Advertisement

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

  • Russell Bradford
  • Changbo Chen
  • James H. Davenport
  • Matthew England
  • Marc Moreno Maza
  • David Wilson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8660)

Abstract

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art.

Keywords

cylindrical algebraic decomposition equational constraint regular chains triangular decomposition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bradford, R., Davenport, J.H.: Towards better simplification of elementary functions. In: Proc. ISSAC 2002, pp. 16–22. ACM (2002)Google Scholar
  2. 2.
    Bradford, R., Davenport, J.H., England, M., McCallum, S., Wilson, D.: Cylindrical algebraic decompositions for boolean combinations. In: Proc. ISSAC 2013, pp. 125–132. ACM (2013)Google Scholar
  3. 3.
    Bradford, R., Davenport, J.H., England, M., McCallum, S., Wilson, D.: Truth table invariant cylindrical algebraic decomposition. Preprint: arXiv:1401.0645Google Scholar
  4. 4.
    Bradford, R., Davenport, J.H., England, M., Wilson, D.: Optimising problem formulations for cylindrical algebraic decomposition. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS, vol. 7961, pp. 19–34. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Algorithms and Computations in Mathematics, vol. 10. Springer (2006)Google Scholar
  6. 6.
    Brown, C.W.: Simplification of truth-invariant cylindrical algebraic decompositions. In: Proc. ISSAC 1998, pp. 295–301. ACM (1998)Google Scholar
  7. 7.
    Brown, C.W.: An overview of QEPCAD B: A program for computing with semi-algebraic sets using CADs. SIGSAM Bulletin 37(4), 97–108 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Brown, C.W., El Kahoui, M., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibria in epidemic modelling. J. Symb. Comp. 41, 1157–1173 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brown, C.W., McCallum, S.: On using bi-equational constraints in CAD construction. In: Proc. ISSAC 2005, pp. 76–83. ACM (2005)Google Scholar
  10. 10.
    Buchberger, B., Hong, H.: Speeding up quantifier elimination by Gröbner bases. Technical report, 91-06. RISC, Johannes Kepler University (1991)Google Scholar
  11. 11.
    Chen, C., Golubitsky, O., Lemaire, F., Maza, M.M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comp. 47(6), 610–642 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, C., Moreno Maza, M.: An incremental algorithm for computing cylindrical algebraic decompositions. In: Proc. ASCM 2012. Springer (2012) (to appear) Preprint: arXiv:1210.5543Google Scholar
  14. 14.
    Chen, C., Moreno Maza, M.: Quantifier elimination by cylindrical algebraic decomposition based on regular chains. In: Proc. ISSAC 2014 (to appear, 2014)Google Scholar
  15. 15.
    Chen, C., Moreno Maza, M., Xia, B., Yang, L.: Computing cylindrical algebraic decomposition via triangular decomposition. In: Proc. ISSAC 2009, pp. 95–102. ACM (2009)Google Scholar
  16. 16.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Proc. 2nd GI Conference on Automata Theory and Formal Languages, pp. 134–183. Springer (1975)Google Scholar
  17. 17.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comp. 12, 299–328 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Davenport, J.H., 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
  19. 19.
    Dolzmann, A., Seidl, A., Sturm, T.: Efficient projection orders for CAD. In: Proc. ISSAC 2004, pp. 111–118. ACM (2004)Google Scholar
  20. 20.
    England, M., Bradford, R., Chen, C., Davenport, J.H., Maza, M.M., Wilson, D.: Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 45–60. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  21. 21.
    England, M., Bradford, R., Davenport, J.H., Wilson, D.: Understanding branch cuts of expressions. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS, vol. 7961, pp. 136–151. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  22. 22.
    England, M., Bradford, R., Davenport, J.H., Wilson, D.: Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 450–457. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  23. 23.
    England, M., Wilson, D., Bradford, R., Davenport, J.H.: Using the Regular Chains Library to build cylindrical algebraic decompositions by projection and lifting. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 458–465. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  24. 24.
    Fotiou, I.A., Parrilo, P.A., Morari, M.: Nonlinear parametric optimization using cylindrical algebraic decomposition. In: Proc. Decision and Control, European Control Conference 2005, pp. 3735–3740 (2005)Google Scholar
  25. 25.
    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
  26. 26.
    McCallum, S.: On projection in CAD-based quantifier elimination with equational constraint. In: Proc. ISSAC 1999, pp. 145–149. ACM (1999)Google Scholar
  27. 27.
    Paulson, L.C.: Metitarski: Past and future. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 1–10. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  28. 28.
    Phisanbut, N., Bradford, R.J., Davenport, J.H.: Geometry of branch cuts. ACM Communications in Computer Algebra 44(3), 132–135 (2010)Google Scholar
  29. 29.
    Schwartz, J.T., Sharir, M.: On the “Piano-Movers” Problem: II. General techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4, 298–351 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Strzeboński, A.: Cylindrical algebraic decomposition using validated numerics. J. Symb. Comp. 41(9), 1021–1038 (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Strzeboński, A.: Computation with semialgebraic sets represented by cylindrical algebraic formulas. In: Proc. ISSAC 2010, pp. 61–68. ACM (2010)Google Scholar
  32. 32.
    Wang, D.: Computing triangular systems and regular systems. J. Symb. Comp. 30(2), 221–236 (2000)CrossRefzbMATHGoogle Scholar
  33. 33.
    Wilson, D., Bradford, R., Davenport, J.H., England, M.: Cylindrical algebraic sub-decompositions. Mathematics in Computer Science 8(2), 263–288 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Russell Bradford
    • 1
  • Changbo Chen
    • 2
  • James H. Davenport
    • 1
  • Matthew England
    • 1
  • Marc Moreno Maza
    • 3
  • David Wilson
    • 1
  1. 1.University of BathBathUK
  2. 2.CIGIT, Chinese Academy of SciencesChongqingChina
  3. 3.University of Western OntarioLondonCanada

Personalised recommendations