Klaus Weihrauch

Computable Analysis

An Introduction

Springer-Verlag Berlin/Heidelberg, 2000
ISBN 3-540-66817-9, 285 pp. 44 figs.

Is the exponential function computable? Are union and intersection of closed subsets of the real plane computable? Are differentiation and integration computable operators? Is zero finding for complex polynomials computable? Is the Mandelbrot set decidable? And in case of computability, what is the computational complexity? Computable analysis supplies exact definitions for these and many other similar questions and tries to solve them.

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.

© Springer-Verlag Berlin/Heidelberg 2000

Contents
  1. Introduction 
    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 
    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 
    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 
    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 
    5.1 Closed Sets and Open Sets  . . . . . . . . . . . . . . . . . . . . . . 123 
5.2 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
  6. Spaces of Continuous Functions 
    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 
    7.1 Complexity of Type-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 
    8.1 Computable Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 237 
8.2 Degrees of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . 244
  9. Other Approaches to Computable Analysis 
    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 Domain Theory  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.6 Markov's Approach  . . . . . . . . . . . . . . . . . . . . . . . . . . 258 
9.7 The real-RAM and Related Models  . . . . . . . . . . . . . . . . . . . 260 
9.8 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
References
Index

© Springer-Verlag Berlin/Heidelberg 2000
Back to my homepage