Kolmogorov Complexity And Algorithmic Randomness

Author : A. Shen,V. A. Uspensky,N. Vereshchagin
Publisher : American Mathematical Society
Page : 511 pages
File Size : 55,9 Mb
Release : 2022-05-18
Category : Mathematics
ISBN : 9781470470647

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Kolmogorov Complexity and Algorithmic Randomness by A. Shen,V. A. Uspensky,N. Vereshchagin Pdf

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 textbook-style 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.

Author : Rodney G. Downey,Denis R. Hirschfeldt
Publisher : Springer Science & Business Media
Page : 855 pages
File Size : 49,8 Mb
Release : 2010-10-29
Category : Computers
ISBN : 9780387684413

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Algorithmic Randomness and Complexity by Rodney G. Downey,Denis R. Hirschfeldt Pdf

Computability and complexity theory are two central areas of research in theoretical computer science. This book provides a systematic, technical development of "algorithmic randomness" and complexity for scientists from diverse fields.

Author : Ming Li,Paul Vitanyi
Publisher : Springer Science & Business Media
Page : 655 pages
File Size : 52,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475726060

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An Introduction to Kolmogorov Complexity and Its Applications by Ming Li,Paul Vitanyi Pdf

Briefly, we review the basic elements of computability theory and prob ability theory that are required. Finally, in order to place the subject in the appropriate historical and conceptual context we trace the main roots of Kolmogorov complexity. This way the stage is set for Chapters 2 and 3, where we introduce the notion of optimal effective descriptions of objects. The length of such a description (or the number of bits of information in it) is its Kolmogorov complexity. We treat all aspects of the elementary mathematical theory of Kolmogorov complexity. This body of knowledge may be called algo rithmic complexity theory. The theory of Martin-Lof tests for random ness of finite objects and infinite sequences is inextricably intertwined with the theory of Kolmogorov complexity and is completely treated. We also investigate the statistical properties of finite strings with high Kolmogorov complexity. Both of these topics are eminently useful in the applications part of the book. We also investigate the recursion theoretic properties of Kolmogorov complexity (relations with Godel's incompleteness result), and the Kolmogorov complexity version of infor mation theory, which we may call "algorithmic information theory" or "absolute information theory. " The treatment of algorithmic probability theory in Chapter 4 presup poses Sections 1. 6, 1. 11. 2, and Chapter 3 (at least Sections 3. 1 through 3. 4).

Author : A. Shen
Publisher : Unknown
Page : 534 pages
File Size : 44,7 Mb
Release : 2017
Category : MATHEMATICS
ISBN : 1470440830

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Kolmogorov Complexity and Algorithmic Randomness by A. Shen Pdf

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 textbook-style exposition of the basic notions of complexity and randomness; the second part cover.

Author : Johanna N. Y. Franklin,Christopher P. Porter
Publisher : Cambridge University Press
Page : 370 pages
File Size : 46,8 Mb
Release : 2020-05-07
Category : Computers
ISBN : 9781108478984

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Algorithmic Randomness by Johanna N. Y. Franklin,Christopher P. Porter Pdf

Surveys on recent developments in the theory of algorithmic randomness and its interactions with other areas of mathematics.

Author : Osamu Watanabe
Publisher : Springer Science & Business Media
Page : 111 pages
File Size : 54,9 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9783642777356

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Kolmogorov Complexity and Computational Complexity by Osamu Watanabe Pdf

The mathematical theory of computation has given rise to two important ap proaches to the informal notion of "complexity": Kolmogorov complexity, usu ally a complexity measure for a single object such as a string, a sequence etc., measures the amount of information necessary to describe the object. Compu tational complexity, usually a complexity measure for a set of objects, measures the compuational resources necessary to recognize or produce elements of the set. The relation between these two complexity measures has been considered for more than two decades, and may interesting and deep observations have been obtained. In March 1990, the Symposium on Theory and Application of Minimal Length Encoding was held at Stanford University as a part of the AAAI 1990 Spring Symposium Series. Some sessions of the symposium were dedicated to Kolmogorov complexity and its relations to the computational complexity the ory, and excellent expository talks were given there. Feeling that, due to the importance of the material, some way should be found to share these talks with researchers in the computer science community, I asked the speakers of those sessions to write survey papers based on their talks in the symposium. In response, five speakers from the sessions contributed the papers which appear in this book.

Author : Bradley S. Tice
Publisher : CRC Press
Page : 98 pages
File Size : 43,5 Mb
Release : 2022-09-01
Category : Science
ISBN : 9781000797152

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Aspects of Kolmogorov Complexity the Physics of Information by Bradley S. Tice Pdf

The research presented in Aspects of Kolmogorov Complexity addresses the fundamental standard of defining randomness as measured by a Martin-Lof level of randomness as found in random sequential binary strings. A classical study of statistics that addresses both a fundamental standard of statistics as well as an applied measure for statistical communication theory. The research points to compression levels in a random state that are greater than is found in current literature. A historical overview of the field of Kolmogorov Complexity and Algorithmic Information Theory, a subfield of Information Theory, is given as well as examples using a radix 3, radix 4, and radix 5 base numbers for both random and non-random sequential strings. The text also examines monochromatic and chromatic symbols and both theoretical and applied aspects of data compression as they relate to the transmission and storage of information. The appendix contains papers on the subject given at conferences and the references are current.ContentsTechnical topics addressed in Aspects of Kolmogorov Complexity include:• Statistical Communication Theory• Algorithmic Information Theory• Kolmogorov Complexity• Martin-Lof Randomness• Compression, Transmission and Storage of Information

Author : Cristian Calude
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 41,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662030493

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Information and Randomness by Cristian Calude Pdf

"Algorithmic information theory (AIT) is the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously", says G.J. Chaitin, one of the fathers of this theory of complexity and randomness, which is also known as Kolmogorov complexity. It is relevant for logic (new light is shed on Gödel's incompleteness results), physics (chaotic motion), biology (how likely is life to appear and evolve?), and metaphysics (how ordered is the universe?). This book, benefiting from the author's research and teaching experience in Algorithmic Information Theory (AIT), should help to make the detailed mathematical techniques of AIT accessible to a much wider audience.

Author : Hector Zenil
Publisher : World Scientific
Page : 439 pages
File Size : 41,6 Mb
Release : 2011
Category : Computers
ISBN : 9789814327749

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Randomness Through Computation by Hector Zenil Pdf

This review volume consists of an indispensable set of chapters written by leading scholars, scientists and researchers in the field of Randomness, including related subfields specially but not limited to the strong developed connections to the Computability and Recursion Theory. Highly respected, indeed renowned in their areas of specialization, many of these contributors are the founders of their fields. The scope of Randomness Through Computation is novel. Each contributor shares his personal views and anecdotes on the various reasons and motivations which led him to the study of the subject. They share their visions from their vantage and distinctive viewpoints. In summary, this is an opportunity to learn about the topic and its various angles from the leading thinkers.

Author : André Nies
Publisher : OUP Oxford
Page : 450 pages
File Size : 45,6 Mb
Release : 2012-03-29
Category : Mathematics
ISBN : 9780191627880

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Computability and Randomness by André Nies Pdf

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability. Both the basics and recent research results are desribed, providing a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

Author : Osamu Watanabe
Publisher : Unknown
Page : 120 pages
File Size : 54,7 Mb
Release : 1992-10-26
Category : Electronic
ISBN : 3642777368

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Kolmogorov Complexity and Computational Complexity by Osamu Watanabe Pdf

There are many ways to measure the complexity of a given object, but there are two measures of particular importance in the theory of computing: One is Kolmogorov complexity, which measures the amount of information necessary to describe an object. Another is computational complexity, which measures the computational resources necessary to recognize (or produce) an object. The relation between these two complexity measures has been studied since the 1960s. More recently, the more generalized notion of resource bounded Kolmogorov complexity and its relation to computational complexity have received much attention. Now many interesting and deep observations on this topic have been established. This book consists of four survey papers concerning these recent studies on resource bounded Kolmogorov complexity and computational complexity. It also contains one paper surveying several types of Kolmogorov complexity measures. The papers are based on invited talks given at the AAAI Spring Symposium on Minimal-Length Encoding in 1990. The book is the only collection of survey papers on this subject and provides fundamental information for researchers in the field.

Author : Vladimir Vovk,Alexander Gammerman,Glenn Shafer
Publisher : Springer Science & Business Media
Page : 344 pages
File Size : 53,8 Mb
Release : 2005-03-22
Category : Computers
ISBN : 0387001522

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Algorithmic Learning in a Random World by Vladimir Vovk,Alexander Gammerman,Glenn Shafer Pdf

Algorithmic Learning in a Random World describes recent theoretical and experimental developments in building computable approximations to Kolmogorov's algorithmic notion of randomness. Based on these approximations, a new set of machine learning algorithms have been developed that can be used to make predictions and to estimate their confidence and credibility in high-dimensional spaces under the usual assumption that the data are independent and identically distributed (assumption of randomness). Another aim of this unique monograph is to outline some limits of predictions: The approach based on algorithmic theory of randomness allows for the proof of impossibility of prediction in certain situations. The book describes how several important machine learning problems, such as density estimation in high-dimensional spaces, cannot be solved if the only assumption is randomness.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 41,7 Mb
Release : 2024-06-01
Category : Electronic
ISBN : 9789814474399

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Randomness and Complexity, From Leibniz to Chaitin by Anonim Pdf

Author : Bradley S. Tice
Publisher : CRC Press
Page : 130 pages
File Size : 45,5 Mb
Release : 2012-10-09
Category : Mathematics
ISBN : 9781466565616

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A Level of Martin-Lof Randomness by Bradley S. Tice Pdf

This work addresses the notion of compression ratios greater than what has been known for random sequential strings in binary and larger radix-based systems as applied to those traditionally found in Kolmogorov complexity. A culmination of the author's decade-long research that began with his discovery of a compressible random sequential string, th

Author : Cristian Calude,Gregory J. Chaitin
Publisher : World Scientific
Page : 466 pages
File Size : 55,8 Mb
Release : 2007
Category : Science
ISBN : 9789812770820

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Randomness and Complexity by Cristian Calude,Gregory J. Chaitin Pdf

The book is a collection of papers written by a selection of eminent authors from around the world in honour of Gregory Chaitin's 60th birthday. This is a unique volume including technical contributions, philosophical papers and essays.