A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability

by Rod Downey, Noam Greenberg

Overview

Computability theory is a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field has developed growing connections in diverse areas of mathematics, with applications in topology, group theory, and other subfields.

In A Hierarchy of Turing Degrees, Rod Downey and Noam Greenberg introduce a new hierarchy that allows them to classify the combinatorics of constructions from many areas of computability theory, including algorithmic randomness, Turing degrees, effectively closed sets, and effective structure theory. This unifying hierarchy gives rise to new natural definability results for Turing degree classes, demonstrating how dynamic constructions become reflected in definability. Downey and Greenberg present numerous construction techniques involving high-level nonuniform arguments, and their self-contained work is appropriate for graduate students and researchers.

Blending traditional and modern research results in computability theory, A Hierarchy of Turing Degrees establishes novel directions in the field.

Product Details

ISBN-13:	9780691199665
Publisher:	Princeton University Press
Publication date:	06/16/2020
Series:	Annals of Mathematics Studies , #206
Pages:	234
Product dimensions:	6.00(w) x 9.00(h) x 0.70(d)

About the Author

Rod Downey and Noam Greenberg are professors of mathematics at Victoria University of Wellington in New Zealand. Downey is the coauthor of Parameterized Complexity, Algorithmic Randomness and Complexity, and Fundamentals of Parameterized Complexity. Greenberg is the author of The Role of True Finiteness in the Admissible Recursively Enumerable Degrees.

Table of Contents

Acknowledgments ix

1 Introduction 1

1.1 Historical context 1

1.2 Background: unifying constructions and natural definability 3

1.3 Toward the hierarchy of totally α-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

2 α-c.a. functions 23

2.1 R-c.a. functions 23

2.2 Canonical well-orderings and strong notations 29

2.3 Weak truth-table jumps and w^αa-c.a. sets and functions 37

3 The hierarchy of totally α-c.a. degrees 55

3.1 Totally R-c.a. degrees 55

3.2 The first hierarchy theorem: totally w^α-c.a. degrees 58

3.3 A refinement of the hierarchy: uniformly totally w^α-c.a. degrees 68

3.4 Another refinement of the hierarchy: totally <w^α-c.a. degrees 74

3.5 Domination properties 80

4 Maximal totally α-c.a. degrees 84

4.1 Existence of maximal totally w^α-c.a. degrees 84

4.2 Limits on further maximality 94

5 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

6 m-topped degrees 134

6.1 Totally w-c.a. degrees are not m-topped 135

6.2 Totally w²-c.a. degrees are not m-topped 140

6.3 Totally < w^w-c.a. degrees are not m-topped 145

7 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

8 Prompt permissions 188

8.1 Prompt classes 188

8.2 Minimal pairs of separating classes 202

8.3 Prompt permission and other constructions 212

Bibliography 215