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12
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...
Optimal Advice
 Theoretical Computer Science
, 1994
"... Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector functio ..."
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Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," usually a polynomialtime deterministic or nondeterministic function, that selects which of any two given inputs is logically no less likely than the other to belong to the given set. Definition 1.1 [HNOS94] Let FC be any class of functions (possibly multivalued or partial). A set A is FCselective if there is a function f 2 FC such that for every x and y, it holds that f(x; y) ` fx; yg, and if fx; yg " A 6= ;, then f(x; y) 6= ; and f(x; y) ` A. Let FCsel denote the class of sets that are FCselective. The class that would be notated FP single\Gammavalued; total sel according to the definition above was defined directly by Selman in 1979 [Sel79]. Henceforward, we refer to these sets ...
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.
A Note on LinearNondeterminism, LinearSized, KarpLipton Advice for the PSelective Sets
, 1998
"... Hemaspaandra and Torenvliet showed that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and quasilinear nondeterminism. We show that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and linea ..."
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Cited by 5 (3 self)
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Hemaspaandra and Torenvliet showed that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and quasilinear nondeterminism. We show that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and linear nondeterminism.
The Complexity of Finding TopTodaEquivalenceClass Members
, 2003
"... We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from Σ^n that the set's Pselector function declar ..."
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Cited by 5 (3 self)
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We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from &Sigma;^n that the set's Pselector function declares to be most likely to belong to the set) is FP computable, and we show that each Pselective set contains a weaklyP rankable subset.
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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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...
Polynomialtime multiselectivity
, 1997
"... We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove ..."
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We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove the first known (and optimal) lower bounds for generalized selectivitylike classes in terms of EL2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara's Pmc (polynomialtime membershipcomparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the Pselective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that PSel, the class of Pselective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the Pselective sets, and yet, in contrast with the Pmc classes, is refined enough to distinguish them.
On the Size of Classes with Weak Membership Properties
"... It is shown that the following classes have resourcebounded measure 0 in E: the class of Pselective sets, the class of Pmultiselective sets, the class of cheatable sets, the class of easily countable sets, the class of easily approximable sets, the class of neartestable sets, the class of nearly ..."
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It is shown that the following classes have resourcebounded measure 0 in E: the class of Pselective sets, the class of Pmultiselective sets, the class of cheatable sets, the class of easily countable sets, the class of easily approximable sets, the class of neartestable sets, the class of nearly neartestable sets, the class of locally selfreducible sets. These are corollaries of a more general result stating that the class of sets that are pisomorphic to Pquasiapproximable sets has measure 0 in E. By considering the recent approach of Allender and Strauss for measuring in subexponential classes, we obtain similar results with respect to P for classes having weak logarithmic time membership properties. Keywords: resourcebounded measure, Pselective sets, Pmultiselective sets, cheatable sets, easily countable sets, easily approximable sets, neartestable sets, nearly neartestable sets, locally selfreducible sets, Pbiimmune sets. 1 Introduction By the deterministic time hie...
LinearNondeterminism Linear Advice for the PSelective Sets
, 1997
"... Hemaspaandra and Torenvliet showed that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and quasilinear nondeterminism. We extend this by showing that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linea ..."
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Hemaspaandra and Torenvliet showed that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and quasilinear nondeterminism. We extend this by showing that each Pselective set can be accepted by a polynomialtime nondeterministic machine using linear advice and linear nondeterminism. 1 Introduction The Pselective sets, sometimes referred to as the semifeasible sets, were introduced by Selman [Sel79] as polynomialtime analogs of the semirecursive sets of recursive function theory. They have played an active role in many facets of complexity theory (see [HNOS96b] and the survey [DHHT94] for references and discussion). By definition, the Pselective sets are those sets that have a polynomialtime function that chooses one of any two given strings, and that never chooses one that is out of the set if the other is in the set. (Informally, the function chooses one that is "no less likely than the other to be in the set.") Definition 1....