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32
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 69 (22 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
ResourceBounded Measure and Randomness
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than ..."
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Cited by 42 (6 self)
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We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory.
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 14 (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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Cited by 12 (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.
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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Cited by 9 (1 self)
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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.
The Communication Complexity of Enumeration, Elimination, and Selection
"... Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, y1) · · · f(xk, yk)) communicating as few bits as possibl ..."
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Cited by 7 (1 self)
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Let k, n ∈ N and f: {0, 1} n × {0, 1} n → {0, 1}. Assume Alice has x1,..., xk ∈ {0, 1} n, Bob has y1,..., yk ∈ {0, 1} n, and they want to compute f k (x1x2 · · · xk, y1y2 · · · yk) = (f(x1, y1), · · · , f(xk, yk)) (henceforth f(x1, 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) Alice and Bob output i ∈ {1, 2} such that if f(x1, y1) = 1 ∨ f(x2, y2) = 1 then f(xi, yi) = 1. a) We devise the Enumeration Conjecture (henceforth ENC) and the Elimination
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 7 (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 the Reducibility of Sets Inside NP to Sets with Low Information Content
, 2002
"... We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism... ..."
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Cited by 4 (2 self)
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We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism...