# Algorithmic Randomness and Complexity

• Rodney G. Downey
• Denis R. Hirschfeldt
Book

Part of the Theory and Applications of Computability book series (THEOAPPLCOM)

1. Front Matter
Pages i-xxviii
2. ### Background

1. Front Matter
Pages 1-1
2. Rodney G. Downey, Denis R. Hirschfeldt
Pages 2-6
3. Rodney G. Downey, Denis R. Hirschfeldt
Pages 7-109
4. Rodney G. Downey, Denis R. Hirschfeldt
Pages 110-153
5. Rodney G. Downey, Denis R. Hirschfeldt
Pages 154-196
6. Rodney G. Downey, Denis R. Hirschfeldt
Pages 197-224
3. ### Notions of Randomness

1. Front Matter
Pages 225-225
2. Rodney G. Downey, Denis R. Hirschfeldt
Pages 226-268
3. Rodney G. Downey, Denis R. Hirschfeldt
Pages 269-322
4. Rodney G. Downey, Denis R. Hirschfeldt
Pages 323-401
4. ### Relative Randomness

1. Front Matter
Pages 403-403
2. Rodney G. Downey, Denis R. Hirschfeldt
Pages 404-463
3. Rodney G. Downey, Denis R. Hirschfeldt
Pages 464-499
4. Rodney G. Downey, Denis R. Hirschfeldt
Pages 500-553
5. Rodney G. Downey, Denis R. Hirschfeldt
Pages 554-591
6. Rodney G. Downey, Denis R. Hirschfeldt
Pages 592-666
5. ### Further Topics

1. Front Matter
Pages 667-667
2. Rodney G. Downey, Denis R. Hirschfeldt
Pages 668-704
3. Rodney G. Downey, Denis R. Hirschfeldt
Pages 705-727

### Introduction

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. Much of this theory can be seen as exploring the relationships between three fundamental concepts: relative computability, as measured by notions such as Turing reducibility; information content, as measured by notions such as Kolmogorov complexity; and randomness of individual objects, as first successfully defined by Martin-Löf. Although algorithmic randomness has been studied for several decades, a dramatic upsurge of interest in the area, starting in the late 1990s, has led to significant advances. This is the first comprehensive treatment of this important field, designed to be both a reference tool for experts and a guide for newcomers. It surveys a broad section of work in the area, and presents most of its major results and techniques in depth. Its organization is designed to guide the reader through this large body of work, providing context for its many concepts and theorems, discussing their significance, and highlighting their interactions. It includes a discussion of effective dimension, which allows us to assign concepts like Hausdorff dimension to individual reals, and a focused but detailed introduction to computability theory. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

### Keywords

Algorithms algorithm complexity complexity theory computability theory computational geometry computer computer science information theory logic modeling

#### Authors and affiliations

• Rodney G. Downey
• 1
• Denis R. Hirschfeldt
• 2
1. 1.School of Mathematics, Statistics &, Operations ResearchVictoria UniversityWellingtonNew Zealand
2. 2., Department of MathematicsUniversity of ChicagoChicagoUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-0-387-68441-3