# Sets computable in polynomial time on average

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## 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 P_{P-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 ⊂ P_{P-comp} ⊂ E. In this paper, we show that P_{P-comp} is not contained in DTIME(2^{cn}) 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 P_{P-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 P_{P-comp} and P_{P-comp} is not in PSPACE with probability 1.

## Keywords

Turing Machine Complexity Core Random Oracle Average Computation Time Existential Operator## Preview

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