Computability theory / Rebecca Weber.
"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 i...
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I. :
American Mathematical Society,
©2012.
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| Series: | Student mathematical library ;
v. 62. |
| Subjects: |
Table of 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.