an:03881880
Zbl 0553.68031
Arnon, Dennis S.; Smith, Scott F.
Towards mechanical solution of the Kahan ellipse problem. I
EN
Computer algebra, EUROCAL '83, Proc. Conf., London 1983, Lect. Notes Comput. Sci. 162, 36-44 (1983).
1983
a
68W30 68T15 03C10 03B35 12L12 12D15
Collins decision algorithm; real closed fields; Kahan ellipse problem
[For the entire collection see Zbl 0532.00010.]
The authors present means for a speed-up of the Collins decision algorithm for the theory of the real closed fields [\textit{G. E. Collins}, Lect. Notes Comput. Sci. 33, 134-183 (1975; Zbl 0318.02051)]. The Collins algorithm may also be applied to solve the Kahan ellipse problem, i.e., to find a, b, c, d for which \((\forall x)(\forall y)((x-c)^ 2/a^ 2+(y-d)^ 2/b^ 2=1\to y^ 2+x^ 2<1)\) holds, but proved to be inefficient in this case. Thus, the authors give a survey of the Collins method and discuss the necessity for a preprocessing of the formulas. Such a preprocessing may consist in the application of a simple rule in predicate logic, namely: \((\forall u)(\forall v)(F(u)=G(v)\to \phi (u,G(v)))\) implies \((\forall u)((\exists v)F(u)=G(v)\to \phi (u,F(u))).\) It is shown, that for the case \(d=0\) the Collins algorithm, applied to Kahan's ellipse problem, may be improved by this rule. Finally the authors propose a priori restrictions on free variables as a mean of improved by this rule. Finally the authors propose a priori restrictions on free variables as a mean of improvement.
A.Leitsch
Zbl 0532.00010; Zbl 0318.02051