Computability and Randomness

Author: André Nies

Publisher: OUP Oxford

ISBN: 0191627887

Category: Philosophy

Page: 456

View: 312

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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.

Computability Theory

Author: Rebecca Weber

Publisher: American Mathematical Soc.

ISBN: 082187392X

Category: Mathematics

Page: 203

View: 3323

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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.

Evolving Computability

11th Conference on Computability in Europe, CiE 2015, Bucharest, Romania, June 29-July 3, 2015. Proceedings

Author: Arnold Beckmann,Victor Mitrana,Mariya Soskova

Publisher: Springer

ISBN: 3319200283

Category: Computers

Page: 363

View: 2127

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This book constitutes the refereed proceedings of the 11th Conference on Computability in Europe, CiE 2015, held in Bucharest, Romania, in June/July 2015. The 26 revised papers presented were carefully reviewed and selected from 64 submissions and included together with 10 invited papers in this proceedings. The conference CiE 2015 has six special sessions: two sessions, Representing Streams and Reverse Mathematics, were introduced for the first time in the conference series. In addition to this, new developments in areas frequently covered in the CiE conference series were addressed in the further special sessions on Automata, Logic and Infinite Games; Bio-inspired Computation; Classical Computability Theory; as well as History and Philosophy of Computing.

Probability And Random Number: A First Guide To Randomness

Author: Sugita Hiroshi

Publisher: World Scientific

ISBN: 981322827X

Category: Mathematics

Page: 140

View: 5154

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This is a book of elementary probability theory that includes a chapter on algorithmic randomness. It rigorously presents definitions and theorems in computation theory, and explains the meanings of the theorems by comparing them with mechanisms of the computer, which is very effective in the current computer age. Random number topics have not been treated by any books on probability theory, only some books on computation theory. However, the notion of random number is necessary for understanding the essential relation between probability and randomness. The field of probability has changed very much, thus this book will make and leave a big impact even to expert probabilists. Readers from applied sciences will benefit from this book because it presents a very proper foundation of the Monte Carlo method with practical solutions, keeping the technical level no higher than 1st year university calculus. Contents: Mathematics of Coin TossingMathematical ModelRandom NumberLimit TheoremMonte Carlo MethodInfinite coin TossesRandom Number: Recursive FunctionKolmogorov Complexity and Random NumberLimit Theorem: Bernoulli's TheoremLaw of Large NumbersDe Moivre–Laplace's TheoremCentral Limit TheoremMathematical StatisticsMonte Carlo Method: Monte Carlo Method as GamblingPseudorandom GeneratorMonte Carlo IntegrationFrom the Viewpoint of Mathematical StatisticsAppendices: Symbols and TermsBinary Numeral SystemLimit of Sequence and FunctionLimits of Exponential Function and LogarithmC Language Program Readership: First year university students to professionals. Keywords: Probability;Probability Theory;Randomness;Random Number;Pseudorandom Number;Monte Carlo Method;Monte Carlo IntegrationReview: Key Features: This is the first book that presents both probability theory and algorithmic randomness for from 1st year university students to experts. It is technically easy but worth reading for experts as wellThis book presents basic limit theorems with proofs that are not seen in usual probability textbooks; for readers should learn that a good solution is not always uniqueThis book rigorously treats the Monte Carlo method. In particular, it presents the random Weyl sampling, which produces pseudorandom numbers for the Monte Carlo integration that act complete substitutes for random numbers

Computability and Complexity

Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday

Author: Adam Day,Michael Fellows,Noam Greenberg,Bakhadyr Khoussainov,Alexander Melnikov,Frances Rosamond

Publisher: Springer

ISBN: 3319500627

Category: Computers

Page: 755

View: 2713

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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.

Turing Computability

Theory and Applications

Author: Robert I. Soare

Publisher: Springer

ISBN: 3642319335

Category: Computers

Page: 263

View: 3834

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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.

Category Theory

Author: Steve Awodey

Publisher: OUP Oxford

ISBN: 0191612553

Category: Philosophy

Page: 328

View: 626

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Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.

Reductive Logic and Proof-search

Proof Theory, Semantics, and Control

Author: David J. Pym,Eike Ritter

Publisher: Oxford University Press on Demand

ISBN: 0198526334

Category: Mathematics

Page: 208

View: 6215

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This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .

Bolzano's Logical System

Author: Ettore Casari

Publisher: Oxford University Press

ISBN: 0198788290

Category:

Page: 368

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This book is focused on the first three parts of Bolzano's Theory of Sciene and introduces a more systematic reconsideration of Bolzano's logial thought. In undertaking this task, the book is intended as an exploration, not so much of the more specifically discursive aspects of Bolzano's logial thought - already amply studied - as muh as on identifying the singularly coherent and systematic nature of the logic presented in Bolzano's work. Casari presents this within a formal system and adopts the approach of the predicate calculus with identity and choice operator by using Hilbert's epsilon calculus (the logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics).

Computability

An Introduction to Recursive Function Theory

Author: Nigel Cutland

Publisher: Cambridge University Press

ISBN: 9780521294652

Category: Computers

Page: 251

View: 7001

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This introduction to recursive theory computability begins with a mathematical characterization of computable functions, develops the mathematical theory and includes a full discussion of noncomputability and undecidability. Later chapters move on to more advanced topics such as degrees of unsolvability and Gödel's Incompleteness Theorem.

Simplicity Theory

Author: Byunghan Kim

Publisher: OUP Oxford

ISBN: 0191511587

Category: Mathematics

Page: 272

View: 7589

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Model theory, a major branch of mathematical logic, plays a key role connecting logic and other areas of mathematics such as algebra, geometry, analysis, and combinatorics. Simplicity theory, a subject of model theory, studies a class of mathematical structures, called simple. The class includes all stable structures (vector spaces, modules, algebraically closed fields, differentially closed fields, and so on), and also important unstable structures such as the random graph, smoothly approximated structures, pseudo-finite fields, ACFA and more. Simplicity theory supplies the uniform model theoretic points of views to such structures in addition to their own mathematical analyses. This book starts with an introduction to the fundamental notions of dividing and forking, and covers up to the hyperdefinable group configuration theorem for simple theories. It collects up-to-date knowledge on simplicity theory and it will be useful to logicians, mathematicians and graduate students working on model theory.

Algorithmic Randomness and Complexity

Author: Rodney G. Downey,Denis R. Hirschfeldt

Publisher: Springer Science & Business Media

ISBN: 0387684417

Category: Computers

Page: 855

View: 2698

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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.

A Study of Logics

Author: John P. Cleave

Publisher: Oxford University Press

ISBN: 0198532113

Category: Literary Criticism

Page: 417

View: 6768

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This is a new systematic study of the principles behind the variety of logical systems in mathematical logic and computer science. The technical work is illuminated by information about its historical and philosophical context.

Change, Choice and Inference

A Study of Belief Revision and Nonmonotonic Reasoning

Author: Hans Rott

Publisher: Oxford University Press

ISBN: 9780198503064

Category: Mathematics

Page: 381

View: 8643

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Change, Choice and Inference unifies lively and significant strands of research in logic, philosophy, economics and artificial intelligence.

Recursively Enumerable Sets and Degrees

A Study of Computable Functions and Computably Generated Sets

Author: Robert I. Soare

Publisher: Springer Science & Business Media

ISBN: 9783540152996

Category: Mathematics

Page: 437

View: 9438

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..."The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees. .... The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists." Acta Scientiarum Mathematicarum, Ungarn 1988 ..."The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be." Zentralblatt für Mathematik, 623.1988

Models of Peano arithmetic

Author: Richard Kaye

Publisher: Oxford University Press, USA

ISBN: N.A

Category: Literary Criticism

Page: 292

View: 9803

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Non-standard models of arithmetic are of interest to mathematicians through the presence of infinite integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s, they have come to play an important role in model theory, and in combinatorics through independence results such as the Paris-Harrington theorem. This book is an introduction to these developments, and stresses the interplay between the first-order theory, recursion-theoretic aspects, and the structural properties of these models. Prerequisites for an understanding of the text have been kept to a minimum, these being a basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets. Consequently, the book is suitable for postgraduate students coming to the subject for the first time, and a number of exercises of varying degrees of difficulty will help to further the reader's understanding.

Higher-Order Computability

Author: John Longley,Dag Normann

Publisher: Springer

ISBN: 3662479923

Category: Computers

Page: 571

View: 6042

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This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers

Lectures on Inductive Logic

Author: Jon Williamson,Professor of Reasoning Inference and Scientific Method Jon Williamson

Publisher: Oxford University Press

ISBN: 0199666474

Category:

Page: 220

View: 4850

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Logic is a field studied mainly by researchers and students of philosophy, mathematics and computing. Inductive logic seeks to determine the extent to which the premisses of an argument entail its conclusion, aiming to provide a theory of how one should reason in the face of uncertainty. It has applications to decision making and artificial intelligence, as well as how scientists should reason when not in possession of the full facts. In this book, Jon Williamson embarks on a quest to find a general, reasonable, applicable inductive logic (GRAIL), all the while examining why pioneers such as Ludwig Wittgenstein and Rudolf Carnap did not entirely succeed in this task. Along the way he presents a general framework for the field, and reaches a new inductive logic, which builds upon recent developments in Bayesian epistemology (a theory about how strongly one should believe the various propositions that one can express). The book explores this logic in detail, discusses some key criticisms, and considers how it might be justified. Is this truly the GRAIL? Although the book presents new research, this material is well suited to being delivered as a series of lectures to students of philosophy, mathematics, or computing and doubles as an introduction to the field of inductive logic