Hard properties with (very) short PCPPs and their applications

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.