Sets computable in polynomial time on average

Abstract

In this paper, we discuss the complexity and properties of the sets which are computable in polynomial-time on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomial-time on average for every reasonable (i.e., polynomial-time computable) distribution on the instances. Let PP-comp denote the class of all those sets which are computable in polynomial-time on average for every polynomial-time computable distribution on the instances. It is known that P ⊂ PP-comp ⊂ E. In this paper, we show that PP-comp is not contained in DTIME(2cn) for any constant c and that it lacks some basic structural properties: for example, it is not closed under many-one reducibility or for the existential operator. From these results, it follows that PP-comp contains P-immune sets but no P-bi-immune sets; it is not included in P/cn for any constant c; and it is different from most of the well-known complexity classes, such as UP, NP, BPP, and PP. Finally, we show that, relative to a random oracle, NP is not included in PP-comp and PP-comp is not in PSPACE with probability 1.