Abstract

We extend Levin's definition 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 timebounds, 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: computational complexity, average time complexity classes, hierarchy, Average-P, logarithmico-exponential ACM Computing R...

Citations