A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry

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,

Section: Related Workmentioning

confidence: 99%

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

Font

Cyr

Richard

et al. 2020

“…The axioms such as congpseudo reflexivity are then straightforward versions of the axioms we already got to know in Figure 3. A current overview of the status of GEOCOQ can be found in [3,4].…”

Section: Axiomatic Geometrymentioning

confidence: 99%

Towards Intuitive Reasoning in Axiomatic Geometry

Doré

Broda

2019

Proving lemmas in synthetic geometry is often a time-consuming endeavour since many intermediate lemmas need to be proven before interesting results can be obtained. Improvements in automated theorem provers (ATP) in recent years now mean they can prove many of these intermediate lemmas.The interactive theorem prover ELFE accepts mathematical texts written in fair English and verifies them with the help of ATP. Geometrical texts can thereby easily be formalized in ELFE, leaving only the cornerstones of a proof to be derived by the user. This allows for teaching axiomatic geometry to students without prior experience in formalized mathematics.

“…For this reason, we focus on synthetic methods. A popular approach is to use Tarski's axioms, which have interesting computational properties [16,45]. However, the geometry taught in high-school is based on Euclide's axioms, which are not trivially correlated to Tarski's.…”

Section: Automated Theorem Provingmentioning

confidence: 99%

Improving QED-Tutrix by Automating the Generation of Proofs

Font

Richard

Gagnon

2018

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.