The study of the complexity of sets encompasses two complementary aims: (1) establishing -- usually via explicit construction of algorithms -- that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...

this paper, if T is time-constructible, then

We extend Levin's definition of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language L belongs to DTIME(T(n)), for some time-bound T(n), then every distributional problem (L;µ) is T on the µ-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem (L;µ) is T on the µ-average. We present hierarchy theorems for average-case complexity, for arbitrary timebounds, that are as tight as the well-known Hartmanis-Stearns [HS65] hierarchy theorem for deterministic complexity. As a consequence, for every time-bound T(n), there are distributional problems (L;µ) that can be solved using only a slight increase in time but that cannot be solved on the µ-average in time T(n). Keywords: computational complexity, average time complexity classes, hierarchy, Average-P, logarithmico-exponential ACM Computing R...

An oracle A is k-cheatable if there is a polynomial-time algorithm to determine the answers to 2 k parallel queries to A from the answers to only k queries to some other oracle B. It is known that 1-cheatable sets cannot be bi-immune for P. In contrast, we construct 2-cheatable sets that are bi-immune for arbitrary time complexity classes. In addition, for each k, we construct a set that is (k + 1)-cheatable, but not k-cheatable; we show that this separation does not hold with biimmunity. We show that if a recursive set A is bi-immune for P then there exists an infinite 1-cheatable set that is polynomial-time mreducible to A. Consequently if NP contains a set that is bi-immune for P then NP contains a set that is not polynomial-time Turingequivalent to a self-reducible set. 1. Introduction Complexity theory deals with how hard problems are. Time, space, and alternation have served as measures of difficulty. Recently, researchers have Research supported by a Fannie and John Hertz ...

Cai and Selman [CS96] proposed a general definition of average computation time that, when applied to polynomials, results in a modification of Levin's [Lev86] notion of average-polynomial-time. The effect of the modification is to control the rate of convergence of the expressions that define average computation time. With this modification, they proved a hierarchy theorem for average-time complexity that is as tight as the Hartmanis-Stearns [HS65] hierarchy theorem for worst-case deterministic time. They also proved that under a fairly reasonable condition on distributions, called condition W, a distributional problem is solvable in average-polynomial-time under the modification exactly when it is solvable in average-polynomial-time under Levin's denition. Various notions of reductions, as defined by Levin [Lev86] and others, play a central role in the study of average-case complexity. However, the class of distributional problems that are solvable in average-polynomial-time under the modification is not closed under the standard reductions. In particular, we prove that there is a distributional problem that is not solvable in average-polynomial-time under the modication but is reducible, by the identity function, t...

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We extend Levin's theory of average polynomial time to arbitrary time bounds and prove that average time complexity classes form as fine a hierarchy as do deterministic time complexity classes. Keywords: computational complexity, average time complexity classes, hierarchy, Average-P, logarithmico-exponential ACM Computing Reviews Subject Category: F.1.3 1 Introduction One of the central issues in complexity theory for any complexity-theoretic measure is the question of fine hierarchies. Here we consider this issue for average case complexity. The average complexity of a problem is, in many cases, a more significant measure than its worst case complexity. This has motivated a rich area in algorithms research, but Levin [Lev86] was the first to advocate the general study of average case complexity. An average case complexity class consists of pairs, called distributional problems. Each pair consists of a decision problem and a probability distribution on problem instances. Most papers...

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