A Symbiosis of Interval Constraint Propagation and Cylindrical Algebraic Decomposition

  • Ulrich Loup
  • Karsten Scheibler
  • Florian Corzilius
  • Erika Ábrahám
  • Bernd Becker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7898)


We present a novel decision procedure for non-linear real arithmetic: a combination of iSAT, an incomplete SMT solver based on interval constraint propagation (ICP), and an implementation of the complete cylindrical algebraic decomposition (CAD) method in the library GiNaCRA . While iSAT is efficient in finding unsatisfiability, on satisfiable instances it often terminates with an interval box whose satisfiability status is unknown to iSAT. The CAD method, in turn, always terminates with a satisfiability result. However, it has to traverse a double-exponentially large search space.

A symbiosis of iSAT and CAD combines the advantages of both methods resulting in a fast and complete solver. In particular, the interval box determined by iSAT provides precious extra information to guide the CAD-method search routine: We use the interval box to prune the CAD search space in both phases, the projection and the construction phase, forming a search “tube” rather than a search tree. This proves to be particularly beneficial for a CAD implementation designed to search a satisfying assignment pointedly, as opposed to search and exclude conflicting regions.


Real Root Construction Phase Satisfying Assignment Bound Model Check Theory Atom 
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.
    Anai, H., Yokoyama, K.: Cylindrical algebraic decomposition via numerical computation with validated symbolic reconstruction. In: Algorithmic Algebra and Logic, pp. 25–30 (2005)Google Scholar
  2. 2.
    Basu, S., Pollack, R., Roy, M.: Algorithms in Real Algebraic Geometry. Springer (2010)Google Scholar
  3. 3.
    Benhamou, F., Granvilliers, L.: Continuous and Interval Constraints. In: Handbook of Constraint Programming, pp. 571–603. Foundations of Artificial Intelligence (2006)Google Scholar
  4. 4.
    Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Advances in Computers 58, 118–149 (2003)CrossRefGoogle Scholar
  5. 5.
    Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)Google Scholar
  6. 6.
    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
  7. 7.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5, 394–397 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Eggers, A., Kruglov, E., Kupferschmid, S., Scheibler, K., Teige, T., Weidenbach, C.: Superposition modulo non-linear arithmetic. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS, vol. 6989, pp. 119–134. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. Journal on Satisfiability, Boolean Modeling, and Computation 1(3-4), 209–236 (2007)Google Scholar
  10. 10.
    Hong, H.: An improvement of the projection operator in cylindrical algebraic decomposition. In: ISSAC 1990, pp. 261–264. ACM (1990)Google Scholar
  11. 11.
    Iwane, H., Yanami, H., Anai, H.: An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems. In: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation, pp. 168–177. ACM (2012)Google Scholar
  12. 12.
    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
  13. 13.
    Mishra, B.: Algorithmic Algebra. Texts and Monographs in Computer Science. Springer (1993)Google Scholar
  14. 14.
    Ratschan, S.: Approximate quantified constraint solving by cylindrical box decomposition. Reliable Computing 8(1), 21–42 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Silva, J.P.M., Sakallah, K.A.: Grasp - a new search algorithm for satisfiability. In: ICCAD, pp. 220–227 (1996)Google Scholar
  16. 16.
    Strzebonski, A.W.: Cylindrical algebraic decomposition using validated numerics. Journal of Symbolic Computation 41(9), 1021–1038 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Tseitin, G.S.: On the complexity of derivations in propositional calculus. In: Slisenko, A. (ed.) Studies in q(1968)Google Scholar
  18. 18.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1-2), 3–27 (1988)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ulrich Loup
    • 1
  • Karsten Scheibler
    • 2
  • Florian Corzilius
    • 1
  • Erika Ábrahám
    • 1
  • Bernd Becker
    • 2
  1. 1.RWTH Aachen UniversityGermany
  2. 2.University of FreiburgGermany

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