About this Title
Rebecca Weber, Dartmouth College, Hanover, NH
Publication: The Student Mathematical Library
Publication Year 2012: Volume 62
ISBNs: 978-0-8218-7392-2 (print); 978-0-8218-8543-7 (online)
MathSciNet review: MR2920681
MSC: Primary 03-01; Secondary 03Dxx
What can we compute—even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory.
The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.
Undergraduate students interested in computability theory and recursion theory.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. Background
- Chapter 3. Defining computability
- Chapter 4. Working with computable functions
- Chapter 5. Computing and enumerating sets
- Chapter 6. Turing reduction and Post’s problem
- Chapter 7. Two hierarchies of sets
- Chapter 8. Further tools and results
- Chapter 9. Areas of research
- Appendix A. Mathematical asides