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Hard Properties with (Very) Short PCPPs and Their Applications

Authors Omri Ben-Eliezer, Eldar Fischer, Amit Levi, Ron D. Rothblum

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ACM Subject Classification
  • Theory of computation → Interactive proof systems
Keywords
  • PCPP
  • Property testing
  • Tolerant testing
  • Erasure resilient testing
  • Randomized encoding
  • Coding theory

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Abstract

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ℓ, we construct a property P^(ℓ)⊆ {0,1}^n satisfying the following: Any testing algorithm for P^(ℓ) requires Ω(n) many queries, and yet P^(ℓ) has a constant query PCPP whose proof size is O(n⋅log^(ℓ)n), where log^(ℓ) denotes the ℓ times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n⋅polylog(n)).
As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ℓ, we construct a property that has a constant-query tester, but requires Ω(n/log^(ℓ)(n)) queries for every tolerant or erasure-resilient tester.

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Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Ron D. Rothblum. Hard Properties with (Very) Short PCPPs and Their Applications. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 9:1-9:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.9

Author Details

Omri Ben-Eliezer
  • Tel Aviv University, Israel
Eldar Fischer
  • Technion - Israel Institute of Technology, Haifa, Israel
Amit Levi
  • University of Waterloo, Canada
Ron D. Rothblum
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

  • Levi, Amit: Supported by the David R. Cheriton Graduate Scholarship. Part of this work was done while the author was visiting the Technion.
  • Rothblum, Ron D.: Supported in part by the Israeli Science Foundation (Grant No. 1262/18), a Milgrom family grant and the Technion Hiroshi Fujiwara cyber security research center and the Israel cyber directorate.

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