A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (AMS-206)

Rod Downey; Noam Greenberg

Published by Princeton University Press 2020

A Hierarchy of Turing Degrees

A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (AMS-206)

Rod Downey and Noam Greenberg

Volume 206 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9780691200217

Overview
Contents

About this book

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 suitable to 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.

Author information

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.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Acknowledgments Publicly Available Download PDF	ix
Chapter One. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Chapter Two. ɑ-c.a. functions Requires Authentication Unlicensed Licensed Download PDF	23
Chapter Three. The hierarchy of totally ɑ-c.a. degrees Requires Authentication Unlicensed Licensed Download PDF	55
Chapter Four. Maximal totally ɑ-c.a. degrees Requires Authentication Unlicensed Licensed Download PDF	84
Chapter Five. Presentations of left-c.e. reals Requires Authentication Unlicensed Licensed Download PDF	106
Chapter Six. m-topped degrees Requires Authentication Unlicensed Licensed Download PDF	134
Chapter Seven. Embeddings of the 1-3-1 lattice Requires Authentication Unlicensed Licensed Download PDF	149
Chapter Eight. Prompt permissions Requires Authentication Unlicensed Licensed Download PDF	188
Bibliography Requires Authentication Unlicensed Licensed Download PDF	215

Language: English
Publisher: Princeton University Press
Audience: General/trade;Professional and scholarly;College/higher education;
Pages
- Main content: 240
Illustrations
- Other: 3 b/w illus.
Keywords: Recursion theory; computable model theory; relative recursive randomness; modern computability theory; pi-zero-one classes; transfinite hierarchy of Turing degrees; mind changes in computability theory; c.e. degrees; c.e. reals; m-topped degrees; lattice embeddings; prompt permissions

eBook

Published: June 16, 2020
ISBN: 9780691200217