On Fast Heuristic Non-deterministic Algorithms and Short Heuristic Proofs

In this paper we study heuristic proof systems and heuristic non-deterministic algorithms. We give an example of a language Y and a polynomial-time samplable distribution D such that the distributional problem (Y,D) belongs to the complexity class HeurNP but Y / ∈ NP if NP 6= coNP, and (Y,D) / ∈ HeurBPP if (NP,PSamp) 6⊆ HeurBPP. For a language L and a polynomial q we define the language padq(L) composed of pairs (x, r) where x is an element of L and r is an arbitrary binary string of length at least q(|x|). If D = {Dn}n=1 is an ensemble of distributions on strings, let D × U q be a distribution on pairs (x, r), where x is distributed according to Dn and r is uniformly distributed on strings of length q(n). We show that for every language L in AM there is a polynomial q such that for every distribution D concentrated on the complement of L the distributional problem (padq(L), D×U q) has a polynomially bounded heuristic proof system. Since graph non-isomorphism (GNI) is in AM, the above result is applicable to GNI.