Turing Computability

Theory and Applications

  • Robert I.¬†Soare

Part of the Theory and Applications of Computability book series (THEOAPPLCOM)

Table of contents

  1. Front Matter
    Pages i-xxxvi
  2. Foundations of Computability

    1. Front Matter
      Pages 1-1
    2. Robert I. Soare
      Pages 3-22
    3. Robert I. Soare
      Pages 23-50
    4. Robert I. Soare
      Pages 51-78
    5. Robert I. Soare
      Pages 79-105
    6. Robert I. Soare
      Pages 107-129
    7. Robert I. Soare
      Pages 131-146
    8. Robert I. Soare
      Pages 147-162
  3. Trees and $$\Pi_1^0$$ Classes

    1. Front Matter
      Pages 163-163
    2. Robert I. Soare
      Pages 165-173
    3. Robert I. Soare
      Pages 175-182
    4. Robert I. Soare
      Pages 183-187
    5. Robert I. Soare
      Pages 189-194
  4. Minimal Degrees

    1. Front Matter
      Pages 195-195
    2. Robert I. Soare
      Pages 203-208
  5. Games in Computability Theory

    1. Front Matter
      Pages 209-209
    2. Robert I. Soare
      Pages 211-216
    3. Robert I. Soare
      Pages 217-219
    4. Robert I. Soare
      Pages 221-224
  6. History of Computability

    1. Front Matter
      Pages 225-225
    2. Robert I. Soare
      Pages 227-249
  7. Back Matter
    Pages 251-263

About this book


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.


Alan Turing Computability theory Computably enumerable (C.E.) sets Turing reducibility Finite injury method Oracle constructions Tree method Minimal degrees Games in computability theory Relative computability Peano arithmetic

Authors and affiliations

  • Robert I.¬†Soare
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2016
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Computer Science
  • Print ISBN 978-3-642-31932-7
  • Online ISBN 978-3-642-31933-4
  • Series Print ISSN 2190-619X
  • Series Online ISSN 2190-6203
  • Buy this book on publisher's site