Better Answers to Real Questions

We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified variables. Implementations of extended quantifier elimination for the quadratic case via virtual substitution have been successfully applied to various problems in science and engineering. So far, the answers produced by these implementations included infinitesimal and infinite numbers, which are hard to interpret in practice. We introduce here a post-processing procedure to convert, for fixed parameters, all answers into standard real numbers. The relevance of our procedure is demonstrated by application of our implementation to various examples from the literature, where it significantly improves the quality of the results.

[1]  Nestan Tsiskaridze,et al.  Conflict Resolution , 2009, CP.

[2]  Markus Eiswirth,et al.  Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems , 2005, J. Symb. Comput..

[3]  Volker Weispfenning,et al.  Simulation and Optimization by Quantifier Elimination , 1997, J. Symb. Comput..

[4]  Thomas Sturm,et al.  Simplification of Quantifier-Free Formulae over Ordered Fields , 1997, J. Symb. Comput..

[5]  Volker Weispfenning,et al.  Parametric linear and quadratic optimization by elimina-tion , 1994 .

[6]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[7]  Thomas Sturm,et al.  Approaches to parallel quantifier elimination , 1998, ISSAC '98.

[8]  Thomas Sturm,et al.  Algorithmic Global Criteria for Excluding Oscillations , 2011, Bulletin of mathematical biology.

[9]  Andreas Seidl,et al.  Boolean Quantification in a First-Order Context , 2006 .

[10]  François Lemaire,et al.  On Proving the Absence of Oscillations in Models of Genetic Circuits , 2007, AB.

[11]  Thomas Sturm,et al.  Real Quantifier Elimination in Practice , 1997, Algorithmic Algebra and Number Theory.

[12]  Cesare Tinelli,et al.  Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL(T) , 2006, JACM.

[13]  Dima Grigoriev,et al.  Efficient Methods to Compute Hopf Bifurcations in Chemical Reaction Networks Using Reaction Coordinates , 2013, CASC.

[14]  Ashish Tiwari,et al.  Verification and synthesis using real quantifier elimination , 2011, ISSAC '11.

[15]  Thomas Sturm,et al.  Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology , 2008, AB.

[16]  Greg Nelson,et al.  Simplification by Cooperating Decision Procedures , 1979, TOPL.

[17]  Rüdiger Loos,et al.  Applying Linear Quantifier Elimination , 1993, Comput. J..

[18]  Thomas Sturm,et al.  Quantifier Elimination in Term Algebras The Case of Finite Languages , 2002 .

[19]  Alkiviadis G. Akritas,et al.  Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials , 2009, J. Univers. Comput. Sci..

[20]  Thomas Sturm,et al.  Reasoning over Networks by Symbolic Methods , 1999, Applicable Algebra in Engineering, Communication and Computing.

[21]  Thomas Sturm,et al.  A New Approach for Automatic Theorem Proving in Real Geometry , 1998, Journal of Automated Reasoning.

[22]  Dima Grigoriev,et al.  Detection of Hopf bifurcations in chemical reaction networks using convex coordinates , 2015, J. Comput. Phys..

[23]  Volker Weispfenning,et al.  Semilinear Motion Planning in REDLOG , 2001, Applicable Algebra in Engineering, Communication and Computing.

[24]  Thomas Sturm,et al.  Towards Conflict-Driven Learning for Virtual Substitution , 2014, SMT.

[25]  Jean-Francois Collard,et al.  Reasoning About Program Transformations , 2002, Springer New York.

[26]  Thomas Sturm,et al.  Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology , 2009, Math. Comput. Sci..

[27]  M'hammed El Kahoui,et al.  Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture , 2000, J. Symb. Comput..

[28]  Thomas Sturm,et al.  Weak quantifier elimination for the full linear theory of the integers , 2007, Applicable Algebra in Engineering, Communication and Computing.

[29]  Thomas Sturm,et al.  REDLOG: computer algebra meets computer logic , 1997, SIGS.

[30]  Thomas Sturm,et al.  Computational Geometry Problems in REDLOG , 1996, Automated Deduction in Geometry.

[31]  Volker Weispfenning,et al.  Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond , 1997, Applicable Algebra in Engineering, Communication and Computing.

[32]  Thomas Sturm,et al.  Linear Problems in Valued Fields , 2000, J. Symb. Comput..

[33]  Thomas Sturm,et al.  Parametric quantified SAT solving , 2010, ISSAC.

[34]  Thomas Sturm,et al.  Real Quantifier Elimination in Geometry 1 , 1999 .

[35]  Thomas Sturm,et al.  An Algebraic Approach to Offsetting and Blending of Solids , 2000 .

[36]  Volker Weispfenning,et al.  Quantifier elimination for real algebra—the cubic case , 1994, ISSAC '94.