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

This must-read textbook presents an essential introduction to Kolmogorov complexity (KC), a central theory and powerful tool in information science that deals with the quantity of information in individual objects. The text covers both the fundamental concepts and the most important practical applications, supported by a wealth of didactic features.This thoroughly revised and enhanced fourth edition includes new and updated material on, amongst other topics, the Miller-Yu theorem, the Gács-Kučera theorem, the Day-Gács theorem, increasing randomness, short lists computable from an input string containing the incomputable Kolmogorov complexity of the input, the Lovász local lemma, sorting, the algorithmic full Slepian-Wolf theorem for individual strings, multiset normalized information distance and normalized web distance, and conditional universal distribution. Topics and features:Describes the mathematical theory of KC, including the theoriesof algorithmic complexity and algorithmic probabilityPresents a general theory of inductive reasoning and its applications, and reviews the utility of the incompressibility methodCovers the practical application of KC in great detail, including the normalized information distance (the similarity metric) and information diameter of multisets in phylogeny, language trees, music, heterogeneous files, and clusteringDiscusses the many applications of resource-bounded KC, and examines different physical theories from a KC point of viewIncludes numerous examples that elaborate the theory, and a range of exercises of varying difficulty (with solutions)Offers explanatory asides on technical issues, and extensive historical sectionsSuggests structures for several one-semester courses in the prefaceAs the definitive textbook on Kolmogorov complexity, this comprehensive and self-contained work is an invaluable resource for advanced undergraduate students, graduate students, and researchers in all fields of science.“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.

Develops Kolmogorov theory in detail, and outlines a wide range of illustrative applications Examines major results from prominent researchers in the field Details the practical application of KC in the similarity metric and information diameter of multisets in phylogeny, language trees, music, heterogeneous files, and clustering Includes new and updated material on the Miller-Yu theorem, the Gács-Kucera theorem, the Day-Gács theorem, the Lovász local lemma, and the Slepian-Wolf theorem Discusses short lists computable from an input string containing the incomputable Kolmogorov complexity of the input Covers topics of increasing randomness, sorting, multiset normalized information distance and normalized web distance, and conditional universal distribution Includes supplementary material: sn.pub/extras