Computability And Randomness

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Computability and Randomness

André Nies
Computability and Randomness Book PDF

Author : André Nies
Publisher : OUP Oxford
Page : 450 pages
File Size : 46,9 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.

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 : 44,5 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.

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 : 55,7 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.

Structure And Randomness In Computability And Set Theory

Douglas Cenzer,Christopher Porter,Jindrich Zapletal
Structure And Randomness In Computability And Set Theory Book PDF

Author : Douglas Cenzer,Christopher Porter,Jindrich Zapletal
Publisher : World Scientific
Page : 387 pages
File Size : 45,6 Mb
Release : 2020-10-02
Category : Mathematics
ISBN : 9789813228245

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Structure And Randomness In Computability And Set Theory by Douglas Cenzer,Christopher Porter,Jindrich Zapletal Pdf

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.

Randomness Through Computation

Hector Zenil
Randomness Through Computation Book PDF

Author : Hector Zenil
Publisher : World Scientific
Page : 439 pages
File Size : 52,7 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.

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 : 43,7 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.

Some Results on Algorithmic Randomness and Computability-theoretic Strength

Anonim
Some Results on Algorithmic Randomness and Computability theoretic Strength Book PDF

Author : Anonim
Publisher : Unknown
Page : 0 pages
File Size : 54,8 Mb
Release : 2014
Category : Electronic
ISBN : OCLC:890708623

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Some Results on Algorithmic Randomness and Computability-theoretic Strength by Anonim Pdf

Algorithmic randomness uses tools from computability theory to give precise formulations for what it means for mathematical objects to be random. When the objects in question are reals (infinite sequences of zeros and ones), it reveals complex interactions between how random they are and how useful they are as computational oracles. The results in this thesis are primarily on interactions of this nature. Chapter 1 provides a brief introduction to notation and basic notions from computability theory. Chapter 2 is on shift-complex sequences, also known as everywhere complex sequences. These are sequences all of whose substrings have uniformly high prefix-free Kolmogorov complexity. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is 0. We refine this result to show that the Martin-Löf random sequences that compute shift-complex sequences compute the halting problem. In the other direction, we answer the question of whether every Martin-Löf random sequence computes a shift-complex sequence in the negative by translating it into a question about diagonally noncomputable (or DNC) functions. The key in this result is analyzing how growth rates of DNC functions affect what they can compute. This is the subject of Chapter 3. Using bushy-tree forcing, we show (with J. Miller) that there are arbitrarily slow-growing (but unbounded) DNC functions that fail to compute a Kurtz random sequence. We also extend Kumabe's result that there is a DNC function of minimal Turing degree by showing that for every oracle X, there is a function f that is DNC relative to X and of minimal Turing degree. Chapter 4 is on how "effective" Lebesgue density interacts with computability-theoretic strength and randomness. Bienvenu, Hölzl, Miller, and Nies showed that if we restrict our attention to the Martin-Löf random sequences, then the positive density sequences are exactly the ones that do not compute the halting problem. We prove several facts around this theorem. For example, one direction of the theorem fails without the assumption of Martin-Löf randomness: Given any sequence X, there is a density-one sequence Y that computes it. Another question we answer is whether a positive density point can have minimal degree. It turns out that every such point is either Martin-Löf random, or computes a 1-generic. In either case, it is nonminimal.

Randomness & Undecidability in Physics

Karl Svozil
Randomness Undecidability in Physics Book PDF

Author : Karl Svozil
Publisher : World Scientific
Page : 314 pages
File Size : 52,6 Mb
Release : 1993
Category : Science
ISBN : 981020809X

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Randomness & Undecidability in Physics by Karl Svozil Pdf

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.

Handbook of Computability and Complexity in Analysis

Vasco Brattka,Peter Hertling
Handbook of Computability and Complexity in Analysis Book PDF

Author : Vasco Brattka,Peter Hertling
Publisher : Springer Nature
Page : 427 pages
File Size : 51,8 Mb
Release : 2021-06-04
Category : Computers
ISBN : 9783030592349

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Handbook of Computability and Complexity in Analysis by Vasco Brattka,Peter Hertling Pdf

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

The Incomputable

S. Barry Cooper,Mariya I. Soskova
The Incomputable Book PDF

Author : S. Barry Cooper,Mariya I. Soskova
Publisher : Springer
Page : 292 pages
File Size : 52,6 Mb
Release : 2017-05-05
Category : Computers
ISBN : 9783319436692

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The Incomputable by S. Barry Cooper,Mariya I. Soskova Pdf

This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions, but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines, and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world. The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness; and the mathematics of emergence and morphogenesis. This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy.

Information and Randomness

Cristian Calude
Information and Randomness Book PDF

Author : Cristian Calude
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 51,9 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 Theory

S. Barry Cooper
Computability Theory Book PDF

Author : S. Barry Cooper
Publisher : CRC Press
Page : 420 pages
File Size : 52,5 Mb
Release : 2017-09-06
Category : Mathematics
ISBN : 9781351991964

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Computability Theory by S. Barry Cooper Pdf

Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.

Ordinal Computability

Merlin Carl
Ordinal Computability Book PDF

Author : Merlin Carl
Publisher : Walter de Gruyter GmbH & Co KG
Page : 343 pages
File Size : 47,8 Mb
Release : 2019-09-23
Category : Mathematics
ISBN : 9783110496154

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Ordinal Computability by Merlin Carl Pdf

Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics are covered.

Turing Computability

Robert I. Soare
Turing Computability Book PDF

Author : Robert I. Soare
Publisher : Springer
Page : 263 pages
File Size : 49,5 Mb
Release : 2016-06-20
Category : Computers
ISBN : 9783642319334

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Turing Computability by Robert I. Soare Pdf

Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part II includes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory. The author has honed the content over decades according to feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic.

Computability Theory

Rebecca Weber
Computability Theory Book PDF

Author : Rebecca Weber
Publisher : American Mathematical Soc.
Page : 203 pages
File Size : 51,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821873922

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Computability Theory by Rebecca Weber Pdf

What can we compute--even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory. The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.