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Continuous First-Order Constraint Satisfaction

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Artificial Intelligence, Automated Reasoning, and Symbolic Computation (AISC 2002, Calculemus 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2385))

Abstract

This paper shows how to use constraint programming techniques for solving first-order constraints over the reals (i.e., formulas in the first-order predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., box-consistency), the paper provides a narrowing operator for first-order constraints that implements a corresponding notion of first-order consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.

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References

  1. S. Basu, R. Pollack, and M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination. In S. Goldwasser, editor, Proceedings fo the 35th Annual Symposium on Foundations of Computer Science, pages 632–641, Los Alamitos, CA, USA, 1994. IEEE Computer Society Press.

    Google Scholar 

  2. F. Benhamou. Interval constraint logic programming. InPodelski [21].

    Google Scholar 

  3. F. Benhamou and F. Goualard. Universally quantified interval constraints. In Proc. of the Sixth Intl. Conf. on Principles and Practice of Constraint Programming (CP’2000), number 1894 in LNCS, Singapore, 2000. Springer Verlag.

    Google Scholar 

  4. F. Benhamou, F. Goualard, L. Granvilliers, and J. F. Puget. Revising hull and box consistency. In Int. Conf. on Logic Programming, pages 230–244, 1999.

    Google Scholar 

  5. F. Benhamou, D. McAllester, and P. V. Hentenryck. CLP (Intervals) Revisited. In International Symposium on Logic Programming, pages 124–138, Ithaca, NY, USA, 1994. MIT Press.

    Google Scholar 

  6. F. Benhamou and W. J. Older. Applying interval arithmetic to real, integer and Boolean constraints. Journal of Logic Programming, 32(1):1–24, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, 1998.

    Google Scholar 

  8. G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Second GI Conf. Automata Theory and Formal Languages, volume 33 of LNCS, pages 134–183. Springer Verlag, 1975. Also in [7].

    Google Scholar 

  9. G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12:299–328, 1991. Also in [7].

    Article  MATH  MathSciNet  Google Scholar 

  10. J. H. Davenport and J. Heintz. Real quantifier elimination is doubly exponential. Journal of Symbolic Computation, 5:29–35, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Granvilliers. A symbolic-numerical branch and prune algorithm for solving nonlinear polynomial systems. Journal of Universal Computer Science, 4(2):125–146, 1998.

    MATH  MathSciNet  Google Scholar 

  12. P. V. Hentenryck, D. McAllester, and D. Kapur. Solving polynomial systems using a branch and prune approach. SI0AM Journal on Numerical Analysis, 34(2), 1997.

    Google Scholar 

  13. P. V. Hentenryck, V. Saraswat, and Y. Deville. The design, implementation, and evaluation of the constraint language cc(FD). In Podelski [21].

    Google Scholar 

  14. H. Hong and V. Stahl. Safe starting regions by fixed points and tightening. Computing, 53:323–335, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Jaulin and É. Walter. Guaranteed tuning, with application to robust control and motion planning. Automatica, 32(8):1217–1221, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Jourdan and T. Sola. The versatility of handling disjunctions as constraints. In M. Bruynooghe and J. Penjam, editors, Proc. of the 5th Intl. Symp. on Programming Language Implementation and Logic Programming, PLILP’93, number 714 in LNCS, pages 60–74. Springer Verlag, 1993.

    Google Scholar 

  17. A. Macintyre and A. Wilkie. On the decidability of the real exponential field. In P. Odifreddi, editor, Kreiseliana—About and Around Georg Kreisel, pages 441–467. A K Peters, 1996.

    Google Scholar 

  18. S. Malan, M. Milanese, and M. Taragna. Robust analysis and design of control systems using interval arithmetic. Automatica, 33(7):1363–1372, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Marriott, P. Moulder, P. J. Stuckey, and A. Borning. Solving disjunctive constraints for interactive graphical applications. In Seventh Intl. Conf. on Principles and Practice of Constraint Programming-CP2001, number 2239 in LNCS, pages 361–376. Springer, 2001.

    Chapter  Google Scholar 

  20. A. Neumaier. Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge, 1990.

    MATH  Google Scholar 

  21. A. Podelski, editor. Constraint Programming: Basics and Trends, volume 910 of LNCS. Springer Verlag, 1995.

    MATH  Google Scholar 

  22. S. Ratschan. Approximate quantified constraint solving (AQCS). http://www.risc.uni-linz.ac.at/research/software/AQCS, 2000. Software package.

  23. S. Ratschan. Applications of real first-order constraint solving —bibliography. http://www.risc.uni-linz.ac.at/people/sratscha/appFOC.html, 2001.

  24. S. Ratschan. Real first-order constraints are stable with probability one. http://www.risc.uni-linz.ac.at/people/sratscha, 2001. Draft.

  25. S. Ratschan. Approximate quantified constraint solving by cylindrical box decomposition. Reliable Computing, 8(1):21–42, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Ratschan. Quantified constraints under perturbations. Journal of Symbolic Computation, 33(4):493–505, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Renegar. On the computational complexity and geometry of the first-order theory of the reals. Journal of Symbolic Computation, 13(3):255–352, March 1992.

    Google Scholar 

  28. D. Richardson. Some undecidable problems involving elementary functions of a real variable. Journal of Symbolic Logic, 33:514–520, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  29. D. Sam-Haroud and B. Faltings. Consistency techniques for continuous constraints. Constraints, 1(1/2):85–118, September 1996.

    Google Scholar 

  30. S. P. Shary. Outer estimation of generalized solution sets to interval linear systems. Reliable Computing, 5:323–335, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Tarski. A Decision Method for Elementary Algebra and Geometry. Univ. of California Press, Berkeley, 1951. Also in [7].

    MATH  Google Scholar 

  32. L. van den Dries. Alfred Tarski’s elimination theory for real closed fields. Journal of Symbolic Logic, 53(1):7–19, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  33. V. Weispfenning. The complexity of linear problems in fields. Journal of Symbolic Computation, 5(1–2):3–27, 1988.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut d’Informatica i Aplicacions, Universitat de Girona, Spain

    Stefan Ratschan

Authors
  1. Stefan Ratschan
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Editor information

Editors and Affiliations

  1. University of Karlsruhe (TH), Am Fasanengarten 5, Postfach 6980, D-76128, Karlsruhe, Germany

    Jacques Calmet

  2. CMI, Université de Provence, 39 rue F. Juliot-Curie, 13453, Marseille Cedex 13, France

    Belaid Benhamou

  3. Research Institute for Symbolic Computation (RISC-Linz), Johannes Kepler University, A-4040, Linz, Austria

    Olga Caprotti

  4. ESIL, Université de la Méditerannée, 163 Avenue de Luminy, Marseille Cedex 09, France

    Laurent Henocque

  5. School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK

    Volker Sorge

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© 2002 Springer-Verlag Berlin Heidelberg

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Ratschan, S. (2002). Continuous First-Order Constraint Satisfaction. In: Calmet, J., Benhamou, B., Caprotti, O., Henocque, L., Sorge, V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC Calculemus 2002 2002. Lecture Notes in Computer Science(), vol 2385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45470-5_18

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  • DOI: https://doi.org/10.1007/3-540-45470-5_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43865-6

  • Online ISBN: 978-3-540-45470-0

  • eBook Packages: Springer Book Archive

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