Recursively Enumerable Sets and Degrees

Recursively Enumerable Sets and Degrees PDF

Author: Robert I. Soare

Publisher: Springer Science & Business Media

Published: 1999-11-01

Total Pages: 460

ISBN-13: 9783540152996

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..."The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees. .... The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists." Acta Scientiarum Mathematicarum, Ungarn 1988 ..."The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be." Zentralblatt für Mathematik, 623.1988

Automorphisms of the Lattice of Recursively Enumerable Sets

Automorphisms of the Lattice of Recursively Enumerable Sets PDF

Author: Peter Cholak

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 166

ISBN-13: 0821826018

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A version of Harrington's [capital Greek]Delta3-automorphism technique for the lattice of recursively enumerable sets is introduced and developed by reproving Soare's Extension Theorem. Then this automorphism technique is used to show two technical theorems: the High Extension Theorem I and the High Extension Theorem II. This is a degree-theoretic technique for constructing both automorphisms of the lattice of r.e. sets and isomorphisms between various substructures of the lattice.

Computability in Analysis and Physics

Computability in Analysis and Physics PDF

Author: Marian B. Pour-El

Publisher: Cambridge University Press

Published: 2017-03-02

Total Pages: 219

ISBN-13: 1107168449

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The first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning.

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees PDF

Author: Noam Greenberg

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 114

ISBN-13: 0821838857

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When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu

Degrees of Unsolvability

Degrees of Unsolvability PDF

Author: Gerald E. Sacks

Publisher: Princeton University Press

Published: 1966

Total Pages: 192

ISBN-13: 9780691079417

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A classic treatment of degrees of unsolvability from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

Turing Computability

Turing Computability PDF

Author: Robert I. Soare

Publisher: Springer

Published: 2016-06-20

Total Pages: 263

ISBN-13: 3642319335

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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.

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