Structural average case complexity

Abstract

Levin introduced an average-case complexity measure among randomized decision problems. We generalize his notion of Average-P into Aver(C, F), a set of randomized decision problems (L,/µ) such that the density function µ is in F and L is computed by a type-C machine in time t (or space t) on µ-average. Mainly studied are two sorts of reductions between randomized problems, average-case many-one and Turing reductions, and structural properties of average-case complexity classes. We give average-case analogs of concepts of classical complexity theory, e.g., the polynomial time hierarchy and self-reducibility.