The complexity of linear problems in fields

Weispfenning, Volker

doi:10.1016/s0747-7171(88)80003-8

Cited by 219 publications

(147 citation statements)

References 6 publications

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“…In 1988 Davenport-Heintz [15] and Weispfenning [72] independently showed that it is doubly exponential. Weispfenning's article actually brought even stronger results: first, it showed that the decision problem is doubly exponential already for linear formulas, where the total degree in all quantified variables does not exceed 1.…”

Section: Historymentioning

confidence: 99%

“…Quantifiers are essentially eliminated one by one. Except for Weispfenning's original linear case [39,72], where there are no products of quantified variables at all, the elimination of a quantifier will in general increase the degrees of other quantified variables. As a consequence, one cannot tell by inspection of the input whether or not quantifier elimination will succeed subject to the current degree bound.…”

Section: Degree Boundsmentioning

confidence: 99%

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A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Sturm

2017

Math.Comput.Sci.

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Effective quantifier elimination procedures for first-order theories provide a powerful tool for generically solving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifier elimination procedures are based on a fixed set of admissible logical symbols with an implicitly fixed semantics. This admits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination for the reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifier elimination we are going to discuss recent results with a subtropical procedure for an existential fragment of the reals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems in chemistry and in the life sciences.

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Section: Historymentioning

confidence: 99%

Section: Degree Boundsmentioning

confidence: 99%

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

Sturm

2017

Math.Comput.Sci.

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“…The project function can be implemented, for instance, by using the Fourier-Motzkin variable elimination algorithm or the algorithm presented in [13].…”

Section: Preliminary Definitionsmentioning

confidence: 99%

A Folding Algorithm for Eliminating Existential Variables from Constraint Logic Programs

2008

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Abstract. The existential variables of a clause in a constraint logic program are the variables which occur in the body of the clause and not in its head. The elimination of these variables is a transformation technique which is often used for improving program efficiency and verifying program properties. We consider a folding transformation rule which ensures the elimination of existential variables and we propose an algorithm for applying this rule in the case where the constraints are linear inequations over rational or real numbers. The algorithm combines techniques for matching terms modulo equational theories and techniques for solving systems of linear inequations. We show that an implementation of our folding algorithm performs well in practice.

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“…However, even whithin the simpler domain of linear constraints, it was not obvious how these conditions can be found except by repeated application of existential quantifier elimination [1,2], an operation that is fundamentally exponential [5]. In this paper we address this problem by providing an alternative and elegant way for computing mutual exclusion conditions over linear constraints by applying Craig interpolation [6], a topic that has attracted much interest recently in program verification [7].…”

Section: Introductionmentioning

confidence: 99%

Mutual Exclusion by Interpolation

Kriener

King

2012

Functional and Logic Programming

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Abstract. The question of what constraints must hold for a predicate to behave as a (partial) function, is key to understanding the behaviour of a logic program. It has been shown how this question can be answered by combining backward analysis, a form of analysis that propagates determinacy requirements against the control flow, with a component for deriving so-called mutual exclusion conditions. The latter infers conditions sufficient to ensure that if one clause yields an answer then another cannot. This paper addresses the challenge of how to compute these conditions by showing that this problem can be reformulated as that of vertex enumeration. Whilst directly applicable in logic programming, the method might well also find application in reasoning about type classes.

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The complexity of linear problems in fields

Cited by 219 publications

References 6 publications

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

A Folding Algorithm for Eliminating Existential Variables from Constraint Logic Programs

Mutual Exclusion by Interpolation

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