Turing Computability
Theory and Applications
(Sprache: Englisch)
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...
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Klappentext zu „Turing Computability “
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 IIincludes 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.
Inhaltsverzeichnis zu „Turing Computability “
Part I Foundations of Computability.- Chap. 1 Defining Computability.- Chap. 2 Computably Enumerable Sets.- Chap. 3 Turing Reducibility.- Chap. 4 The Arithmetical Hierarchy.- Chap. 5 Classifying C.E. Sets.- Chap. 6 Oracle Constructions and Forcing.- Chap. 7 The Finite Injury Method.- Part II Trees and 01 Classes.- Chap. 8 Open and Closed Classes.- Chap. 9 Basis Theorems.- Chap. 10 Peano Arithmetic and 01-Classes.- Chap. 11 Randomness and 01-Classes.- Part III Minimal Degrees.- Chap. 12 Minimal Degrees Below Ø .- Chap. 13 Minimal Degrees Below Ø .- Part IV Games in Computability Theory.- Chap. 14 Banach-Mazur Games.- Chap. 15 Gale-Stewart Games.- Chap. 16 More Lachlan Games.- Part V History of Computability.- Chap. 17 History of Computability.- References.- Index.Autoren-Porträt von Robert I. Soare
Prof. Robert Irving Soare has been a leading researcher and teacher in this field since the 1970s. He received his PhD from Cornell University in 1967, he has been a full professor of mathematics at The University of Chicago since 1975, and has been the Paul Snowden Russell Distinguished Service Professor of Mathematics and Computer Science at the university since 1994. Among other duties he was the editor of the Annals of Pure and Applied Logic from 1997 to 2009, and among many publications he wrote the primary reference on computability theory for students and researchers: "Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets" (Springer, 1987). He has been a plenary or invited speaker at the top events in his field, including the International Congress of Mathematicians, the International Congress for Logic, Methodology and Philosophy of Science, the Centennial Meeting of the Association for Symbolic Logic, and the Computability in Europe (CiE) Conference on Computation and Logic in the Real World.
Bibliographische Angaben
- Autor: Robert I. Soare
- 2016, 1st ed., XXXVI, 263 Seiten, Masse: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 3642319327
- ISBN-13: 9783642319327
- Erscheinungsdatum: 28.06.2016
Sprache:
Englisch
Pressezitat
"At a time when computability theory is enjoying remarkable activity and fruitfulness, and benefiting from having a large number of students and young researchers, there is no question that the subject is ready for a new standard introductory text. The present book shares all the features that helped its predecessor become such a standard thirty years ago, and at the same time, it is modern, and it is relevant to today's state of the field. The subject will be well-served by it." (Damir D. Dzhafarov, Bulletin of Symbolic Logic, Vol. 23 (1), March, 2017)
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