Structural Average Case Complexity

Levin introduced an average case complexity measure based on a notion of polynomial on average and de ned average case polynomial time many one reducibility among randomized decision problems We generalize his notions of average case complexity classes Random NP and Average P Ben David et al use the notation of hC Fi to denote the set of randomized de cision problems L such that L is a set in C and is a probability density function in F This paper introduces AverhC Fi as the class of randomized decision problems L such that L is computed by a type C machine on average and is a density function in F These notations capture all known average case complexity classes as for example Random NP hNP P compi and Average P AverhP i where P comp denotes the set of density functions whose distribu tions are computable in polynomial time and denotes the set of all density functions Mainly studied are polynomial time reductions between randomized decision problems many one de terministic Turing and nondeterministic Turing reductions and the average case versions of them Based on these reducibilities structural properties of average case complexity classes are discussed We give average case analogues of concepts in worst case complexity theory in par ticular the polynomial time hierarchy and Turing self reducibility and we show that all known complete sets for Random NP are Turing self reducible A new notion of real polynomial time computations is introduced based on average polynomial time computations for arbitrary dis tributions from a xed set and it is used to characterize the worst case complexity classes p k and p k of the polynomial time hierarchy