# Fine separation of average time complexity classes

## Abstract

We extend Levin's theory of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language *L* belongs to DTIME(*T(n)*), for some time-bound *T(n)*, then every distributional problem (*L,μ*) is *T* on the *μ*-average. (3) If *L* does not belong to DTIME(*T(n)*) *almost everywhere*, then no distributional problem (*L, μ*) *is T on* the *μ*-average.

We present hierarchy theorems for average-case complexity, for arbitrary time-bounds, that are as tight as the well-known Hartmanis-Stearns [HS65] hierarchy theorem for deterministic complexity. As a consequence, for every time-bound T(*n*), there are distributional problems (*L, μ*) that can be solved using only a slight increase in time but that cannot be solved on the *μ*-average in time *T(n)*.

## Keywords

Polynomial Time Turing Machine Complexity Class Time Bound Distributional Problem## Preview

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