Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing Reed-Muller codes. IEEE Transactions on Information Theory 51(11), 4032–4039 (2005)
Austin, T., Tao, T.: On the testability and repair of hereditary hypergraph properties. Random Structures and Algorithms (2008) (to appear), http://arxiv.org/abs/0801.2179
Bhattacharyya, A., Chen, V., Sudan, M., Xie, N.: Testing linear-invariant non-linear properties. In: Symposium on Theoretical Aspects of Computer Science, pp. 135–146 (2009)
Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1(1), 3–40 (1991)
Bhattacharyya, A., Grigorescu, E., Shapira, A.: A unified framework for testing linear-invariant properties. To appear in FOCS (2010)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comp. Sys. Sci. 47, 549–595 (1993); Earlier version in STOC 1990
Bhattacharyya, A., Xie, N.: Lower bounds for testing triangle-freeness in boolean functions. In: Proc. 21st ACM-SIAM Symposium on Discrete Algorithms, pp. 87–98 (2010)
Fox, J.: A new proof of the graph removal lemma. Technical report (June 2010), http://arxiv.org/abs/1006.1300
Frankl, P., Rödl, V.: Extremal problems on set systems. Random Structures and Algorithms 20(2), 131–164 (2002)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM 45, 653–750 (1998)
Gowers, W.T.: Hypergraph regularity and the multidimensional Szemerédi theorem. Annals of Mathematics 166(3), 897–946 (2007)
Goldreich, O., Ron, D.: On proximity oblivious testing. In: Proc. 41st Annual ACM Symposium on the Theory of Computing, pp. 141–150 (2009)
Green, B.: A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal. 15(2), 340–376 (2005)
Green, B., Tao, T.: Linear equations in primes. Annals of Mathematics (2006) (to appear)
Jutla, C.S., Patthak, A.C., Rudra, A., Zuckerman, D.: Testing low-degree polynomials over prime fields. In: Proc. 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 423–432 (2004)
Kaufman, T., Ron, D.: Testing polynomials over general fields. SIAM J. on Comput. 36(3), 779–802 (2006)
Kaufman, T., Sudan, M.: Algebraic property testing: the role of invariance. In: Proc. 40th Annual ACM Symposium on the Theory of Computing, pp. 403–412. ACM, New York (2008)
Král’, D., Serra, O., Vena, L.: A removal lemma for systems of linear equations over finite fields (2008)
Král, D., Serra, O., Vena, L.: A combinatorial proof of the removal lemma for groups. Journal of Combinatorial Theory 116(4), 971–978 (2009)
Nagle, B., Rödl, V., Schacht, M.: The counting lemma for regular k-uniform hypergraphs. Random Structures and Algorithms 28(2), 113–179 (2006)
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. on Comput. 25, 252–271 (1996)
Rödl, V., Skokan, J.: Regularity lemma for k-uniform hypergraphs. Random Structures and Algorithms 25(1), 1–42 (2004)
Shapira, A.: Green’s conjecture and testing linear-invariant properties. In: Proc. 41st Annual ACM Symposium on the Theory of Computing, pp. 159–166 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bhattacharyya, A., Chen, V., Sudan, M., Xie, N. (2010). Testing Linear-Invariant Non-linear Properties: A Short Report. In: Goldreich, O. (eds) Property Testing. Lecture Notes in Computer Science, vol 6390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16367-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-16367-8_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16366-1
Online ISBN: 978-3-642-16367-8
eBook Packages: Computer ScienceComputer Science (R0)