eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-01-06
9:1
9:27
10.4230/LIPIcs.ITCS.2020.9
article
Hard Properties with (Very) Short PCPPs and Their Applications
Ben-Eliezer, Omri
1
Fischer, Eldar
2
Levi, Amit
3
Rothblum, Ron D.
2
Tel Aviv University, Israel
Technion - Israel Institute of Technology, Haifa, Israel
University of Waterloo, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol151-itcs2020/LIPIcs.ITCS.2020.9/LIPIcs.ITCS.2020.9.pdf
PCPP
Property testing
Tolerant testing
Erasure resilient testing
Randomized encoding
Coding theory