Softcover ISBN:  9781470470647 
Product Code:  SURV/220.S 
List Price:  $124.00 
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AMS Member Price:  $99.20 
Electronic ISBN:  9781470440831 
Product Code:  SURV/220.E 
List Price:  $124.00 
MAA Member Price:  $111.60 
AMS Member Price:  $99.20 

Book DetailsMathematical Surveys and MonographsVolume: 220; 2017; 511 ppMSC: 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 textbookstyle 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.ReadershipGraduate 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

MartinLöf 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


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 Book Details
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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 textbookstyle 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.

Chapters

What is this book about?

Plain Kolmogorov complexity

Complexity of pairs and conditional complexity

MartinLöf 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