797 research outputs found
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
Improving QED-Tutrix by Automating the Generation of Proofs
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.Comment: In Proceedings ThEdu'17, arXiv:1803.0072
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
Generalizing Morley’s and other theorems with automated realization
A new approach is shown that mechanically proves various theorems in plane geometry by recasting them in terms of constraint satisfaction. A Python 3 implementation called GEOPAR affords transparent proofs of well-known theorems as well as new ones, including a generalization of Morley’s Theorem
Automated Theorem Proving in GeoGebra: Current Achievements
GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way
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