. This paper closes a gap in the foundations of the theory of average case complexity. First, we clarify the notion of a feasible solution for a search problem and prove its robustness. Second, we give a general and usable notion of many-one randomizing reductions of search problems and prove that it has desirable properties. All reductions of search problems to search problems in the literature on average case complexity can be viewed as such many-one randomizing reductions; this includes those reductions in the literature that use iterations and therefore do not look many-one. As an illustration, we present a careful proof in our framework of a theorem of Impagliazzo and Levin. Key words. Average case, search problems, reduction, randomization. 1. Introduction and results. Reduction theory for average case computational complexity was pioneered by Leonid Levin [?]. Recently, one of us wrote a survey on the subject [?], and we refer the reader there for a general background. However,...

We show that every language in NP has a probabilistically checkable proof of proximity (i.e., proofs asserting that an instance is “close ” to a member of the language), where the verifier’s running time is polylogarithmic in the input size and the length of the probabilistically checkable proof is only polylogarithmically larger that the length of the classical proof. (Such a verifier can only query polylogarithmically many bits of the input instance and the proof. Thus it needs oracle access to the input as well as the proof, and cannot guarantee that the input is in the language — only that it is close to some string in the language.) If the verifier is restricted further in its query complexity and only allowed q queries, then the proof size blows up by (log n)c/q a factor of 2 where the constant c depends only on the language (and is independent of q). Our results thus give efficient (in the sense of running time) versions of the shortest known PCPs, due to Ben-Sasson et al. (STOC ’04) and Ben-Sasson and Sudan (STOC ’05), respectively. The time complexity of the verifier and the size of the proof were the original emphases in the definition of holographic proofs, due to Babai et al.

We say that a distribution µ is reasonable if there exists a constant s ≥ 0 such that µ({x | |x | ≥ n}) = Ω ( 1 ns). We prove the following result, which suggests that all DistNP-complete problems have reasonable distributions. If NP contains a DTIME(2 n)-bi-immune set, then every DistNP-complete set has a reasonable distribution. It follows from work of Mayordomo [May94] that the consequent holds if the p-measure of NP is not zero. Cai and Selman [CS96] defined a modification and extension of Levin’s notion of average polynomial time to arbitrary time-bounds and proved that if L is P-bi-immune, then L is distributionally hard, meaning, that for every polynomial-time computable distribution µ, the distributional problem (L, µ) is not polynomial on the µ-average. We prove the following results, which suggest that distributional hardness is closely related to more traditional notions of hardness. 1. If NP contains a distributionally hard set, then NP contains a P-immune set. 2. There exists a language L that is distributionally hard but not P-bi-immune if and only if P contains a set that is immune to all P-printable sets. The following corollaries follow readily 1. If the p-measure of NP is not zero, then there exists a language L that is distributionally hard but not P-bi-immune. 2. If the p2-measure of NP is not zero, then there exists a language L in NP that is distributionally hard but not P-bi-immune. 1

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analysis of cardinal characteristics of the continuum. Its morphisms have been used in describing reductions between search problems in complexity theory. We describe this category and how it arises in these various contexts. We also show how these contexts suggest certain new multiplicative connectives for linear logic. Perhaps the most interesting of these is a sequential composition suggested by the set-theoretic application.

In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of ”efficiency-onaverage”, simple distributions and efficiency-on-average preserving reductions, Levin developed a theory analogous to the theory of N P-completeness. In particular, he showed that there exists a simple distributional problem that is complete under these reductions. But since then very few distributional problems were shown to be complete in this sense. In this paper we show a simple sufficient condition for an N P-complete decision problem to have a distributional version that is complete under these reductions (and thus to be ”hard on the average ” with respect to some simple probability distribution). Apparently all known N P-complete decision problems meet this condition.

We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the average-case complexity of NP-complete problems. Gurevich showed that a problem with a flat distribution is not DistNP complete (for deterministic reductions), unless DEXPTime<F NaN> 6= NEXPTime. We express the known results concerning fixed size and fixed density distributions for CNF in the framework of average-case complexity and show that all these distributions are flat. We introduce the family of symmetric distributions, which generalizes those mentioned before, and show that bounded symmetric distributions on ordered tuples of clauses (CNFTuples) and on k-CNF (sets of k-literal-clauses), are flat. This eliminates all these distributions as candidates for 'provably hard' (i.e. DistNP complete) distributions for SAT, if one considers only deterministic reductions. Given the (presumed) naturalness and generality o...

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...

The modular group occupies a central position in many branches of mathematical sciences. In this paper we give average polynomial-time algorithms for the unbounded and bounded membership problems for finitely generated subgroups of the modular group. The latter result affirms a conjecture of Gurevich [5]. 1 Introduction 1.1 The Modular Group The modular group \Gamma is a remarkable mathematical object. It has several equivalent characterizations: (i) SL 2 ()= \Sigma I, the quotient of the group SL 2 () of 2 \Theta 2 integer matrices with determinant 1 modulo its central subgroup f\SigmaI g; (ii) the group of complex fractional linear transformations z 7! az + b cz + d with integer coefficients satisfying ad \Gamma bc = 1; (iii) the free product of cyclic groups of order 2 and 3; i.e., the group presented by generators R; S and relations R 2 j S 3 j 1; (iv) the group of automorphisms of a certain regular tesselation of the hyperbolic plane (Figure 1); Proc. 35th IEEE Symp...

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