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An Introduction to Kolmogorov Complexity and Its Applications [Hardcover]

Ming Li , Paul M.B. Vitányi
5.0 out of 5 stars  See all reviews (5 customer reviews)
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Book Description

Nov. 21 2008 0387339981 978-0387339986 3rd ed. 2008

“The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field.” Journal of Symbolic Logic

Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, Kolmogorov–Loveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.


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Review

From the reviews of the second edition:

"We are indeed in the information age and the scientific exploration of information and the laws that govern its behavior has taken center stage in the dramatic development of sciences. Kolmogorov complexity is a central concept and a powerful tool in the understanding of the quantitative nature of information and its processing and transmission. Li and Vitanyi's book beautifully captures the elegance of these ideas, their relevance to more of computer science and their theoretical as well as practical applications. The basic concepts of Kolmogorov complexity should be understood by any technically educated person, and they should be studied by all computer scientists. Li and Vitanyi have provided an ideal book for the exploration of a deep, beautiful and important part of the computer science."

Juris Hartmanis, (Turing Award Winner 1993), NSF, Washington D.C.

"Special attention is paid to the theory underlying inductive inference and its potential applications. The book is likely to remain the standard treatment of Kolmogorov complexity for a long time."

Jorma J. Rissanen, IBM Research, California

"The book of Li and Vitanyi is unexcelled."

Ray J. Solomonoff, Oxbridge Research, Cambridge, Massachusetts

"The book is outstanding . . . the authors did their job unbelievably well...necessary reading for all kinds of readers from undergraduate students to top authorities in the field."

Vladimir A. Uspensky and Alexander K. Shen, Journal of Symbolic Logic

"It is clear that this book will become 'the' Kolmogorov complexity book."

Marius Zimand, Mathematical Reviews

From the reviews of the third edition:

"Kolmogorov complexity, algorithmic information theory, minimum description length, and other information-based disciplines have experienced a phenomenal explosion in the last decade. … is this third edition worth reading? Yes, it is. The authors have added an extra 204 pages, distributed throughout the book … . Eight new figures were also added. Most impressively, 301 new references were added, bringing the total to 820. … It is sure to maintain its reputation … ." (Jacques Carette, ACM Computing Reviews, April, 2009)

From the Back Cover

This ongoing bestseller, now in its third edition, is considered the standard reference on Kolmogorov complexity, a modern theory of information that is concerned with information in individual objects.

New key features and topics in the 3rd edition:

* New results on randomness

* Kolmogorov's structure function, model selection, and MDL

* Incompressibility method: counting unlabeled graphs, Shellsort, communication complexity

* Derandomization

* Kolmogorov complexity versus Shannon information, rate distortion, lossy compression, denoising

* Theoretical results on information distance

* The similarity metric with applications to genomics, phylogeny, clustering, classification, semantic meaning, question-answer systems

*Quantum Kolmogorov complexity

Written by two experts in the field, this book is ideal for advanced undergraduate students, graduate students, and researchers in all fields of science. It is self-contained: it contains the basic requirements from mathematics, probability theory, statistics, information theory, and computer science. Included are history, theory, new developments, a wide range of applications, numerous (new) problem sets, comments, source references, and hints to solutions of problems. This is the only comprehensive treatment of the central ideas of Kolmogorov complexity and their applications.

``Li and Vitányi have provided an ideal book for the exploration of a deep, beautiful and important part of computer science.''

-- Juris Hartmanis, Turing Award Winner 1993, Cornell University, Ithaca, NY.

``The book is likely to remain the standard treatment of Kolmogorov complexity for a long time.''

-- Jorma J. Rissanen, IBM Research, California.

``The book of Li and Vitányi is unexcelled.''

-- Ray J. Solomonoff, Oxbridge Research, Cambridge, Massachusetts

"The book is outstanding...the authors did their job unbelievably well...necessary reading for all kinds of readers from undergraduate students to top authorities in the field."

-- Vladimir A. Uspensky and Alexander K. Shen, Journal of Symbolic Logic [Review]

``Careful and clear introduction to a subtle and deep field.''

--David G. Stork, Ricoh Innovations, California, Amazon [Review]

``THE book on Kolmogorov Complexity.''

--Lance Fortnow, University of Chicago, IL, Amazon [Review]


Inside This Book (Learn More)
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Front Cover | Copyright | Table of Contents | Excerpt | Index
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Most helpful customer reviews
2 of 2 people found the following review helpful
5.0 out of 5 stars THE book on Kolmogorov Complexity Oct. 13 1998
Format:Hardcover
When is an object "random"? Kolmogorov (and others) argue that one could measure randomness by the shortest description, i.e. computer program, that generates it.
This simple idea leads to a beautiful mathematical theory and a powerful tool as one can show that random objects have several interesting properties.
Li and Vitanyi have written this wonderful monograph on the area covering the depth of theory and applications not seen anywhere else. They give a clear and complete descriptions of many of the important concepts in the book. I have used this book twice in teaching graduate courses on the topic.
This book is a must have for anyone interested in a serious mathematical treatment of Kolmogorov complexity.
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1 of 1 people found the following review helpful
5.0 out of 5 stars The only one of its kind.... Sept. 22 2001
Format:Hardcover
The theory of Kolmogorov complexity attempts to define randomness in terms of the complexity of the program used to compute it. The authors give an excellent overview of this theory, and even discuss some of its philosophical ramifications, but they are always careful to distinguish between mathematical rigor and philosophical speculation. And, interestingly, the authors choose to discuss information theory in physics and the somewhat radical idea of reversible computation. The theory of Kolmogorov complexity is slowly making its way into applications, these being coding theory and computational intelligence, and network performance optimization, and this book serves as a fine reference for those readers interested in these applications. Some of the main points of the book I found interesting include: 1. A very condensed but effective discussion of Turing machines and effective computability. 2. The historical motivation for defining randomness and its defintiion using Kolmogorov complexity. 3. The discussion of coding theory and its relation to information theory. The Shannon-Fano code is discussed, along with prefix codes, Kraft's inequality, the noiseless coding theorem, and universal codes for infinite source word sets. 4. The treatment of algorithmic complexity. The authors stress that the information content of an object must be intrinsic and independent of the means of description. 5. The discussion of the explicit universal randomness test. 6. The discussion (in an exercise) of whether a probabilistic machine can perform a task that is impossible on a deterministic machine. 7. The notion of incompressibility of strings. 8. Read more ›
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0 of 1 people found the following review helpful
5.0 out of 5 stars Excellent if you have the math... Aug. 13 2002
By Zentao
Format:Hardcover
to understand it. This book is intended for serious students of computer science or those who have some similar training - it is definitely set up as a textbook. However, that being said, if you have the background the authors' delivery is fist-class and very clear.
The reviews below give more than enough information so I won't belabour the Kolmogorov complexity here. Suffice it to say you won't find the subject detailed more fully in any other reference work in existence today.
However, this book does need to be revised and updated. There has been a lot of development in the field and the sections overviewing Solomonoff's work, in particular, could be expanded. Also, I found it hard to believe that nothing about the 'philosophical' importance of the whole induction question - this is at the core of many very important questions and should not be treated trivially.
There should also be some overview of two other areas that, in combination with the theory outlined in this text, are starting to form the nexus of a "new kind of science" (definitely not Wolfram's pathetic attempt). I refer to some information regarding non-classical logical systems as well as anticipatory computing systems. Both will, I predict, become core areas in addition to extensions to Kolmogorov/Chaitin complexity in the future.
All textbooks should be as clear and concise as this example.
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0 of 1 people found the following review helpful
Format:Hardcover
Li and Vitanyi have done an admirable job at clarifying some very subtle and deep issues in computational complexity. The organization is clear and natural, and the notation good. While a superb undergraduate might learn from it, I suspect the greatest benefits are to advanced students and practicing professionals.
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0 of 1 people found the following review helpful
5.0 out of 5 stars Comprehensive and Excellent July 30 1999
By A Customer
Format:Hardcover
This is one of the best-written mathematical texts I've read. It builds up the theory from basic principles, and illustrates it with numerous examples and applications. A definitive work.
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