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A Poly-algorithmic Quantifier Elimination Package in Maple

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Maple in Mathematics Education and Research (MC 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1125))

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Abstract

The problem of Quantifier Elimination (QE) in Computer Algebra is that of eliminating all quantifiers from a statement featuring polynomial constraints. This problem is known to be worst case time complexity worst case doubly exponential in the number of variables. As such implementations are sometimes seen as undesirable to use, despite problems arising in algebraic geometry and even economics lending themselves to formulations as QE problems. This paper largely concerns discussion of current progress of a package QuantifierElimination written using Maple that uses a poly-algorithm between two well known algorithms to solve QE: Virtual Term Substitution (VTS), and Cylindrical Algebraic Decomposition (CAD). While mitigation of efficiency concerns is the main aim of the implementation, said implementation being built in Maple reconciles with an aim of providing rich output to users to make use of algorithms to solve QE valuable. We explore the challenges and scope such an implementation gives in terms of the desires of the Satisfiability Modulo Theory (SMT) community, and other frequent uses of QE, noting Maple’s status as a Mathematical toolbox.

Thanks to my supervisor James Davenport.

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References

  1. Alvandi, P., Chen, C., Lemaire, F., Maza, M., Xie, Y.: The RegularChains Library. http://www.regularchains.org/

  2. Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bull. 37(4), 97–108 (2003). https://doi.org/10.1145/968708.968710

    Article  MATH  Google Scholar 

  3. Brown, C.W.: Fast simplifications for Tarski formulas based on monomial inequalities. J. Symb. Comput. 47(7), 859–882 (2012). https://doi.org/10.1016/j.jsc.2011.12.012

    Article  MathSciNet  MATH  Google Scholar 

  4. Brown, C.W., Davenport, J.H.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, ISSAC 2007, pp. 54–60. ACM, New York, NY, USA (2007). https://doi.org/10.1145/1277548.1277557

  5. Chen, C., Maza, M.M.: Simplification of cylindrical algebraic formulas. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 119–134. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24021-3_9

    Chapter  Google Scholar 

  6. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07407-4_17

    Chapter  Google Scholar 

  7. Davenport, J., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1), 29–35 (1988). https://doi.org/10.1016/S0747-7171(88)80004-X

    Article  MathSciNet  MATH  Google Scholar 

  8. England, M., Wilson, D., Bradford, R., Davenport, J.H.: Using the regular chains library to build cylindrical algebraic decompositions by projecting and lifting. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 458–465. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_69

    Chapter  Google Scholar 

  9. Hong, H., Collins, G.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 299–328 (1991). https://doi.org/10.1016/S0747-7171(08)80152-6

    Article  MathSciNet  Google Scholar 

  10. Košta, M.: New concepts for real quantifier elimination by virtual substitution. Ph.D. thesis, Universität des Saarlandes (2016). https://doi.org/10.22028/D291-26679

  11. Maplesoft: Maple Programming Guide, pp. 360–372. https://www.maplesoft.com/documentation_center/maple2019/ProgrammingGuide.pdf

  12. McCallum, S.: On projection in CAD-based quantifier elimination with equational constraint. In: Proceedings ISSAC 1999, pp. 145–149 (1999). https://doi.org/10.1145/309831.309892

  13. McCallum, S.: On propagation of equational constraints in CAD-based quantifier elimination. In: Proceedings ISSAC 2001, pp. 223–231 (2001). https://doi.org/10.1145/384101.384132

  14. McCallum, S., Parusiński, A., Paunescu, L.: Validity proof of Lazard’s method for CAD construction. J. Symb. Comput. 92, 52–69 (2019). https://doi.org/10.1016/j.jsc.2017.12.002

    Article  MathSciNet  MATH  Google Scholar 

  15. Strzebonski, A.: Real Polynomial Systems, Wolfram Mathematica. https://reference.wolfram.com/language/tutorial/RealPolynomialSystems.html

  16. Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. Univ. Cal. Press (1951). Reprinted in Quantifier Elimination and Cylindrical Algebraic Decomposition (ed. B.F. Caviness & J.R. Johnson), pp. 24–84. Springer, Wein-New York (1998). https://doi.org/10.1007/978-3-7091-9459-1_3

    Google Scholar 

  17. Weispfenning, V.: The complexity of linear problems in fields. J. Symb. Comput. 5(1), 3–27 (1988). https://doi.org/10.1016/S0747-7171(88)80003-8

    Article  MathSciNet  MATH  Google Scholar 

  18. Yanami, H., Anai, H.: SyNRAC: a maple toolbox for solving real algebraic constraints. ACM Commun. Comput. Algebra 41(3), 112–113 (2007). https://doi.org/10.1145/1358190.1358205

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Bath, Bath, Somerset, BA2 7AY, UK

    Zak Tonks

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  1. Zak Tonks
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Correspondence to Zak Tonks .

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Editors and Affiliations

  1. Maplesoft, Waterloo, ON, Canada

    Jürgen Gerhard

  2. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

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Tonks, Z. (2020). A Poly-algorithmic Quantifier Elimination Package in Maple. In: Gerhard, J., Kotsireas, I. (eds) Maple in Mathematics Education and Research. MC 2019. Communications in Computer and Information Science, vol 1125. Springer, Cham. https://doi.org/10.1007/978-3-030-41258-6_13

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  • DOI: https://doi.org/10.1007/978-3-030-41258-6_13

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-41258-6

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