Abstract
We present an extension of constraint logic programming, where the admissible constraints are arbitrary first-order formulas over various domains: real numbers with ordering, linear constraints over p-adic numbers, complex numbers, linear constraints over the integers with ordering and congruences (parametric Presburger Arithmetic), quantified propositional calculus (parametric qsat), term algebras. Our arithmetic is always exact. For ℝ are ℂ there are no restrictions on the polynomial degree of admissible constraints. Constraint solving is realized by effective quantifier elimination. We have implemented our methods in our system clp(rl). A number of computation examples with clp(rl) are given in order to illustrate the conceptual generalizations provided by our approach and to demonstrate its feasibility.
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References
Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Berlin (1998)
Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: Dooley, S. (ed.) Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (ISSAC 1999), Vancouver, BC, pp. 129–136. ACM Press, New York (1999)
Sturm, T., Weispfenning, V.: Quantifier elimination in term algebras. The case of finite languages. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2002, pp. 285–300. TUM München (2002)
Hong, H.: RISC-CLP(Real): Constraint logic programming over real numbers. In: Benhamou, F., Colmerauer, A. (eds.) Constraint Logic Programming: Selected Research. MIT Press, Cambridge (1993)
Colmerauer, A., et al.: Workshop on first-order constraints, Marseille, France. Abstracts of talks (2001)
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31, 2–9 (1997)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12, 299–328 (1991)
Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5, 3–27 (1988)
Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8, 85–101 (1997)
Becker, E., Wörmann, T.: On the trace formula for quadratic forms. In: Jacob, W.B., Lam, T.Y., Robson, R.O. (eds.) Recent Advances in Real Algebraic Geometry and Quadratic Forms. Contemporary Mathematics, vol. 155, pp. 271–291. American Mathematical Society, Providence (1994); Proceedings of the RAGSQUAD Year, Berkeley, (1990–1991)
Pedersen, P., Roy, M.F., Szpirglas, A.: Counting real zeroes in the multivariate case. In: Eysette, F., Galigo, A. (eds.) Computational Algebraic Geometry, Berlin. Progress in Mathematics, vol. 109, pp. 203–224. Birkhäuser, Boston (1993); Proceedings of the MEGA 1992
Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation, pp. 376–392. Springer, Wien (1998)
Sturm, T.: Linear problems in valued fields. Journal of Symbolic Computation 30, 207–219 (2000)
Weispfenning, V.: Comprehensive Gröbner bases. Journal of Symbolic Computation 14, 1–29 (1992)
Weispfenning, V.: Complexity and uniformity of elimination in Presburger arithmetic. In: Küchlin, W.W. (ed.) Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (ISSAC 1997), Maui, HI, pp. 48–53. ACM Press, New York (1997)
Seidl, A., Sturm, T.: Boolean quantification in a first-order context. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing. Proceedings of the CASC 2003, pp. 345–356. TUM München (2003)
Dincbas, M., Van Hentenryck, P., Simonis, H., Aggoun, A., Graf, T., Berthier, F.: The constraint logic programming language CHIP. In: Proceedings of the International Conference on Fifth Generation Computer Systems, Tokyo, Japan, Decemeber 1988, pp. 693–702. Ohmsha Publishers, Tokyo (1988)
Jaffar, J., Michaylov, S., Stuckey, P.J., Yap, R.H.C.: The CLP(R) language and system. ACM Transactions on Programming Languages and Systems 14, 339–395 (1992)
Colmerauer, A.: Prolog III. Communications of the ACM 33, 70–90 (1990)
Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24, 209–231 (1997)
Frühwirth, T., Abdennadher, S.: Essentials of Constraint Programming. Springer, Heidelberg (2003)
Kameny, S.L.: Roots: A reduce root finding package. In: Hearn, A.C., Codemist, Ltd. (eds.) Reduce User’s and Contributed Packages Manual Version 3.7, pp. 513–518. Anthony C. Hearn and Codemist Ltd. (1999)
Sturm, T., Weispfenning, V.: Computational geometry problems in Redlog. In: Wang, D. (ed.) ADG 1996. LNCS(LNAI), vol. 1360, pp. 58–86. Springer, Heidelberg (1998)
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Sturm, T. (2005). Quantifier Elimination for Constraint Logic Programming. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_36
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DOI: https://doi.org/10.1007/11555964_36
Publisher Name: Springer, Berlin, Heidelberg
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