Structure in Average Case Complexity

In Schapire gave an equivalent characterization of Levin s notion of functions that are polynomial on average This characterization gives a very smooth translation from worst case complexity to average case complexity of the notions for time and space complexity We prove tight space and time hierarchy theorems and discuss the structure of deterministic and nondeterministic average case complexity classes In particular we show for polynomial and exponential time bounded classes that nondeterministic average case is equivalent to deterministic average case if this is true in worst case complexity We consider tally encodings of randomized decision problems and show that there are tally randomized decision problems in average nondeterministic polynomial time which are not in average deterministic polynomial time if and only if average deterministic exponential time is di erent from average nondeterministic exponential time