Improving QED-Tutrix by Automating the Generation of Proofs

Abstract

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.

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

“…In some moderately difficult problems, the HPDIC graph can become quite big, making its exploration time consuming. In a previous version of the prover [9], we used all the available inferences, stored their results, then started again, until no new result is found. This algorithm had the major drawback of duplicating inferences: at step N, the inferences found at step N-1 are still valid, and are therefore found again, increasing the complexity of the algorithm to quadratic on the number of nodes in the final graph.…”

“…This paper presents the advancement of our work on QED-Tutrix. In one of our previous papers [9], we presented our preliminary results, including a review of research work on tutoring systems and automated theorem proving relevant to this project. Since the basis for our work has not changed, this review is quite similar to the previous one with, however, the addition of some recent work that was not published at the time.…”

“…In one of our previous papers [9], we documented several challenges that arose during the development of the solver. One of the most difficult stems from the fact that the computer must have complete and exact information to infer new results.…”

“…For this reason, our system requires a much more extensive referential that includes those properties that we cannot generate to be able to solve the problems. This limitation adds another layer of complexity concerning the challenge of the level of granularity discussed in [9].…”

“…first, a structure that contains all the possible proofs, called the HPDIC graph 1 [15], and, second, the generation of such a structure for every problem that is offered in QED-Tutrix [9]. However, even quite simple problems can have a huge amount of possible proofs when considering all the slight variations.…”