A hierarchy of Turing degrees: a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability
[alpha]-c.a. functions -- The hierarchy of totally [alpha]-c.a. degrees -- Maximal totally [alpha]-c.a. degrees -- Presentations of left-c.e. reals -- m-topped degrees -- Embeddings of the 1-3-1 lattice -- Prompt permissions
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton and Oxford
Princeton University Press
2020
|
| Schriftenreihe: | Annals of mathematics studies
number 206 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | [alpha]-c.a. functions -- The hierarchy of totally [alpha]-c.a. degrees -- Maximal totally [alpha]-c.a. degrees -- Presentations of left-c.e. reals -- m-topped degrees -- Embeddings of the 1-3-1 lattice -- Prompt permissions "This book presents new results in computability theory, a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field's connections with disparate areas of mathematical logic and mathematics more generally have grown deeper, and now have a variety of applications in topology, group theory, and other subfields. This monograph establishes new directions in the field, blending classic results with modern research areas such as algorithmic randomness. The significance of the book lies not only in the depth of the results contained therein, but also in the fact that the notions the authors introduce allow them to unify results from several subfields of computability theory"-- |
| Beschreibung: | Includes bibliographical references |
| Beschreibung: | viii, 222 Seiten Illustrationen |
| ISBN: | 9780691199658 9780691199665 |
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Datensatz im Suchindex
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| adam_text | A Hierarchy of Turing Degrees
A Transfinite Hierarchy of Lowness Notions in the
Computably Enumerable Degrees, Unifying Classes,
and Natural Definability
Rod Downey
Noam Greenberg
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
Universitäts- und
Landesbibliothek
Darmstadt
Contents
Acknowledgments ix
Introduction 1
1 1 Historical context 1
1 2 Background: unifying constructions and
natural definability 3
1 3 Toward the hierarchy of totally a-c a degrees 8
1 4 The contents of this monograph 14
1 5 An application to admissible computability 16
1 6 Notation and general definitions 17
a-c a functions 23
2 1 Tl-c s functions 23
2 2 Canonical well-orderings and strong notations 29
2 3 Weak truth-table jumps and oi“-c a sets
and functions 37
The hierarchy of totally a-c a degrees 55
3 1 Totally Ti-c iL degrees 55
3 2 The first hierarchy theorem: totally u“-c a degrees 58
33A refinement of the hierarchy: uniformly totally
wQ-c a degrees 68
3 4 Another refinement of the hierarchy: totally
iuQ-c a degrees 74
3 5 Domination properties 80
Maximal totally a-c a degrees 84
4 1 Existence of maximal totally u“-c a degrees 84
4 2 Limits on further maximality 94
Presentations of left-c e reals 106
5 1 Background 106
5 2 Presentations of c e reals and non-total
w-c a permitting 110
5 3 Total w-c a anti-permitting 123
CONTENTS
m- topped degrees 134
6 1 Totally w-c a degrees are not m-topped 135
6 2 Totally o)2-c a degrees are not m-topped 140
6 3 Totally cn^-c a degrees are not m-topped 145
Embeddings of the 1-3-1 lattice 149
7 1 Embedding the 1-3-1 lattice 150
7 2 Non-embedding critical triples 167
7 3 Defeating two gates 176
7 4 The general construction 184
Prompt permissions 188
8 1 Prompt classes 188
8 2 Minimal pairs of separating classes 202
8 3 Prompt permission and other constructions 212
215
|
| adam_txt |
A Hierarchy of Turing Degrees
A Transfinite Hierarchy of Lowness Notions in the
Computably Enumerable Degrees, Unifying Classes,
and Natural Definability
Rod Downey
Noam Greenberg
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
Universitäts- und
Landesbibliothek
Darmstadt
Contents
Acknowledgments ix
Introduction 1
1 1 Historical context 1
1 2 Background: unifying constructions and
natural definability 3
1 3 Toward the hierarchy of totally a-c a degrees 8
1 4 The contents of this monograph 14
1 5 An application to admissible computability 16
1 6 Notation and general definitions 17
a-c a functions 23
2 1 Tl-c s functions 23
2 2 Canonical well-orderings and strong notations 29
2 3 Weak truth-table jumps and oi“-c a sets
and functions 37
The hierarchy of totally a-c a degrees 55
3 1 Totally Ti-c iL degrees 55
3 2 The first hierarchy theorem: totally u“-c a degrees 58
33A refinement of the hierarchy: uniformly totally
wQ-c a degrees 68
3 4 Another refinement of the hierarchy: totally
iuQ-c a degrees 74
3 5 Domination properties 80
Maximal totally a-c a degrees 84
4 1 Existence of maximal totally u“-c a degrees 84
4 2 Limits on further maximality 94
Presentations of left-c e reals 106
5 1 Background 106
5 2 Presentations of c e reals and non-total
w-c a permitting 110
5 3 Total w-c a anti-permitting 123
CONTENTS
m- topped degrees 134
6 1 Totally w-c a degrees are not m-topped 135
6 2 Totally o)2-c a degrees are not m-topped 140
6 3 Totally cn^-c a degrees are not m-topped 145
Embeddings of the 1-3-1 lattice 149
7 1 Embedding the 1-3-1 lattice 150
7 2 Non-embedding critical triples 167
7 3 Defeating two gates 176
7 4 The general construction 184
Prompt permissions 188
8 1 Prompt classes 188
8 2 Minimal pairs of separating classes 202
8 3 Prompt permission and other constructions 212
215 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-03T15:16:06Z |
| indexdate | 2024-07-10T08:56:16Z |
| institution | BVB |
| isbn | 9780691199658 9780691199665 |
| language | English |
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| publishDate | 2020 |
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| spelling | Downey, R. G. 1957- Verfasser (DE-588)1052125417 aut A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability Rod Downey, Noam Greenberg Princeton and Oxford Princeton University Press 2020 viii, 222 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies number 206 Includes bibliographical references [alpha]-c.a. functions -- The hierarchy of totally [alpha]-c.a. degrees -- Maximal totally [alpha]-c.a. degrees -- Presentations of left-c.e. reals -- m-topped degrees -- Embeddings of the 1-3-1 lattice -- Prompt permissions "This book presents new results in computability theory, a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field's connections with disparate areas of mathematical logic and mathematics more generally have grown deeper, and now have a variety of applications in topology, group theory, and other subfields. This monograph establishes new directions in the field, blending classic results with modern research areas such as algorithmic randomness. The significance of the book lies not only in the depth of the results contained therein, but also in the fact that the notions the authors introduce allow them to unify results from several subfields of computability theory"-- Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Unsolvability (Mathematical logic) Computable functions Recursively enumerable sets Mathematische Logik (DE-588)4037951-6 s DE-604 Greenberg, Noam 1974- Verfasser (DE-588)173835864 aut Erscheint auch als Online-Ausgabe 978-0-691-20021-7 Annals of mathematics studies number 206 (DE-604)BV000000991 206 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032285972&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Downey, R. G. 1957- Greenberg, Noam 1974- A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability Annals of mathematics studies Mathematische Logik (DE-588)4037951-6 gnd |
| subject_GND | (DE-588)4037951-6 |
| title | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability |
| title_auth | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability |
| title_exact_search | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability |
| title_exact_search_txtP | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability |
| title_full | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability Rod Downey, Noam Greenberg |
| title_fullStr | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability Rod Downey, Noam Greenberg |
| title_full_unstemmed | A hierarchy of Turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability Rod Downey, Noam Greenberg |
| title_short | A hierarchy of Turing degrees |
| title_sort | a hierarchy of turing degrees a transfinite hierarchy of lowness notions in the computably enumerable degrees unifying classes and natural definability |
| title_sub | a transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability |
| topic | Mathematische Logik (DE-588)4037951-6 gnd |
| topic_facet | Mathematische Logik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032285972&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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