Simplification of Cylindrical Algebraic Formulas

  • Changbo ChenEmail author
  • Marc Moreno Maza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)


For a set S of cells in a cylindrical algebraic decomposition of ℝ n , we introduce the notion of generalized cylindrical algebraic formula (GCAF) associated with S. We propose a multi-level heuristic algorithm for simplifying the cylindrical algebraic formula associated with S into a GCAF. The heuristic strategies are motivated by solving examples coming from the application of automatic loop transformation. While the algorithm works well on these examples, its effectiveness is also illustrated by examples from other application domains.


Child Node Atomic Formula Disjunctive Normal Form Heuristic Strategy Tree Data Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition II: an adjacency algorithm for the plane. SIAM J. Comput. 13(4), 878–889 (1984)Google Scholar
  2. 2.
    Brown, C.W.: Solution Formula Construction for Truth Invariant CAD’s. PhD thesis, University of Delaware (1999)Google Scholar
  3. 3.
    Brown, C.W.: Fast simplifications for tarski formulas based on monomial inequalities. Journal of Symbolic Computation 47(7), 859–882 (2012)Google Scholar
  4. 4.
    Brown, C.W., Strzeboński, A.: Black-box/white-box simplification and applications to quantifier elimination. In: Proc. of ISSAC 2010, pp. 69–76 (2010)Google Scholar
  5. 5.
    Chen, C., Davenport, J.H., May, J., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. In: Watt, S.M. (ed.) Proceedings ISSAC 2010, pp. 187–194 (2010)Google Scholar
  6. 6.
    Chen, C., Moreno Maza, M.: An incremental algorithm for computing cylindrical algebraic decompositions. In: Computer Mathematics: Proc. of ASCM 2012, pp. 199–222 (2014)Google Scholar
  7. 7.
    Chen, C., Moreno Maza, M.: Quantifier elimination by cylindrical algebraic decomposition based on regular chains. In: Proc. of ISSAC 2014, pp. 91–98 (2014)Google Scholar
  8. 8.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Springer Lecture Notes in Computer Science 33, 515–532 (1975)Google Scholar
  9. 9.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition. Journal of Symbolic Computation 12(3), 299–328 (1991)Google Scholar
  10. 10.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulas over ordered fields. Journal of Symbolic Computation 24, 209–231 (1995)Google Scholar
  11. 11.
    Größlinger, A.: Scanning index sets with polynomial bounds using cylindrical algebraic decomposition. Number MIP-0803 (2008)Google Scholar
  12. 12.
    Größlinger, A., Griebl, M., Lengauer, C.: Quantifier elimination in automatic loop parallelization. J. Symb. Comput. 41(11), 1206–1221 (2006)Google Scholar
  13. 13.
    Iwane, H., Higuchi, H., Anai, H.: An effective implementation of a special quantifier elimination for a sign definite condition by logical formula simplification. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 194–208. Springer, Heidelberg (2013)Google Scholar
  14. 14.
    Strzeboński, A.: Computation with semialgebraic sets represented by cylindrical algebraic formulas. In: Proc. of ISSAC 2010, pp. 61–68. ACM (2010)Google Scholar
  15. 15.
    Strzeboński, A.: Solving polynomial systems over semialgebraic sets represented by cylindrical algebraic formulas. In: Proc. of ISSAC 2012, pp. 335–342. ACM (2012)Google Scholar
  16. 16.
    Wilson, D.J.: Real geometry and connectedness via triangular description: Cad example bank (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent TechnologyChinese Academy of SciencesChongqingChina
  2. 2.ORCCA, University of Western OntarioLondonCanada

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