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Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 70 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
On Membership Comparable Sets
 Journal of Computer and System Sciences
, 1999
"... A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then Unique ..."
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Cited by 15 (1 self)
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A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then UniqueSAT 2 P. This extends the work of Ogihara; Beigel, Kummer, and Stephan; and Agrawal and Arvind [Ogi94, BKS94, AA94], and answers in the affirmative an open question suggested by Buhrman, Fortnow, and Torenvliet [BFT97]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2 o(n) . Our main technical tool is an algorithm of Ar et al. [ALRS92] to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.
SemiMembership Algorithms: Some Recent Advances
 SIGACT News
, 1994
"... A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990 ..."
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Cited by 12 (8 self)
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A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semimembership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semimembership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semimembership algorithms have been studied in a number of settings. Recursive semimembership algorithms (and the associated semirecursive setsthose sets having recursive semimembership algorithms) were introduced in the 1...
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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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.
Commutative Queries
, 1999
"... 7 We consider polynomialtime Turing machines that have access to two oracles and investigate when the order of oracle queries is significant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries doe ..."
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Cited by 8 (1 self)
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7 We consider polynomialtime Turing machines that have access to two oracles and investigate when the order of oracle queries is significant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries does not matter. This improves upon the previous result of Hemaspaandra, Hemaspaandra and Hempel, who showed that the order of the queries does not matter if the base machine asks only one query to each oracle. On the other hand, we prove that, for computing functions, the order of oracle queries does matter, unless PH collapses. 4 Address: Department of Electrical Engineering and Computer Science, University of Illinois at Chicago, 851 South Morgan Street (M/C 154), Chicago, IL 606077053. Supported in part by the National Science Foundation under grants CCR9415410 & CCR9700417 and by NASA under grant NAG 52895. Research performed while this author was at the University of Maryland Human...
On Sets Bounded TruthTable Reducible to Pselective Sets
, 1996
"... We show that if every NP set is polynomialtime bounded truthtable reducible to some Pselective set, then NP is contained in DTIME(2 n O(1= p log n) ). In the proof, we implement a recursive procedure that reduces the number of nondeterministic steps of a given nondeterministic computation. ..."
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We show that if every NP set is polynomialtime bounded truthtable reducible to some Pselective set, then NP is contained in DTIME(2 n O(1= p log n) ). In the proof, we implement a recursive procedure that reduces the number of nondeterministic steps of a given nondeterministic computation.
NPhard sets are superterse unless NP is small
 Information Processing Letters
, 1997
"... Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonunif ..."
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Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonuniform complexity measure, how easy is NP in the uniform complexity measure? Let P T (SPARSE) be the class of languages that are polynomial time Turing reducible to some sparse sets. Then it is well known that P T (SPARSE) = P=poly. Hence the above question is equivalent to the following question. NP `?PT (SPARSE): It has been shown by Wilson [18] that thi
Reducibility Classes of Pselective Sets
, 1995
"... A set is Pselective [Sel79] if there is a polynomialtime semidecision algorithm for the setan algorithm that given any two strings decides which is \more likely" to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to Pselective sets. 1 I ..."
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Cited by 4 (2 self)
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A set is Pselective [Sel79] if there is a polynomialtime semidecision algorithm for the setan algorithm that given any two strings decides which is \more likely" to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to Pselective sets. 1 Introduction Given the large number of important problems that do not appear to have easy solutions, researchers have explored more exible approaches to ecient set recognition (or nearrecognition): almost polynomial time [MP79], average polynomial time (see [Gur91]), implicit membership testability [HH91], neartestability [GHJY91], Pcloseness [Sch86,Yes83], Pselectivity [Sel79], and others. One such notion, that of the Pselective sets, has proven useful in many contexts, such as characterizing P [BvHT93] and understanding whether SAT may have unique solutions [HNOSb]. Intuitively, a set is Pselective if there is a 2ary polynomialtime function that chooses which of its inputs is \more...
The Communication Complexity of Enumeration, Elimination, and Selection
"... y1) f(xk , yk )) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1 , y1 ), then f(x2 , y2 ), etc.) is, roughly speaking, the best. This conjecture arose in the study of cir ..."
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Cited by 4 (1 self)
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y1) f(xk , yk )) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1 , y1 ), then f(x2 , y2 ), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 #= NC 2 . We consider two related problems. Enumeration: Alice and Bob output e # 2 k  1 elements of {0, 1} k , one of which is f(x1 , y1) f(xk , yk ). Elimination: Alice and Bob output # b such that # b #= f(x1 , y1 ) f(xk , yk ). Selection: (k = 2) Al
PolynomialTime SemiRankable Sets
 Special Issue: Proceedings of the 8th International Conference on Computing and Information
, 1995
"... We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join ..."
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Cited by 3 (3 self)
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We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join with P sets, and closure under Pisomorphism. While P=poly is equal to the closure of Pselective sets under polynomialtime Turing reductions, we build a tally set that is not polynomialtime reducible to any Psr set. We also show that though Psr falls between the Prankable and the weaklyPrankable sets in its inclusiveness, it equals neither of these classes. Key words: semifeasible sets, Pselectivity, ranking, closure properties, NNT. 1 Introduction In the late 1970s, Selman [Sel79] defined the semifeasible (i.e., Pselective) sets, which are the polynomialtime analog of the Jockusch's [Joc68] semirecursive sets. Recently, there has been an intense renewal of interest in the P...