Proofs and computations / Helmut Schwichtenberg, Stanley S. Wainer.

Format:

Book

Author/Creator:

Schwichtenberg, Helmut, 1942- author.

Wainer, S. S., author.

Contributor:

Association for Symbolic Logic, issuing body.

Series:

Perspectives in logic.

Perspectives in logic

Language:

English

Subjects (All):

Computable functions.

Proof theory.

Physical Description:

1 online resource (xiii, 465 pages) : digital, PDF file(s).

Edition:

1st ed.

Other Title:

Proofs & Computations

Place of Publication:

Cambridge : Cambridge University Press, 2012.

Language Note:

English

Summary:

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.

Contents:

1.4. Soundness and completeness of the classical fragment1.4.1. Models.; 1.4.2. Soundness of classical logic.; 1.4.3. Completeness of classical logic.; 1.4.4. Compactness and Löwenheim-Skolem theorems.; 1.5. Tait calculus; 1.6. Notes; Chapter 2: RECURSION THEORY; 2.1. Register machines; 2.1.1. Programs.; 2.1.2. Program constructs.; 2.1.3. Register machine computable functions.; 2.2. Elementary functions; 2.2.1. Definition and simple properties.; 2.2.2. Elementary relations.; 2.2.3. The class ε.; 2.2.4. Closure properties of ε.; 2.2.5. Coding finite lists.; 2.3. Kleene's normal form theorem

2.3.1. Program numbers.2.3.2. Normal form.; 2.3.3. Σo1-definable relations and μ-recursive functions.; 2.3.4. Computable functions.; 2.3.5. Undecidability of the halting problem.; 2.4. Recursive definitions; 2.4.1. Least fixed points of recursive definitions.; 2.4.2. The principles of finite support and monotonicity, and the effective index property.; 2.4.3. Recursion theorem.; 2.4.4. Recursive programs and partial recursive functions.; 2.4.5. Relativized recursion.; 2.5. Primitive recursion and for-loops; 2.5.1. Primitive recursive functions.; 2.5.2. Loop-programs.

2.5.3. Reduction to primitive recursion.2.5.4. A complexity hierarchy for Prim.; 2.6. The arithmetical hierarchy; 2.6.1. Kleene's second recursion theorem.; 2.6.2. Characterization of Σ01-definable and recursive relations.; 2.6.3. Arithmetical relations.; 2.6.4. Closure properties.; 2.6.6. Σ0r-complete relations.; 2.7. The analytical hierarchy; 2.7.1. Analytical relations.; 2.7.2. Closure properties.; 2.7.3. Universal Σ1r+1-definable relations.; 2.7.4. Σ1r-complete relations.; 2.8. Recursive type-2 functionals and well-foundedness; 2.8.1. Computation trees.; 2.8.2. Ordinal assignments

recursive ordinals.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-107-22428-4

1-280-48485-3

1-139-22192-2

9786613579836

1-139-21710-0

1-139-21403-9

1-139-22363-1

1-139-22020-9

1-139-03190-2

OCLC:

775869745