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Composition of Low-Error 2-Query PCPs Using Decodable PCPs

  • Chapter
Property Testing

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6390))

Abstract

The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specific PCPs that were being composed), resulting in complicated constructions of PCPs. Furthermore, until recently, composition in the low error regime suffered from incurring an extra ‘consistency’ query, resulting in PCPs that are not ‘two-query’ and hence, much less useful for hardness-of-approximation reductions.

In a recent breakthrough, Moshkovitz and Raz [In Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS), 2008] constructed almost linear-sized low-error 2-query PCPs for every language in NP. Indeed, the main technical component of their construction is a novel composition of certain specific PCPs. We give a modular and simpler proof of their result by repeatedly applying the new composition theorem to known PCP components.

To facilitate the new modular composition, we introduce a new variant of PCP, which we call a decodable PCP (dPCP). A dPCP is an encoding of an NP witness that is both locally checkable and locally decodable. The dPCP verifier in addition to verifying the validity of the given proof like a standard PCP verifier, also locally decodes the original NP witness. Our composition is generic in the sense that it works regardless of the way the component PCPs are constructed.

A full version of this paper appears in the Electronic Colloquium on Computational Complexity [DH09]. The current extended abstract is a modification of the introduction of the full version for the purposes of the ICS mini-workshop on propert testing.

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science, Israel

    Irit Dinur

  2. Tata Institute of Fundamental Research, India

    Prahladh Harsha

Authors
  1. Irit Dinur
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  2. Prahladh Harsha
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Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 76100, Rehovot, Israel

    Oded Goldreich

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Dinur, I., Harsha, P. (2010). Composition of Low-Error 2-Query PCPs Using Decodable PCPs. In: Goldreich, O. (eds) Property Testing. Lecture Notes in Computer Science, vol 6390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16367-8_21

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  • DOI: https://doi.org/10.1007/978-3-642-16367-8_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16366-1

  • Online ISBN: 978-3-642-16367-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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