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.

Preliminaries.- Algorithmic Complexity.- Algorithmic Prefix Complexity.- Algorithmic Probability.- Inductive Reasoning.- The Incompressibility Method.- Resource-Bounded Complexity.- Physics, Information, and Computation.

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]

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 thisthird 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)

Develops Kolmogorov theory in detail and outlines the wide range of illustrative applications Presents recent, major results in the field; topics include, Omega numbers, Kolmogorov-Loveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information Introduces results from prominent researchers in the field, such as those found by Slaman-Kucera, Downey, Hirschfeldt, Nies, Solovay, Miller and many more Includes applications to hierarchical clustering, phylogeny in genomics, classification of music, Google, and the World Wide Web Additional exercises on new results Improved sections on high-probability properties and Kolmogorov’s structure function, information distance Includes supplementary material: sn.pub/extras