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Proofs and Computation

 

Helmut Schwichtenberg and Stanley S. Wainer

 

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.

Year: 2012
Price:$90.00 ISBN-13: 9780521517690
480 pages. Hardcover.
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Table of Contents

Part 1. Basic proof theory and computability
Chapter 1. Logic
Chapter 2. Recursion Theory
Chapter 3. Gödel's Theorems

Part 2. Provable recursion in classical systems
Chapter 4. The Provably Recursive Functions of Arithmetic
Chapter 5. Accessible Recursive Functions, ID < ω and Π11–CA0

Part 3. Constructive logic and complexity
Chapter 6. Computability in Higher Types
Chapter 7. Extracting Computational Content From Proofs
Chapter 8. Linear Two-Sorted Arithmetic  

 

 

 

 





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