On average P vs. average NP

Structures of polynomial-time computable distributions and polynomial-time many-one reductions on randomized decision problems are investigated. The scope is widened from the most studied class DNP (distributional-NP) to the class ANP (average-NP), which consists of randomized decision problems accepted by nondeterministic Turing machines in average polynomial time. Results indicate that the structures of average-case complexity classes are very different from their counterparts in worst-case complexity due to the complex structures of probability distributions and distribution functions