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Proofs and Computations


Part of Perspectives in Logic

  • Date Published: December 2011
  • availability: Available
  • format: Hardback
  • isbn: 9780521517690

$ 85.00

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About the Authors
  • 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.

    • The first single source for students and researchers in this area
    • Authors are acknowledged experts in the field
    • Covers an important area of mathematics at the interface of logic and computer science
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    Product details

    • Date Published: December 2011
    • format: Hardback
    • isbn: 9780521517690
    • length: 480 pages
    • dimensions: 236 x 160 x 30 mm
    • weight: 0.86kg
    • contains: 8 b/w illus.
    • availability: Available
  • Table of Contents

    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<ω and Π11–CA0
    Part III. Constructive Logic and Complexity:
    6. Computability in higher types
    7. Extracting computational content from proofs
    8. Linear two-sorted arithmetic

  • Authors

    Helmut Schwichtenberg, Ludwig-Maximilians-Universität Munchen
    Helmut Schwichtenberg is an Emeritus Professor of Mathematics at Ludwig-Maximilians-Universität München. 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, University of Leeds
    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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