Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Interactive proofs and the hardness of approximating cliques
Journal of the ACM (JACM)
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Lower Bounds for Approximations by Low Degree Polynomials Over Z_m
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Inverse conjecture for the gowers norm is false
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
IEEE Transactions on Information Theory
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We consider the problem of testing if a given function f : Fn2 → F2 is close to any degree d polynomial in n variables, also known as the problem of testing Reed-Muller codes. We are interested in determining the query-complexity of distinguishing with constant probablity between the case where f is a degree d polynomial and the case where f is ω(1)-far from all degree d polynomials. Alon et al. [AKK+05] proposed and analyzed a natural 2d+1 -query test T0, and showed that it accepts every degree d polynomial with probability 1, while rejecting functions that are ω(1)-far with probability ω(1/(d2d)). This leads to a O(d4d)-query test for degree d Reed-Muller codes. We give an asymptotically optimal analysis of T0, showing that it rejects functions that are ω(1)-far with ω(1)-probability (so the rejection probability is a universal constant independent of d and n). In particular, this implies that the query complexity of testing degree d Reed-Muller codes is O(2d). Our proof works by induction on n, and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping Fn2 to F2. Our results also imply a "query hierarchy" result for property testing of affine-invariant properties: For every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [GKNR08] for graph properties. This is a brief overview of the results in the paper [BKS+09].