Proofs and Computations

This major graduate-level text provides a detailed, self-contained coverage of proof theory.

Helmut Schwichtenberg (Author), Stanley S. Wainer (Author)

9780521517690, Cambridge University Press

Hardback, published 15 December 2011

480 pages, 8 b/w illus.
23.6 x 16 x 3 cm, 0.86 kg

"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

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 ?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.

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.

Subject Areas: Computer architecture & logic design [UYF], Programming & scripting languages: general [UMX], Mathematical logic [PBCD]