License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2020.9
URN: urn:nbn:de:0030-drops-116949
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11694/
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Ben-Eliezer, Omri ; Fischer, Eldar ; Levi, Amit ; Rothblum, Ron D.

Hard Properties with (Very) Short PCPPs and Their Applications

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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.

BibTeX - Entry

@InProceedings{beneliezer_et_al:LIPIcs:2020:11694,
  author =	{Omri Ben-Eliezer and Eldar Fischer and Amit Levi and Ron D. Rothblum},
  title =	{{Hard Properties with (Very) Short PCPPs and Their Applications}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{9:1--9:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Thomas Vidick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11694},
  URN =		{urn:nbn:de:0030-drops-116949},
  doi =		{10.4230/LIPIcs.ITCS.2020.9},
  annote =	{Keywords: PCPP, Property testing, Tolerant testing, Erasure resilient testing, Randomized encoding, Coding theory}
}

Keywords: PCPP, Property testing, Tolerant testing, Erasure resilient testing, Randomized encoding, Coding theory
Collection: 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Issue Date: 2020
Date of publication: 06.01.2020

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