Testing Linear-Invariant Non-linear Properties: A Short Report

The rich collection of successes in property testing raises a natural question: Why are so many different properties turning out to be locally testable? Are there some broad "features" of properties that make them testable? Kaufman and Sudan (STOC 2008) proposed the study of the relationship between the invariances satisfied by a property and its testability. Particularly, they studied properties that were invariant under linear transformations of the domain and gave a characterization of testability in certain settings. However, the properties that they examined were also linear. This led us to investigate linear-invariant properties that are not necessarily linear. Here we describe some of the resulting works which consider natural linear-invariant properties, specifically properties that are described by forbidden patterns of values that a function can take, and show testability under various settings.

[1]  Noga Alon,et al.  Testing Reed-Muller codes , 2005, IEEE Transactions on Information Theory.

[2]  Dana Ron,et al.  Testing polynomials over general fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[3]  Terence Tao,et al.  Testability and repair of hereditary hypergraph properties , 2008, Random Struct. Algorithms.

[4]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[5]  B. Green.  A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[6]  D. Král,et al.  A removal lemma for systems of linear equations over finite fields , 2008, 0809.1846.

[7]  Vojtech Rödl,et al.  Extremal problems on set systems , 2002, Random Struct. Algorithms.

[8]  Madhu Sudan,et al.  Algebraic property testing: the role of invariance , 2008, Electron. Colloquium Comput. Complex..

[9]  Vojtech Rödl,et al.  Regularity Lemma for k‐uniform hypergraphs , 2004, Random Struct. Algorithms.

[10]  Dana Ron,et al.  On proximity oblivious testing , 2009, STOC '09.

[11]  Asaf Shapira.  Green's conjecture and testing linear-invariant properties , 2009, STOC '09.

[12]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[13]  Asaf Shapira,et al.  A Unified Framework for Testing Linear-Invariant Properties , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[14]  Atri Rudra,et al.  Testing low-degree polynomials over prime fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[15]  Ben Green,et al.  Linear equations in primes , 2006, math/0606088.

[16]  Arnab Bhattacharyya,et al.  Lower bounds for testing triangle-freeness in Boolean functions , 2010, SODA '10.

[17]  Daniel Král,et al.  A combinatorial proof of the Removal Lemma for Groups , 2008, J. Comb. Theory, Ser. A.

[18]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[19]  Jacob Fox,et al.  A new proof of the graph removal lemma , 2010, ArXiv.

[20]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[21]  Madhu Sudan,et al.  Testing Linear-Invariant Non-Linear Properties , 2011, Theory Comput..

[22]  Vojtech Rödl,et al.  The counting lemma for regular k‐uniform hypergraphs , 2006, Random Struct. Algorithms.

[23]  Terence Tao,et al.  Testability and repair of hereditary hypergraph properties , 2010 .

[24]  W. T. Gowers,et al.  Hypergraph regularity and the multidimensional Szemerédi theorem , 2007, 0710.3032.

[25]  V. Rödl,et al.  The counting lemma for regular k-uniform hypergraphs , 2006 .