Hard Properties with (Very) Short PCPPs and Their Applications

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

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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

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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)


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.

Subject Classification

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


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