Automated Deduction in Geometry: 9th International Workshop, by Michael Beeson (auth.), Tetsuo Ida, Jacques Fleuriot (eds.)

By Michael Beeson (auth.), Tetsuo Ida, Jacques Fleuriot (eds.)

This ebook constitutes the completely refereed post-workshop lawsuits of the ninth foreign Workshop on automatic Deduction in Geometry, ADG 2012, held in Edinburgh, united kingdom, in September 2012. the ten revised complete papers offered including 2 invited papers have been rigorously chosen in the course of rounds of reviewing and development from the lectures given on the workshop. The convention represents a discussion board to replace rules and perspectives, to give study effects and growth, and to illustrate software program instruments on the intersection among geometry and automatic deduction; the scope of the ADG 2012 additionally has been improved to hide issues in dynamic geometry.

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Additional resources for Automated Deduction in Geometry: 9th International Workshop, ADG 2012, Edinburgh, UK, September 17-19, 2012. Revised Selected Papers

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To put this method into practice, we need to answer two questions: Exactly how will we convert from geometry to algebra, and exactly how will we make the required computations? Among the ways to convert geometry to algebra, we mention the ordinary introduction of coordinates, and Wu’s method [35], and Chou’s area method [8]. Among the ways to compute, we mention Gr¨obner bases and the Collins CAD algorithm [7,6]. While theoretically, any geometry problem can be solved by CAD, since it is a decision procedure for real-closed fields, in practice, it breaks down on problems with five or six (number) variables, so a geometry problem with four points is likely to be intractable, and geometry problems with fewer than four points are rare.

It turned out that using Otter was not as efficient in finding formal proofs as we had hoped, many human hours were also required. Our hope that every theorem in [29] would be a single, easy run with Otter turned out not to be justified; while that was true of the simpler theorems, every theorem complex enough to require a diagram required several runs, case distinctions made by hand, points defined, and the use of resonators made from lemmas or partial results. As mentioned above, Coq does not produce first-order proofs, and it is probably not easy to extract them from Coq proofs.

2 Computation for n = 3 We computed the Gr¨ obner basis for the ideal {fi , gi , hi } using Maple 14, applying “lexdeg” group ordering in the Maple Gr¨ obner package so that the variables were eliminated in the appropriate order. We note that Equations (1) and (2) expressing the Descartes circle theorem were independently computed. Hence, we consider the following three polynomial ideals using each monomial ordering: (i) JR = (f1 , f2 , f3 , h1 , h2 , h3 ) , [x1 , x2 , x3 , y1 , y2 , y3 ] [R, r1 , r2 , r3 ] The Gr¨ obner basis in this case contains a polynomial ϕ3 (R, r1 , r2 , r3 ) that appears in Equation (8).

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