Towards mechanical solution of the Kahan ellipse problem I
Tarski (1948) gave a quantifier elimination procedure for elementary algebra and geometry which, when implemented, was found to be impractical. Collins (1975) gave a different and far more efficient algorithm, which has revived interest in the feasibility of quantifier elimination as a means of solving nontrivial problems in elementary algebra and geometry. Collins' algorithm has recently been implemented (1981). Experience so far obtained with the software indicates that, while some worthwhile problems can now be solved, it is desirable to find methods for improving performance. In this paper we report on several such methods, that we have developed in the course of applying the Collins algorithm to the Kahan ellipse problem. These methods have yielded substantial performance improvements; in fact, they have made feasible computations relating to the Kahan problem that initially could not be carried out at all in our computing environment. We are pursuing the further development of these techniques, with the goal of eventually producing a solution to the Kahan problem using the Collins algorithm.
Unable to display preview. Download preview PDF.
- Arn81a.DS Arnon, “Algorithms for the geometry of semi-algebraic sets,” Technical Report #436, Computer Science Dept., University of Wisconsin, Madison, Wisconsin(1981). (Ph.D. thesis)Google Scholar
- Arn82a.DS Arnon, GE Collins, and S McCallum, “Cylindrical algebraic decomposition I: the basic algorithm,” Technical Report CSD TR-427, Computer Science Dept., Purdue University(December, 1982).Google Scholar
- Arn82b.DS Arnon and S McCallum, “Cylindrical algebraic decomposition by quantifier elimination,” pp. 215–222 in Proceedings of the European Computer Algebra Conference (EUROCAM '82), ed. J Calmet, Lecture Notes in Computer Science, 144, Springer-Verlag, Berlin(1982).Google Scholar
- Col75a.GE Collins, “Quantifier elimination for real closed fields by cylindrical algebraic decomposition,” pp. 134–163 in Proceedings of the Second GI Conference on Automata and Formal Languages, Lecture notes in Computer Science, 33, Springer-Verlag, Berlin(1975).Google Scholar
- Col80a.GE Collins, “SAC-2 and ALDES now available,” SIGSAM Bulletin (of the Association for Computing Machinery) 14, p. 19 (1980).Google Scholar
- Kah75a.W Kahan, “Problem #9: An ellipse problem,” SIGSAM Bulletin of the Assoc. Comp. Mach. 9, p. 11 (1975).Google Scholar
- Kre67a.G Kreisel and JL Krivine, Elements of mathematical logic (model theory), North-Holland, Amsterdam(1967).Google Scholar
- Lau77a.M Lauer, “A solution to Kahan's problem (SIGSAM problem no. 9),” SIGSAM Bulletin of the Assoc. Comp. Mach. 11, pp. 16–20 (1977).Google Scholar
- Tar51a.A Tarski, A decision method for elementary algebra and geometry, University of California Press(1951). (second revised edition)Google Scholar