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Kolmogorov Complexity and Algorithmic Randomness

A. Shen LIRMM CRNS, Université de Montpellier, Montpellier, France
V. A. Uspensky Lomonosov Moscow State University, Moscow, Russia
N. Vereshchagin Lomonosov Moscow State University, Moscow, Russia
Available Formats:
Softcover ISBN: 978-1-4704-7064-7
Product Code: SURV/220.S
List Price: $124.00 MAA Member Price:$111.60
AMS Member Price: $99.20 Electronic ISBN: 978-1-4704-4083-1 Product Code: SURV/220.E List Price:$124.00
MAA Member Price: $111.60 AMS Member Price:$99.20
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $186.00 MAA Member Price:$167.40
AMS Member Price: $148.80 Click above image for expanded view Kolmogorov Complexity and Algorithmic Randomness A. Shen LIRMM CRNS, Université de Montpellier, Montpellier, France V. A. Uspensky Lomonosov Moscow State University, Moscow, Russia N. Vereshchagin Lomonosov Moscow State University, Moscow, Russia Available Formats:  Softcover ISBN: 978-1-4704-7064-7 Product Code: SURV/220.S  List Price:$124.00 MAA Member Price: $111.60 AMS Member Price:$99.20
 Electronic ISBN: 978-1-4704-4083-1 Product Code: SURV/220.E
 List Price: $124.00 MAA Member Price:$111.60 AMS Member Price: $99.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$186.00 MAA Member Price: $167.40 AMS Member Price:$148.80
• Book Details

Mathematical Surveys and Monographs
Volume: 2202017; 511 pp
MSC: Primary 68; 03; 60; 97;

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.

Readership

Graduate students and researchers interested in topics related to an algorithmic approach to complexity and randomness.

• Table of Contents

• Chapters
• 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
• Additional Material

• Reviews

• This book is an excellent reference for the working mathematician and would also serve well as a text for a graduate course. The exercises fit well into the text and are at an appropriate level.

Johanna N. Y. Franklin, Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2202017; 511 pp
MSC: Primary 68; 03; 60; 97;

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.

Readership

Graduate students and researchers interested in topics related to an algorithmic approach to complexity and randomness.

• Chapters
• 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
• This book is an excellent reference for the working mathematician and would also serve well as a text for a graduate course. The exercises fit well into the text and are at an appropriate level.

Johanna N. Y. Franklin, Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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