Quantifier Elimination for Real Algebra -- the Quadratic Case and Beyond

Weispfenning, Volker

doi:10.1007/s002000050055

Cited by 171 publications

(140 citation statements)

References 13 publications

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“…Nevertheless we will encounter significantly higher degrees, because VS implementations are supplemented with powerful simplification strategies; see, e.g., [20] and [36,Section 5]. The nonlinear bound of 2 is based on work by Weispfenning in the early 1990s, which also theoretically discussed arbitrary degree bounds [73]. Only recently, Košta has theoretically improved Weispfenning's framework for higher degrees and provided a generic implementation in Redlog, which can be instantiated with substitution tables up to a chosen degree bound [35].…”

Section: Historymentioning

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%

“…Extended quantifier elimination [73] does not form a disjunction over the virtual substitution results. Instead, they are kept separate and associated with formal assignments of the substituted test points:…”

Section: Extended Quantifier Eliminationmentioning

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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“…Any algebraic constraints on the vector field can be transferred into the first-order formula over the ordered field of the reals expressing Muldowney's criteria. Simple conservation constraints stating that the sum of certain state variables is constant-conditions that are commonly found in chemical reaction systems or in epidemiological models-will not induce a failure of the degree limited virtual substitution methods [22] for quantifier elimination, if these were successful on the unconstrained system.…”

Section: Muldowney's Extensions Of the Bendixson-dulac Criterion To Hmentioning

confidence: 99%

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

Errami

Seiler

Sturm

et al. 2011

Computer Algebra in Scientific Computing

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Abstract. We study Muldowney's extension of the classical BendixsonDulac criterion for excluding periodic orbits to higher dimensions for polynomial vector fields. Using the formulation of Muldowney's sufficient criteria for excluding periodic orbits of the parameterized vector field on a convex set as a quantifier elimination problem over the ordered field of the reals we provide case studies of some systems arising in the life sciences. We discuss the use of simple conservation constraints and the use of parametric constraints for describing simple convex polytopes on which periodic orbits can be excluded by Muldowney's criteria.

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“…Improvements to qepcad are described in [7], which yield much "simpler" output formulas than before. Various other algorithms for the elimination of quantifiers for first-order formulas over the field of the reals have also been implemented over the last years, see e. g. [31,30,19]. Their realizations are available in the Redlog system [9], which is implemented in Reduce.…”

Section: The Quantifier-elimination Program Qepcadmentioning

confidence: 99%

Extending a Java Based Framework for Scientific Software-Components

Göbel

Küchlin

Müller

et al. 1999

Computer Algebra in Scientific Computing CASC’99

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Abstract.A prototypical framework, which was used for building software components for symbolic computation, is extended as follows. First, we demonstrate that the server components can be accessed from other frameworks for collaborative scientific computing, too. Specifically, we incorporate access from the PROGRESS system. Second, we discuss several design issues that arise when encapsulating existing systems as services into the framework. Many of these issues are of a general nature but become relevant in our effort to incorporate the invariant package of MAS and the quantifier elimination package qepcad into our framework.

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Quantifier Elimination for Real Algebra -- the Quadratic Case and Beyond

Cited by 171 publications

References 13 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

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

Extending a Java Based Framework for Scientific Software-Components

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