Polynomially low error PCPs with polyloglog n queries via modular composition

Irit Dinur, Prahladh Harsha, Guy Kindler

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations

Abstract

We show that every language in NP has a PCP verifier that tosses O(log n) random coins, has perfect completeness, and a soundness error of at most 1/poly(n), while making O(poly log log n) queries into a proof over an alphabet of size at most n1/poly log log n. Previous constructions that obtain 1/poly(n) soundness error used either poly log n queries or an exponential alphabet, i.e. of size 2nc for some c > 0. Our result is an exponential improvement in both parameters simultaneously. Our result can be phrased as polynomial-gap hardness for approximate CSPs with arity poly log log n and alphabet size n1/poly log log n. The ultimate goal, in this direction, would be to prove polynomial hardness for CSPs with constant arity and polynomial alphabet size (aka the sliding scale conjecture for inverse polynomial soundness error). Our construction is based on a modular generalization of previous PCP constructions in this parameter regime, which involves a composition theorem that uses an extra 'consistency' query but maintains the inverse polynomial relation between the soundness error and the alphabet size. Our main technical/conceptual contribution is a new notion of soundness, which we refer to as distributional soundness, that replaces the previous notion of "list decoding soundness", and allows us to invoke composition a super-constant number of times without incurring a blow-up in the soundness error.

Original languageAmerican English
Title of host publicationSTOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Pages267-276
Number of pages10
ISBN (Electronic)9781450335362
DOIs
StatePublished - 14 Jun 2015
Event47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States
Duration: 14 Jun 201517 Jun 2015

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume14-17-June-2015
ISSN (Print)0737-8017

Conference

Conference47th Annual ACM Symposium on Theory of Computing, STOC 2015
Country/TerritoryUnited States
CityPortland
Period14/06/1517/06/15

Bibliographical note

Publisher Copyright:
© Copyright 2015 ACM.

Keywords

  • Composition
  • Decodable PCPs
  • PCPs
  • Sliding scale conjecture

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