Computable Analysis

An Introduction

By: Klaus Weihrauch, K. Weihrauch

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Hardcover | 14 September 2000

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304 Pages

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24.13 x 15.88 x 2.54

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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 basis 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.

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Introduction	p. 1
The Aim of Computable Analysis	p. 1
Why a New Introduction?	p. 2
A Sketch of TTE	p. 3
A Model of Computation	p. 3
A Naming System for Real Numbers	p. 4
Computable Real Numbers and Functions	p. 4
Subsets of Real Numbers	p. 7
The Space C[0; 1] of Continuous Functions	p. 8
Computational Complexity of Real Functions	p. 9
Prerequisites and Notation	p. 10
Computability on the Cantor Space	p. 13
Type-2 Machines and Computable String Functions	p. 14
Computable String Functions are Continuous	p. 27
Standard Representations of Sets of Continuous String Func- tions	p. 33
E ective Subsets	p. 43
Naming Systems	p. 51
Continuity and Computability Induced by Naming Systems	p. 51
Admissible Naming Systems	p. 62
Constructions of New Naming Systems	p. 75
Computability on the Real Numbers	p. 85
Various Representations of the Real Numbers	p. 85
Computable Real Numbers	p. 101
Computable Real Functions	p. 108
Computability on Closed, Open and Compact Sets	p. 123
Closed Sets and Open Sets	p. 123
Compact Sets	p. 143
Spaces of Continuous Functions	p. 153
Various representations	p. 153
Computable Operators on Functions, Sets and Numbers	p. 163
Zero-Finding	p. 173
Di erentiation and Integration	p. 182
Analytic Functions	p. 190
Computational Complexity	p. 195
Complexity of Type-2 Machine Computations	p. 195
Complexity Induced by the Signed Digit Representation	p. 204
The Complexity of Some Real Functions	p. 218
Complexity on Compact Sets	p. 230
Some Extensions:	p. 237
Computable Metric Spaces	p. 237
Degrees of Discontinuity	p. 244
Other Approaches to Computable Analysis	p. 249
Banach/Mazur Computability	p. 249
Grzegorczyk's Characterizations	p. 250
The Pour-El/Richards Approach	p. 252
Ko's Approach	p. 254
Domain Theory	p. 256
Markov's Approach	p. 258
The real-RAM and Related Models	p. 260
Comparison	p. 266
References	p. 269
Index	p. 277
Table of Contents provided by Publisher. All Rights Reserved.

Computable Analysis

An Introduction

At a Glance

Hardcover

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Paperback

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