Computable Structure Theory

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Computable Structure Theory

Author : Antonio Montalbán
Publisher : Cambridge University Press
ISBN 13 : 1108534422
Total Pages : 214 pages
Book Rating : 4.1/5 (85 download)

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Book Synopsis Computable Structure Theory by : Antonio Montalbán

Download or read book Computable Structure Theory written by Antonio Montalbán and published by Cambridge University Press. This book was released on 2021-06-24 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Structures and the Hyperarithmetical Hierarchy

Author : C.J. Ash
Publisher : Elsevier
ISBN 13 : 0080529526
Total Pages : 363 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Computable Structures and the Hyperarithmetical Hierarchy by : C.J. Ash

Download or read book Computable Structures and the Hyperarithmetical Hierarchy written by C.J. Ash and published by Elsevier. This book was released on 2000-06-16 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).

Computable Structure Theory

Author : Antonio Montalbán
Publisher : Cambridge University Press
ISBN 13 : 1108423299
Total Pages : 213 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis Computable Structure Theory by : Antonio Montalbán

Download or read book Computable Structure Theory written by Antonio Montalbán and published by Cambridge University Press. This book was released on 2021-06-24 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents main results and techniques in computable structure theory together in a coherent framework for the first time in 20 years.

Computability Theory

Author : S. Barry Cooper
Publisher : CRC Press
ISBN 13 : 1420057561
Total Pages : 420 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Computability Theory by : S. Barry Cooper

Download or read book Computability Theory written by S. Barry Cooper and published by CRC Press. This book was released on 2017-09-06 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Computability, Forcing and Descriptive Set Theory

Author : Douglas Cenzer
Publisher : World Scientific Publishing Company
ISBN 13 : 9789813228221
Total Pages : 200 pages
Book Rating : 4.2/5 (282 download)

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Book Synopsis Computability, Forcing and Descriptive Set Theory by : Douglas Cenzer

Download or read book Computability, Forcing and Descriptive Set Theory written by Douglas Cenzer and published by World Scientific Publishing Company. This book was released on 2019-12-31 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Turing's Legacy

Author : Rod Downey
Publisher : Cambridge University Press
ISBN 13 : 1139916831
Total Pages : 540 pages
Book Rating : 4.1/5 (399 download)

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Book Synopsis Turing's Legacy by : Rod Downey

Download or read book Turing's Legacy written by Rod Downey and published by Cambridge University Press. This book was released on 2014-05-01 with total page 540 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A Hierarchy of Turing Degrees

Author : Rod Downey
Publisher : Princeton University Press
ISBN 13 : 0691199663
Total Pages : 234 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis A Hierarchy of Turing Degrees by : Rod Downey

Download or read book A Hierarchy of Turing Degrees written by Rod Downey and published by Princeton University Press. This book was released on 2020-06-16 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: [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.

Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles

Author : Denis R Hirschfeldt
Publisher : World Scientific
ISBN 13 : 9814612634
Total Pages : 231 pages
Book Rating : 4.8/5 (146 download)

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Book Synopsis Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles by : Denis R Hirschfeldt

Download or read book Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles written by Denis R Hirschfeldt and published by World Scientific. This book was released on 2014-07-18 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions.

Computability

Author : B. Jack Copeland
Publisher : MIT Press
ISBN 13 : 0262018993
Total Pages : 373 pages
Book Rating : 4.2/5 (62 download)

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Book Synopsis Computability by : B. Jack Copeland

Download or read book Computability written by B. Jack Copeland and published by MIT Press. This book was released on 2013-06-07 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Turing Computability

Author : Robert I. Soare
Publisher : Springer
ISBN 13 : 3642319335
Total Pages : 289 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Turing Computability by : Robert I. Soare

Download or read book Turing Computability written by Robert I. Soare and published by Springer. This book was released on 2016-06-20 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Handbook of Computability and Complexity in Analysis

Author : Vasco Brattka
Publisher : Springer Nature
ISBN 13 : 3030592340
Total Pages : 427 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Handbook of Computability and Complexity in Analysis by : Vasco Brattka

Download or read book Handbook of Computability and Complexity in Analysis written by Vasco Brattka and published by Springer Nature. This book was released on 2021-06-04 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Introduction to Computable General Equilibrium Models

Author : Mary E. Burfisher
Publisher : Cambridge University Press
ISBN 13 : 1107132207
Total Pages : 443 pages
Book Rating : 4.1/5 (71 download)

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Book Synopsis Introduction to Computable General Equilibrium Models by : Mary E. Burfisher

Download or read book Introduction to Computable General Equilibrium Models written by Mary E. Burfisher and published by Cambridge University Press. This book was released on 2016 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a hands-on introduction to computable general equilibrium (CGE) models, written at an accessible, undergraduate level.

Computability Theory

Author : Herbert B. Enderton
Publisher : Academic Press
ISBN 13 : 0123849594
Total Pages : 193 pages
Book Rating : 4.1/5 (238 download)

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Book Synopsis Computability Theory by : Herbert B. Enderton

Download or read book Computability Theory written by Herbert B. Enderton and published by Academic Press. This book was released on 2010-12-30 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computability Theory: An Introduction to Recursion 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 text 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 way. - Frequent historical information presented throughout - More extensive motivation for each of the topics than other texts currently available - Connects with topics not included in other textbooks, such as complexity theory

Computability and Randomness

Author : André Nies
Publisher : OUP Oxford
ISBN 13 : 0191627887
Total Pages : 450 pages
Book Rating : 4.1/5 (916 download)

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Book Synopsis Computability and Randomness by : André Nies

Download or read book Computability and Randomness written by André Nies and published by OUP Oxford. This book was released on 2012-03-29 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Connecting with Computability

Author : Liesbeth De Mol
Publisher : Springer
ISBN 13 : 9783030800482
Total Pages : 506 pages
Book Rating : 4.8/5 (4 download)

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Book Synopsis Connecting with Computability by : Liesbeth De Mol

Download or read book Connecting with Computability written by Liesbeth De Mol and published by Springer. This book was released on 2021-07-02 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 17th Conference on Computability in Europe, CiE 2021, organized by the University of Ghent in July 2021. Due to COVID-19 pandemic the conference was held virtually. The 48 full papers presented in this volume were carefully reviewed and selected from 50 submissions. CiE promotes the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences, such as physics and biology, as well as related fields, such as philosophy and history of computing. CiE 2021 had as its motto Connecting with Computability, a clear acknowledgement of the connecting and interdisciplinary nature of the conference series which is all the more important in a time where people are more than ever disconnected from one another due to the COVID-19 pandemic.

Computable Analysis

Author : Klaus Weihrauch
Publisher : Springer Science & Business Media
ISBN 13 : 9783540668176
Total Pages : 312 pages
Book Rating : 4.6/5 (681 download)

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Book Synopsis Computable Analysis by : Klaus Weihrauch

Download or read book Computable Analysis written by Klaus Weihrauch and published by Springer Science & Business Media. This book was released on 2000-09-14 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Structure And Randomness In Computability And Set Theory

Author : Douglas Cenzer
Publisher : World Scientific
ISBN 13 : 9813228245
Total Pages : 387 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis Structure And Randomness In Computability And Set Theory by : Douglas Cenzer

Download or read book Structure And Randomness In Computability And Set Theory written by Douglas Cenzer and published by World Scientific. This book was released on 2020-10-02 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.