Polynomial-time reducibilities and “almost all” oracle sets

AbstractIt is shown for every k>0 and for almost every tally setT, {A|A ⩽Pk−ttT} ≠ {A|A ⩽P(k+1)−ttT}. In contrast, it is shown that for every set A, the following holds: (a) for almost every set B,A ⩽ Pm B if and only if A ⩽ P(logn)−T B; and (b) for almost every set B, A ⩽Ptt B if and only ifA ⩽PTB

On adaptive versus nonadaptive bounded query machines

AbstractThe polynomial-time adaptive (Turing) and nonadaptive (truth-table) bounded query machines are compared with respect to sparse oracles. A k-query adaptive machine has been found which, relative to a sparse oracle, cannot be simulated by any (2k−2)-query nonadaptive machine, even with a different sparse oracle. Conversely, there is a (3·2k−2)-query nonadaptive machine which, relative to a sparse oracle, cannot be simulated by any k-query adaptive machine, with any sparse oracle

Structural Average Case Complexity

AbstractLevin introduced an average-case complexity measure, based on a notion of “polynomial on average,” and defined “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-Davidet al. use the notation of 〈C, F〉 to denote the set of randomized decision problems (L, μ) such thatLis a set in C andμis a probability density function in F. This paper introduces Aver〈C, F〉 as the class of randomized decision problems (L, μ) such thatLis computed by a type-C machine onμ-average andμis a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= 〈NP, P-comp〉 and Average-P=Aver〈P, ∗〉, where P-comp denotes the set of density functions whose distributions 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, deterministic 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 particular, 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 distributions from a fixed set, and it is used to characterize the worst-case complexity classesΔpkandΣpkof the polynomial-time hierarchy