Computable Structure Theory
Computable Structure Theory

Author: Antonio Montalbán

Publisher: Cambridge University Press

Published: 2021-06-24

Total Pages: 214

ISBN-13: 1108534422

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In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming second volume will study structures beyond arithmetic.

Computable Structure Theory
Computable Structure Theory

Author: Antonio Montalbán

Publisher: Cambridge University Press

Published: 2021-06-24

Total Pages: 213

ISBN-13: 1108423299

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Presents main results and techniques in computable structure theory together in a coherent framework for the first time in 20 years.

Computable Structures and the Hyperarithmetical Hierarchy
Computable Structures and the Hyperarithmetical Hierarchy

Author: C.J. Ash

Publisher: Elsevier

Published: 2000-06-16

Total Pages: 363

ISBN-13: 0080529526

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This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties).

Computability and Complexity
Computability and Complexity

Author: Adam Day

Publisher: Springer

Published: 2016-11-30

Total Pages: 788

ISBN-13: 3319500627

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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.

Computability, Forcing and Descriptive Set Theory
Computability, Forcing and Descriptive Set Theory

Author: Douglas Cenzer

Publisher: World Scientific Publishing Company

Published: 2019-12-31

Total Pages: 200

ISBN-13: 9789813228221

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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. Contents: Limits of the Kucerea-Gacs Coding Method (George Barmpalias and Andrew Lewis-Pye);Infinitary partition properties of sums of selective ultrafilters (Andreas Blass);Semiselective Coideals and Ramsey Sets (Carlos DiPrisco and Leonardo Pacheco);Survey on Topological Ramsey Spaces Dense in Forcings (Natasha Dobrinen);Higher Computability in the Reverse Mathematics of Borel Determinacy (Sherwood Hachtman);Computability and Definability (Valentina Harizanov);A Ramsey Space of Infinite Polyhedra and the Random Polyhedron (Jose G Mijares Palacios and Gabriel Padilla);Computable Reducibility for Cantor Space (Russell G Miller);Information vs Dimension - An Algorithmic Perspective (Jan Reimann); Readership: Graduate students and researchers interested in the interface between set theory and computability.

Computability Theory
Computability Theory

Author: S. Barry Cooper

Publisher: CRC Press

Published: 2017-09-06

Total Pages: 420

ISBN-13: 1420057561

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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.

Turing's Legacy
Turing's Legacy

Author: Rod Downey

Publisher: Cambridge University Press

Published: 2014-05-01

Total Pages: 540

ISBN-13: 1139916831

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Alan Turing was an inspirational figure who is now recognised as a genius of modern mathematics. In addition to leading the Allied forces' code-breaking effort at Bletchley Park in World War II, he proposed the theoretical foundations of modern computing and anticipated developments in areas from information theory to computer chess. His ideas have been extraordinarily influential in modern mathematics and this book traces such developments by bringing together essays by leading experts in logic, artificial intelligence, computability theory and related areas. Together, they give insight into this fascinating man, the development of modern logic, and the history of ideas. The articles within cover a diverse selection of topics, such as the development of formal proof, differing views on the Church–Turing thesis, the development of combinatorial group theory, and Turing's work on randomness which foresaw the ideas of algorithmic randomness that would emerge many years later.

Structure And Randomness In Computability And Set Theory
Structure And Randomness In Computability And Set Theory

Author: Douglas Cenzer

Publisher: World Scientific

Published: 2020-10-02

Total Pages: 387

ISBN-13: 9813228245

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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.

Computability Theory and Some Applications
Computability Theory and Some Applications

Author: Michael Deveau

Publisher:

Published: 2019

Total Pages: 104

ISBN-13:

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We explore various areas of computability theory, ranging from applications in computable structure theory primarily focused on problems about computing isomorphisms, to a number of new results regarding the degree-theoretic notion of the bounded Turing hierarchy. In Chapter 2 (joint with Csima, Harrison-Trainor, Mahmoud), the set of degrees that are computably enumerable in and above $\mathbf{0}^{(\alpha)}$ are shown to be degrees of categoricity of a structure, where $\alpha$ is a computable limit ordinal. We construct such structures in a particularly useful way: by restricting the construction to a particular case (the limit ordinal $\omega$) and proving some additional facts about the widgets that make up the structure, we are able to produce a computable prime model with a degree of categoricity as high as is possible. This then shows that a particular upper bound on such degrees is exact. In Chapter 3 (joint with Csima and Stephenson), a common trick in computable structure theory as it relates to degrees of categoricity is explored. In this trick, the degree of an isomorphism between computable copies of a rigid structure is often able to be witnessed by the clever choice of a computable set whose image or preimage through the isomorphism actually attains the degree of the isomorphism itself. We construct a pair of computable copies of $(\omega,

A Hierarchy of Turing Degrees
A Hierarchy of Turing Degrees

Author: Rod Downey

Publisher: Princeton University Press

Published: 2020-06-16

Total Pages: 234

ISBN-13: 0691199663

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[Alpha]-c.a. functions -- The hierarchy of totally [alpha]-c.a. degrees -- Maximal totally [alpha]-c.a. degrees -- Presentations of left-c.e. reals -- m-topped degrees -- Embeddings of the 1-3-1 lattice -- Prompt permissions.