Proofs and Computations

Proofs and Computations

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Description

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 Goedel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to 11-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 11-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.
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Product details

  • Hardback | 480 pages
  • 160 x 236 x 30mm | 860g
  • Cambridge, United Kingdom
  • English
  • 8 Line drawings, unspecified
  • 0521517699
  • 9780521517690
  • 1,530,250

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< and 11-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.
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Review quote

"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
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About Helmut Schwichtenberg

Helmut Schwichtenberg is an Emeritus Professor of Mathematics at Ludwig-Maximilians-Universitat Munchen. 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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