Abstract
In this paper, we report how certain AI techniques can be used to speed up an algebraic algorithm for deciding the satisfiability of a system of polynomial equations, dis-equations, and inequalities. We begin by viewing the algebraic algorithm (Cylindrical Algebraic Decomposition) as a search procedure which searches for a solution of the given system. When viewed in this way, the algebraic algorithm is non-deterministic, in the sense that it can often achieve the same goal, but following different search paths requiring different amounts of computing times. Obviously one wishes to follow the least time-consuming path. However, in practice it is not possible to determine such an optimal path. Thus it naturally renders itself to the heuristic search techniques of AI. In particular we experimented with Best-First strategy. We also study a way to prune the search space, which proves to be useful especially when the given polynomial system is not satisfiable. The experimental results indicate that such AI techniques can often help in speeding up the algebraic method, sometimes dramatically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.S. Arnon, Algorithms for the geometry of semi-algehraic sets, Ph.D. Thesis, Technical Report 436, Computer Sciences Dept., Univ. of Wisconsin-Madison (1981).
D.S. Arnon, A bibliography of quantifier elimination for real closed fields, Journal of Symbolic Computation 5(1,2) (1988) 267–274.
D.S. Arnon, A cluster-based cylindrical algebraic decomposition algorithm, Journal of Symbolic Computation 5(1,2) (1988) 189–212.
M. Ben-Or, D. Kozen and J.H. Reif, The complexity of elementary algebra and geometry, J. Comput. System Sci. 32(2) (1986) 251–264.
B. Buchberger and H. Hong, Speeding-up quantifier elimination by Groebner bases, Technical Report 91–06.0, Research Institute for Symbolic Computation, Johannes Kepler University A-4040 Linz, Austria (1991).
J. Canny, Some algebraic and geometric computations in PSPACE, in: Proceedings of the 20th Annual ACM Symposium on the Theory of Computing (1988) pp. 460–467.
G.E. Collins, Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, in: Lecture Notes in Computer Science, Vol. 33 (Springer-Verlag, Berlin, 1975) pp. 134–183.
G.E. Collins and H. Hong, Partial cylindrical algebraic decomposition for quantifier elimination, Journal of Symbolic Computation 12(3) (September 1991) 299–328.
G.E. Collins and R. Loos, The SAC-2 Computer Algebra System (Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria A-4040).
N. Fitchas, A. Galligo and J. Morgenstern, Algorithmes repides en séquential et en parallele pour l' élimination de quantificateurs en géométrie élémentaire, Technical Report, UER de Mathématiques Universite de Paris VII (1987). To appear in: Séminaire Structures Algébriques Ordonnées.
D.Yu. Grigor'ev and N.N. Vorobjov, Jr., Solving systems of polynomial inequalities in subexponential time, Journal of Symbolic Computation 5(1,2) (1988) 37–64.
J. Heintz, M.-F. Roy and P. Solernó, On the complexity of semialgebraic sets, in: Proc. IFIP (1989) pp. 293–298.
H. Hong, An improvement of the projection operator in cylindrical algebraic decomposition, in: International Symposium of Symbolic and Algebraic Computation ISSAC-90 (1990) pp. 261–264.
H. Hong, Improvements in CAD-based quantifier elimination, Ph.D. Thesis, The Ohio State University (1990).
H. Hong, Comparison of several decision algorithms for the existential theory of the reals, Technical Report 91–41.0, Research Institute for Symbolic Computation, Johannes Kepler University A-4040 Linz, Austria (1991).
H. Hong, Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination, in: International Conference on Symbolic and Algebraic Computation ISSAC-92 (1992) pp. 177–188.
H. Hong, Parallelization of quantifier elimination on a workstation network, in: Proceedings of AAECC 10 (Puerto Rico) (1993).
H. Hong, Quantifier elimination for formulas constrained by quadratic equations, in: International Conference on Symbolic and Algebraic Computation ISSAC-93, Kiev (ACM, July 1993).
H. Hong, Quantifier elimination for formulas constrained by quadratic equations via slope resultants, The Computer Journal 36(5) (1993).
L. Langemyr, The cylindrical algebraic decomposition algorithm and multiple algebraic extensions, in: Proc. 9th IMA Conference on the Mathematics of Surfaces (September 1990).
D. Lazard, An improved projection for cylindrical algebraic decomposition. Unpublished manuscript (1990).
R.G.K. Loos, The algorithm description language ALDES (Report), ACM SIGSAM Bull. 10(1) (1976) 15–39.
S. McCallum, An improved projection operator for cylindrical algebraic decomposition, Ph.D. Thesis, University of Wisconsin-Madison (1984).
S. McCallum, Solving polynomial strict inequalities using cylindrical algebraic decomposition, Technical Report 87–25.0, RISC-LINZ, Johannes Kepler University, A-4040 Linz, Austria (1987).
J. Pearl, Heuristics (Intelligent Search Strategies for Computer Problem Solving) (Addison-Wesley, 1984).
J. Renegar, On the computational complexity and geometry of the first-order theory of the reals (part I), Technical Report 853, Cornell University, Ithaca, New York 14853–7501 USA (July 1989).
A. Tarski, A Decision Method for Elementary Algebra and Geometry (Univ. of California Press, Berkeley, 2nd edn., 1951).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hong, H. Heuristic search and pruning in polynomial constraints satisfaction. Annals of Mathematics and Artificial Intelligence 19, 319–334 (1997). https://doi.org/10.1023/A:1018911907086
Issue Date:
DOI: https://doi.org/10.1023/A:1018911907086