Reasoning By Analogy: A Progress Report

This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract number N00014-75-C-0643. The views expressed are necessarily (and perhaps only) those of the author.Rather.MIT Artificial Intelligence Laboratory Department of Defense Advanced Research Projects Agenc

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

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

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

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

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.Comment: In Proceedings ThEdu'19, arXiv:2002.1189

Decision-making and problem-solving methods in automation technology

The state of the art in the automation of decision making and problem solving is reviewed. The information upon which the report is based was derived from literature searches, visits to university and government laboratories performing basic research in the area, and a 1980 Langley Research Center sponsored conferences on the subject. It is the contention of the authors that the technology in this area is being generated by research primarily in the three disciplines of Artificial Intelligence, Control Theory, and Operations Research. Under the assumption that the state of the art in decision making and problem solving is reflected in the problems being solved, specific problems and methods of their solution are often discussed to elucidate particular aspects of the subject. Synopses of the following major topic areas comprise most of the report: (1) detection and recognition; (2) planning; and scheduling; (3) learning; (4) theorem proving; (5) distributed systems; (6) knowledge bases; (7) search; (8) heuristics; and (9) evolutionary programming

The effect of proof format on reading comprehension of geometry proof:The case of Indonesian prospective mathematics teachers

This study aims to investigate the effects of the use of multiple geometry proof formats on Indonesian students’ reading comprehension of geometry proof (RCGP). Four classes of prospective secondary mathematics teachers (N=125), aged 18 to 19 years, participated in this quasi-experimental study. While the experimental group was instructed in three proof formats (paragraph, two-column and flow-chart proof), the control group was instructed in only the two-column proof format. Similar pre- and post-tests, based on Yang and Lin’s (2008) RCGP test, were administered to both groups. N-Gain scores were used to determine the improvement of both groups. The N-Gain scores showed significantly more improvement of students’ RCGP in the experimental group. More detailed analysis indicated that the use of multiple proof formats supports the students’ understanding of the facets of logical status of statements and the critical ideas in the proof. This study shows the benefits of offering multiple proof formats to support prospective mathematics teachers’ RCGP

Spatial Aggregation: Theory and Applications

Visual thinking plays an important role in scientific reasoning. Based on the research in automating diverse reasoning tasks about dynamical systems, nonlinear controllers, kinematic mechanisms, and fluid motion, we have identified a style of visual thinking, imagistic reasoning. Imagistic reasoning organizes computations around image-like, analogue representations so that perceptual and symbolic operations can be brought to bear to infer structure and behavior. Programs incorporating imagistic reasoning have been shown to perform at an expert level in domains that defy current analytic or numerical methods. We have developed a computational paradigm, spatial aggregation, to unify the description of a class of imagistic problem solvers. A program written in this paradigm has the following properties. It takes a continuous field and optional objective functions as input, and produces high-level descriptions of structure, behavior, or control actions. It computes a multi-layer of intermediate representations, called spatial aggregates, by forming equivalence classes and adjacency relations. It employs a small set of generic operators such as aggregation, classification, and localization to perform bidirectional mapping between the information-rich field and successively more abstract spatial aggregates. It uses a data structure, the neighborhood graph, as a common interface to modularize computations. To illustrate our theory, we describe the computational structure of three implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the spatial aggregation generic operators by mixing and matching a library of commonly used routines.Comment: See http://www.jair.org/ for any accompanying file

Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations

We prove that local and global parabolic BMO spaces are equal thus extending the classical result of Reimann and Rychener. Moreover, we show that functions in parabolic BMO are exponentially integrable in a general class of space-time cylinders. As a corollary, we establish global integrability for positive supersolutions to a wide class of doubly nonlinear parabolic equations.Comment: 19 page

O método do ângulo completo no sistema OpenGeoProver

Dissertação de Mestrado em Matemática apresentada à Faculdade de Ciências e Tecnologia da Universidade de CoimbraO método do ângulo completo para geometria euclideana construtiva foi proposto por Chou, Gao e Zhang no início dos anos 1990. Este método, uma extensão do método da área proposto pelos mesmos autores, produz demonstrações legíveis e de um modo eficiente demonstra muitos teoremas não triviais. Pode ser considerado como um dos métodos mais interessante e de maior sucesso na demonstração de teoremas em geometria e, possivelmente, o mais bem sucedido na produção de demonstrações automáticas legíveis. Nesta dissertação de mestrado faz-se a apresentação do mêtodo do ângulo completo e demonstram-se muitos dos seus lemas. Descreve-se ainda a planificação da implementação, em código livre, do método do ângulo completo.The full-angle method for euclidean constructive geometry was proposed by Chou, Gao, Zhang in early 1990’s. The method, an extension of the area method proposed by the same authors, produces humanreadable proofs and can efficiently prove many non-trivial theorems. It can be considered as one of the most interesting and most successful methods in geometry theorem proving and probably the most successful in the domain of automated production of readable proofs. In this master thesis a presentation of the full-angle method is made and several of its lemmas are proved. A plannification of the implementation, in open source code, of the full-angle method is also described