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Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Published: 08 November 2016 Publication History

Abstract

The PCP theorem [Arora et al. 1998; Arora and Safra 1998] says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, that is, how long is the PCP compared to the original NP-proof? The state-of-the-art work of Ben-Sasson and Sudan [2008] and Dinur [2007] shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to nε, then one can construct PCPs of length O(n) for circuit-SAT, and PCPs of length O(tlog t) for any language in NTIME(t).
More specifically, for any ε > 0, we present (nonuniform) probabilistically checkable proofs (PCPs) of length 2O(1/ε) · n that can be checked using nε queries for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).
Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations that are crucial in earlier high-rate algebraic PCP constructions. Using this observation, we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this article for the first time for every message length, building on an earlier construction for infinitely many message lengths by Stichtenoth [2006].

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cover image Journal of the ACM
Journal of the ACM  Volume 63, Issue 4
November 2016
365 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/2997039
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 08 November 2016
Accepted: 01 March 2016
Revised: 01 November 2015
Received: 01 March 2014
Published in JACM Volume 63, Issue 4

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Author Tags

  1. PCPs
  2. algebraic-geometric codes
  3. error-correcting codes
  4. probabilistic proof systems
  5. sublinear time algorithms

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  • Refereed

Funding Sources

  • European Community's Seventh Framework Programme (FP7/2007-2013)
  • National Science Foundation
  • Sloan Fellowship
  • US-Israel Binational Science Foundation
  • Israeli Science Foundation
  • Tubitak

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