Solving Existentially Quantified Constraints with One Equality and Arbitrarily Many Inequalities

  • Stefan Ratschan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2833)

Abstract

This paper contains the first algorithm that can solve disjunctions of constraints of the form \(\exists{y}\!\in\! B \; [ f=0 \;\wedge\; g_1\geq 0\wedge\dots\wedge g_k\geq 0 ]\) in free variables x, terminating for all cases when this results in a numerically well-posed problem. Here the only assumption on the terms f, g1,..., gn is the existence of a pruning function, as given by the usual constraint propagation algorithms or by interval evaluation. The paper discusses the application of an implementation of the resulting algorithm on problems from control engineering, parameter estimation, and computational geometry.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, p. 67. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Benhamou, F., McAllester, D., Hentenryck, P.V.: CLP(Intervals) Revisited. In: International Symposium on Logic Programming, Ithaca, NY, USA, pp. 124–138. MIT Press, Cambridge (1994)Google Scholar
  3. 3.
    Benhamou, F., Older, W.J.: Applying interval arithmetic to real, integer and Boolean constraints. Journal of Logic Programming 32(1), 1–24 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bordeaux, L., Monfroy, E.: Beyond NP: Arc-consistency for quantified constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, p. 371. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien (1998)MATHGoogle Scholar
  6. 6.
    Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Caviness and Johnson [5], pp. 134– 183Google Scholar
  7. 7.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5, 29–35 (1988)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Davis, E.: Constraint propagation with interval labels. Artificial Intelligence 32(3), 281–331 (1987)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gardeñes, E., Sainz, M.Á., Jorba, L., Calm, R., Estela, R., Mielgo, H., Trepat, A.: Modal intervals. Reliable Computing 7(2), 77–111 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Granvilliers, L.: On the combination of interval constraint solvers. Reliable Computing 7(6), 467–483 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
  12. 12.
    Hickey, T.J., Ju, Q., van Emden, M.H.: Interval arithmetic: from principles to implementation. Journal of the ACM 48(5), 1038–1068 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hong, H.: Improvements in CAD-based Quantifier Elimination. PhD thesis, The Ohio State University (1990)Google Scholar
  14. 14.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, Berlin (2001)MATHGoogle Scholar
  15. 15.
    Jaulin, L., Walter, E.: Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis. Mathematics and Computers in Simulation 35(2), 123–137 (1993)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36(5), 450–462 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mayer, G.: Epsilon-inflation in verification algorithms. Journal of Computational and Applied Mathematics 60, 147–169 (1994)CrossRefGoogle Scholar
  18. 18.
    Milanese, M., Vicino, A.: Estimation theory for nonlinear models and set membership uncertainty. Automatica (Journal of IFAC) 27(2), 403–408 (1991)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge (1990)MATHGoogle Scholar
  20. 20.
    Nise, N.S.: Control Systems Engineering, 3rd edn. John Wiley & Sons, Chichester (2000)Google Scholar
  21. 21.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations. Academic Press, London (1970)MATHGoogle Scholar
  22. 22.
    Ratschan, S.: Applications of quantified constraint solving over the reals— bibliography (2001), http://www.mpi-sb.mpg.de/~ratschan/appqcs.html
  23. 23.
    Ratschan, S.: Continuous first-order constraint satisfaction. In: Calmet, J., Benhamou, B., Caprotti, O., Hénocque, L., Sorge, V. (eds.) AISC 2002 and Calculemus 2002. LNCS (LNAI), vol. 2385, pp. 181–195. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  24. 24.
    Ratschan, S.: Continuous first-order constraint satisfaction with equality and disequality constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 680–685. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  25. 25.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers (2002), submitted for publication, http://www.mpi-sb.mpg.de/~ratschan/preprints.html
  26. 26.
    Ratschan, S.: Quantified constraints under perturbations. Journal of Symbolic Computation 33(4), 493–505 (2002)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Rump, S.M.: A note on epsilon-inflation. Reliable Computing 4, 371–375 (1998)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Shary, S.P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Computing 8, 321–418 (2002)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. Univ. of California Press, Berkeley (1951); Also in [5]MATHGoogle Scholar
  30. 30.
    Walsh, T.: Stochastic constraint programming. In: Proc. of ECAI (2002)Google Scholar
  31. 31.
    Walter, E., Pronzato, L.: Identification of Parametric Models from Experimental Data. Springer, Heidelberg (1997)MATHGoogle Scholar
  32. 32.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1–2), 3–27 (1988)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Yorke-Smith, N., Gervet, C.: On constraint problems with incomplete or erroneous data. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 732–737. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Stefan Ratschan
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations