SIAM Digital Library
 
 
 

You are not logged in Logged Out Log In

SIAM J. on Computing / Volume 38 / Issue 2

SIAM J. Comput. 38, pp. 551-607 (57 pages)

Short PCPs with Polylog Query Complexity

Eli Ben-Sasson and Madhu Sudan

Full Text: Download PDF | Buy PDF (US$25) | View Cart
We give constructions of probabilistically checkable proofs (PCPs) of length $n \cdot polylog n$ proving satisfiability of circuits of size $n$ that can be verified by querying $polylog n$ bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping $n$ information bits to $n\cdot polylog n$ bit long codewords that are testable with $polylog n$ queries. Our constructions rely on new techniques revolving around properties of codes based on relatively high-degree polynomials in one variable, i.e., Reed–Solomon codes. In contrast, previous constructions of short PCPs, beginning with [L. Babai, L. Fortnow, L. Levin, and M. Szegedy, Checking computations in polylogarithmic time, in Proceedings of the 23rd ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 21–31] and until the recent [E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan, Robust PCPs of proximity, shorter PCPs, and applications to coding, in Proceedings of the 36th ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 13–15], relied extensively on properties of low-degree polynomials in many variables. We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed–Solomon codeword, i.e., a univariate polynomial of specified degree. This reduction also gives an alternative to using the “sumcheck protocol” [C. Lund, L. Fortnow, H. Karloff, and N. Nisan, J. ACM, 39 (1992), pp. 859–868]. We then give a new PCP for the special task of proving that a function is close to being a Reed–Solomon codeword. The resulting PCPs are not only shorter than previous ones but also arguably simpler. In fact, our constructions are also more natural in that they yield locally testable codes first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite direction of getting locally testable codes from PCPs.

© 2008 Society for Industrial and Applied Mathematics

RELATED DATABASES

To view database links for this article, you need to log in.

KEYWORDS

AMS Subject Headings

68Q17

PUBLICATION DATA

ISSN

0097-5397 (print)  
1095-7111 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received November 30, 2005
Accepted December 05, 2006
Published online May 23, 2008
Digital Object Identifier
http://dx.doi.org/10.1137/050646445

For access to fully linked references, you need to log in.

For access to citing articles, you need to log in.


Close

Your Recent History Recent History Help

Recently Viewed

  1. Short PCPs with Polylog Query Complexity
  2. The Round Complexity of Two-Party Random Selection
  3. Worst‐Case to Average‐Case Reductions Based on Gaussian Measures
  4. Algebras with Polynomial Identities and Computing the Determinant
  5. Complexity of Self‐Assembled Shapes
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close