Problem Formulation for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

  • Matthew England
  • Russell Bradford
  • Changbo Chen
  • James H. Davenport
  • Marc Moreno Maza
  • David Wilson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8543)


Cylindrical algebraic decompositions (CADs) are a key tool for solving problems in real algebraic geometry and beyond. We recently presented a new CAD algorithm combining two advances: truth-table invariance, making the CAD invariant with respect to the truth of logical formulae rather than the signs of polynomials; and CAD construction by regular chains technology, where first a complex decomposition is constructed by refining a tree incrementally by constraint. We here consider how best to formulate problems for input to this algorithm. We focus on a choice (not relevant for other CAD algorithms) about the order in which constraints are presented. We develop new heuristics to help make this choice and thus allow the best use of the algorithm in practice. We also consider other choices of problem formulation for CAD, as discussed in CICM 2013, revisiting these in the context of the new algorithm.


cylindrical algebraic decomposition truth table invariance regular chains triangular decomposition problem formulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnon, D., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: The basic algorithm. SIAM J. Comput. 13, 865–877 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnon, D.S., Mignotte, M.: On mechanical quantifier elimination for elementary algebra and geometry. J. Symb. Comp. 5(1-2), 237–259 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bradford, R., Chen, C., Davenport, J.H., England, M., Moreno Maza, M., Wilson, D.: Truth table invariant cylindrical algebraic decomposition by regular chains (submitted, 2014), Preprint:
  4. 4.
    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
  5. 5.
    Bradford, R., Davenport, J.H., England, M., McCallum, S., Wilson, D.: Truth table invariant cylindrical algebraic decomposition (submitted, 2014), Preprint:
  6. 6.
    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
  7. 7.
    Brown, C.W., Davenport, J.H.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In: Proc. ISSAC 2007, pp. 54–60. ACM (2007)Google Scholar
  8. 8.
    Brown, C.W., El Kahoui, M., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibria in epidemic modelling. J. Symbolic Computation 41, 1157–1173 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, C., Moreno Maza, M.: An incremental algorithm for computing cylindrical algebraic decompositions. In: Proc. ASCM 2012. Springer (2012) (to appear), Preprint: arXiv:1210.5543v1Google Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    Collins, G.E.: 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
  13. 13.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comp. 12, 299–328 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Dolzmann, A., Seidl, A., Sturm, T.: Efficient projection orders for CAD. In: Proc. ISSAC 2004, pp. 111–118. ACM (2004)Google Scholar
  16. 16.
    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 (LNAI), vol. 7961, pp. 136–151. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    England, M.: An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance. Uni. Bath, Dept. Comp. Sci. Tech. Report Series, 2013-04 (2013),
  18. 18.
    Fotiou, I.A., Parrilo, P.A., Morari, M.: Nonlinear parametric optimization using cylindrical algebraic decomposition. In: Proc. CDC-ECC 2005, pp. 3735–3740 (2005)Google Scholar
  19. 19.
    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
  20. 20.
    Kahan, W.: Problem #9: an ellipse problem. SIGSAM Bull. 9(3), 11–12 (1975)CrossRefGoogle Scholar
  21. 21.
    McCallum, S.: On projection in CAD-based quantifier elimination with equational constraint. In: Proc. ISSAC 1999, pp. 145–149. ACM (1999)Google Scholar
  22. 22.
    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
  23. 23.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Strzeboński, A.: Cylindrical algebraic decomposition using validated numerics. J. Symb. Comp. 41(9), 1021–1038 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Matthew England
    • 1
  • Russell Bradford
    • 1
  • Changbo Chen
    • 2
  • James H. Davenport
    • 1
  • Marc Moreno Maza
    • 3
  • David Wilson
    • 1
  1. 1.University of BathBathU.K.
  2. 2.Chongqing Key Laboratory of Automated Reasoning and CognitionChongqing Institute of Green and Intelligent Technology, CASChongqingChina
  3. 3.University of Western OntarioLondonCanada

Personalised recommendations