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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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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 th ..."
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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...
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.
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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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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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.
R SN 1−tt(NP) distinguishes robust manyone and Turing completeness
 Theory of Computing Systems
, 1998
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