Skip to main content
Springer Nature Link
Log in

Interval Constraints: Results and Perspectives

  • Conference paper
  • First Online:
New Trends in Constraints (WC 1999)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1865))

Included in the following conference series:

Abstract

Reliably solving non-linear real constraints on computer is a challenging task due to the approximation induced by the resort to floating-point numbers. Interval constraints have consequently gained some interest from the scientific community since they are at the heart of complete algorithms that permit enclosing all solutions with an arbitrary accuracy. Yet, soundness is beyond reach of present-day interval constraint-based solvers, while it is sometimes a strong requirement. What is more, many applications involve constraint systems with some quantified variables these solvers are unable to handle. Basic facts on interval constraints and local consistency algorithms are first surveyed in this paper; then, symbolic and numerical methods used to compute inner approximations of real relations and to solve constraints with quantified variables are briefly presented, and directions for extending interval constraint techniques to solve these problems are pointed out.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. S. Abrams and P. K. Allen. Computing camera viewpoints in an active robot work-cell. Technical Report IBM Research Report: RC 20987, IBM Research Division, 1997.

    Google Scholar 

  2. B. D. O. Anderson, N. K. Bose, and E. I. Jury. Output feedback stabilization and related problems —solution via decision methods. IEEE Transactions on Automatic Control, AC-20(1), 1975.

    Google Scholar 

  3. J. Armengol, L. Travé-Massuyés, J. Lluís de la Rosa, and J. Vehí. Envelope generation for interval systems. In M. Toro, editor, Actas del seminario sobre técnicas cualitativas. VII Congreso de la Asociación Española para la Inteligencia Artificial (CAEPIA 97), pages 33–48, Málaga, 1997.

    Google Scholar 

  4. F. Benhamou, F. Goualard, L. Granvilliers, and J.-F. Puget. Revising hull and box consistency. In Proceedings of the sixteenth International Conference on Logic Programming (ICLP’99), pages 230–244, Las Cruces, USA, 1999. The MIT Press. ISBN 0-262-54104-1.

    Google Scholar 

  5. F. Benhamou, F. Goualard, É. Languénou, and M. Christie. An algorithm to com-pute inner approximations for interval constraints. In Proceeding of the Andrei Ershov Third International Conference on “Perspectives of System Informatics” (PSI’99), Lecture Notes in Computer Science, Novosibirsk, Akademgorodok, Russia, 1999. Springer-Verlag.

    Google Scholar 

  6. F. Benhamou and L. Granvilliers. Automatic Generation of Numerical Redundan-cies for Non-Linear Constraint Solving. Reliable Computing, 3(3):335–344, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Benhamou, D. McAllester, and P. Van Hentenryck. CLP(Intervals) Revisited. In Proceedings of International Symposium on Logic Programming (ILPS’94), pages 1–21, Ithaca, NY, USA, 1994. MIT Press.

    Google Scholar 

  8. 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 

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

    Google Scholar 

  10. H. Collavizza, F. Delobel, and M. Rueher. Comparing partial consistencies. Reliable Computing, 5:1–16, 1999.

    Article  Google Scholar 

  11. G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proceedings of the Second GI Conf. Automata Theory and Formal Languages, volume 33 of Lecture Notes in Computer Science, pages 134–183, Kaiserslauten, 1975. Springer.

    Google Scholar 

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

    Article  Google Scholar 

  13. S. M. Drucker. Intelligent Camera Control for Graphical Environments. PhD thesis, PhD Thesis MIT Media Lab, 1994.

    Google Scholar 

  14. J. Garloff and B. Graf. Solving strict polynomial inequalities by bernstein ex-pansion. In N. Munro, editor, The Use of Symbolic Methods in Control System Analysis and Design, pages 339–352. The Institution of Electrical Engineers, London, England, 1999.

    Google Scholar 

  15. IEEE. IEEE Standard for Binary Floating-Point Arithmetic. Technical Report IEEE Std 754-1985, Institute of Electrical and Electronics Engineers, 1985. Reaffirmed 1990.

    Google Scholar 

  16. J. Jaffar and J.-L. Lassez. Constraint Logic Programming. In Proceedings 14th ACM Symposium on Principles of Programming Languages (POPL’87), pages 111–119, Munich, Germany, 1987. ACM.

    Google Scholar 

  17. E. W. Kaucher. Interval analysis in the extended interval space \( \mathbb{I}\mathbb{R} \). Computing. Supplementum, 2:33–49, 1980.

    MathSciNet  Google Scholar 

  18. T. Kutsia and J. Schicho. Numerical solving of constraints of multivariate polynomial strict inequalities. Manuscript, October 1999.

    Google Scholar 

  19. O. Lhomme. Consistency techniques for numeric CSPs. In R. Bajcsy, editor, Proceedings of International Joint Conference on Artificial Intelligence (IJCAI’93), pages 232–238, Chambéry, France, 1993. IEEE Computer Society Press.

    Google Scholar 

  20. A. Mackworth. Consistency in Networks of Relations. Artificial Intelligence, 8(1):99–118, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Markov. On directed interval arithmetic and its applications. Journal of Universal Computer Science, 1(7):514–526, 1995.

    MATH  MathSciNet  Google Scholar 

  22. U. Montanari. Networks of Constraints: Fundamental Properties and Applications to Picture Processing. Information Science, 7(2):95–132, 1974.

    Article  MathSciNet  Google Scholar 

  23. R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1966.

    MATH  Google Scholar 

  24. E. H. Garde nes and H. Mielgo. Modal intervals: reasons and ground semantics. In Interval Mathematics, number 212 in Lecture Notes in Computer Science. Springer-Verlag, 1986.

    Google Scholar 

  25. W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

    Google Scholar 

  26. P. Pau and J. Schicho. Quantifier elimination for trigonometric polynomials by cylindrical trigonometric decomposition. Technical report, Research Institute for Symbolic Computation, March 1999.

    Google Scholar 

  27. Prolog IA. Prolog IV: Reference manual and User’s guide. Technical report, 1994.

    Google Scholar 

  28. J.-F. Puget and P. Van Hentenryck. A constraint satisfaction approach to a circuit design problem. Journal of Global Optimization, 13:75–93, 1998.

    Article  MATH  Google Scholar 

  29. H. Ratschek. Centered forms. SIAM Journal on Numerical Analysis, 17(5):656–662, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Sam. Constraint Consistency Techniques for Continuous Domains. Phd. thesis, École polytechnique fédérale de Lausanne, 1995.

    Google Scholar 

  31. J. Sam-Haroud and B. V. Faltings. Consistency techniques for continuous con-straints. Constraints, 1:85–118, 1996.

    Article  MathSciNet  Google Scholar 

  32. S. P. Shary. Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of Kaucher arithmetic. Reliable Computing, 2(1):3–33, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  33. K. Tarabanis. Automated sensor planning and modeling for robotic vision tasks. Technical Report CUCS-045-90, University of Columbia, 1990.

    Google Scholar 

  34. M. H. VanEmden. Canonical extensions as common basis for interval constraints and interval arithmetic. In Proceedings of French Conference on Logic Programming and Constraint Programming (JFPLC’97), pages 71–83. HERMES, 1997.

    Google Scholar 

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

    Google Scholar 

  36. P. Van Hentenryck, L. Michel, and Y. Deville. Numerica: A Modeling Language for Global Optimization. MIT Press, 1997.

    Google Scholar 

  37. A. C. Ward, T. Lozano-Pérez, and W. P. Seering. Extending the constraint propagation of intervals. In Proceedings of IJCAI’89, pages 1453–1458, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRIN, Université de Nantes, B.P. 92208, F-44322, Nantes Cedex 3

    Frédéric Benhamou, Laurent Granvilliers & Frédéric Goualard

Authors
  1. Frédéric Benhamou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Granvilliers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frédéric Goualard
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. CWI, Kruislaan 413, 1090, GB Amsterdam, The Netherlands

    Krzysztof R. Apt  & Eric Monfroy  & 

  2. Department of Computer Science, University of Cyprus, 75 Kallipoleos Str., 1678, Nicosia, Cyprus

    Antonis C. Kakas

  3. Department of Pure and Applied Mathematics, University of Padua, Via G. B. Belzoni 7, 35131, Padua, Italy

    Francesca Rossi

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Benhamou, F., Granvilliers, L., Goualard, F. (2000). Interval Constraints: Results and Perspectives. In: Apt, K.R., Monfroy, E., Kakas, A.C., Rossi, F. (eds) New Trends in Constraints. WC 1999. Lecture Notes in Computer Science(), vol 1865. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44654-0_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-44654-0_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67885-4

  • Online ISBN: 978-3-540-44654-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics