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33
A complexity theory for feasible closure properties. In:
 Structure in Complexity Theory Conference.
, 1991
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Computing Solutions Uniquely Collapses the Polynomial Hierarchy
 SIAM Journal on Computing
, 1993
"... Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterm ..."
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Cited by 41 (25 self)
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Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to its second level. As the existence of such a function is known to be equivalent to the statement "every multivalued NP function has a singlevalued NP refinement," our result provides the strongest evidence yet that multivalued NP functions cannot be refined. We prove our result via theorems of independent interest. We say that a set A is NPSVselective (NPMVselective) if there is a 2ary partial function in NPSV (NPMV, respectively) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursiontheoretic notion of the semirecursive sets and the extant complexitythe...
A Relationship between Difference Hierarchies and Relativized Polynomial Hierarchies
, 1993
"... Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP c ..."
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Cited by 40 (9 self)
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Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)tt j NP , the class of sets recognized in polynomial time with k \Gamma 1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has p m complete sets and is closed under p conj  and NP m reductions (alternatively, closed under p disj  and coNP m reductions), if the difference hierarchy over C collapses to level k, then PH C = i P NP (k\Gamma1)tt j C . Then we show that the exact counting class C=P is closed under p disj  and coNP m  reductions. Consequently, if the difference hiera...
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m j ..."
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Cited by 32 (5 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)tt reducible to a Pselective set, then the set is polynomialtime (1 + c) log f(n)membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
Upper bounds for the Complexity of Sparse and Tally Descriptions
 Mathematical Systems Theory
, 1996
"... We investigate the complexity of computing small descriptions for sets in various reduction classes to sparse sets. For example, we show that if a set A and its complement conjunctively reduce to some sparse set, then they also are conjunctively reducible to a P(A \Phi SAT)printable tally set. As ..."
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Cited by 17 (8 self)
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We investigate the complexity of computing small descriptions for sets in various reduction classes to sparse sets. For example, we show that if a set A and its complement conjunctively reduce to some sparse set, then they also are conjunctively reducible to a P(A \Phi SAT)printable tally set. As a consequence, the class IC[log; poly] of sets with low instance complexity is contained in the EL \Sigma 1 level of the extended low hierarchy. By refining our techniques, we also show that all worddecreasing selfreducible sets in IC[log; poly] are in NP " coNP and therefore low for NP. We derive similar results for sets in R p d (SPARSE)) and R p hd (R p c (SPARSE)), as well as in some nondeterministic reduction classes to sparse sets. Parts of this work have been presented at ISAAC'92, MFCS'93, and CIAC'94 [AKM92b, AKM93, Mu94]. y Work done while visiting Universitat Ulm. Supported in part by an Alexander von Humboldt research fellowship. 1 1 Introduction Sparse sets play...
The Structure of Logarithmic Advice Complexity Classes
 Theoretical Computer Science
, 1992
"... A nonuniform class called here FullP/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated b ..."
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Cited by 12 (4 self)
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A nonuniform class called here FullP/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomialtime deterministic reductions. Several characterizations of FullP/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets. Partially supported by the E.U. through the ESPRIT Long Term Research Project 20244 (ALCOMIT) and through the HCM Network CHRXCT930415 (COLORET); by the Spanish DGICYT through project PB950787 (KOALA), and by Acciones Integradas HispanoAl...
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.
Locating P/poly Optimally in the Extended Low Hierarchy
, 1993
"... The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hier ..."
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Cited by 11 (0 self)
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The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hierarchy. The location of P=poly in EL P;\Theta 3 is optimal since, as shown by Allender and Hemachandra (1992), there exist sparse sets that are not contained in the next lower level EL P;\Sigma 2 . As a consequence of our result, all NP sets in P=poly are relocated from the third \Sigmalevel L P;\Sigma 3 (Ko and Schoning, 1985) to the third \Thetalevel L P;\Theta 3 of the low hierarchy.