Computational Logic and Set Theory: Applying Formalized Logic to Analysis

ISBN-13: 9781447160182 / Angielski / Miękka / 2014 / 416 str.

Jacob T Schwartz;Courant Institute Domenico Cantone (Univ;Eugenio G Omodeo

Opis

As computer software becomes more complex, the question of how its correctness can be assured grows ever more critical. Formal logic embodied in computer programs is an important part of the answer to this problem.This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the AEtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Taking a systematic approach, the book begins with a survey of traditional branches of logic before describing in detail the underlying design of the AEtnaNova system. This system is then used to derive several of the main classical results on undecidability and unsolvability. Readers do not require great knowledge of formal logic in order to follow the text, although a good understanding of standard programming techniques, and a familiarity with the mathematics of definitions and proof scenarios is assumed.Topics and features: with a Foreword by Dr. Martin Davis, Professor Emeritus of the Courant Institute of Mathematical Sciences, New York University; describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn's lemma.This ground-breaking work is essential reading for researchers and advanced graduates of computer science.

Recenzje

From the book reviews:

"This is a text defining new formulae, theorems, lemmas, and sublanguages. Partition calculus and subset theory is well developed. ... I recommend this book to all students and logicians." (Joseph J. Grenier, Amazon.com, August, 2014)

"The contents of the book makes it inspiring and interesting both to mathematicians and to computer scientists. ... This is one of a few recent books which promise to both make mathematicians aware of the possibilities opened by the recent progress in automated theorem proving and draw the attention of the researchers working in logic and automated theorem proving to the challenges and possibilities raised by interesting problems in mathematics." (Viorica Sofronie-Stokkermans, Zentralblatt MATH, Vol. 1246, 2012)

Spis treści

Introduction

Propositional and Predicate-Calculus Preliminaries

A Survey of Inference Mechanisms

More on the Structure of the Verifier System

A Closer Examination of the Sequence of Definitions and Theorems Presented in this Book

Undecidability and Unsolvability

Blurb

As computer software becomes more complex, the question of how its correctness can be assured grows ever more critical. Formal logic embodied in computer programs is an important part of the answer to this problem.

This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the ÆtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Taking a systematic approach, the book begins with a survey of traditional branches of logic before describing in detail the underlying design of the ÆtnaNova system. Major classical results on undecidability and unsolvability are then recast for this system. Readers do not require great knowledge of formal logic in order to follow the text, but a good understanding of standard programming techniques, and a familiarity with mathematical definitions and proofs reflecting the usual levels of rigor is assumed.

Topics and features:

• With a Foreword by Dr. Martin Davis, Professor Emeritus of the Courant Institute of Mathematical Sciences, New York University
• Describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics
• Presents an unique system for automated proof verification on the large scale
• Integrates important proof-engineering issues, reflecting the goals of large-scale verifiers
• Includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn’s lemma

This ground-breaking work is essential reading for researchers and advanced graduates of computer science.