Automated Development of Fundamental Mathematical Theories / Edition 1

by Art Quaife

Overview

The author provides an introduction to automated reasoning, and in particular to resolution theorem proving using the prover OTTER. He presents a new clausal version of von Neumann-Bernays-Gödel set theory, and lists over 400 theorems proved semiautomatically in elementary set theory. He presents a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem.
The author next develops Peano's arithmetic, and gives more than 1200 definitions and theorems in elementary number theory. He gives part of the proof of the fundamental theorem of arithmetic (unique factorization), and gives and OTTER-generated proof of Euler's generalization of Fermat's theorem.
Next he develops Tarski's geometry within OTTER. He obtains proofs of most of the challenge problems appearing in the literature, and offers further challenges. He then formalizes the modal logic calculus K4, in order to obtain very high level automated proofs of Löb's theorem, and of Gödel's two incompleteness theorems. Finally he offers thirty-one unsolved problems in elementary number theory as challenge problems.

Product Details

ISBN-13:	9780792320210
Publisher:	Springer Netherlands
Publication date:	11/30/1992
Series:	Automated Reasoning Series , #2
Edition description:	1993
Pages:	273
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

Preface: A Personal View of Automated Reasoning Research. 1. Introduction to Automated Reasoning. 2. Von Neumann-Bernays-Gödel Set Theory. 3. Peano's Arithmetic. 4. Tarski's Geometry. 5. Löb's Theorem and Gödel's Two Incompleteness Theorems. 6. Unsolved Problems in Elementary Number Theory. Appendix 1: Gödel's Axioms for Set Theory. Appendix 2: Theorems Proved in NBG Set Theory. Appendix 3: Theorems Proved in Peano's Arithmetic. Bibliography. Index of Names. Index of Subjects.

Editorial Reviews

‘ Quaife's work represents a breakthrough in automated reasoning for demonstrating that proofs of deep, well-known theorems of set theory and number theory can be mechanically derived, in a practical way, from the most elegant, simple, and general purpose foundations: set theory (a finite, first order axiomatization) and resolution (as implemented in the Otter system). '
Robert Boyer, University of Texas at Austin

From the Publisher

To me this book was fascinating. After reading it, I found myself turning over in my mind the question of the role played in theorem proving by understanding, a quality that Penrose regards as a prime desideratum of consciousness....Art Quaife writes clearly, honestly and with infectious enthusiasm. His interesting book is a compilation of research papers which retains the excitement and focus on detail that his creative use of OTTER engendered. I can't help liking a scholarly book that carries the ancient Greek or southern Californian dedication: To the pursuit and realization of unlimited pleasure.
— Psyche

Desmond Fearnley-Sander