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18
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility 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 de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility 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 selfreducible. 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. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
Power from Random Strings
 IN PROCEEDINGS OF THE 43RD IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2002
"... 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 nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let ..."
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Cited by 36 (15 self)
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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 nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete 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 NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Reductions to Sets of Low Information Content
, 1992
"... this paper was coauthored by K. Wagner) ..."
On Bounded TruthTable, Conjunctive, and Randomized Reductions to Sparse Sets
 IN PROC. 12TH CONFERENCE ON THE FOUNDATIONS OF SOFTWARE TECHNOLOGY & THEORETICAL COMPUTER SCIENCE
, 1992
"... 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 NPcomplete set is boundedtruthtable reducible to a set that conjunctively reduces to ..."
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Cited by 17 (8 self)
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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 NPcomplete set is boundedtruthtable reducible to a set that conjunctively reduces to a sparse set then P = NP. Relatedly, we show that if an NPcomplete set is boundedtruthtable reducible to a set that corp 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 manyone reductions to sparse and cosparse sets. We prove that if a coNPcomplete set reduces via a nondeterministic polynomial time manyone reduction to a cosparse set then PH = \Theta p 2 . On the other hand, we show that nondeterministic polynomial ...
Languages that are Easier than their Proofs
, 1991
"... 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 reduc ..."
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Cited by 13 (7 self)
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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, randomselfreducibility, 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 selfreducibility. Under assumptions about tripleexponential time, we construct incoherent sets in NP....
On the Structure of Low Sets
 PROC. 10TH STRUCTURE IN COMPLEXITY THEORY CONFERENCE, IEEE
, 1995
"... 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 ..."
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Cited by 10 (2 self)
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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 polynomialsize circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.
Designing LDPC Codes Using BitFilling
 in Proc. Int. Conf. Communications (ICC
, 2001
"... 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 checkdegree, the bitdegree, and the girth, we consider the problem of constructing bipartite graphs wit ..."
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Cited by 10 (1 self)
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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 checkdegree, the bitdegree, 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 simpletoimplement heuristic BITFILLING algorithm for this problem. As a benchmark, our algorithm yields codes better or comparable to those in MacKay [1]. I.
Sparse Sets versus Complexity Classes
, 1996
"... 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 k ..."
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Cited by 8 (2 self)
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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...
SpaceEfficient Recognition Of Sparse SelfReducible Languages
, 1994
"... . Mahaney and others have shown that sparse selfreducible sets have timeecient 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 complet ..."
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Cited by 5 (3 self)
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. Mahaney and others have shown that sparse selfreducible sets have timeecient 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 selfreducible sets have spaceecient algorithms, and in many cases, even have timespaceecient algorithms. We conclude that NL, NC k , AC k , LOG(DCFL), LOG(CFL), and P lack complete (or even Turinghard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, k , or NP) has a polylogsparse logspacehard set, then NL SC (respectively P SC, k SC, or PH SC), and if P has subpolynomially sparse logspacehard sets, then P 6= PSPACE. Subject classications. 68Q15, 03D15. 1. Introduction Complete sets are the quintessences of their complexity cla...