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The Logic Programming Paradigm in Numerical Computation

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The Logic Programming Paradigm

Part of the book series: Artificial Intelligence ((AI))

Summary

Although CLP(R) is a promising application of the logic programming paradigm to numerical computation, it has not addressed what has long been known as “the pitfalls of [numerical] computation” [12]. These show that rounding errors induce a severe correctness problem wherever floating-point computation is used. Independently of logic programming, constraint processing has been applied to problems in terms of real-valued variables. By using the techniques of interval arithmetic, constraint processing can be regarded as a computer-generated proof that a certain real-valued solution lies in a narrow interval. In this paper we propose a method for interfacing this technique with CLP(R). This is done via a real-valued analogy of Apt’s proof-theoretic framework for constraint processing.

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References

  1. K.R. Apt. A proof theoretic view of constraint programming.Fundamenta Informaticae, 33(3):263–293, 1998.

    MathSciNet  Google Scholar 

  2. K.R. Apt. Prom chaotic iteration to constraint propagation. In Proceedings of the 24th International Colloquium on Automata, Languages, and Programming (ICALP’97), 1997.

    Google Scholar 

  3. Frédéric Benhamou, Pascal Bouvier, Alain Colmerauer, Henri Garetta, Bruno Giletta, Jean-Luc Massat, Guy Alain Narboni, Stéphane N’Dong, Robert Pasero, Jean-François Pique, Touraïvane, Michel Van Caneghem, Eric Vétillard. Le manuel de Prolog IV. Technical report, PrologIA, Parc Tech- nologique de Luminy, Marseille, France, 1996.

    Google Scholar 

  4. Frédéric Benhamou, William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1–24, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  5. BNR. BNR Prolog user guide and reference manual. Version 3.1 for Macintosh, 1988.

    Google Scholar 

  6. N. Bourbaki . Théorie des Ensembles (Fascicule de Résultats). Hermann, 1939.

    Google Scholar 

  7. J.G. Cleary. Logical arithmetic. Future Computing Systems, 2:125–149, 1987.

    Google Scholar 

  8. A. Colmerauer, H. Kanoui, R. Paséro, P. Roussel. Un système de communication homme-machine en français. Technical report, Groupe d’Intelligence Artificielle, Université d’Aix-Marseille II, 1972.

    Google Scholar 

  9. E. Davis. Constraint propagation with labels. Artificial Intelligence, 32:281–331, 1987.

    MATH  Google Scholar 

  10. M. Dincbas, P. Van Hentenryck, H. Simonis, A. Aggoun, T. Graf, F. Berthier. The constraint programming language CHIP. In Proc. Int. Conf on Fifth Generation Computer Systems, 1988.

    Google Scholar 

  11. H.B. Enderton. A Mathematical Introduction to Logic. Fletcher and Sons, Ltd, 1972.

    MATH  Google Scholar 

  12. George E. Forsythe. Pitfalls of computation, or why a math book isn’t enough. Amer. Math. Monthly, 77:931–956, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  13. Andrzej Grzegorczyk.An Outline of Mathematical Logic: Fundamental Results and Notions Explained with All Details. D. Reidel, 1974.

    Google Scholar 

  14. P. Van Hentenryck, L. Michel, F. Benhamou. Newton: Constraint programming over nonlinear constraints. Science of Computer Programming, 1996.

    Google Scholar 

  15. Pascal Van Hentenryck, Laurent Michel, Yves Deville. Numerica: A Modeling Language for Global Optimization. MIT Press, 1997.

    Google Scholar 

  16. T. Hickey, Q. Ju, M. van Emden. Using the IEEE floating-point standard for implementing interval arithmetic. In preparation.

    Google Scholar 

  17. Patricia Hill and John Lloyd. The Gödel Programming Language. MIT Press, 1994.

    MATH  Google Scholar 

  18. Joxan Jaffar, Michael J. Maher. Constraint logic programming: A survey. Journal of Logic Programming, 19/20:503–582, 1994.

    Article  MathSciNet  Google Scholar 

  19. JW Lloyd Foundations of Logic Programming. Springer-Verlag, 2nd edition, 1987.

    MATH  Google Scholar 

  20. Ugo Montanari, Francesca Rossi. Constraint relaxation may be perfect. Artificial Intelligence, 48:143–170, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ramon E. Moore. Interval Analysis. Prentice-Hall, 1966.

    Google Scholar 

  22. D. Stott Parker. Monte Carlo arithmetic: an effective way to improve upon floating-point arithmetic. Technical Report CSD-970002, Computer Science Department, University of California at Los Angeles, 1997.

    Google Scholar 

  23. Jean-François Puget, Pascal Van Hentenryck. A constraint satisfaction approach to a circuit design problem. Journal of Global Optimization, 13(1), 1998.

    Google Scholar 

  24. Alexander L. Semenov. Solving optimization problems with help of the Unicalc solver. In R. Baker Kearfott, Vladik Kreinovich, editors, Application of Interval Computations, pages 211–224. Kluwer Academic Publishers, 1996.

    Google Scholar 

  25. Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, 1967.

    Google Scholar 

  26. M.H. van Emden. Value constraints in the CLP Scheme. Constraints, 2:163–183, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. van Wijngaarden. Numerical analysis as an independent science. BIT, 6:66–81, 1966.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Victoria, P.O. Box 3055, Victoria, B.C., V8W 3P6, Canada

    Maarten H. van Emden

Authors
  1. Maarten H. van Emden
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Editor information

Editors and Affiliations

  1. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098, Amsterdam, SJ, The Netherlands

    Krzysztof R. Apt

  2. Department of Computer Science, University of Kentucky, 773 Anderson Hall, 40506-0046, Lexington, KY, USA

    Victor W. Marek  & Mirek Truszczynski  & 

  3. Computer Science Department, University at Stony Brook, Stony Brook, 11794-4400, NY, USA

    David S. Warren

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© 1999 Springer-Verlag Berlin Heidelberg

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van Emden, M.H. (1999). The Logic Programming Paradigm in Numerical Computation. In: Apt, K.R., Marek, V.W., Truszczynski, M., Warren, D.S. (eds) The Logic Programming Paradigm. Artificial Intelligence. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60085-2_11

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  • DOI: https://doi.org/10.1007/978-3-642-60085-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64249-4

  • Online ISBN: 978-3-642-60085-2

  • eBook Packages: Springer Book Archive

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