Proofs and computations

Helmut Schwichtenberg, 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 Godel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to PI11-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 PI11-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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  • 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<omega and PI11-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.

「Nielsen BookData」より


書名 Proofs and computations
著作者等 Association for Symbolic Logic
Schwichtenberg, Helmut
Wainer, S. S
Wainer Stanley S. (University of Leeds)
シリーズ名 Perspectives in logic
出版元 Cambridge University Press
刊行年月 2012
ページ数 xiii, 465 p.
大きさ 25 cm
ISBN 9780521517690
NCID BB08096296
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言語 英語
出版国 イギリス