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Recursively Enumerable Sets and Degrees

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

Book Synopsis Recursively Enumerable Sets and Degrees by : Robert I. Soare

Download or read book Recursively Enumerable Sets and Degrees written by Robert I. Soare and published by Springer Science & Business Media. This book was released on 1999-11-01 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: ..."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

递归可枚举集和图灵度/[英文本]/可计算函数与可计算生成集研究/国外数学名著系列/Recursively enumerable sets and degerees

Author: R.I.·索尔 (美)

Publisher:

Published: 1987

Total Pages: 437

ISBN-13: 9787030182951

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Book Synopsis 递归可枚举集和图灵度/[英文本]/可计算函数与可计算生成集研究/国外数学名著系列/Recursively enumerable sets and degerees by : R.I.·索尔 (美)

Download or read book 递归可枚举集和图灵度/[英文本]/可计算函数与可计算生成集研究/国外数学名著系列/Recursively enumerable sets and degerees written by R.I.·索尔 (美) and published by . This book was released on 1987 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Degree Theoretic Definitions on the Low2 Recursively Enumerable Sets

Author: Rod G. Downey

Publisher:

Published: 1992

Total Pages: 64

ISBN-13:

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Book Synopsis Degree Theoretic Definitions on the Low2 Recursively Enumerable Sets by : Rod G. Downey

Download or read book Degree Theoretic Definitions on the Low2 Recursively Enumerable Sets written by Rod G. Downey and published by . This book was released on 1992 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Uniformity in the Recursively Enumerable Degrees and Infima in the Degrees of the Differences of Recursively Enumberable Sets

Author: Deborah Suzanne Kaddah

Publisher:

Published: 1992

Total Pages: 226

ISBN-13:

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Book Synopsis Uniformity in the Recursively Enumerable Degrees and Infima in the Degrees of the Differences of Recursively Enumberable Sets by : Deborah Suzanne Kaddah

Download or read book Uniformity in the Recursively Enumerable Degrees and Infima in the Degrees of the Differences of Recursively Enumberable Sets written by Deborah Suzanne Kaddah and published by . This book was released on 1992 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Turing and Truth Table Degrees of 1-generic and Recursively Enumerable Sets

Author: Christine Ann Haught

Publisher:

Published: 1985

Total Pages: 192

ISBN-13:

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Book Synopsis Turing and Truth Table Degrees of 1-generic and Recursively Enumerable Sets by : Christine Ann Haught

Download or read book Turing and Truth Table Degrees of 1-generic and Recursively Enumerable Sets written by Christine Ann Haught and published by . This book was released on 1985 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Degrees of Unsolvability

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.

Book Synopsis Degrees of Unsolvability by : Gerald E. Sacks

Download or read book Degrees of Unsolvability written by Gerald E. Sacks and published by Princeton University Press. This book was released on 1966 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Computability in Analysis and Physics

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.

Book Synopsis Computability in Analysis and Physics by : Marian B. Pour-El

Download or read book Computability in Analysis and Physics written by Marian B. Pour-El and published by Cambridge University Press. This book was released on 2017-03-02 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning.

Automorphisms of the Lattice of Recursively Enumerable Sets

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.

Book Synopsis Automorphisms of the Lattice of Recursively Enumerable Sets by : Peter Cholak

Download or read book Automorphisms of the Lattice of Recursively Enumerable Sets written by Peter Cholak and published by American Mathematical Soc.. This book was released on 1995 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

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

Book Synopsis The Role of True Finiteness in the Admissible Recursively Enumerable Degrees by : Noam Greenberg

Download or read book The Role of True Finiteness in the Admissible Recursively Enumerable Degrees written by Noam Greenberg and published by American Mathematical Soc.. This book was released on 2006 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Turing Computability

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.

Book Synopsis Turing Computability by : Robert I. Soare

Download or read book Turing Computability written by Robert I. Soare and published by Springer. This book was released on 2016-06-20 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.