Degree Spectra of Relations on a Cone

Let \mathcal A be a mathematical structure with an additional relation R. The author is interested in the degree spectrum of R, either among computable copies of \mathcal A when (\mathcal A,R) is a "natural" structure, or (to make this rigorous) among copies of (\mathcal A,R) computable in...

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Main Author: Harrison-Trainor, Matthew
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2018
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Angles (Geometry)
Angles (Geometry) > Measurement
Conic sections
Unsolvability (Mathematical logic)
degree spectrum of a relation
cone of Turing degrees
computable structure
ISBN:1470428393, 9781470428396
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Abstract Let \mathcal A be a mathematical structure with an additional relation R. The author is interested in the degree spectrum of R, either among computable copies of \mathcal A when (\mathcal A,R) is a "natural" structure, or (to make this rigorous) among copies of (\mathcal A,R) computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on \mathcal A and R, if R is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.
AbstractList Let $\mathcal A$ be a mathematical structure with an additional relation $R$. The author is interested in the degree spectrum of $R$, either among computable copies of $\mathcal A$ when $(\mathcal A,R)$ is a ``natural'' structure, or (to make this rigorous) among copies of $(\mathcal A,R)$ computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on $\mathcal A$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.
Let
Author Harrison-Trainor, Matthew
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Keywords degree spectrum of a relation
cone of Turing degrees
computable structure
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Notes Includes bibliographical references and index
May 2018, volume 253, number 1208 (third of 7 numbers)
OCLC 1038078986
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Snippet Let
Let \mathcal A be a mathematical structure with an additional relation R. The author is interested in the degree spectrum of R, either among computable copies...
Let $\mathcal A$ be a mathematical structure with an additional relation $R$. The author is interested in the degree spectrum of $R$, either among computable...
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SubjectTerms Angles (Geometry)
Angles (Geometry) -- Measurement
Conic sections
Unsolvability (Mathematical logic)
TableOfContents Introduction -- Preliminaries -- Degree Spectra between the C.E. Degrees and the D.C.E. Degrees -- Degree Spectra of Relations on the Naturals -- A “Fullness” Theorem for 2-CEA\xspace Degrees -- Further Questions -- Relativizing Harizanov’s Theorem on C.E. Degrees
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- 2.1. Computability Theory -- 2.2. Computable Structure Theory -- 2.3. Relativizing to a Cone -- Chapter 3. Degree Spectra between the C.E. Degrees and the D.C.E. Degrees -- 3.1. Necessary and Sufficient Conditions to be Intrinsically of C.E. Degree -- 3.2. Incomparable Degree Spectra of D.C.E. Degrees -- Chapter 4. Degree Spectra of Relations on the Naturals -- Chapter 5. A "Fullness" Theorem for 2-\cea Degrees -- 5.1. Approximating a 2-\cea Set -- 5.2. Basic Framework of the Construction -- 5.3. An Informal Description of the Construction -- 5.4. The Game Gs and the Final Condition -- 5.5. Basic Plays and the Basic Game -- 5.6. The Construction -- Chapter 6. Further Questions -- Appendix A. Relativizing Harizanov's Theorem on C.E. Degrees -- A.1. Framework of the Proof -- A.2. The First Two Cases -- A.3. The Third Case -- Bibliography -- Index of Notation and Terminology -- Back Cover
Title Degree Spectra of Relations on a Cone
URI https://www.ams.org/memo/1208/
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