Turing Computability : Theory and Applications / by Robert I. Soare.

Format:

Book

Author/Creator:

Soare, R. I. (Robert Irving), 1940- author.

Contributor:

SpringerLink (Online service)

Series:

Computer Science (Springer-11645)

Theory and Applications of Computability, In cooperation with the association Computability in Europe,. 2190-619X

Theory and Applications of Computability, In cooperation with the association Computability in Europe, 2190-619X

Language:

English

Subjects (All):

Computers.

Computer science--Mathematics.

Computer science.

Logic, Symbolic and mathematical.

Theory of Computation.

Mathematics of Computing.

Mathematical Logic and Foundations.

Local Subjects:

Theory of Computation.

Mathematics of Computing.

Mathematical Logic and Foundations.

Physical Description:

1 online resource (XXXVI, 263 pages) : 4 illustrations.

Edition:

First edition 2016.

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2016.

System Details:

text file PDF

Summary:

Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part II includes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory. The author is a leading authority on the topic and he has taught the subject using the book content over decades, honing it according to experience and feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic.

Contents:

Part I Foundations of Computability

Chap. 1 Defining Computability

Chap. 2 Computably Enumerable Sets

Chap. 3 Turing Reducibility

Chap. 4 The Arithmetical Hierarchy

Chap. 5 Classifying C.E. Sets

Chap. 6 Oracle Constructions and Forcing

Chap. 7 The Finite Injury Method

Part II Trees and Π01 Classes

Chap. 8 Open and Closed Classes

Chap. 9 Basis Theorems

Chap. 10 Peano Arithmetic and Π01-Classes

Chap. 11 Randomness and Π01-Classes

Part III Minimal Degrees

Chap. 12 Minimal Degrees Below Øʹʹ

Chap. 13 Minimal Degrees Below Øʹ

Part IV Games in Computability Theory

Chap. 14 Banach-Mazur Games

Chap. 15 Gale-Stewart Games

Chap. 16 More Lachlan Games

Part V History of Computability

Chap. 17 History of Computability

References

Index.

Other Format:

Printed edition:

ISBN:

978-3-642-31933-4

9783642319334

Access Restriction:

Restricted for use by site license.