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Computability Theory

Rebecca Weber
Format
Book
Published
Providence, R.I. : American Mathematical Society, c2012.
Language
English
Series
Student mathematical library
ISBN
9780821873922 (alk. paper), 082187392X (alk. paper)
Summary
"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." --Publisher description.
Contents
  • 1. Introduction. Approach ; Some history ; Notes on use of the text ; Acknowledgements and references
  • 2. Background. First-order logic ; Sets ; Relations ; Bijection and isomorphism ; Recursion and introduction ; Some notes on proofs and abstractions
  • 3. Defining computability. Functions, sets, and sequences ; Turing machines ; Partial recursive functions ; Coding and countability ; A universal Turing machine ; The Church-Turing thesis ; Other definitions of computability
  • 4. Working with computable functions. The halting problem ; The "three contradictions" ; Parametrization ; The recursive theorem ; Unsolvability
  • 5. Computing and enumerating sets. Dovetailing ; Computing and enumerating ; Aside : enumeration and incompleteness ; Enumerating noncomputable sets
  • 6.Turing reduction and Post's problem. Reducibility of sets ; Finite injury priority arguments ; Notes on approximation
  • 7. Two hierarchies of sets. Turing degrees and relativization ; The arithmetical hierarchy ; Index sets and arithmetical completeness
  • 8. Further tools and results. The limit Lemma ; The Arslanov completeness criterion ; ε modulo finite difference
  • 9. Areas of research. Computably enumerable sets and degrees ; Randomness ; Some model theory ; Computable model theory ; Reverse mathematics
  • Appendix A : Mathematical asides. The Greek alphabet ; Summations ; Cantor's cardinality proofs.
Description
vii, 203 p. : ill. ; 22 cm.
Notes
Includes bibliographical references (p. 193-197) and index.
Series Statement
Student mathematical library ; v. 62
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