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An Introduction to Kolmogorov Complexity and Its Applications (Texts in Computer Science) Hardcover – 1 Mar 1997


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Product details

  • Hardcover: 637 pages
  • Publisher: Springer; 2nd ed. edition (1 Mar 1997)
  • Language: French
  • ISBN-10: 0387948686
  • ISBN-13: 978-0387948683
  • Product Dimensions: 3.8 x 18.4 x 24.8 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 4,157,612 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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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)

--This text refers to an out of print or unavailable edition of this title.

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]

--This text refers to an out of print or unavailable edition of this title.

Inside This Book (Learn More)
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Most Helpful Customer Reviews

8 of 8 people found the following review helpful By A Customer on 13 Oct 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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5 of 5 people found the following review helpful By A Customer on 30 July 1999
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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2 of 2 people found the following review helpful By C. Rogers on 17 Oct 2010
Format: Hardcover
Aswell as an excellent textbook, this is a reference book for researchers. The third edition has recent updates of the progress that has been made in the field of Kolmogorov complexity.

The best thing about Li and Vitanyi's book is that it is very interesting. For example, it contains quotes about randomness from famous researchers like Laplace which deepens the reader's understanding of Kolmogorov complexity and makes the topic of randomess even more fascinating.

The book explains subjects very well. Many readers will gain a much deeper understanding of the areas of physics covered in this book than they would from a physics textbook. The book has interesting backgrounds to all the subjects which are covered in the book.
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 9 reviews
81 of 82 people found the following review helpful
Biggest return for the biggest investment 7 May 2005
By Randall Helzerman - Published on Amazon.com
Format: Hardcover
This was the second-hardest book I ever read. Honestly, it took me years and years to get through it. I even had to buy a 2nd copy, because I kept getting frustrated and throwing the first copy across the room until it was destroyed. So yes, this book requires a substantial effort to read.

But the payback!! I've gotten more return on investment from this book than from any other book I've ever read. If you dilligently read and master this book, you will be able to analyze and solve problems your collegues just can't.

The basic idea behind Kolmogorov complexity is straighforward: a good measure of the complexity of an object is the length of the shortest computer program which will construct that object. From this basic idea an amazing variety of insights and powerful techniques have been developed, and this book is quite comprehensive in cataloging and explaining them.

For computer scientists and working programmers, probably the most useful result of Kolmogorov complexity would be the "Incompressibility Method", which is a powerful technique for the analysis of the runtime of algorithms. Typically, it is relatively easy to figure out what the best case or the worst case runtime of an algorithm is. Until now, it was hard to calculate the average runtime of an algorithm, because it usually involved a tricky counting problem, to enumerate all possible runs of the the algorithm and summing over them. The incompressibility method eliminates the need for doing these complicated enumerations, by letting you perform the analysis on a single run of the algorithm which is guarunteed to be representative of the average runtime of the algorithm. If you program for a living like I do, this will give you an edge, because if you can accurately predict that the worst-case runtimes almost never happen, you can usually simplify and streamline your programs by optimizing it for the average case. If your competitors are wasting time optimizing for a worst case which almost never happens--at the expense of _not_ optimizing for the average case, you win bigtime.

For philosophers of science and AI/knowledge representation folks, the most useful results of Kolmogorov complexity are probably the contributions of Kolmogorov complexity to Baysianism. To be a Baysian is to follow a two step process: (STEP 1) for every possible sentence, assign to it a number between 0 and 1 which represents how certain you are that that sentence is true. This initial assignment should be a probability distribution over all possible sentences. It should be a "good" probability distrubution, but of course it won't be perfect, since you don't know everything. (STEP 2) when confronted with new evidence, e.g. an observation, update your current "good" degrees of belief by using Bayes' law, to yield a new "better" set of degrees of belief.

The Baysians always had a good story for Step 2--just use Bayes law. But until now, they were mostly hand-waving on Step 1--what would constitude a "good" initial probability distribution? There were many proposals (e.g. maximum entropy) but all proposals had benefits and drawbacks. What Kolmogorov complexity provides is the so-called "universal" distribution, which is guarunteed to be a "good" initial distirbution. This book devotes much time to explaining and exploring this, and shows how previous techniques, like maximum entropy, minimum description length, etc all can be seen as computable approximations to the (unfortunately uncomputable) universal distribution. This really gives a nice framework for evalutating and formulating good prior distributions.

After remarking on how hard this book was to read, I should emphasize that this is not due to bad writing on the part of the authors! Indeed, after throwing the book across the room, I was always drawn back by Li & Vitanyi's most engaging writing style to pick the book back up, dust it off, and have another go at it. If it were not for their wonderul ability to expain a very complicated subject matter, I never would have gotten through it.

An unsung hero of this book is Peter Gacs, who wrote a set of lecture notes which really could be considered to be an Urtext for this book. If you tackle this book, I highly recommend that you also get ahold of these notes, because it is sometimes very useful, when trying to puzzle out a difficult argument, to get another description/explaination of it from a different point of view. These notes are available on the web, just google for "Lecture note on descriptional complexity and randomness" by Peter Gacs.

If you're up to the challange, then buy this book, dilligently read it, swear at it--then swear by it.
24 of 26 people found the following review helpful
The only one of its kind.... 23 Sep 2001
By Dr. Lee D. Carlson - Published on Amazon.com
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. The discussion of randomness in the Diophantine equations; it is shown that the set of indices of the Diophantine equations with infinitely many different solutions is not recursively enumerable; with the initial segment of length n in the characteristic sequence having Kolmogorov complexity n. 9. The discussion on algorithmic probability, especially the test for randomness by martingales. 10. The Solomonoff theory of prediction and its ability to solve the problem of induction. 11. The treatment of Pac-learning and the resultant formalization of Occam's razor. 12. The discussion of compact routing; the optimal space to represent routing schemes in communication networks on the average for all static networks. 13. Computational complexity and its connection to resource-bounded complexity. 14. The notion of logical depth, i.e. the time required by a universal computer to compute the object from its compressed original description. 15. The connection between algorithmic complexity and Shannon's entropy. 16. The discussion on reversible computation, i.e. logically reversible computers that do not dissipate heat. 17. The treatment of information distance, i.e. for two strings, the minimal quantity of information sufficient to translate from one to the other.
16 of 17 people found the following review helpful
THE book on Kolmogorov Complexity 13 Oct 1998
By Lance Fortnow (fortnow@cs.uchicago.edu) - Published on Amazon.com
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.
9 of 9 people found the following review helpful
Excellent if you have the math... 13 Aug 2002
By Zentao - Published on Amazon.com
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.
9 of 9 people found the following review helpful
A must 29 Oct 2003
By Volker Nannen - Published on Amazon.com
Format: Hardcover
The book provides all the tools needed for a productive use of the theory. Written by leading experts in the field, the book is both a fascinating introduction as well as a comprehensive reference for experts.
The authors are careful to place the development of the theory in its historical context, give a face to the main players in the field and explore frictions with other lines of thought. But the main storyline is the mathematical world of Kolmogorov complexity. Neccessary background knowledge is provided, most proofs are given and the open problems are presented. Most chapters are more or less self sufficient, making it possible to skip those that are of less relevance to you. In the later chapters much thought is given to the different fields of application.
A third edition is in the making which will include recent advances. But since the authors make new discoveries available on the web, the present edition will continue for a long time to hold a prominent place in the book shelves of many computer scientist.
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