We obtain several results that distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not self-reducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992--Dec. 1992. Current address: Department of Computer Science, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan. -- Department of Computer Science, State University of New York at Buffalo, 226...

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let

This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

this paper was coauthored by K. Wagner)

A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....

Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomial-size circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.

The study of sparse hard sets and sparse complete sets has been a central research area in complexity theory for nearly two decades. Recently new results using unexpected techniques have been obtained. They provide new and easier proofs of old theorems, proofs of new theorems that unify previously known results, resolutions of old conjectures, and connections to the fascinating world of randomization and derandomization. In this article we give an exposition of this vibrant research area. 1 Introduction Complexity theory is concerned with the quantitative limitation and power of computation. During the past several decades computational complexity theory developed gradually from its initial awakening [Rab59, Yam62, HS65, Cob65] to the current edifice of a scientific discipline that is rich in beautiful results, powerful techniques, fascinating research topics and conjectures, deep connections to other mathematical subjects, and of critical importance to everyday computing. The buildin...

Bipartite graphs of bit nodes and parity check nodes arise as Tanner graphs corresponding to low density parity check codes. Given graph parameters such as the number of check nodes, the maximum check-degree, the bit-degree, and the girth, we consider the problem of constructing bipartite graphs with the largest number of bit nodes, that is, the highest rate. We propose a simple-to-implement heuristic BIT-FILLING algorithm for this problem. As a benchmark, our algorithm yields codes better or comparable to those in MacKay [1]. I.

. Mahaney and others have shown that sparse self-reducible sets have time-ecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complete sets for feasible classes until now. This paper shows that sparse self-reducible sets have space-ecient algorithms, and in many cases, even have time-space-ecient algorithms. We conclude that NL, NC k , AC k , LOG(DCFL), LOG(CFL), and P lack complete (or even Turing-hard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, k , or NP) has a polylog-sparse logspace-hard set, then NL SC (respectively P SC, k SC, or PH SC), and if P has subpolynomially sparse logspace-hard sets, then P 6= PSPACE. Subject classications. 68Q15, 03D15. 1. Introduction Complete sets are the quintessences of their complexity cla...

In this paper we study the consequences of the existence of sparse hard sets for different complexity classes under certain types of deterministic, randomized and nondeterministic reductions. We show that if an NP-complete set is bounded-truthtable reducible to a set that conjunctively reduces to a sparse set then P = NP. Relatedly, we show that if an NP-complete set is bounded-truth-table reducible to a set that co-rp reduces to some set that conjunctively reduces to a sparse set then RP = NP. We also prove similar results under the (apparently) weaker assumption that some solution of the promise problem (1SAT; SAT) reduces via the mentioned reductions to a sparse set. Finally we consider nondeterministic polynomial time many-one reductions to sparse and co-sparse sets. We prove that if a coNP-complete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then PH = \Theta p 2 . On the other hand, we show that nondeterministic polynomial time many-o...