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Sub-constant error low degree test of almost-linear size

Published: 21 May 2006 Publication History

Abstract

Given a function f:Fm→F over a finite field F, a low degree tester tests its agreement with an m-variate polynomial of total degree at most d over F. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies.We wish to minimize two parameters of a tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester's queries.Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (PCPs) and locally testable codes (LTCs). The error of the low degree tester is related to the soundness of the PCP and its size is related to the size of the PCP (or the length of the LTC).We design and analyze new low degree testers that have both sub-constant error o(1) and almost-linear size n1+o(1) (where n=|F|m). Previous constructions of sub-constant error testers had polynomial size [13, 16]. These testers enabled the construction of PCPs with sub-constant soundness, but polynomial size [13, 16, 9]. Previous constructions of almost-linear size testers obtained only constant error [13, 7]. These testers were used to construct almost-linear size LTCs and almost-linear size PCPs with constant soundness [13, 7, 5, 6, 8].

References

[1]
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. JACM, 45(3):501--555, 1998.
[2]
S. Arora and S. Safra. Probabilistic checking of proofs: a new characterization of NP. JACM, 45(1):70--122, 1998.
[3]
S. Arora and M. Sudan. Improved low-degree testing and its applications. Combin., 23(3):365--426, 2003.
[4]
L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. In Proc. 31st IEEE FOCS, pages 16--25, 1990.
[5]
E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan. Robust PCPs of proximity, shorter pcps and applications to coding. In Proc. 36th ACM STOC, pages 1--10, 2004.
[6]
E. Ben-Sasson and M. Sudan. Simple PCPs with poly-log rate and query complexity. In Proc. 37th ACM STOC, pages 266--275, 2005.
[7]
E. Ben-Sasson, M. Sudan, S. P. Vadhan, and A. Wigderson. Randomness-efficient low degree tests and short PCPs via epsilon-biased sets. In Proc. 34th ACM STOC, pages 612--621, 2003.
[8]
I. Dinur. The PCP theorem by gap amplification. In Proc. 38th ACM STOC, 2006.
[9]
I. Dinur, E. Fischer, G. Kindler, R. Raz, and S. Safra. PCP characterizations of NP: Towards a polynomially-small error-probability. In Proc. 31st ACM STOC, pages 29--40, 1999.
[10]
U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. JACM, 43(2):268--292, 1996.
[11]
K. Ford. The distribution of integers with a divisor in a given interval. 2004.
[12]
K. Friedl and M. Sudan. Some improvements to low degree tests. In 3rd ISTCS, 1995.
[13]
O. Goldreich and M. Sudan. Locally testable codes and PCPs of almost-linear length. In Proc. 43rd IEEE FOCS, pages 13--22, 2002.
[14]
R. Hall and G. Tenenbaum. Divisors, volume 90 of Cambridge Tracts in Mathematics. Cambridge University Press, 1988.
[15]
D. Moshkovitz and R. Raz. Sub-constant error low degree test of almost-linear size. Technical Report TR05-086, ECCC, 2005.
[16]
R. Raz and S. Safra. A sub-constant error-probability low-degree test and a sub-constant error-probability PCP characterization of NP. In Proc. 29th ACM STOC, pages 475--484, 1997.
[17]
R. Rubinfeld and M. Sudan. Robust characterizations of polynomials with applications to program testing. SICOMP, 25(2):252--271, 1996.

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cover image ACM Conferences
STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
May 2006
786 pages
ISBN:1595931341
DOI:10.1145/1132516
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Published: 21 May 2006

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Author Tags

  1. locally testable codes
  2. low degree testing
  3. plane vs. point test
  4. probabilistically checkable proofs

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STOC06: Symposium on Theory of Computing
May 21 - 23, 2006
WA, Seattle, USA

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