Average Time Complexity

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;) is T on the-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem (L;) is T on the-average. We present a hierarchy theorem for average-case complexity, for arbitrary time-bounds, that is 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;) that can be solved using only a slight increase in time but that cannot be solved on the-average in time T (n). We demonstrate that our deenition is natural and is as justiied for arbitrary time-bounds as is Levin's deenition for polynomial time-bounds. We critique an earlier proposal of a deenition of average case complexity for arbitrary time-bounds BDCGL92] by demonstrating that it does not satisfy our general principles. Nevertheless, we obtain a ne hierarchy, for the earlier deenition, for distributional problems whose running time is bounded by a polynomial. Our proofs use techniques of convexity, HH older's inequality, and properties of Hardy's class of logarithmico-exponential functions.