Computability and Randomness

The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability.

Author: André Nies

Publisher: OUP Oxford

ISBN: 0191627887

Category: Philosophy

Page: 456

View: 909

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.

Logic Computability and Randomness

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Page: 557

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Algorithmic Randomness and Complexity

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: Rodney G. Downey

Publisher: Springer Science & Business Media

ISBN: 0387684417

Category: Computers

Page: 855

View: 139

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.

Degrees of Computability and Randomness

Author: Anthony William Morphett

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Page: 139

View: 558


Structure And Randomness In Computability And Set Theory

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.

Author: Douglas Cenzer

Publisher: World Scientific

ISBN: 9813228245

Category: Mathematics

Page: 388

View: 310

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.

Algorithmic Randomness and Complexity

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: Rodney G. Downey

Publisher: Springer

ISBN: 9780387955674

Category: Computers

Page: 855

View: 376

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

This volume surveys much of the recent work that has not been included in published volumes until now.

Author: Johanna N. Y. Franklin

Publisher: Cambridge University Press

ISBN: 1108808271

Category: Mathematics

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View: 714

The last two decades have seen a wave of exciting new developments in the theory of algorithmic randomness and its applications to other areas of mathematics. This volume surveys much of the recent work that has not been included in published volumes until now. It contains a range of articles on algorithmic randomness and its interactions with closely related topics such as computability theory and computational complexity, as well as wider applications in areas of mathematics including analysis, probability, and ergodic theory. In addition to being an indispensable reference for researchers in algorithmic randomness, the unified view of the theory presented here makes this an excellent entry point for graduate students and other newcomers to the field.

Computability Complexity and Randomness

Author: Rod G. Downey

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Page: 178

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Special Issue on Computability Complexity and Randomness

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Page: 154

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Some Results on Algorithmic Randomness and Computability theoretic Strength

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.

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Page: 93

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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.

Special Issue on Computability Complexity and Randomness

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On Randomness Determinism and Computability

This paper discusses the meaning and relationship of randomness and determinism. A fundamental development of chaotic dynamical systems is given with examples.

Author: Edward J. Wegman

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Page: 21

View: 936

This paper discusses the meaning and relationship of randomness and determinism. A fundamental development of chaotic dynamical systems is given with examples. Such systems are seen to exhibit randomness in the usual sense of unpredictability. The formal definition of randomness in terms of algorithmic incompressibility is also discussed. The role of recursion in computability and randomness is also discussed. Keywords: Random; Chaos; Chaotic dynamics; Computability; Recursive functions; Unsolvability; Degrees of Randomness; Random number generators. (Author).

Handbook of Computability and Complexity in Analysis

This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field.

Author: Vasco Brattka

Publisher: Springer

ISBN: 9783030592332

Category: Computers

Page: 427

View: 948

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.

Information and Randomness

Remarkably, however, the text is so self-contained and coherent that the book may also serve as a textbook. All proofs are given in the book and, thus, it is not necessary to consult other sources for classroom instruction.

Author: Cristian S. Calude

Publisher: Springer Science & Business Media

ISBN: 3662049783

Category: Mathematics

Page: 468

View: 911

The first edition of the monograph Information and Randomness: An Algorithmic Perspective by Crist ian Calude was published in 1994. In my Foreword I said: "The research in algorithmic information theory is already some 30 years old. However, only the recent years have witnessed a really vigorous growth in this area. . . . The present book by Calude fits very well in our series. Much original research is presented. . . making the approach richer in consequences than the classical one. Remarkably, however, the text is so self-contained and coherent that the book may also serve as a textbook. All proofs are given in the book and, thus, it is not necessary to consult other sources for classroom instruction. " The vigorous growth in the study of algorithmic information theory has continued during the past few years, which is clearly visible in the present second edition. Many new results, examples, exercises and open prob lems have been added. The additions include two entirely new chapters: "Computably Enumerable Random Reals" and "Randomness and Incom pleteness". The really comprehensive new bibliography makes the book very valuable for a researcher. The new results about the characterization of computably enumerable random reals, as well as the fascinating Omega Numbers, should contribute much to the value of the book as a textbook. The author has been directly involved in these results that have appeared in the prestigious journals Nature, New Scientist and Pour la Science.

Patterns and Probabilities

This dissertation connects the theory of algorithmic randomness--a branch of computability theory--with the foundations of induction.

Author: Francesca Zaffora Blando

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This dissertation connects the theory of algorithmic randomness--a branch of computability theory--with the foundations of induction. Algorithmic randomness provides a mathematical analysis of the notion of a sequence displaying no effective regularities. In my dissertation, I investigate the role that algorithmic randomness plays in inductive learning when randomness is taken to be a property of sequences of observations (or data streams) and the learners are computationally limited. In the first chapter, I show that the algorithmically random data streams are exactly the ones that ensure that a computable Bayesian agent's beliefs will asymptotically converge to the truth. In the second chapter, I show that algorithmic randomness leads to Bayesian merging of opinions. When two computable Bayesian agents perform the same experiment, agreeing on which data streams are algorithmically random suffices to guarantee that they will eventually reach a consensus. In the third and final chapter, I study a learning-theoretic approach--in the spirit of formal learning theory--for modelling algorithmic randomness itself. My main finding is that, in this context, the algorithmically random data streams can be systematically shown to coincide with the ones from which no computable qualitative learning method can extrapolate any patterns.

Topics in Algorithmic Randomness and Computability Theory

Author: Michael Patrick McInerney

Publisher:

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Category: Computable functions

Page: 238

View: 377


Computability and Complexity

The volume contains several surveys that make the various areas accessible to non-specialists while also including some proofs that illustrate the flavor of the fields.

Author: Adam Day

Publisher: Springer

ISBN: 3319500627

Category: Computers

Page: 755

View: 108

This Festschrift is published in honor of Rodney G. Downey, eminent logician and computer scientist, surfer and Scottish country dancer, on the occasion of his 60th birthday. The Festschrift contains papers and laudations that showcase the broad and important scientific, leadership and mentoring contributions made by Rod during his distinguished career. The volume contains 42 papers presenting original unpublished research, or expository and survey results in Turing degrees, computably enumerable sets, computable algebra, computable model theory, algorithmic randomness, reverse mathematics, and parameterized complexity, all areas in which Rod Downey has had significant interests and influence. The volume contains several surveys that make the various areas accessible to non-specialists while also including some proofs that illustrate the flavor of the fields.

Randomness Through Computation

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 ...

Author: Hector Zenil

Publisher: World Scientific

ISBN: 9814327743

Category: Computers

Page: 419

View: 123

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.

Randomness Computability and Algebraic Specifications

Author: Bakhadyr Khoussainov

Publisher:

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Effective Mathematics of the Uncountable

A comprehensive introduction to eight major approaches to computation on uncountable mathematical domains.

Author: Noam Greenberg

Publisher: Cambridge University Press

ISBN: 1107014514

Category: Mathematics

Page: 204

View: 461

A comprehensive introduction to eight major approaches to computation on uncountable mathematical domains.