Author: Heinz-Dieter Ebbinghaus
Publisher: Springer
ISBN: 3540395962
Category : Mathematics
Languages : en
Pages : 427

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Book Description

Author: Chi Tat Chong
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110275643
Category : Mathematics
Languages : en
Pages : 322

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Book Description
This monograph presents recursion theory from a generalized point of view centered on the computational aspects of definability. A major theme is the study of the structures of degrees arising from two key notions of reducibility, the Turing degrees and the hyperdegrees, using techniques and ideas from recursion theory, hyperarithmetic theory, and descriptive set theory. The emphasis is on the interplay between recursion theory and set theory, anchored on the notion of definability. The monograph covers a number of fundamental results in hyperarithmetic theory as well as some recent results on the structure theory of Turing and hyperdegrees. It also features a chapter on the applications of these investigations to higher randomness.

Author: Andrea Sorbi
Publisher: CRC Press
ISBN: 1482269759
Category : Mathematics
Languages : en
Pages : 380

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Book Description
"Integrates two classical approaches to computability. Offers detailed coverage of recent research at the interface of logic, computability theory, nd theoretical computer science. Presents new, never-before-published results and provides informtion not easily accessible in the literature."

Author: Marat M. Arslanov
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110807483
Category : Mathematics
Languages : en
Pages : 248

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Book Description
The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.

Author:
Publisher: Elsevier
ISBN: 9780080533698
Category : Computers
Languages : en
Pages : 619

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Book Description
Recursive Model Theory

Author: S. Barry Cooper
Publisher: CRC Press
ISBN: 1420057561
Category : Mathematics
Languages : en
Pages : 420

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Book Description
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.

Author: André Nies
Publisher: OUP Oxford
ISBN: 0191627887
Category : Mathematics
Languages : en
Pages : 450

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Book Description
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.

Author:
Publisher: Elsevier
ISBN: 0080533701
Category : Computers
Languages : en
Pages : 799

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Book Description
Recursive Algebra, Analysis and Combinatorics

Author: Arnold Beckmann
Publisher: Springer Science & Business Media
ISBN: 3540694056
Category : Computers
Languages : en
Pages : 613

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Book Description
This book constitutes the refereed proceedings of the 4th International Conference on Computability in Europe, CiE 2008, held in Athens, Greece, in June 2008. The 36 revised full papers presented together with 25 invited tutorials and lectures were carefully reviewed and selected from 108 submissions. Among them are papers of 6 special sessions entitled algorithms in the history of mathematics, formalising mathematics and extracting algorithms from proofs, higher-type recursion and applications, algorithmic game theory, quantum algorithms and complexity, and biology and computation.

Author: Peter Cholak
Publisher: American Mathematical Soc.
ISBN: 0821819224
Category : Mathematics
Languages : en
Pages : 338

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Book Description
This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A. Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A. Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong and Y. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A. Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. Ambos-Spies and A. Kucera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh).