AUTOMATED THEOREM PROVING IN PROJECTIVE GEOMETRY WITH BRACKET ALGEBRA

Abstract

We present a method which can produce readable proofs for theorems of constructive type involving points, lines and conics in projective geometry. The method extends Wu's method to bracket algebra and develops the area method of Chou, Gao and Zhang in the framework of projective geometry. Bracket algebra is an important computational tool for projective geometry and has various applications in geometric reasoning, computer vision and robotics. Bracket algebra has been used to prove projective geometric theorems mechanically by several authors in the last decade. Sturmfels and Whitely (1991) proved that the algorithm of straightening laws (Doubilet, Rota and Stein, 1974) is a special algorithm to compute Grobner bases for polynomials of brackets. So the method of automated theorem proving by Grobner bases on the coordinate algebra level can be extended to theorem proving by straightening laws on the bracket algebra level. Mourrain and Stolfi (1994) proposed a theorem proving method extending Wu's method from coordinate algebra to bracket algebra. The main feature of their method is delicate applications of the Cramer's rule in linear algebra. Crapo and Richter-Gebert (1994), Richter-Gebert (1995) proposed a method based on bi-quadratic final polynomials. Chou, Gao and Zhang's area method (1994) can also be used to prove theorems in projective geometry, and the areas can be understood as brackets. The method of straightening laws can be applied to all projective geometric theorems, while the latter three methods are applied mainly to incidence theorems. However, proofs produced by the latter three methods are remarkably short, both in the number of steps taken and in the length of polynomials occurred; in one word, the proofs are readable. Can we get even shorter proofs? Can we extend the methods to theorems involving projective conics? The two problems are the focus of our research on applying bracket algebra to theorem proving. The following are what we have achieved so far: first, we further develop the elimination rules in the area method to produce shorter proofs for incidence theorems; second, we propose a set of elimination rules for theorems involving conics, which turns out to be a method producing specific elimination rules for every specific theorem involving conics; third, we propose a powerful technique called contraction to reduce the size of polynomials of brackets. All these innovations are put together to form a theorem proving method featuring automated production of delicate elimination rules and simplification by contraction. The method extends Wu's method to bracket algebra and develops the area method in the framework of projective geometry.