Automating the Generation of High School Geometry Proofs using Prolog in an Educational Context

Abstract

When working on intelligent tutor systems designed for mathematics education and its specificities, an interesting objective is to provide relevant help to the students by anticipating their next steps. This can only be done by knowing, beforehand, the possible ways to solve a problem. Hence the need for an automated theorem prover that provide proofs as they would be written by a student. To achieve this objective, logic programming is a natural tool due to the similarity of its reasoning with a mathematical proof by inference. In this paper, we present the core ideas we used to implement such a prover, from its encoding in Prolog to the generation of the complete set of proofs. However, when dealing with educational aspects, there are many challenges to overcome. We also present the main issues we encountered, as well as the chosen solutions.1 Context The QED-Tutrix softwareThe QED-Tutrix software [15,19] provides an environment where a high-school student can solve geometry proof problems. One of its key features is that it allows the student to provide proof elements in any order, not limiting them to forward-or backward-chaining. For instance, when solving the simple problem "prove that a quadrilateral with three right angles is a rectangle", the student can provide any element of any possible proof, such as a direct consequence of the hypotheses ("if two lines are perpendicular to a third, they are parallel"), a necessary premise for the conclusion ("a rectangle is a quadrilateral that has four right angles"), or anything in between ("the quadrilateral ABCD is a parallelogram"). A second key feature is the tutoring aspect. When the student is stuck is the resolution, the software is able to provide them with relevant messages. In the previous example, if the student entered "the quadrilateral ABCD is a parallelogram" and is stuck afterwards, the software identifies that they are working on a proof using parallelogram properties, and will provide them messages such as "what is the definition of a parallelogram?" or "is there a relation between parallelogram and rectangle?"These features, the flexibility in exploration and the tutoring, are very interesting from a mathematics education perspective, but come with a cost. Indeed, to allow this behavior, the software must know, first, all the various mathematical elements that can be used at any step of any proof for the problem at hand, and second, how these elements are used in the various possible proofs of the problem. This requires,

Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor

“…With dynamically generated strategies it could be possible to add points introduced by students without a high increase in CPU time. QED-Tutrix is a geometry tutor that uses a different approach by first extending a problem figure into a super-figure that contains possible useful points and segments (Font et al 2020). A student who uses this system can only use points and segments in the figure and the super-figure.…”

Theorem Proving as Constraint Solving with Coherent Logic

“…For this corpus, Larus is used with options: no case splits, start by looking for a proof of at most 8 steps, and check the generated proof using Coq . 28…”

“…Having available all proofs of a theorem (within some logical framework) can be useful, for instance, in the context of education: "intelligent tutor systems" for mathematics education should be able to know all possible ways to solve a problem so they can help the students by anticipating their next steps (one such system, developed by Font et. al, based on a form of exhaustive search and implemented in Prolog, generates all proofs of geometry theorems using a custom set of axioms [28]).…”