ISBN: 3540668179
TITLE: Computable Analysis
AUTHOR: Weihrauch, Klaus
TOC:

Contents 1
Introduction 1
1.1 The Aim of Computable Analysis 1
1.2 Why a New Introduction? 2
1.3 A Sketch of TTE 3
1.3.1 A Model of Computation 3
1.3.2 A Naming System for Real Numbers 4
1.3.3 Computable Real Numbers and Functions 4
1.3.4 Subsets of Real Numbers 7
1.3.5 The Space C[0; 1] of Continuous Functions 8
1.3.6 Computational Complexity of Real Functions 9
1.4 Prerequisites and Notation 10
2. Computability on the Cantor Space 13
2.1 Type-2 Machines and Computable String Functions 14
2.2 Computable String Functions are Continuous 27
2.3 Standard Representations of Sets of Continuous String Functions 33
2.4 Effective Subsets 43
3. Naming Systems 51
3.1 Continuity and Computability Induced by Naming Systems 51
3.2 Admissible Naming Systems 62
3.3 Constructions of New Naming Systems 75
4. Computability on the Real Numbers 85
4.1 Various Representations of the Real Numbers 85
4.2 Computable Real Numbers 101
4.3 Computable Real Functions 108
5. Computability on Closed, Open and Compact Sets 123
5.1 Closed Sets and Open Sets 123
5.2 Compact Sets 143
6. Spaces of Continuous Functions 153
6.1 Various representations 153
6.2 Computable Operators on Functions, Sets and Numbers 163
6.3 Zero-Finding 173
6.4 Differentiation and Integration 182
6.5 Analytic Functions 190
7. Computational Complexity 195
7.1 Complexity ofType-2 Machine Computations 195
7.2 Complexity Induced by the Signed Digit Representation 204
7.3 The Complexity of Some Real Functions 218
7.4 Complexity on Compact Sets 230
8. Some Extensions 237
8.1 Computable Metric Spaces 237
8.2 Degrees of Discontinuity 244
9. Other Approaches to Computable Analysis 249
9.1 Banach/Mazur Computability 249
9.2 Grzegorczyk's Characterizations 250
9.3 The Pour-El/Richards Approach 252
9.4 Ko's Approach 254
9.5 DomainTheory 256
9.6 Markov's Approach 258
9.7 The real-RAM and Related Models 260
9.8 Comparison 266
References 269
Index 277
END
