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On the Relative Complexity of Approximate Counting Problems
, 2000
"... Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reduc ..."
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Cited by 60 (19 self)
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Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically dened subclass of #P. Research Report 370, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. This work was supported in part by the EPSRC Research Grant \Sharper Analysis of Randomised Algorithms: a Computational Approach" and by the ESPRIT Projects RANDAPX and ALCOMFT. y dyer@scs.leeds.ac.uk, School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. z leslie@dcs.warwick.ac.uk, http://www.dcs.warw...
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
01 Laws for Fragments of Existential SecondOrder Logic: A Survey
"... The probability of a property on the collection of all finite relational structures is the limit as n ! 1 of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that the 01 law holds for every property expressible in firstorder logic, i.e., th ..."
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The probability of a property on the collection of all finite relational structures is the limit as n ! 1 of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that the 01 law holds for every property expressible in firstorder logic, i.e., the probability of every such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable. In this survey, we consider fragments of existential secondorder logic in which we restrict the patterns of firstorder quantifiers. We focus on fragments in which the firstorder part belongs to a prefix class. We show that the classifications of prefix classes of firstorder logic with equality according to the solvability of the finite satisfiability problem and according to the 01 law for the corresponding 1 1 fragments are identical, but the classifications are different without equality.