The Automation of Mathematical Reasoning

2009

IEEE Potentials, 1992

HAL (Le Centre pour la Communication Scientifique Directe), 2022

HAL (Le Centre pour la Communication Scientifique Directe), 2022

The international Erasmus+ European Project: "ARC-Automated Reasoning in the Class", running from 2019 to 2022 is a partnership of universities from Austria, France, Germany, Hungary, and Romania, and has the purpose of developing advanced material for teaching subjects related to Computational Logic by using Automated Reasoning. The material includes a comprehensive textbook treating the necessary theoretical background (selected topics in Mathematical Logic), but mostly the practical methods from Automated Theorem Proving, as well as the description of the basic programming paradigms and the associated languages, in relation to their logical aspects. Furthermore, we address the most important applications, like program verification and testing, semantic representation of information, algorithm synthesis, etc. One of the main goals of the approach is to improve the logical background of the software professionals in order to motivate them to use formal methods for certification of complex systems and thus to avoid costly failures.

Indonesian Journal of Computer Science

This article takes a look at the progress and advancement of automated reasoning and its applications in the 21st century. Reasoning refers to the method of reaching logical conclusions. The construction of computing systems that automate this process over some knowledge bases is the focus of automatic reasoning. Automated Reasoning is frequently regarded as a subfield of machine learning. It is also studied in theoretical computer science and philosophy. Some of the applications of automated reasoning include but not limited to Tableau-style systems, Automatic Theorem Proving, Superposition and Saturation, benchmarks and Classical First-Order Logic. The development of formal led to the development of artificial intelligence, which was essential in the development of artificial intelligence for reasoning.

Springer International Publishing eBooks, 2022

The European Erasmus+ project ARC-Automated Reasoning in the Class aims at improving the academic education in disciplines related to Computational Logic by using Automated Reasoning tools. We present the technical aspects of the tools as well as our education experiments, which took place mostly in virtual lectures due to the COVID pandemics. Our education goals are: to support the virtual interaction between teacher and students in the absence of the blackboard, to explain the basic Computational Logic algorithms, to study their implementation in certain programming environments, to reveal the main relationships between logic and programming, and to develop the proof skills of the students. For the introductory lectures we use some programs in C and in Mathematica in order to illustrate normal forms, resolution, and DPLL (Davis-Putnam-Logemann-Loveland) with its Chaff version, as well as an implementation of sequent calculus in the Theorema system. Furthermore we developed special tools for SAT (propositional satisfiability), some based on the original methods from the partners, including complex tools for SMT (Satisfiability Modulo Theories) that allow the illustration of various solving approaches. An SMT related approach is natural-style proving in Elementary Analysis, for which we developed and interesting set of practical heuristics. For more advanced lectures on rewrite systems we use the Coq programming and proving environment, in order on one hand to demonstrate programming in functional style and on the other hand to prove properties of programs. Other advanced approaches used in some lectures are the deduction based synthesis of algorithms and the techniques for program transformation.

Inaugural Lecture, School of Computer Science and IT …, 2001

The modern world is highly dependent on the reliable functioning of computer software. The sheer scale of software systems makes their design and implementation a highly demanding intellectual activity. Meeting these demands has inspired a revolution in the way that mathematics, the art of effective reasoning, is conducted and presented. Continued effort is needed in education and research in the mathematical construction of programs, based on the controlled manipulation of mathematical formulae.

1985

We discuss the automation of mathematical reasoning, surveying the abilities displayed by human mathematicians and the computational techniques available for automating these abilities. We argue the importance of the simultaneous study of these techniques, because problems inherent in one technique can often be solved if it is able to interact with others.

1999

In these days, logics are playing an important role not only in Philosophy and Mathematics but also in Computer Science, Artificial Intelligence, and in other areas such as Aesthetics, which used to be considered in the opposite position to logic. In these areas many kinds of logical systems have been proposed and studied for coping with a wide variety of problem domains. In order to deal with these logics, various types of reasoning systems have been developed. Automated theorem provers have been investigated to pursue the mechanisms to prove the practical theorems automatically. Proof checkers aim to help the users with creating rigorous proofs by providing the means of checking the correctness of the proofs given by them. lll I would like to give my special thanks to my elder brother Tetsuhiro Minami, who is always willing to support me and has given me continuous encouragement and valuable advice. Finally, my deepest thanks are due to my parents and my family. My mother, Fujie, and my late father, Fumio, have been supporting and encouraging me with pride and affection in my whole life. My wife, Tomoko, and my daughters, Mariko and Emiko, have been with me and given me supports and energies to working during the period of the research in this thesis.

Logical reasoning in any form is an important aspect of life; it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper, which is published in an international mathematics journal, presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony.