Results 1  10
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13
Computing Functions with Parallel Queries to NP
, 1993
"... The class \Theta p 2 of languages polynomialtime truthtable reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP l ..."
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Cited by 39 (1 self)
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The class \Theta p 2 of languages polynomialtime truthtable reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP log , and FP NP k are obtained. In contrast to the language case the function classes seem to all be different. We give evidence in support of this fact by showing that FL NP log coincides with any of the other classes then L = P, and that the equality of the classes FP NP log and FP NP k would imply that the number of nondeterministic bits needed for the computation of any problem in NP can be reduced by a polylogarithmic factor, and that the problem can be computed deterministically with a subexponential time bound of order 2 n O(1= log log n) . 1 Introduction The study of nondeterministic computation is a central topic in structural complexity theory. The acceptance mechanism of...
Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
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Cited by 18 (0 self)
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Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
An Unambiguous Class Possessing a Complete Set
, 1996
"... In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to ..."
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Cited by 15 (3 self)
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In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to compare determinism with nondeterminism. Our inability to exhibit the precise relationship between these two features motivates the investigation of intermediate features such as symmetry or unambiguity. In this paper we will concentrate on the notion of unambiguity. Unfortunately, unambiguity of a device or of a language is in general an undecidable property. Unambiguous classes are not defined by a `syntactical' machine property but rather by a `semantical' restriction. A nasty consequence is the apparent lack of complete sets. In the case of time bounded computations there are relativizations of unambiguity which provably have no complete problem ([10]). For space bounded computations t...
Compendium of Parameterized Problems
, 2001
"... This document is mainly based on "A Compendium of Parameterized Complexity Results", version 2.0 (May 22, 1996), by Michael T. Hallett and H. Todd Wareham, and on Downey and Fellows' book [53]. However, this document includes several new results that have been published in the last few years ..."
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Cited by 9 (0 self)
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This document is mainly based on "A Compendium of Parameterized Complexity Results", version 2.0 (May 22, 1996), by Michael T. Hallett and H. Todd Wareham, and on Downey and Fellows' book [53]. However, this document includes several new results that have been published in the last few years
R SN 1tt (NP) distinguishes robust manyone and Turing completeness
 In Proceedings of the 3rd Italian Conference on Algorithms and Complexity
, 1998
"... Do complexity classes have manyone complete sets if and only if they have Turingcomplete sets? We prove that there is a relativized world in which a relatively natural complexity class—namely a downward closure of NP, RSN 1tt (NP)—has Turingcomplete sets but has no manyone complete sets. In fact ..."
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Cited by 7 (4 self)
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Do complexity classes have manyone complete sets if and only if they have Turingcomplete sets? We prove that there is a relativized world in which a relatively natural complexity class—namely a downward closure of NP, RSN 1tt (NP)—has Turingcomplete sets but has no manyone complete sets. In fact, we show that in the same relativized world this class has 2truthtable complete sets but lacks 1truthtable complete sets. As part of the groundwork for our result, we prove that RSN 1tt(NP) has many equivalent forms having to do with ordered and parallel access to NP and NP ∩ coNP. 1
The Complexity of Obtaining Solutions for Problems in NP and NL
, 1998
"... We review some of the known results about the complexity of computing solutions or proofs of membership for problems in NP. Trying to capture the complexity of this problem, we consider the classes of functions FP NP , FP NP [f ] (for certain bounded functions f ), NPSV, and FP NP tt and prov ..."
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Cited by 7 (0 self)
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We review some of the known results about the complexity of computing solutions or proofs of membership for problems in NP. Trying to capture the complexity of this problem, we consider the classes of functions FP NP , FP NP [f ] (for certain bounded functions f ), NPSV, and FP NP tt and provide some examples of NP problems with search functions in these classes. We also consider whether NPcomplete problems can have such proofs of membership. We use the problem of obtaining solutions to compare the relative powers of the function classes above . Finally, we consider the situation in the nondeterministic logarithmic space setting, showing how the complexity of obtaining solutions for NL sets compares with the NP case. 1 Introduction Problems in the class NP have traditionally been studied from a decisional point of view. This has been so mainly because in all natural cases an algorithm providing a yes/no answer to an NP problem can be used to obtain a solution for the problem, ...
Looking for an analogue of Rice's Theorem in circuit complexity theory
 Mathematical Logic Quarterly
, 1989
"... Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a ..."
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Cited by 5 (1 self)
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Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UPhard, NPhard, generators.
Simplicity and Strong Reductions
, 2000
"... A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit ..."
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Cited by 2 (0 self)
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A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit a relativized world in which there is an NPsimple set that is complete under Turing (Cook) reductions, even conjunctive reductions. This raises the questions whether the result by Hartmanis, Li and Yesha generalizes to reductions of intermediate strength. We show that NPsimple sets are not complete for NP under positive bounded truthtable reductions unless UP # SUBEXP. In fact, NPsimple sets cannot be complete for NP under bounded truthtable reductions under the stronger assumption that UP # coUP ## SUBEXP (while there is an oracle relative to which there is an NPsimple set conjuntively complete for NP). We present several other results for di#erent types of reductions, a...
A Note on Quadratic Residuosity and UP
, 2004
"... UP is the class of languages accepted by polynomialtime nondeterministic Turing machines that have at most one accepting path. We show that the quadratic residue problem belongs to UP intersect coUP. This answers afirmatively an open problem, discussed in Theory of Computational Complexity (Du and ..."
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Cited by 1 (0 self)
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UP is the class of languages accepted by polynomialtime nondeterministic Turing machines that have at most one accepting path. We show that the quadratic residue problem belongs to UP intersect coUP. This answers afirmatively an open problem, discussed in Theory of Computational Complexity (Du and Ko, 2000), of whether the quadratic nonresidue problem is in NP. We generalize to higher powers and show the higher power residue problem also belongs to UP intersect coUP.
Deciding the Winner in Parity Games Is in UP. . .
 INFORMATION PROCESSING LETTERS
, 1998
"... We observe that the problem of deciding the winner in mean payoff games is in the complexity class UP # coUP. We also show a simple reduction from parity games to mean payoff games. From this it follows that deciding the winner in parity games and the modal µcalculus model checking are in UP ..."
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We observe that the problem of deciding the winner in mean payoff games is in the complexity class UP # coUP. We also show a simple reduction from parity games to mean payoff games. From this it follows that deciding the winner in parity games and the modal µcalculus model checking are in UP # coUP.