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Applying Machine Learning to the Problem of Choosing a Heuristic to Select the Variable Ordering for Cylindrical Algebraic Decomposition

  • Zongyan Huang
  • Matthew England
  • David Wilson
  • James H. Davenport
  • Lawrence C. Paulson
  • James Bridge
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8543)

Abstract

Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.

Keywords

machine learning support vector machine symbolic computation cylindrical algebraic decomposition problem formulation 

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References

  1. 1.
    Akbarpour, B., Paulson, L.: MetiTarski: An automatic theorem prover for real-valued special functions. Journal of Automated Reasoning 44(3), 175–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alpaydin, E.: Introduction to machine learning. MIT Press (2004)Google Scholar
  3. 3.
    Arnon, D., Collins, G., McCallum, S.: Cylindrical algebraic decomposition I: The basic algorithm. SIAM Journal of Computing 13, 865–877 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baldi, P., Brunak, S., Chauvin, Y., Andersen, C.A., Nielsen, H.: Assessing the accuracy of prediction algorithms for classification: an overview. Bioinformatics 16(5), 412–424 (2000)CrossRefGoogle Scholar
  5. 5.
    Basu, S.: Algorithms in real algebraic geometry: A survey (2011), www.math.purdue.edu/~sbasu/raag_survey2011_final.pdf
  6. 6.
    Boyan, J., Freitag, D., Joachims, T.: A machine learning architecture for optimizing web search engines. In: AAAI Workshop on Internet Based Information Systems, pp. 1–8 (1996)Google Scholar
  7. 7.
    Bradford, R., Davenport, J., England, M., McCallum, S., Wilson, D.: Cylindrical algebraic decompositions for boolean combinations. In: Proc. ISSAC 2013, pp. 125–132. ACM (2013)Google Scholar
  8. 8.
    Bradford, R., Davenport, J.H., England, M., Wilson, D.: Optimising problem formulation for cylindrical algebraic decomposition. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 19–34. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Bridge, J.P.: Machine learning and automated theorem proving. University of Cambridge Computer Laboratory Technical Report UCAM-CL-TR-792 (2010), http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-792.pdf
  10. 10.
    Bridge, J., Holden, S., Paulson, L.: Machine learning for first-order theorem proving. Journal of Automated Reasoning, 1–32 (2014)Google Scholar
  11. 11.
    Brown, C.: Improved projection for cylindrical algebraic decomposition. Journal of Symbolic Computation 32(5), 447–465 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brown, C.: QEPCAD B: A program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bulletin 37(4), 97–108 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Brown, C.: Companion to the Tutorial: Cylindrical algebraic decomposition. Presented at ISSAC 2004 (2004), www.usna.edu/Users/cs/wcbrown/research/ISSAC04/handout.pdf
  14. 14.
    Brown, C., Davenport, J.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In: Proc. ISSAC 2007, pp. 54–60. ACM (2007)Google Scholar
  15. 15.
    Brown, C., Kahoui, M.E., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibria in epidemic modelling. Journal of Symbolic Computation 41, 1157–1173 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carette, J.: Understanding expression simplification. In: Proc. ISSAC 2004, pp. 72–79. ACM (2004)Google Scholar
  17. 17.
    Chen, C., Maza, M.M., Xia, B., Yang, L.: Computing cylindrical algebraic decomposition via triangular decomposition. In: Proc. ISSAC 2009, pp. 95–102. ACM (2009)Google Scholar
  18. 18.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  19. 19.
    Collins, G.: Quantifier elimination by cylindrical algebraic decomposition – 20 years of progress. In: Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts & Monographs in Symbolic Computation, pp. 8–23. Springer (1998)Google Scholar
  20. 20.
    Collins, G., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12, 299–328 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cristianini, N., Shawe-Taylor, J.: An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press (2000)Google Scholar
  22. 22.
    Davenport, J., Bradford, R., England, M., Wilson, D.: Program verification in the presence of complex numbers, functions with branch cuts etc. In: Proc. SYNASC 2012, pp. 83–88. IEEE (2012)Google Scholar
  23. 23.
    Dolzmann, A., Seidl, A., Sturm, T.: Efficient projection orders for CAD. In: Proc. ISSAC 2004, pp. 111–118. ACM (2004)Google Scholar
  24. 24.
    Dolzmann, A., Sturm, T.: REDLOG: Computer algebra meets computer logic. SIGSAM Bulletin 31(2), 2–9 (1997)CrossRefGoogle Scholar
  25. 25.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Algorithmic Algebra and Number Theory, pp. 221–247. Springer (1998)Google Scholar
  26. 26.
    England, M.: An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance. University of Bath Department of Computer Science Technical Report 2013-04 (2013), http://opus.bath.ac.uk/35636/
  27. 27.
    Forsyth, R., Rada, R.: Machine learning: Applications in expert systems and information retrieval. Halsted Press (1986)Google Scholar
  28. 28.
    Fotiou, I., Parrilo, P., Morari, M.: Nonlinear parametric optimization using cylindrical algebraic decomposition. In: 2005 European Control Conference on Decision and Control, CDC-ECC 2005, pp. 3735–3740 (2005)Google Scholar
  29. 29.
    Hong, H.: An improvement of the projection operator in cylindrical algebraic decomposition. In: Proc. ISSAC 1990, pp. 261–264. ACM (1990)Google Scholar
  30. 30.
    Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Networks 2(5), 359–366 (1989)CrossRefGoogle Scholar
  31. 31.
    Huang, Z., Paulson, L.: An application of machine learning to rcf decision procedures. In: Proc. 20th Automated Reasoning Workshop (2013)Google Scholar
  32. 32.
    Hsu, C., Chang, C., Lin, C.: A practical guide to support vector classification (2003)Google Scholar
  33. 33.
    Iwane, H., Yanami, H., Anai, H., Yokoyama, K.: An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination. In: Proc. SNC 2009, pp. 55–64 (2009)Google Scholar
  34. 34.
    Joachims, T.: Making large-scale SVM learning practical. In: Advances in Kernel Methods - Support Vector Learning, pp. 169–184. MIT Press (1999)Google Scholar
  35. 35.
    Joachims, T.: A support vector method for multivariate performance measures. In: Proc. 22nd Intl. Conf. on Machine Learning, pp. 377–384. ACM (2005)Google Scholar
  36. 36.
    Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 339–354. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  37. 37.
    McCallum, S.: An improved projection operation for cylindrical algebraic decomposition. In: Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts & Monographs in Symbolic Computation, pp. 242–268. Springer (1998)Google Scholar
  38. 38.
    McCallum, S.: On projection in CAD-based quantifier elimination with equational constraint. In: Proc. ISSAC 1999, pp. 145–149. ACM (1999)Google Scholar
  39. 39.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics 5(4), 115–133 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychological Review 65(6), 386 (1958)CrossRefGoogle Scholar
  41. 41.
    Schölkopf, B., Tsuda, K., Vert, J.-P.: Kernel methods in computational biology. MIT Press (2004)Google Scholar
  42. 42.
    Sebastiani, F.: Machine learning in automated text categorization. ACM Computing Surveys (CSUR) 34(1), 1–47 (2002)CrossRefGoogle Scholar
  43. 43.
    Shawe-Taylor, J., Cristianini, N.: Kernel methods for pattern analysis. Cambridge University Press (2004)Google Scholar
  44. 44.
    Stone, P., Veloso, M.: Multiagent systems: A survey from a machine learning perspective. Autonomous Robots 8(3), 345–383 (2000)CrossRefGoogle Scholar
  45. 45.
    Strzeboński, A.: Cylindrical algebraic decomposition using validated numerics. Journal of Symbolic Computation 41(9), 1021–1038 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Strzeboński, A.: Solving polynomial systems over semialgebraic sets represented by cylindrical algebraic formulas. In: Proc. ISSAC 2012, pp. 335–342. ACM (2012)Google Scholar
  47. 47.
    Tarski, A.: A decision method for elementary algebra and geometry. In: Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation, pp. 24–84. Springer (1998)Google Scholar
  48. 48.
    Wilson, D., Bradford, R., Davenport, J.: A repository for CAD examples. ACM Communications in Computer Algebra 46(3), 67–69 (2012)zbMATHGoogle Scholar
  49. 49.
    Wilson, D., Davenport, J., England, M., Bradford, R.: A “piano movers” problem reformulated. In: Proc. SYNASC 2013. IEEE (2013)Google Scholar
  50. 50.
    The benchmarks used in solving nonlinear arithmetic. New York University (2012), http://cs.nyu.edu/~dejan/nonlinear/

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Zongyan Huang
    • 1
  • Matthew England
    • 2
  • David Wilson
    • 2
  • James H. Davenport
    • 2
  • Lawrence C. Paulson
    • 1
  • James Bridge
    • 1
  1. 1.University of Cambridge Computer LaboratoryCambridgeU.K.
  2. 2.Department of Computer ScienceUniversity of BathBathU.K.

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