Computable Structure Theory
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Computable Structure Theory
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Author : Antonio Montalbán
language : en
Publisher: Cambridge University Press
Release Date : 2021-06-24
Computable Structure Theory written by Antonio Montalbán 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 2021-06-24 with Mathematics categories.
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
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Author : C.J. Ash
language : en
Publisher: Elsevier
Release Date : 2000-06-16
Computable Structures And The Hyperarithmetical Hierarchy written by C.J. Ash and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-06-16 with Mathematics categories.
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 Theory
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Author : S. Barry Cooper
language : en
Publisher: CRC Press
Release Date : 2017-09-06
Computability Theory written by S. Barry Cooper and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-06 with Mathematics categories.
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 Computability
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Author : Robert I. Soare
language : en
Publisher: Springer
Release Date : 2016-06-20
Turing Computability written by Robert I. Soare and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-20 with Computers categories.
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.
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.
Computability
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Author : B. Jack Copeland
language : en
Publisher: MIT Press
Release Date : 2013-06-07
Computability written by B. Jack Copeland and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-07 with Computers categories.
Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments. In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding.Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics.ContributorsScott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani
A Hierarchy Of Turing Degrees
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Author : Rod Downey
language : en
Publisher: Princeton University Press
Release Date : 2020-06-16
A Hierarchy Of Turing Degrees written by Rod Downey and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-06-16 with Computers categories.
[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.
Computable Structure Theory
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Author : Rodney G. Downey
language : en
Publisher: Springer
Release Date : 2025-06-06
Computable Structure Theory written by Rodney G. Downey and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-06-06 with Computers categories.
This is the first book which gives a unified theory for countable and uncountable computable structures. The work treats computable linear orderings, graphs, groups and Boolean algebras unified with computable metric and Banach spaces, profinite groups, and the like. Further, it provides the first account of these that exploits effective versions of dualities, such as Stone and Pontryagin dualities. The themes are effective classification and enumeration. Topics and features: !-- [if !supportLists]--· !--[endif]--Delivers a self-contained, gentle introduction to priority arguments, directly applying them in algebraic contexts !-- [if !supportLists]--· !--[endif]--Includes extensive exercises that both cement and amplify the materials !-- [if !supportLists]--· !--[endif]--Provides complete introduction to the basics of computable analysis, particularly in the context of computable structures !-- [if !supportLists]--· !--[endif]--Offers the first monograph treatment of computable Polish groups, effective profinite groups via Stone duality, and effective abelian groups via Pontryagin duality !-- [if !supportLists]--· !--[endif]--Presents the first book treatment of Friedberg enumerations of structures This unique volume is aimed at graduate students and researchers in computability theory, as well as mathematicians seeking to understand the algorithmic content of structure theory. Being self-contained, it provides ample opportunity for self-study.