A Hierarchy of Computably Enumerable Degrees

Downey, Rodney G.; Greenberg, Noam

doi:10.1017/bsl.2017.41

Cited by 6 publications

(5 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These proofs use constructions which involve high levels of nonuniformity. They are discussed in detail in [16]. Ambos-Spies and Losert [5] have given an example of a 7 element lattice (Fig.…”

Section: Theorem 34 ([22]mentioning

confidence: 99%

“…The goal of this paper is to look at the hierarchy of Turing degrees introduced by Downey and Greenberg [16,22]. Whilst we concentrate on recent work not covered in either the monograph [22] or the BSL survey [16], we will give enough background material for the reader to understand the issues addressed by the hierarchy. We will give some proofs and proof sketches so the the reader can understand the techniques involved.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper. As we have said, along with [16], the goal of this paper is to report on the current results of a program introduced by Downey, Greenberg and some co-authors, which seeks to understand the fine structure of relationship between dynamic properties of sets and functions, their definability, and their algorithmic complexity. Many of the resuls and details can be found in Downey and Greenberg's monograph [24].…”

mentioning

confidence: 99%

“…The reader might well ask why we need yet another hierarchy in computability theory. As the first two authors claim in [16,22], the new hierarchy gives insight into (i) A new methodology for classifying and unifying the combinatorics of a number of constructions from the literature. (ii) New natural definability results in the c.e.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Hierarchy of Computably Enumerable Degrees II

2021

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Theorem 34 ([22]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Hierarchy of Computably Enumerable Degrees II

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…This concludes the proof of (i) ⇒ (iii) and of Theorem 4.3.1. multiple permitting strength present; for this we refer the reader to the monograph [11] or the survey paper [10].…”

Section: Verificationmentioning

confidence: 99%

On equivalence relations and bounded turing degrees

Yu¹

View full text Add to dashboard Cite

This thesis examines three areas in computability theory. In Chapter 2 we look at certain classes of equivalence relations under computable reducibility. In Chapters 3 and 4 we examine the bounded jump operator and the notion of highness for the bounded jump. In Chapter 5 we look at different ways of effectivizing the properties of a dominant function. In Chapter 2 we will examine the α-c.e. and ∆ 0 2 equivalence relations, for α ≤ ω, under computable reducibility. We investigate questions such as the existence of a universal element of each class, the issue of having an enumeration of the class, as well as structural properties such as upward and downward density of the class. In Chapter 3 we prove that the analogue of Sacks jump inversion for the weak truth-table degrees and the bounded jump does not hold. This contrasts with an earlier result [1] that the analogue of Shoenfield jump inversion does hold. Our result answers a question in [1]. In Chapter 4 we characterize the c.e. sets that are bounded high as those sets that compute the Halting problem with a Turing functional where the use is bounded by an ω-c.e. function. This means that every c.e. bounded high set is already Turing complete, and certainly cannot be low, answering a question in [2]. In Chapter 5 we suggest different ways of strengthening the property of a c.e. set being dominant. We provide separations of these different notions and show that together with bounded highness, these different properties form a proper hierarchy.

show abstract