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Efficient Interval Linear Equality Solving in Constraint Logic Programming

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Abstract

Existing interval constraint logic programming languages, such as BNR Prolog, work under the framework of interval narrowing and are deficient in solving systems of linear constraints over real numbers, which constitute an important class of problems in engineering and other applications. In this paper, we suggest to separate linear equality constraint solving from inequality and non-linear constraint solving. The implementation of an efficient interval linear constraint solver, which is based on the preconditioned interval Gauss-Seidel method, is proposed. We show how the solver can be adapted to incremental execution and incorporated into a constraint logic programming language already equipped with a non-linear solver based on interval narrowing. The two solvers share common interval variables, interact and cooperate in a round-robin fashion during computation, resulting in an efficient interval constraint arithmetic language CIAL. The CIAL prototypes, based on CLP(R), are constructed and compared favorably against several major interval constraint logic programming languages.

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References

  1. Aggoun, A. and Beldiceanu, N.: Overview of the CHIP Compiler System, in: Proceedings of the Eighth International Conference on Logic Programming, Paris, France, 1991, pp. 775–789.

  2. Aiba, A., Sakai, K., Sato, Y., Hawley, D. J., and Hasegawa, R.: Constraint Logic Programming Language CAL, in: Proceedings of the International Conference on Fifth Generation Computer Systems 1988, Tokyo, Japan, 1988, pp. 263–276.

  3. Aït-Kaci, H.: Warren's Abstract Machine: A Tutorial Reconstruction, The MIT Press, 1991.

  4. Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, 1983.

  5. Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., and Van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial & Applied Mathematics, 1993.

  6. Bell-Northern Research Ltd.: BNR Prolog Reference Manual, 1988.

  7. Benhamou, F., McAllester, D., and Van Hentenryck, P.: CLP(Intervals) revisited, in: Logic Programming: Proceedings of the 1994 International Symposium, Ithaca, USA, 1994, pp. 124–138.

  8. Benhamou, F. and Older, W. J.: Applying Interval Arithmetic to Real, Integer and Boolean Constraints, Journal of Logic Programming 32 (1997), pp. 1–24.

    Google Scholar 

  9. Bessiere, C.: Arc-Consistency and Arc-Consistency Again, AI Journal 65 (1) (1994), pp. 179–190.

    Google Scholar 

  10. Buchberger, B.: Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, in: Bose, N. K. (ed.), Recent Trends in Multidimensional Systems Theory, chapter 6, D. Riedel Publ. Comp., 1983.

  11. Buchberger, B. and Hong, H.: Speeding-up Quantifier Elimination by Gröbner Bases, Technical Report 91–06.0, Research Institute for Symbolic Computation, Johannes Kepler University, A-4040 Linz, 1991.

  12. Chiu, C. K.: Interval Linear Constraint Solving in Constraint Logic Programming, Master's thesis, Department of Computer Science, The Chinese University of Hong Kong, Shatin, Hong Kong, 1994.

    Google Scholar 

  13. Chiu, C. K. and Lee, J. H. M.: Interval Linear Constraint Solving Using the Preconditioned Interval Gauss-Seidel Method, in: Proceedings of the 12th International Conference on Logic Programming, Kanagawa, Japan, 1995, pp. 17–31.

  14. Chiu, C. K. and Lee, J. H. M.: Towards Practical Interval Constraint Solving in Logic Programming, in: Logic Programming: Proceedings of the 1994 International Symposium, Ithaca, USA, 1994, pp. 109–123.

  15. Cleary, J. G.: Logical Arithmetic, Future Computing Systems 2 (2) (1987), pp. 125–149.

    Google Scholar 

  16. Colmerauer, A.: An Introduction to Prolog III, Communications of the ACM 33 (7) (1990), pp. 69–90.

    Google Scholar 

  17. Cormen, T. H., Leiserson, C. E., and Rivest, R. L.: Introduction to Algorithms, The MIT Press, eighth edition, 1992.

  18. Davis, E.: Constraint Propagation with Interval Labels, Artificial Intelligence 32 (1987), pp. 281–331.

    Google Scholar 

  19. Dincbas, M., Van Hentenryck, P., Simonis, H., Aggoun, A., Graf, T., and Berthier, F.: The Constraint Logic Programming Language CHIP, in: Proceedings of the International Conference on Fifth Generation Computer Systems, Tokyo, Japan, 1988, pp. 693–702.

  20. Gay, D. M.: Solving Interval Linear Equations, SIAM Journal on Numerical Analysis 19 (4) (1982), pp. 858–870.

    Google Scholar 

  21. Hansen, E. R.: Bounding the Solution of Interval Linear Equations, SIAM Journal on Numerical Analysis 29 (5) (1992), pp. 1493–1503.

    Google Scholar 

  22. Hansen, E. R.: Global Optimization Using Interval Analysis, Marcel Dekker, 1992.

  23. Hansen, E. R.: Interval Arithmetic in Matrix Computations, SIAM Journal on Numerical Analysis 2 (1965), pp. 308–320.

    Google Scholar 

  24. Havens, W. S., Sidebottom, S., Jones, J., Cuperman, M., and Davison, R.: Echidna Constraint Reasoning System: Next-Generation Expert System Technology, Technical Report CSS-IS TR 90–09, Centre for Systems Science, Simon Fraser University, Burnaby, 1990.

    Google Scholar 

  25. Heintze, N. C., Jaffar, J., Michaylov, S., Stuckey, P. J., and Yap, R. H. C.: The CLP(R) Programmer's Manual Version 1.2, IBM Thomas J Watson Research Center, 1992.

  26. Heintze, N., Michaylov, S., and Stuckey, P.: CLP(R) and Some Electrical Engineering Problems, Journal of Automated Reasoning 9 (2) (1992), pp. 231–260.

    Google Scholar 

  27. Hong, H.: Improvements in CAD-Based Quantifier Elimination, PhD thesis, The Ohio State University, 1990.

  28. Hong, H.: Non-Linear Real Constraints in Constraint Logic Programming, in: Proceedings of the Second International Conference on Algebraic and Logic Programming, pp. 201–212, 1992.

  29. Jaffar, J. and Lassez, J.-L.: Constraint Logic Programming, in: Proceedings of the 14th ACM POPL Conference, Munich, 1987, pp. 111–119.

  30. Jaffar, J., Michaylov, S., Stuckey, P. J., and Yap, R. H. C.: The CLP(R) Language and System, in: ACM Transactions on Programming Languages and Systems, volume 14, 1992, pp. 339–395.

    Google Scholar 

  31. Journal of Logic Programming Referee C, Personal communication, 1995.

  32. Kearfott, R. B.: Rigorous Global Search, Continuous Problems, Kluwer Academic Publishers, 1996.

  33. Lee, J. H. M.: Numerical Computation as Deduction in Constraint Logic Programming, PhD thesis, Department of Computer Science, Logic Programming Laboratory, University of Victoria, Victoria, 1992.

    Google Scholar 

  34. Lee, J. H. M. and Lee, T. W.: A WAM-Based Abstract Machine for Interval Constraint Logic Programming, in: Proceedings of the Sixth IEEE International Conference on Tools with Artificial Intelligence, New Orleans, 1994, pp. 122–128.

  35. Lee, J. H. M. and Van Emden, M. H.: Adapting CLP(R) to Floating-Point Arithmetic, in: Proceedings of the International Conference on Fifth Generation Computer Systems 1992, volume 16, Tokyo, 1992, pp. 996–1003.

    Google Scholar 

  36. Lee, J. H. M. and Van Emden, M. H.: Interval Computation as Deduction in CHIP, Journal of Logic Programming 16 (1993), pp. 255–276.

    Google Scholar 

  37. Lhomme, O.: Consistency Techniques for Numeric CSPs, in: Proceedings of the 13th International Joint Conference on Artificial Intelligence, 1993.

  38. Lloyd, J. W.: Foundations of Logic Programming, Springer-Verlag, second edition, 1987.

  39. Mackworth, A. K.: Consistency in Networks of Relations, AI Journal 8 (1) (1977), pp. 99–118.

    Google Scholar 

  40. Mackworth, A. K., Mulder, J. A., and Havens, W. S.: Hierarchical arc Consistency: Exploiting Structured Domains in Constraint Satisfaction Problems, Computational Intelligence 1 (1985), pp. 118–126.

    Google Scholar 

  41. Michaylov, S.: Design and Implementation of Practical Constraint Logic Programming Systems, PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1992.

    Google Scholar 

  42. Moore, R. E.: Interval Analysis, Prentice Hall, Englewood Cliffs, 1966.

    Google Scholar 

  43. Moore, R. E.: Methods and Applications of Interval Analysis, SIAM, 1979.

  44. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.

  45. Neumaier, A.: Overestimation in Linear Interval Equations, SIAM Journal on Numerical Analysis 24 (1) (1987), pp. 207–214.

    Google Scholar 

  46. Older, W.: Constraints in BNR Prolog, Technical Report Draft 01, Software Engineering Centre, Bell-Northern Research, Ottawa, 1993.

    Google Scholar 

  47. Older, W.: The Application of Relational Arithmetic to X-Ray Diffraction Crystallography, Technical Report 89001, Software Engineering Centre, Bell-Northern Research, Ottawa, 1989.

    Google Scholar 

  48. Older, W. and Vellino, A.: Constraint Arithmetic on Real Intervals, in: Colmerauer, A. and Benhamou, F. (eds), Constraint Logic Programming: Selected Research, MIT Press, 1992, pp. 175–196.

  49. Older, W. and Vellino, A.: Extending Prolog with Constraint Arithmetic on Real Intervals, in: Proceedings of the Canadian Conference on Computer & Electrical Engineering, Ottawa, 1990.

  50. Sidebottom, G. A.: A Language for Optimizing Constraint Propagation, PhD thesis, School of Computer Science, Simon Fraser University, 1993.

  51. Sidebottom, G. and Havens, W. S.: Hierarchical arc Consistency for Disjoint Real Intervals in Constraint Logic Programming, Computational Intelligence 8 (4) (1992), pp. 601–623.

    Google Scholar 

  52. Sidebottom, S., Havens, W., and Kindersley, S.: Echidna Constraint Reasoning System (Version 1): Programming Manual, Expert Systems Laboratory, Simon Fraser University, 2.0 edition, 1992.

  53. Sterling, L. and Shapiro, E.: The Art of Prolog, The MIT Press, second edition, 1994.

  54. Van Hentenryck, P.: A General Introduction to Numerica, Artificial Intelligence 103 (1–2) (1998), pp. 209–235.

    Google Scholar 

  55. Van Hentenryck, P.: Constraint Satisfaction in Logic Programming, The MIT Press, London, 1989.

    Google Scholar 

  56. Van Hentenryck, P., McAllester, D., and Kapur, D.: Solving Polynomial Systems Using a Branch and Prune Approach, SIAM Journal on Numerical Analysis 34 (2) (1997).

  57. Van Hentenryck, P., Michel, L., and Kapur, D.: Numerica: A Modeling Language for Global Optimization, The MIT Press, 1997.

  58. Vasey, P.: Qualified Answers and Their Application to Transformation, in: Proceedings of the Third International Logic Programming Conference, 1986, pp. 425–432.

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  1. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, P.R. China

    Chong-Kan Chiu & Jimmy Ho-Man Lee

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  1. Chong-Kan Chiu
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  2. Jimmy Ho-Man Lee
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Chiu, CK., Lee, J.HM. Efficient Interval Linear Equality Solving in Constraint Logic Programming. Reliable Computing 8, 139–174 (2002). https://doi.org/10.1023/A:1014754106275

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