Reliable Computing

, Volume 8, Issue 1, pp 21–42

# Approximate Quantified Constraint Solving by Cylindrical Box Decomposition

• Stefan Ratschan
Article

## Abstract

This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols <, = and function symbols +, ×, are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantified constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.

## Keywords

Mathematical Modeling Real Number Approximation Method Computational Mathematic Industrial Mathematic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Anai, H. andWeispfenning, V.: Deciding Linear-Trigonometric Problems, in: Proc. ISAAC 2000, International Symposium on Symbolic and Algebraic Computation, ACM Press, 2000.Google Scholar
2. 2.
Arnon, D. S., Collins, G. E., and McCallum, S.: Cylindrical Algebraic Decomposition I: The Basic Algorithm, SIAM Journal of Computing 13 (4) (1984), pp. 865–877, also in [7].Google Scholar
3. 3.
Beltran, M., Castillo, G., and Kreinovich, V.: Algorithms That Still Produce a Solution (Maybe Not Optimal) Even When Interrupted: Shary's Idea Justified, Reliable Computing 4 (1) (1998), pp. 39–53.Google Scholar
4. 4.
Benhamou, F. and Goualard, F.: Universally Quantified Interval Constraints, in: Proceedings of the Sixth International Conference on Principles and Practice of Constraint Programming (CP'2000), Lecture Notes in Computer Science, Springer-Verlag, Singapore, 2000.Google Scholar
5. 5.
Bodnár, G., Pau, P., and Schicho, J.: Exact Real Computation in Computer Algebra, Technical Report 00–33, RISC-Linz, 2000.Google Scholar
6. 6.
Brown, Ch.: Simple CAD Construction and Its Applications, Journal of Symbolic Computation (2001), to appear.Google Scholar
7. 7.
Caviness, B. F. and Johnson, J. R. (eds): Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, Springer, 1998.Google Scholar
8. 8.
Collins, G. E.: Quantifier Elimination for the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition, in: Second GI Conf. Automata Theory and Formal Languages, Lecture Notes in Computer Science 33, Springer-Verlag, Berlin, 1975, pp. 134–183, also in [7].Google Scholar
9. 9.
Collins, G. E. and Hong, H.: Partial Cylindrical Algebraic Decomposition for Quantifier Elimination, Journal of Symbolic Computation 12 (1991), pp. 299–328, also in [7].Google Scholar
10. 10.
Collins, G. E. and Krandick, W.: A Hybrid Method for High Precision Calculation of Polynomial Real Roots, in: Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation ISSAC'93, ACM Press, Kiev, Ukraine, 1993, pp. 47–52.Google Scholar
11. 11.
Dorato, P.: Quantified Multivariate Polynomial Inequalities, IEEE Control Systems Magazine (2000), pp. 48–58.Google Scholar
12. 12.
Dorato, P., Yang, W., and Abdallah, C.: Robust Multi-Objective Feedback Design by Quantifier Elimination, Journal of Symbolic Computation 24 (1997), pp. 153–159.Google Scholar
13. 13.
Ebbinghaus, H.-D., Flum, J., and Thomas, W.: Mathematical Logic, Springer-Verlag, 1984.Google Scholar
14. 14.
Edelsbrunner, H.: A New Approach to Rectangle Intersections I, II, Internat. J. Comput. Math. 13 (3–4) (1983), pp. 209–219, 221–229.Google Scholar
15. 15.
Engl, H. W.: Regularization Methods for the Stable Solution of Inverse Problems, Surv. Math. Ind. 3 (1993), pp. 71–143.Google Scholar
16. 16.
Fiorio, G., Malan, S., Milanese, M., and Taragna, M.: Robust Performance Design of Fixed Structure Controllers with Uncertain Parameters, in: Proceedings of the 32nd IEEE Conf. Decision and Control, 1993.Google Scholar
17. 17.
Gardeñes, E., Mielgo, H., and Trepat, A.: Modal Intervals: Reason and Ground Semantics, in: Nickel, K. (ed.), Interval Mathematics 1985, Lecture Notes in Computer Science 212, Springer-Verlag, Berlin, Heidelberg, 1968, pp. 27–35.Google Scholar
18. 18.
Gardeñes, E., Sainz, M. Á., Jorba, L., Calm, R., Estela, R., Mielgo, H., and Trepat, A.: Modal Intervals, Reliable Computing 7 (2) (2001), pp. 77–111.Google Scholar
19. 19.
Garloff, J. and Graf, B.: Solving Strict Polynomial Inequalities by Bernstein Expansion, in: Munro, N. (ed.), The Use of Symbolic Methods in Control System Analysis and Design, The Institution of Electrical Engineers (IEE), London, 1999, pp. 339–352.Google Scholar
20. 20.
Gonzalez-Vega, L.: A Combinatorial Algorithm Solving Some Quantifier Elimination Problems, in [7].Google Scholar
21. 21.
22. 22.
Hammer, R., Hocks, M., Kulisch, U., and Ratz, D.: Nonlinear Equations in One Variable, in: Numerical Toolbox for Verified Computing I, Series in Computational Mathematics 21, Springer-Verlag, 1993.Google Scholar
23. 23.
Hong, H.: Heuristic Search and Pruning in Polynomial Constraints Satisfaction, Ann.Math. Artif. Intell. 19 (3–4) (1997), pp. 319–334.Google Scholar
24. 24.
Hong, H.: Heuristic Search Strategies for Cylindrical Algebraic Decomposition, in: Calmet, J. et al. (eds), Proceedings of Artificial Intelligence and Symbolic Mathematical Computing, Lecture Notes in Computer Science 737, Springer-Verlag, 1992, pp. 152–165.Google Scholar
25. 25.
Hong, H.: Improvements in CAD-Based Quantifier Elimination, PhD thesis, The Ohio State University, 1990.Google Scholar
26. 26.
Hong, H.: Quantifier Elimination for Formulas Constrained by Quadratic Equations via Slope Resultants, The Computer Journal 36 (5) (1993), pp. 440–449.Google Scholar
27. 27.
Hong, H.: Symbolic-Numeric Methods for Quantified Constraint Solving, in: International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN-95, 1995, invited talk.Google Scholar
28. 28.
Hong, H.: The Exact Region of Stability for MacCormack Scheme, Computing 56 (4) (1996).Google Scholar
29. 29.
Hong, H., Liska, R., and Steinberg, S.: Testing Stability by Quantifier Elimination, Journal of Symbolic Computation 24 (2) (1997), pp. 161–187.Google Scholar
30. 30.
Hong, H. and Neubacher, A.: Approximate Quantifier Elimination, in: Proceedings of IMACSACA' 96, 1996.Google Scholar
31. 31.
Hong, H., Neubacher, A., and Stahl, V.: The STURM Library Manual-A C++ Library for Symbolic Computation, Technical Report 94–30, RISC Linz, 1994.Google Scholar
32. 32.
Hong, H. and Stahl, V.: Safe Starting Regions by Fixed Points and Tightening, Computing 53 (1994), pp. 323–335.Google Scholar
33. 33.
Jaulin, L. and Walter, É.: Guaranteed Tuning, with Application to Robust Control and Motion Planning, Automatica 32 (8) (1996), pp. 1217–1221.Google Scholar
34. 34.
Jirstrand, M.: Cylindrical Algebraic Decomposition-An Introduction, Technical Report, Automatic Control Group, Linköping, 1985.Google Scholar
35. 35.
Jirstrand, M.: Nonlinear Control System Design by Quantifier Elimination, Journal of Symbolic Computation 24 (2) (1997), pp. 137–152.Google Scholar
36. 36.
Johnson, J. R.: Real Algebraic Number Computation Using Interval Arithmetic, in: International Conference on Symbolic and Algebraic Computation (ISSAC'92), ACM Press, Berkeley, CA, 1992, pp. 195–205.Google Scholar
37. 37.
Malan, S., Milanese, M., and Taragna, M.: Robust Analysis and Design of Control Systems Using Interval Arithmetic, Automatica 33 (7) (1997), pp. 1363–1372.Google Scholar
38. 38.
McCallum, S.: Solving Polynomial Strict Inequalities Using Cylindrical Algebraic Decomposition, The Computer Journal 36 (5) (1993), pp. 432–438.Google Scholar
39. 39.
Mishra, B.: Algorithmic Algebra, Springer-Verlag, 1993.Google Scholar
40. 40.
Moore, R. E.: Interval Analysis, Prentice Hall, Englewood Cliffs, 1966.Google Scholar
41. 41.
Moore, R. E.: Parameter Sets for Bounded-Error Data, Mathematics and Computer in Simulation 34 (1992), pp. 113–119.Google Scholar
42. 42.
Neubacher, A.: Parametric Robust Stability by Quantifier Elimination, PhD thesis, Research Institute for Symbolic Computation-Universität Linz, 1997.Google Scholar
43. 43.
44. 44.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
45. 45.
Pau, P. and Schicho, J.: Quantifier Elimination for Trigonometric Polynomials by Cylindrical Trigonometric Decomposition, Journal of Symbolic Computation 29 (6) (2000).Google Scholar
46. 46.
Preparata, F. P. and Shamos, M. I.: Computational Geometry: An Introduction, Springer-Verlag, 1985.Google Scholar
47. 47.
Ratschan, S.: Approximate Quantified Constraint Solving (AQCS), 2000, software package http://www.risc.uni-linz.ac.at/research/software/AQCS.Google Scholar
48. 48.
Ratschan, S.: Convergence of Quantified Constraint Solving by Approximate Quantifiers, Technical Report 00–23, Research Institute for Symbolic Computation (RISC)-Linz, 2000, submitted.Google Scholar
49. 49.
Ratschan, S.: Uncertainty Propagation in Heterogeneous Algebras for Approximate Quantified Constraint Solving, Journal of Universal Computer Science 6 (9) (2000).Google Scholar
50. 50.
Richardson, D.: Some Undecidable Problems Involving Elementary Functions of a Real Variable, Journal of Symbolic Logic 33 (1968), pp. 514–520.Google Scholar
51. 51.
Shary, S. P.: Algebraic Approach to the Interval Linear Static Identification, Tolerance, and Control Problems, or One More Application of Kaucher Arithmetic, Reliable Computing 2 (1) (1996), pp. 3–33.Google Scholar
52. 52.
Shary, S. P.: On Optimal Solution of Interval Linear Equations, SIAM Journal on Numerical Analysis 32 (1995), pp. 610–630.Google Scholar
53. 53.
Shary, S. P.: Outer Estimation of Generalized Solution Sets to Interval Linear Systems, Reliable Computing 5 (3) (1999), pp. 323–335.Google Scholar
54. 54.
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, Univ. of California Press, Berkeley, 1951, also in [7].Google Scholar
55. 55.
Weihrauch, K.: Introduction to Computable Analysis, Texts in Theoretical Computer Science, Springer-Verlag, 2000.Google Scholar
56. 56.
Weispfenning, V.: The Complexity of Linear Problems in Fields, Journal of Symbolic Computation 5 (1–2) (1988), pp. 3–27.Google Scholar
57. 57.
Zilberstein, Sh. and Russell, S. J.: Approximate Reasoning Using Anytime Algorithms, in: Natarajan, S. (ed.), Imprecise and Approximate Computation, Kluwer Academic Publishers, 1995.Google Scholar