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book: A Hierarchy of Turing Degrees
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A Hierarchy of Turing Degrees

A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability
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Language: English
Published/Copyright: 2020

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 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.

Author / Editor 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

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    Frontmatter

    i
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    Contents

    vii
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    Acknowledgments

    ix
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    Chapter One. Introduction

    1
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    Chapter Two. ɑ-c.a. functions

    23
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    Chapter Three. The hierarchy of totally ɑ-c.a. degrees

    55
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    Chapter Four. Maximal totally ɑ-c.a. degrees

    84
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    Chapter Five. Presentations of left-c.e. reals

    106
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    Chapter Six. m-topped degrees

    134
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    Chapter Seven. Embeddings of the 1-3-1 lattice

    149
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    Chapter Eight. Prompt permissions

    188
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    Bibliography

    215
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Publishing information
Pages and Images/Illustrations in book
eBook published on:
September 7, 2020
eBook ISBN:
9780691200217
Pages and Images/Illustrations in book
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https://doi.org/10.1515/9780691200217
eBook ISBN: 9780691200217
Keywords for this book
Requirement; Computation; Turing degree; Recursively enumerable set; Turing machine; Turing reduction; Theorem; Maximal set; Approximation; Mathematical induction; Computability theory; Ordinal arithmetic; Model of computation; Maximal element; Gödel's incompleteness theorems; Word problem (mathematics); Computable function; Embedding; Ordinal analysis; Restriction (mathematics); Reverse mathematics; Set theory; I0; Algorithmic learning theory; Limit ordinal; Combinatorics; Computable number; Arithmetic; Identity function; Recursion (computer science); Computability; Elaboration; Summation; Mathematics; Entscheidungsproblem; Iteration; Algorithm; Tree (data structure); Order by; Propositional calculus; Halting problem; N0; Result; Join and meet; Counterexample; Enumeration; Notation; Scientific notation; Decision problem; Open set; Pairwise; Truth table; Modular lattice; Limit superior and limit inferior; Partially ordered set; Characteristic function (probability theory); Unit interval; Word problem for groups; Distributive lattice; AND gate; Dyadic rational; Binary number; Combination; Quantity; Subset; Natural number; Arbitrarily large; Discrete space; Computable analysis; Vertex cover
Audience(s) for this book
For a non-specialist adult audience
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