Computable analysis : an introduction
Author(s)
Bibliographic Information
Computable analysis : an introduction
（Texts in theoretical computer science, an EATCS series）
Springer, c2000
- : [pbk.]
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Note
Includes bibliographical references (p. [269]-276) and index
"Softcover reprint of the hardcover 1st edition 2000"--T.p. verso of [pbk.]
Description and Table of Contents
Description
Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.
Table of Contents
- 1. Introduction.- 1.1 The Aim of Computable Analysis.- 1.2 Why a New Introduction?.- 1.3 A Sketch of TTE.- 1.3.1 A Model of Computation.- 1.3.2 A Naming System for Real Numbers.- 1.3.3 Computable Real Numbers and Functions.- 1.3.4 Subsets of Real Numbers.- 1.3.5 The Space C[O
- 1] of ContinuouS Functions.- 1.3.6 Computational Complexity of Real Functions.- 1.4 Prerequisites aud Notation.- 2. Computability on the Cantor Space.- 2.1 Type-2 Machines and Computable String Functions.- 2.2 Computable String Functions are Continuous.- 2.3 Standard Representations of Sets of Continuous String Functions.- 2.4 Effective Subsets.- 3. Naming Systems.- 3.1 Continuity and Computability Induced by Naming Systems.- 3.2 Admissible Naming Systems.- 3.3 Constructions of New Naming Systems.- 4. Computability on the Real Numbers.- 4.1 Various Representations of the Real Numbers.- 4.2 Computable Real Numbers.- 4.3 Computable Real Functions.- 5. Computability on Closed, Open and Compact Sets.- 5.1 Closed Sets and Open Sets.- 5.2 Compact Sets.- 6. Spaces of Continuous Functions.- 6.1 Various representations.- 6.2 Computable Operators on Functions. Sets and Numbers.- 6.3 Zero-Finding.- 6.4 Differentiation and Integration.- 6.5 Analytic Functions.- 7. Computational Complexity.- 7.1 Complexity of Type-2 Machine Computations.- 7.2 Complexity Induced by the Signed Digit Representation.- 7.3 The Complexity of Some Real Functions.- 7.4 Complexity on Compact Sets.- 8. Some Extensions.- 8.1 Computable Metric Spaces.- 8.2 Degrees of Discontinuity.- 9. Other Approaches to Computable Analysis.- 9.1 Banach/Mazur Computability.- 9.2 Grzegorczyk's Characterizations.- 9.3 The Pour-El/Richards Approach.- 9.4 Ko's Approach.- 9.5 Domain Theory.- 9.6 Markov's Approach.- 9.7 The real-RAM and Related Models.- 9.8 Comparison.- References.
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