Abstract
This paper contains the first algorithm that can solve disjunctions of constraints of the form \(\exists{y}\!\in\! B \; [ f=0 \;\wedge\; g_1\geq 0\wedge\dots\wedge g_k\geq 0 ]\) in free variables x, terminating for all cases when this results in a numerically well-posed problem. Here the only assumption on the terms f, g 1,..., g n is the existence of a pruning function, as given by the usual constraint propagation algorithms or by interval evaluation. The paper discusses the application of an implementation of the resulting algorithm on problems from control engineering, parameter estimation, and computational geometry.
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
Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, p. 67. Springer, Heidelberg (2000)
Benhamou, F., McAllester, D., Hentenryck, P.V.: CLP(Intervals) Revisited. In: International Symposium on Logic Programming, Ithaca, NY, USA, pp. 124–138. MIT Press, Cambridge (1994)
Benhamou, F., Older, W.J.: Applying interval arithmetic to real, integer and Boolean constraints. Journal of Logic Programming 32(1), 1–24 (1997)
Bordeaux, L., Monfroy, E.: Beyond NP: Arc-consistency for quantified constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, p. 371. Springer, Heidelberg (2002)
Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Wien (1998)
Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Caviness and Johnson [5], pp. 134– 183
Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5, 29–35 (1988)
Davis, E.: Constraint propagation with interval labels. Artificial Intelligence 32(3), 281–331 (1987)
Gardeñes, E., Sainz, M.Á., Jorba, L., Calm, R., Estela, R., Mielgo, H., Trepat, A.: Modal intervals. Reliable Computing 7(2), 77–111 (2001)
Granvilliers, L.: On the combination of interval constraint solvers. Reliable Computing 7(6), 467–483 (2001)
Hickey, T.J.: smathlib, http://interval.sourceforge.net/interval/C/smathlib/README.html
Hickey, T.J., Ju, Q., van Emden, M.H.: Interval arithmetic: from principles to implementation. Journal of the ACM 48(5), 1038–1068 (2001)
Hong, H.: Improvements in CAD-based Quantifier Elimination. PhD thesis, The Ohio State University (1990)
Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, Berlin (2001)
Jaulin, L., Walter, E.: Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis. Mathematics and Computers in Simulation 35(2), 123–137 (1993)
Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36(5), 450–462 (1993)
Mayer, G.: Epsilon-inflation in verification algorithms. Journal of Computational and Applied Mathematics 60, 147–169 (1994)
Milanese, M., Vicino, A.: Estimation theory for nonlinear models and set membership uncertainty. Automatica (Journal of IFAC) 27(2), 403–408 (1991)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge (1990)
Nise, N.S.: Control Systems Engineering, 3rd edn. John Wiley & Sons, Chichester (2000)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations. Academic Press, London (1970)
Ratschan, S.: Applications of quantified constraint solving over the reals— bibliography (2001), http://www.mpi-sb.mpg.de/~ratschan/appqcs.html
Ratschan, S.: Continuous first-order constraint satisfaction. In: Calmet, J., Benhamou, B., Caprotti, O., Hénocque, L., Sorge, V. (eds.) AISC 2002 and Calculemus 2002. LNCS (LNAI), vol. 2385, pp. 181–195. Springer, Heidelberg (2002)
Ratschan, S.: Continuous first-order constraint satisfaction with equality and disequality constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 680–685. Springer, Heidelberg (2002)
Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers (2002), submitted for publication, http://www.mpi-sb.mpg.de/~ratschan/preprints.html
Ratschan, S.: Quantified constraints under perturbations. Journal of Symbolic Computation 33(4), 493–505 (2002)
Rump, S.M.: A note on epsilon-inflation. Reliable Computing 4, 371–375 (1998)
Shary, S.P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Computing 8, 321–418 (2002)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry. Univ. of California Press, Berkeley (1951); Also in [5]
Walsh, T.: Stochastic constraint programming. In: Proc. of ECAI (2002)
Walter, E., Pronzato, L.: Identification of Parametric Models from Experimental Data. Springer, Heidelberg (1997)
Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1–2), 3–27 (1988)
Yorke-Smith, N., Gervet, C.: On constraint problems with incomplete or erroneous data. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 732–737. Springer, Heidelberg (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ratschan, S. (2003). Solving Existentially Quantified Constraints with One Equality and Arbitrarily Many Inequalities. In: Rossi, F. (eds) Principles and Practice of Constraint Programming – CP 2003. CP 2003. Lecture Notes in Computer Science, vol 2833. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45193-8_42
Download citation
DOI: https://doi.org/10.1007/978-3-540-45193-8_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20202-8
Online ISBN: 978-3-540-45193-8
eBook Packages: Springer Book Archive