"Computability Theory" by Rebecca Weber

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"Computability Theory" by Rebecca Weber

"Computability Theory" by Rebecca Weber
American Mathematical Society. Student mathematical library, volume 62
AMS | 2012 | ISBN: 082187392X 082188543X 9780821873922 9780821885437 | 217 pages | PDF | 8 MB

The goal of this book is to give the reader a fi rm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness.
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.

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.

Contents
Preface
1. Introduction
2. Background
3. Defining computability
4. Working with computable functions
5. Computing and enumerating sets
6. Turing reduction and Post's problem
7. Two hierarchies of sets
8. Further tools and results
9. Areas of research
A. Mathematical asides
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