated in recent years in the context of locally testable codes, starting with the work of [KS08] (see [Sud10] for a recent survey). The role the automorphisms of AG codes play in constructing locally correctable and decodable codes has been recently considered in [BGK + 13].…”

Section: Ag Arithmetizationmentioning

confidence: 99%

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Ben–Sasson

Kaplan

Kopparty

et al. 2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n ε , then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(t log t) for any language in NTIME(t).More specifically, for any ε > 0 we present (non-uniform) probabilistically checkable proofs (PCPs) of length 2 O(1/ε) · n that can be checked using n ε queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this paper for the first time for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth [Trans. Information Theory 2006].

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“…A property P of Boolean functions F n 2 → {−1, 1} is called linear-invariant if a function f satisfies P , then f • A is also in P for any square matrix A, where (f • A)(x) := f (Ax) for any x ∈ F n 2 . For a thorough discussion of linear-invariant properties, we refer the reader to Sudan's survey on the subject [10].…”

Section: Introductionmentioning

confidence: 99%

Testing Linear-Invariant Function Isomorphism

Wimmer

Yoshida

2013

Automata, Languages, and Programming

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Abstract. A function f : F n 2 → {−1, 1} is called linear-isomorphic to g if f = g • A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism with o(log s) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing gisomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.

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Invariance in Property Testing

Cited by 27 publications

References 63 publications

A Unified Framework for Testing Linear-Invariant Properties

A Unified Framework for Testing Linear-Invariant Properties

Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Testing Linear-Invariant Function Isomorphism

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