Heuristic time hierarchies via hierarchies for sampling distributions

Abstract

We introduce a new framework for proving the time hierarchy theorems for heuristic classes. The main ingredient of our proof is a hierarchy theorem for sampling distributions recently proved by Watson [11]. Class Heur∈FBPP consists of functions with distributions on their inputs that can be computed in randomized polynomial time with bounded error on all except _ fraction of inputs. We prove that for every a, δ and integer k there exists a function F : {0, 1}∗ → {0, 1, . . . , k − 1} such that (F,U) (Formula presented.) Heur∈FBPP for all ∈ > 0 and for every ensemble of distributions Dn samplable in na steps, (F,D) ∈ Heur1−1/k−δFBPTime[na]. This extends a previously known result for languages with uniform distributions proved by Pervyshev [9] by handling the case k > 2. We also prove that P[formula presented] Heur-∈BPTime[nk] if one-way functions exist. We also show that our technique may be extended for time hierarchies in some other heuristic classes.