Dealing with Degeneracies in Automated Theorem Proving in Geometry

Abstract

We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components.

Can I Bring My Calculator to the Exam? Some Reflections on the Abstraction Level of Computer Algebra Systems

“…There are very interesting steps in this line of research that start from a dynamic geometry system, like in the earlier [30], even automatizing the whole process [31]. But automated theorem proving is not the key topic of this talk.…”

Theorem Verification of the Quantifier-Guided Dominance Degree with the Mean Operator for Additive Preference Relations

“…A theorem is a proposition that can be proved mathematically and lacks a contradictory case that would allow its invalidation [10]. Every mathematical theorem starts from an assumption, called a hypothesis, and a rationale, called a thesis [11]. Consequently, this study proposes a mathematical theorem to verify that the GDM system works correctly when it selects the best alternative(s) by maximising the quantifier-guided dominance degree (QGDD) with the mean operator [12,13].…”