Proofs and Computations
Author:   Helmut Schwichtenberg (Ludwig-Maximilians-Universität Munchen) ,  Stanley S. Wainer (University of Leeds)
Publisher:   Cambridge University Press
ISBN:  

9781139031905


Publication Date:   05 January 2012
Format:   Undefined
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Proofs and Computations


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Overview

Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Full Product Details

Author:   Helmut Schwichtenberg (Ludwig-Maximilians-Universität Munchen) ,  Stanley S. Wainer (University of Leeds)
Publisher:   Cambridge University Press
Imprint:   Cambridge University Press (Virtual Publishing)
ISBN:  

9781139031905


ISBN 10:   1139031902
Publication Date:   05 January 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Undefined
Publisher's Status:   Active
Availability:   Available To Order   Availability explained
We have confirmation that this item is in stock with the supplier. It will be ordered in for you and dispatched immediately.

Table of Contents

Preface; Preliminaries; Part I. Basic Proof Theory and Computability: 1. Logic; 2. Recursion theory; 3. Godel's theorems; Part II. Provable Recursion in Classical Systems: 4. The provably recursive functions of arithmetic; 5. Accessible recursive functions, ID<ω and Π11–CA0; Part III. Constructive Logic and Complexity: 6. Computability in higher types; 7. Extracting computational content from proofs; 8. Linear two-sorted arithmetic; Bibliography; Index.

Reviews

Written by two leading practitioners in the area of formal logic, the book provides a panoramic view of the topic. This reference volume is a must for the bookshelf of every practitioner of formal logic and computer science. Prahladavaradan Sampath, Computing Reviews


Written by two leading practitioners in the area of formal logic, the book provides a panoramic view of the topic. This reference volume is a must for the bookshelf of every practitioner of formal logic and computer science. Prahladavaradan Sampath, Computing Reviews


Author Information

Helmut Schwichtenberg is an Emeritus Professor of Mathematics at Ludwig-Maximilians-Universität München. He has recently developed the 'proof-assistant' MINLOG, a computer-implemented logic system for proof/program development and extraction of computational content. Stanley S. Wainer is an Emeritus Professor of Mathematics at the University of Leeds and a past-President of the British Logic Colloquium.

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