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Author: Piergiorgio Odifreddi
Publisher:
ISBN: 9780444589439
Category : Recursion theory
Languages : en
Pages : 668
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Book Description
Author: Piergiorgio Odifreddi
Publisher:
ISBN: 9780444589439
Category : Recursion theory
Languages : en
Pages : 668
Get Book Here
Book Description
Author: Herbert B. Enderton
Publisher: Academic Press
ISBN: 0123849594
Category : Mathematics
Languages : en
Pages : 193
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Book Description
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
Author: Peter Bürgisser
Publisher: Springer Science & Business Media
ISBN: 3662033380
Category : Mathematics
Languages : en
Pages : 630
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Book Description
The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under standing of the intrinsic computational difficulty of problems.
Author: Gerald E. Sacks
Publisher: Cambridge University Press
ISBN: 1107168430
Category : Computers
Languages : en
Pages : 361
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Book Description
This almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field.
Author: Ljubomir Lalov Ivanov
Publisher:
ISBN:
Category : Recursion theory
Languages : en
Pages : 268
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Book Description
Author: Chi Tat Chong
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 311038129X
Category : Mathematics
Languages : en
Pages : 409
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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: Nigel Cutland
Publisher: Cambridge University Press
ISBN: 9780521294652
Category : Computers
Languages : en
Pages : 268
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Book Description
What can computers do in principle? What are their inherent theoretical limitations? The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function - a function whose values can be calculated in an automatic way.
Author: Robert L. Causey
Publisher: Jones & Bartlett Learning
ISBN: 9780763737849
Category : Computers
Languages : en
Pages : 536
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Book Description
The new Second Edition incorporates a wealth of exercise sets, allowing students to test themselves and review important topics discussed throughout the text."--Jacket.
Author: Samuel Eilenberg
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 112
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Book Description
Author: Robert I. Soare
Publisher: Springer
ISBN: 3642319335
Category : Computers
Languages : en
Pages : 289
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Book Description
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.
