Proofs and Computations

Erscheinungsjahr 2013
ISBN: 978-0-521-51769-0
Verlag: Cambridge University Press

Buch, Englisch, 480 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 969 g

Reihe: Perspectives in Logic

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.

Autoren/Hrsg.

Schwichtenberg, Helmut

Wainer, Stanley S.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik

Inhaltsverzeichnis

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 [PI]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.

Über Autor(innen)

Wainer, Stanley S.
Stanley S. Wainer is an emeritus professor of mathematics at the University of Leeds and a past-President of the British Logic Colloquium.

Schwichtenberg, Helmut
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.

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 is an emeritus professor of mathematics at the University of Leeds and a past-President of the British Logic Colloquium.