Average Case Complexity Theory

Informally, Average case Complexity theory considers the average complexity with respect to a probability distribution over the set of inputs, instead of the complexity on the worst case input, as a criterion for the hardness of a problem. With regard to this theory, we explore the notion of a natural distribution. In particular, we consider different bounds on the minentropy of such distributions over input strings of a fixed length, and generalize the incompleteness theorem (under deterministic reductions) for distributional problems using techniques based on the work by Gurevich[1991] . We also present some weak restrictions on randomized reductions under which problems with a uniform distribution are average case complete. These restrictions have generalizations to the case when the distribution is not uniform but still has an upper bound on its min entropy, over strings of a given length. We also describe our attempts to study the effect of unnatural encodings on average case completeness.