Computable Analysis: An Introduction

@article{Weihrauch2014ComputableAA,
  title={Computable Analysis: An Introduction},
  author={Klaus Weihrauch},
  journal={Computable Analysis},
  year={2014}
}
  • K. Weihrauch
  • Published 12 March 2014
  • Education
  • Computable Analysis
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… 
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1,284 Citations
Highly Influential Citations
134
Background Citations
456
Methods Citations
230
Results Citations
21

1,284 Citations

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References

SHOWING 1-10 OF 165 REFERENCES

Introduction to Automata Theory, Languages and Computation

Type 2 Recursion Theory

  • K. Weihrauch
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1985

Constructive Mathematics: A Foundation for Computable Analysis

Chapter 13 - The Structure of Computability in Analysis and Physical Theory: An Extension of Church's Thesis

Complexity Theory of Real Functions

  • K. Ko
  • Computer Science, Mathematics
    Progress in Theoretical Computer Science
  • 1991
" polynomial complexity theory extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems.

Mathematical theory of domains

The Hemaspaandra-Ogihara book is clearly intended for people who are already comfortable with the field, and are looking for a new perspective, and this is a book that flows easily and quickly from topic to topic.

Order‐free Recursion on the Real Numbers

  • V. Brattka
  • Mathematics, Computer Science
    Math. Log. Q.
  • 1997
These functions, called order‐free recursive, can be shown to include a large class of computable functions and provide an effective version of the Stone‐Weierstraß Approximation Theorem, as well as recursive partitions of unity.

Outline of a Mathematical Theory of Computation

However, Scott does realize that the approach argued for above is simply an argument for an approach that accomodates human understanding of computation and that the operational approach must not be…

Computational complexity of real functions and real numbers

Computability and complexity of real functions and real numbers are studied in a model where methods of recursive function theory as well as methods of numerical analysis can be used at a very low…

A Domain-Theoretic Approach to Computability on the Real Line

...