Abstract
We are concerned with interval constraints: solving constraints among real unknowns in such a way that soundness is not affected by rounding errors
The contraction operator for the constraint x +y = z can simply be expressed in terms of interval arithmetic. An attempt to use the analogous definition for x * y = z fails if the usual definitions of interval arithmetic are used. We propose an alternative to the interval arithmetic definition of interval division so that the two constraints can be handled in an analogous way. This leads to a unified treatment of both interval constraints and interval arithmetic that makes it easy to derive formulas for other constraint contraction operators
We present a theorem that justifies simulating interval arithmetic evaluation of complex expressions by means of constraint propagation. A naive implementation of this simulation is inefficient. We present a theorem that justifies what we call the totality optimization. It makes simulation of expression evaluation by means of constraint propagation as efficient as in interval arithmetic. It also speeds up the contraction operators for primitive constraints.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Götz Alefeld and Jürgen Herzberger. Introduction to Interval Computations. Academic Press, 1983.
F. Benhamou, D. McAllester, and P. Van Hentenryck. CLP(Intervals) revisited. In Logic Programming: Proc. 1994 International Symposium, pages 124–138, 1994.
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, and Eric Vétillard. Le manuel de Prolog IV. Technical report, PrologIA, Parc Technologique de Luminy, Marseille, France, 1996.
Fréd éeric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1–24, 1997.
BNR. BNR Prolog user guide and reference manual. 1988.
J.G. Cleary. Logical arithmetic. Future Computing Systems, 2:125–149, 1987.
E. Davis. Constraint propagation with labels. Artificial Intelligence, 32:281–331, 1987.
M. Dincbas, P. Van Hentenryck, H. Simonis, A. Aggoun, T. Graf, and F. Berthier. The constraint programming language CHIP. In Proc. Int. Conf. on Fifth Generation Computer Systems, 1988.
Eldon Hansen. Global Optimization Using Interval Analysis. Marcel Dekker, 1992.
Pascal Van Hentenryck, Laurent Michel, and Yves Deville. Numerica: A Modeling Language for Global Optimization. MIT Press, 1997.
T. Hickey and M. van Emden. Using the IEEE floating-point standard for implementing interval arithmetic. In preparation.
E. Hyvönen. Constraint reasoning based on interval arithmetic. In Proceedings of the Eleventh International Joint Conference on Artificial Intelligence, pages 1193–1198, Detroit, USA, 1989.
W.M. Kahan. A more complete interval arithmetic. Technical report, University of Toronto, Canada, 1968.
Seymour Lipschutz. General Topology. Schaum’s Outline Series, 1965.
Ramon E. Moore. Interval Analysis. Prentice-Hall, 1966.
Arnold Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, 1990.
M. H. van Emden. Canonical extensions as common basis for interval constraints and interval arithmetic. In Proceedings of the Sixth French Conference on Logic and Constraint Programming, Orléans, France, 1997.
M. H. van Emden. Value constraints in the CLP Scheme. Constraints, 2:163–183, 1997.
D. Waltz. Understanding line drawings in scenes with shadows. In Patrick Henry Winston, editor, The Psychology of Computer Vision, pages 19–91. McGraw-Hill, 1975.
Huan Wu. Defining and implementing a unified framework for interval constraints and interval arithmetic. Master’s thesis. In preparation.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hickey, T.J., van Emden, M.H., Wu, H. (1998). A Unified Framework for Interval Constraints and Interval Arithmetic. In: Maher, M., Puget, JF. (eds) Principles and Practice of Constraint Programming — CP98. CP 1998. Lecture Notes in Computer Science, vol 1520. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49481-2_19
Download citation
DOI: https://doi.org/10.1007/3-540-49481-2_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65224-3
Online ISBN: 978-3-540-49481-2
eBook Packages: Springer Book Archive