Proofs and Computations

Proofs and Computations

By (author)  , By (author) 

Free delivery worldwide

Available. Dispatched from the UK in 5 business days
When will my order arrive?


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.
show more

Product details

  • Hardback | 480 pages
  • 160 x 236 x 30mm | 860g
  • Cambridge, United Kingdom
  • English
  • 8 Line drawings, unspecified
  • 0521517699
  • 9780521517690
  • 1,871,705

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.
show more

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
show more

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.
show more