Algorithmic Randomness And Complexity

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Algorithmic Randomness and Complexity

Rodney G. Downey,Denis R. Hirschfeldt
Algorithmic Randomness and Complexity Book PDF

Author : Rodney G. Downey,Denis R. Hirschfeldt
Publisher : Springer Science & Business Media
Page : 855 pages
File Size : 41,9 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.

Kolmogorov Complexity and Algorithmic Randomness

A. Shen,V. A. Uspensky,N. Vereshchagin
Kolmogorov Complexity and Algorithmic Randomness Book PDF

Author : A. Shen,V. A. Uspensky,N. Vereshchagin
Publisher : American Mathematical Society
Page : 511 pages
File Size : 49,6 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.

Algorithmic Randomness

Johanna N. Y. Franklin,Christopher P. Porter
Algorithmic Randomness Book PDF

Author : Johanna N. Y. Franklin,Christopher P. Porter
Publisher : Cambridge University Press
Page : 370 pages
File Size : 49,7 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.

An Introduction to Kolmogorov Complexity and Its Applications

Ming Li,Paul Vitanyi
An Introduction to Kolmogorov Complexity and Its Applications Book PDF

Author : Ming Li,Paul Vitanyi
Publisher : Springer Science & Business Media
Page : 655 pages
File Size : 42,8 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).

Information and Randomness

Cristian Calude
Information and Randomness Book PDF

Author : Cristian Calude
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 48,6 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.

Computability and Randomness

André Nies
Computability and Randomness Book PDF

Author : André Nies
Publisher : OUP Oxford
Page : 450 pages
File Size : 54,5 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.

Kolmogorov Complexity and Algorithmic Randomness

A. Shen
Kolmogorov Complexity and Algorithmic Randomness Book PDF

Author : A. Shen
Publisher : Unknown
Page : 534 pages
File Size : 43,7 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.

Computational Complexity

Sanjeev Arora,Boaz Barak
Computational Complexity Book PDF

Author : Sanjeev Arora,Boaz Barak
Publisher : Cambridge University Press
Page : 609 pages
File Size : 52,5 Mb
Release : 2009-04-20
Category : Computers
ISBN : 9780521424264

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Computational Complexity by Sanjeev Arora,Boaz Barak Pdf

New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

Randomness and Complexity

Cristian Calude,Gregory J. Chaitin
Randomness and Complexity Book PDF

Author : Cristian Calude,Gregory J. Chaitin
Publisher : World Scientific
Page : 466 pages
File Size : 54,9 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.

Information, Randomness & Incompleteness

Gregory J. Chaitin
Information Randomness Incompleteness Book PDF

Author : Gregory J. Chaitin
Publisher : World Scientific
Page : 292 pages
File Size : 46,6 Mb
Release : 1987
Category : Computers
ISBN : 9971504790

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Information, Randomness & Incompleteness by Gregory J. Chaitin Pdf

The papers gathered in this book were published over a period of more than twenty years in widely scattered journals. They led to the discovery of randomness in arithmetic which was presented in the recently published monograph on ?Algorithmic Information Theory? by the author. There the strongest possible version of G”del's incompleteness theorem, using an information-theoretic approach based on the size of computer programs, was discussed. The present book is intended as a companion volume to the monograph and it will serve as a stimulus for work on complexity, randomness and unpredictability, in physics and biology as well as in metamathematics.

Pairwise Independence and Derandomization

Michael Luby,Michael George Luby,Avi Wigderson
Pairwise Independence and Derandomization Book PDF

Author : Michael Luby,Michael George Luby,Avi Wigderson
Publisher : Now Publishers Inc
Page : 80 pages
File Size : 43,7 Mb
Release : 2006
Category : Computers
ISBN : 9781933019222

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Pairwise Independence and Derandomization by Michael Luby,Michael George Luby,Avi Wigderson Pdf

Pairwise Independence and Derandomization gives several applications of the following paradigm, which has proven extremely powerful in algorithm design and computational complexity. First, design a probabilistic algorithm for a given problem. Then, show that the correctness analysis of the algorithm remains valid even when the random strings used by the algorithm do not come from the uniform distribution, but rather from a small sample space, appropriately chosen. In some cases this can be proven directly (giving "unconditional derandomization"), and in others it uses computational assumptions, like the existence of 1-way functions (giving "conditional derandomization"). Pairwise Independence and Derandomization is self contained, and is a prime manifestation of the "derandomization" paradigm. It is intended for scholars and graduate students in the field of theoretical computer science interested in randomness, derandomization and their interplay with computational complexity.

Theory of Computational Complexity

Ding-Zhu Du,Ker-I Ko
Theory of Computational Complexity Book PDF

Author : Ding-Zhu Du,Ker-I Ko
Publisher : John Wiley & Sons
Page : 511 pages
File Size : 49,7 Mb
Release : 2011-10-24
Category : Mathematics
ISBN : 9781118031162

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Theory of Computational Complexity by Ding-Zhu Du,Ker-I Ko Pdf

A complete treatment of fundamentals and recent advances in complexity theory Complexity theory studies the inherent difficulties of solving algorithmic problems by digital computers. This comprehensive work discusses the major topics in complexity theory, including fundamental topics as well as recent breakthroughs not previously available in book form. Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography. It also examines the theory of nonuniform computational complexity, including the computational models of decision trees and Boolean circuits, and the notion of polynomial-time isomorphism. The theory of probabilistic complexity, which studies complexity issues related to randomized computation as well as interactive proof systems and probabilistically checkable proofs, is also covered. Extraordinary in both its breadth and depth, this volume: * Provides complete proofs of recent breakthroughs in complexity theory * Presents results in well-defined form with complete proofs and numerous exercises * Includes scores of graphs and figures to clarify difficult material An invaluable resource for researchers as well as an important guide for graduate and advanced undergraduate students, Theory of Computational Complexity is destined to become the standard reference in the field.

Randomness Through Computation

Hector Zenil
Randomness Through Computation Book PDF

Author : Hector Zenil
Publisher : World Scientific
Page : 439 pages
File Size : 43,6 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.

Algorithmic Learning in a Random World

Vladimir Vovk,Alexander Gammerman,Glenn Shafer
Algorithmic Learning in a Random World Book PDF

Author : Vladimir Vovk,Alexander Gammerman,Glenn Shafer
Publisher : Springer Science & Business Media
Page : 344 pages
File Size : 49,6 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.

Complexity and Randomness in Group Theory

Frédérique Bassino,Ilya Kapovich,Markus Lohrey,Alexei Miasnikov,Cyril Nicaud,Andrey Nikolaev,Igor Rivin,Vladimir Shpilrain,Alexander Ushakov,Pascal Weil
Complexity and Randomness in Group Theory Book PDF

Author : Frédérique Bassino,Ilya Kapovich,Markus Lohrey,Alexei Miasnikov,Cyril Nicaud,Andrey Nikolaev,Igor Rivin,Vladimir Shpilrain,Alexander Ushakov,Pascal Weil
Publisher : Walter de Gruyter GmbH & Co KG
Page : 386 pages
File Size : 40,5 Mb
Release : 2020-06-08
Category : Mathematics
ISBN : 9783110667028

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Complexity and Randomness in Group Theory by Frédérique Bassino,Ilya Kapovich,Markus Lohrey,Alexei Miasnikov,Cyril Nicaud,Andrey Nikolaev,Igor Rivin,Vladimir Shpilrain,Alexander Ushakov,Pascal Weil Pdf

This book shows new directions in group theory motivated by computer science. It reflects the transition from geometric group theory to group theory of the 21st century that has strong connections to computer science. Now that geometric group theory is drifting further and further away from group theory to geometry, it is natural to look for new tools and new directions in group theory which are present.