, Volume 14, Issue 1, pp 117–135 | Cite as

Efficient handling of universally quantified inequalities

  • Alexandre Goldsztejn
  • Claude Michel
  • Michel Rueher


This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to tackle these continuous CSPs with parametric constraints, where the pruning and the solution identification processes are dedicated to universally quantified inequalities. Special rules are proposed to handle the parameter domains of the constraints. The originality of our framework lies in the fact that it solves the QCSP as a non-quantified CSP where the quantifiers are handled locally, at the level of each constraint. Experiments show that our algorithm outperforms the state of the art methods based on constraint techniques.


Universally quantified inequalities Branch and bound Interval analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benhamou, F., & Goualard, F. (2000). Universally quantified interval constraints. In Proceedings of international conference on principles and practice of constraint programming, LNCS, (Vol. 1894, pp. 67–82).Google Scholar
  2. 2.
    Benhamou, F., Goualard, F., Languenou, E., & Christie, M. (2004). Interval constraint solving for camera control and motion planning. ACM Transactions on Computational Logic, 5(4), 732–767.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Benhamou, F., McAllester, D. A., & Van Hentenryck, P. (1994). CLP (intervals) revisited. In SLP (pp. 124–138).Google Scholar
  4. 4.
    Benhamou, F., & Older, W. (1997). Applying interval arithmetic to real, integer and Boolean constraints. Journal of Logic Programming, 6, 1–24.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Boerner, F., Bulatov, A., Chen, H., Jeavons, P., & Krokhin, A. (2003). The complexity of constraint satisfaction games and qcsp. In Proc. of computer science logic, LNCS, (Vol. 2803/2003, pp. 58–70).Google Scholar
  6. 6.
    Bordeaux, L., Cadoli, M., & Mancini, T. (2005). Csp properties for quantified constraints: Definitions and complexity. In Proc. of amer. conf. on artificial intelligence (AAAI) (pp. 360–365).Google Scholar
  7. 7.
    Cleary, J. G. (1987). Logical arithmetic. Future Computing Systems (pp. 125–149).Google Scholar
  8. 8.
    Collavizza, H., Delobel, F., & Rueher, M. (1999). Extending consistent domains of numeric CSP. In Proceedings of IJCAI 1999.Google Scholar
  9. 9.
    Collavizza, H., Delobel, F., & Rueher, M. (1999). Comparing partial consistencies. Reliable Computing, 1, 1–16.Google Scholar
  10. 10.
    Dorato, P. (2000). Quantified multivariate polynomial inequalities. IEEE Control Systems Magazine, 20(5), 48–58.CrossRefGoogle Scholar
  11. 11.
    Dorato, P., Yang, W., & Abdallah, C. (1997). Robust multi-objective feedback design by quantifier elimination. Journal of Symbolic Computations, 24, 153–159.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fiorio, G., Malan, S., Milanese, M., & Taragna, M. (1993). Robust performance design of fixed structure controllers for systems with uncertain parameters. In Proceedings of the 32st IEEE conference on decision and control (Vol. 4, pp. 3029–3031).Google Scholar
  13. 13.
    Goldsztejn, A. (2006) A branch and prune algorithm for the approximation of non-linear AE-solution sets. In SAC ’06: Proceedings of the 2006 ACM symposium on applied computing (pp. 1650–1654).Google Scholar
  14. 14.
    Goldsztejn, A., & Jaulin, L. (2006). Inner and outer approximations of existentially quantified equality constraints. In Proceedings of CP 2006, LNCS, (Vol. 4204/2006, pp. 198–212).Google Scholar
  15. 15.
    Goldsztejn, A., Michel, C., & Rueher, M. (2008). An efficient algorithm for a sharp approximation of universally quantified inequalities. In Proceedings of ACM SAC 2008. Fortaleza, Brazil.Google Scholar
  16. 16.
    Goualard, F., & Granvilliers, L. (2005). Controlled propagation in continuous numerical constraint networks. In Proceedings of the 2005 ACM symposium on applied computing (pp. 377–382).Google Scholar
  17. 17.
    Hayes, B. (2003). A lucid interval. American Scientist, 91(6), 484–488.Google Scholar
  18. 18.
    Jaulin, L., Braems, I., & Walter, E. (2002). Interval methods for nonlinear identification and robust control. In In Proceedings of the 41st IEEE conference on decision and control (Vol. 4, pp. 4676–4681).Google Scholar
  19. 19.
    Jaulin, L., Kieffer, M., Didrit, O., & Walter, E. (2001). Applied interval analysis with examples in parameter and state estimation, robust control and robotics. Springer-Verlag.Google Scholar
  20. 20.
    Jaulin, L., & Walter, E. (1996). Guaranteed tuning, with application to robust control and motion planning. Automomatica, 32(8), 1217–1221.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Jirstrand, M. (1997). Nonlinear control system design by quantifier elimination. Journal of Symbolic Computation, 24(2), 137–152.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kearfott, R. B. (1996). Interval computations: Introduction, uses, & resources. Euromath Bulletin, 2(1), 95–112.MathSciNetGoogle Scholar
  23. 23.
    Lhomme, O. (1993). Consistency techniques for numeric CSPs. In Proceedings of IJCAI 1993 (pp. 232–238).Google Scholar
  24. 24.
    Mackworth, A. K. (1977). Consistency in networks of relations. Artificial Intelligence, 8, 99–118.zbMATHCrossRefGoogle Scholar
  25. 25.
    Malan, S., Milanese, M., & Taragna, M. (1997). Robust analysis and design of control systems using interval arithmetic. Automatica, 33(7), 1363–1372.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Neumaier, A. (1990). Interval methods for systems of equations. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  27. 27.
    Ratschan, S. (2002). Approximate quantified constraint solving by cylindrical box decomposition. Reliable Computing, 8(1), 21–42.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ratschan, S. (2006). Efficient solving of quantified inequality constraints over the real numbers. ACM Transactions on Computational Logic, 7(4), 723–748.MathSciNetGoogle Scholar
  29. 29.
    Ratschan, S. (2008). Applications of quantified constraint solving over the reals bibliography.
  30. 30.
    Vu, X. H., Sam-Haroud, D., & Silaghi, M.-C. (2002). Approximation techniques for non-linear problems with continuum of solutions. In Proceedings of the 5th international symposium on abstraction, reformulation and approximation, LNAI (Vol. 2371, pp. 224–241). Springer-Verlag.Google Scholar
  31. 31.
    Vu, X.-H., Silaghi, M., Sam-Haroud, D., & Faltings, B. (2006). Branch-and-prune search strategies for numerical constraint solving. Technical Report LIA-REPORT-2006-007, Swiss Federal Institute of Technology (EPFL).Google Scholar
  32. 32.
    Zettler, M., & Garloff, J. (1998). Robustness analysis of polynomials with polynomial parameterdependency using bernstein expansion. IEEE Transactions on Automatic Control, 43(3), 425–431.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Alexandre Goldsztejn
    • 1
  • Claude Michel
    • 2
  • Michel Rueher
    • 2
  1. 1.LINA, University of NantesCNRSNantesFrance
  2. 2.I3S-CNRSUniversity of Nice-Sophia AntipolisNiceFrance

Personalised recommendations