Quantifier elimination by cylindrical algebraic decomposition based on regular chains

A quantifier elimination algorithm by cylindrical algebraic decomposition based on regular chains is presented. The main idea is to refine a complex cylindrical tree until the signs of polynomials appearing in the tree are sufficient to distinguish the true and false cells. We report on an implementation of our algorithm in the RegularChains library in Maple and illustrate its effectiveness by examples.

[1]  P. Sadayappan,et al.  High-performance code generation for stencil computations on GPU architectures , 2012, ICS '12.

[2]  Michel Coste,et al.  Thom's Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi-Algebraic Sets , 1988, J. Symb. Comput..

[3]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[4]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[5]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[6]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[7]  Changbo Chen,et al.  Cylindrical Algebraic Decomposition in the RegularChains Library , 2014, ICMS.

[8]  Cédric Bastoul,et al.  Code generation in the polyhedral model is easier than you think , 2004, Proceedings. 13th International Conference on Parallel Architecture and Compilation Techniques, 2004. PACT 2004..

[9]  Hoon Hong.  Special Issue Editorial: Computational Quantifier Elimination , 1993, Comput. J..

[10]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[11]  Matthew England,et al.  Cylindrical algebraic decompositions for boolean combinations , 2013, ISSAC '13.

[12]  Martin Griebl,et al.  Quantifier elimination in automatic loop parallelization , 2006, J. Symb. Comput..

[13]  Christopher W. Brown,et al.  Solution formula construction for truth invariant cad's , 1999 .

[14]  Changbo Chen,et al.  Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains , 2014, CASC.

[15]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[16]  James H. Davenport,et al.  The complexity of quantifier elimination and cylindrical algebraic decomposition , 2007, ISSAC '07.

[17]  Saugata Basu,et al.  New results on quantifier elimination over real closed fields and applications to constraint databases , 1999, JACM.

[18]  Changbo Chen,et al.  An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions , 2012, ASCM.

[19]  Changbo Chen,et al.  Quantifier elimination by cylindrical algebraic decomposition based on regular chains , 2016, J. Symb. Comput..

[20]  Changbo Chen,et al.  Real Quantifier Elimination in the RegularChains Library , 2014, ICMS.

[21]  Larry Carter,et al.  Determining the idle time of a tiling , 1997, POPL '97.

[22]  Hoon Hong,et al.  Improvements in cad-based quantifier elimination , 1990 .

[23]  Hoon Hong,et al.  Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination , 1992, ISSAC '92.

[24]  Changbo Chen,et al.  Comprehensive Triangular Decomposition , 2007, CASC.

[25]  Algorithmic Algebra and Logic. Proceedings of the A3L 2005, April 3-6, Passau, Germany; Conference in Honor of the 60th Birthday of Volker Weispfenning , 2005, Algorithmic Algebra and Logic.

[26]  Changbo Chen,et al.  Computing cylindrical algebraic decomposition via triangular decomposition , 2009, ISSAC '09.

[27]  Mohab Safey El Din,et al.  Variant quantifier elimination , 2012, J. Symb. Comput..

[28]  Leonid Khachiyan.  Fourier-Motzkin Elimination Method , 2009, Encyclopedia of Optimization.