A Level Of Martin Lof Randomness

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Randomness And Undecidability In Physics

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Author : Karl Svozil
language : en
Publisher: World Scientific
Release Date : 1993-10-05

Randomness And Undecidability In Physics written by Karl Svozil and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-10-05 with Science categories.

Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua.Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the physical properties of its hardware. Computers are a medium to virtual realities. The foreseeable importance of such virtual realities stimulates the investigation of an “inner description”, a “virtual physics” of these universes of computation. Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery.One motive of this book is the recognition that what is often referred to as “randomness” in physics might actually be a signature of undecidability for systems whose evolution is computable on a step-by-step basis. To give a flavour of the type of questions envisaged: Consider an arbitrary algorithmic system which is computable on a step-by-step basis. Then it is in general impossible to specify a second algorithmic procedure, including itself, which, by experimental input-output analysis, is capable of finding the deterministic law of the first system. But even if such a law is specified beforehand, it is in general impossible to predict the system behaviour in the “distant future”. In other words: no “speedup” or “computational shortcut” is available. In this approach, classical paradoxes can be formally translated into no-go theorems concerning intrinsic physical perception.It is suggested that complementarity can be modelled by experiments on finite automata, where measurements of one observable of the automaton destroys the possibility to measure another observable of the same automaton and it vice versa.Besides undecidability, a great part of the book is dedicated to a formal definition of randomness and entropy measures based on algorithmic information theory.

Algorithmic Randomness And Complexity

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Author : Rodney G. Downey
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-29

Algorithmic Randomness And Complexity written by Rodney G. Downey and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-29 with Computers categories.

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.

Information And Randomness

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Author : Cristian Calude
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

Information And Randomness written by Cristian Calude and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Computers categories.

"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.

Algorithmic Randomness

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Author : Johanna N. Y. Franklin
language : en
Publisher: Cambridge University Press
Release Date : 2020-05-07

Algorithmic Randomness written by Johanna N. Y. Franklin and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-07 with Computers categories.

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

Randomness Through Computation Some Answers More Questions

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Author : Hector Zenil
language : en
Publisher: World Scientific
Release Date : 2011-02-11

Randomness Through Computation Some Answers More Questions written by Hector Zenil and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-02-11 with Computers categories.

This review volume consists of a set of chapters written by leading scholars, most of them founders of their fields. It explores the connections of Randomness to other areas of scientific knowledge, especially its fruitful relationship to Computability and Complexity Theory, and also to areas such as Probability, Statistics, Information Theory, Biology, Physics, Quantum Mechanics, Learning Theory and Artificial Intelligence. The contributors cover these topics without neglecting important philosophical dimensions, sometimes going beyond the purely technical to formulate age old questions relating to matters such as determinism and free will.The scope of Randomness Through Computation is novel. Each contributor shares their personal views and anecdotes on the various reasons and motivations which led them to the study of Randomness. Using a question and answer format, they share their visions from their several distinctive vantage points.

Introductory Statistics And Random Phenomena

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Author : Manfred Denker
language : en
Publisher: Birkhäuser
Release Date : 2017-09-16

Introductory Statistics And Random Phenomena written by Manfred Denker and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-16 with Computers categories.

This textbook integrates traditional statistical data analysis with new computational experimentation capabilities and concepts of algorithmic complexity and chaotic behavior in nonlinear dynamic systems. This was the first advanced text/reference to bring together such a comprehensive variety of tools for the study of random phenomena occurring in engineering and the natural, life, and social sciences. The crucial computer experiments are conducted using the readily available computer program Mathematica® Uncertain Virtual WorldsTM software packages which optimize and facilitate the simulation environment. Brief tutorials are included that explain how to use the Mathematica® programs for effective simulation and computer experiments. Large and original real-life data sets are introduced and analyzed as a model for independent study. This is an excellent classroom tool and self-study guide. The material is presented in a clear and accessible style providing numerous exercises and bibliographical notes suggesting further reading. Topics and Features Comprehensive and integrated treatment of uncertainty arising in engineering and scientific phenomena – algorithmic complexity, statistical independence, and nonlinear chaotic behavior Extensive exercise sets, examples, and Mathematica® computer experiments that reinforce concepts and algorithmic methods Thorough presentation of methods of data compression and representation Algorithmic approach to model selection and design of experiments Large data sets and 13 Mathematica®-based Uncertain Virtual WorldsTM programs and code This text is an excellent resource for all applied statisticians, engineers, and scientists who need to use modern statistical analysis methods to investigate and model their data. The present, softcover reprint is designed to make this classic textbook available to a wider audience.

Abrahamic Reflections On Randomness And Providence

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Author : Kelly James Clark
language : en
Publisher: Springer Nature
Release Date : 2021-09-27

Abrahamic Reflections On Randomness And Providence written by Kelly James Clark and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-27 with Religion categories.

This open access book addresses the question of how God can providentially govern apparently ungovernable randomness. Medieval theologians confidently held that God is provident, that is, God is the ultimate cause of or is responsible for everything that happens. However, scientific advances since the 19th century pose serious challenges to traditional views of providence. From Darwinian evolution to quantum mechanics, randomness has become an essential part of the scientific worldview. An interdisciplinary team of Muslim, Christian and Jewish scholars—biologists, physicists, philosophers and theologians—addresses questions of randomness and providence.

Structure And Randomness In Computability And Set Theory

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Author : Douglas Cenzer
language : en
Publisher: World Scientific
Release Date : 2020-10-02

Structure And Randomness In Computability And Set Theory written by Douglas Cenzer and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-02 with Mathematics categories.

This volume presents some exciting new developments occurring on the interface between set theory and computability as well as their applications in algebra, analysis and topology. These include effective versions of Borel equivalence, Borel reducibility and Borel determinacy. It also covers algorithmic randomness and dimension, Ramsey sets and Ramsey spaces. Many of these topics are being discussed in the NSF-supported annual Southeastern Logic Symposium.

An Introduction To Kolmogorov Complexity And Its Applications

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Author : Ming Li
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

An Introduction To Kolmogorov Complexity And Its Applications written by Ming Li and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.

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).

Kolmogorov Complexity And Computational Complexity

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Author : Osamu Watanabe
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Kolmogorov Complexity And Computational Complexity written by Osamu Watanabe and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Computers categories.

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.