Effective Mathematics of the Uncountable

Effective Mathematics of the Uncountable

Author: Noam Greenberg

Publisher: Cambridge University Press

Published: 2013-10-31

Total Pages: 205

ISBN-13: 110751200X

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Classical computable model theory is most naturally concerned with countable domains. There are, however, several methods – some old, some new – that have extended its basic concepts to uncountable structures. Unlike in the classical case, however, no single dominant approach has emerged, and different methods reveal different aspects of the computable content of uncountable mathematics. This book contains introductions to eight major approaches to computable uncountable mathematics: descriptive set theory; infinite time Turing machines; Blum-Shub-Smale computability; Sigma-definability; computability theory on admissible ordinals; E-recursion theory; local computability; and uncountable reverse mathematics. This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages. It is ideal as both an introductory text for graduate and advanced undergraduate students and a source of interesting new approaches for researchers in computability theory and related areas.


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

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

Author: Denis R Hirschfeldt

Publisher: World Scientific

Published: 2014-07-18

Total Pages: 231

ISBN-13: 9814612634

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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.


Ordinal Computability

Ordinal Computability

Author: Merlin Carl

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2019-09-23

Total Pages: 344

ISBN-13: 3110496151

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Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics are covered.


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.


Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem

Author: Denis R. Hirschfeldt

Publisher: American Mathematical Soc.

Published: 2017-09-25

Total Pages: 114

ISBN-13: 1470426579

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Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.


Pursuit of the Universal

Pursuit of the Universal

Author: Arnold Beckmann

Publisher: Springer

Published: 2016-06-13

Total Pages: 388

ISBN-13: 3319401890

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This book constitutes the refereed proceedings of the 12th Conference on Computability in Europe, CiE 2016, held in Paris, France, in June/July 2016. The 18 revised full papers and 19 invited papers and invited extended abstracts were carefully reviewed and selected from 40 submissions. The conference CiE 2016 has six special sessions – two sessions, cryptography and information theory and symbolic dynamics, are organized 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 following sessions: computable and constructive analysis; computation in biological systems; history and philosophy of computing; weak arithmetic.


The Nature of Computation: Logic, Algorithms, Applications

The Nature of Computation: Logic, Algorithms, Applications

Author: Paola Bonizzoni

Publisher: Springer

Published: 2013-06-03

Total Pages: 462

ISBN-13: 3642390536

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This book constitutes the refereed proceedings of the 9th Conference on Computability in Europe, CiE 2013, held in Milan, Italy, in July 2013. The 48 revised papers presented together with 1 invited lecture and 2 tutorials were carefully reviewed and selected with an acceptance rate of under 31,7%. Both the conference series and the association promote the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences such as physics and biology, and also including the promotion of related non-scientific fields such as philosophy and history of computing.


Turing's Legacy

Turing's Legacy

Author: Rod Downey

Publisher: Cambridge University Press

Published: 2014-05

Total Pages: 540

ISBN-13: 1107043484

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A collection of essays celebrating the influence of Alan Turing's work in logic, computer science and related areas.


Unconventional Computation

Unconventional Computation

Author: Christian S. Calude

Publisher: Springer Science & Business Media

Published: 2009-08-28

Total Pages: 312

ISBN-13: 3642037445

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This book constitutes the refereed proceedings of the 8th International Conference on Unconventional Computation, UC 2009, held in Ponta Delgada, Portugal, in September 2009. The 18 revised full papers presented together with 8 invited talks, 3 tutorials and 5 posters were carefully reviewed and selected from 40 submissions. The papers are devoted to all aspects of unconventional computation ranging from theoretical and experimental aspects to various applications. Typical topics are: natural computing including quantum; cellular, molecular, neural and evolutionary computing; chaos and dynamical system-based computing; and various proposals for computational mechanisms that go beyond the Turing model.