Kolmogorov Complexity and Algorithmic Randomness
About this Title
A. Shen, LIRMM CRNS, Université de Montpellier, Montpellier, France, V. A. Uspensky, Lomonosov Moscow State University, Moscow, Russia and N. Vereshchagin, Lomonosov Moscow State University, Moscow, Russia
Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 220
ISBNs: 978-1-4704-3182-2 (print); 978-1-4704-4083-1 (online)
MathSciNet review: MR3702040
MSC: Primary 03-02; Secondary 03D32, 60F15, 62A01, 62B10, 68-02, 68Q30
Looking at a sequence of zeros and ones, we often feel that it is not random, that is, it is not plausible as an outcome of fair coin tossing. Why? The answer is provided by algorithmic information theory: because the sequence is compressible, that is, it has small complexity or, equivalently, can be produced by a short program. This idea, going back to Solomonoff, Kolmogorov, Chaitin, Levin, and others, is now the starting point of algorithmic information theory.
The first part of this book is a textbook-style exposition of the basic notions of complexity and randomness; the second part covers some recent work done by participants of the “Kolmogorov seminar” in Moscow (started by Kolmogorov himself in the 1980s) and their colleagues.
This book contains numerous exercises (embedded in the text) that will help readers to grasp the material.
Graduate students and researchers interested in topics related to an algorithmic approach to complexity and randomness.
Table of Contents
- What is this book about?
- Plain Kolmogorov complexity
- Complexity of pairs and conditional complexity
- Martin-Löf randomness
- A priori probability and prefix complexity
- Monotone complexity
- General scheme for complexities
- Shannon entropy and Kolmogorov complexity
- Some applications
- Frequency and game approaches to randomness
- Inequalities for entropy, complexity, and size
- Common information
- Multisource algorithmic information theory
- Information and logic
- Algorithmic statistics
- Complexity and foundations of probability
- Four algorithmic faces of randomness