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 ≠ coNP, and (Y, D) ∉ HeurBPP if (NP, PSamp) ⊆ 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 × Uq 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 × Uq) has a polynomially bounded heuristic proof system. Since graph non-isomorphism (GNI) is in AM, the above result is applicable to GNI.