TOTALLY Ω-Computably ENUMERABLE DEGREES AND BOUNDING CRITICAL TRIPLES

Downey, Rodney G.; Greenberg, Noam; Weber, Rebecca

doi:10.1142/s0219061307000640

Cited by 30 publications

(23 citation statements)

References 24 publications

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“…It follows from this fact and Corollary 4.8 that there is a set D of array computable degree d such that D is not wtt-reducible to any ranked set. Downey, Greenberg, and Weber [12] introduced a closely related class of c.e. degrees, called the totally ω-c.e.…”

Section: Construction Of Dmentioning

confidence: 99%

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Π₁⁰ classes and strong degree spectra of relations

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Section: Construction Of Dmentioning

confidence: 99%

“…On the other hand, it was shown in [12] that the array computable c.e. degrees are properly contained in the totally ω-c.e.…”

Section: Construction Of Dmentioning

confidence: 99%

Π₁⁰ classes and strong degree spectra of relations

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“…The proof follows the style of the "anti-permitting" arguments of [15,12,13]. The general idea is that we globally construct some function Γ(D).…”

Section: 2mentioning

confidence: 99%

“…degrees was defined and investigated by Downey, Greenberg and Weber in [15] and later in [12]. These are the c.e.…”

Section: Introductionmentioning

confidence: 99%

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Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Barmpalias

Downey

Greenberg

2009

Trans. Amer. Math. Soc.

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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every left-c.e. real of that degree is reducible in a computable Lipschitz way to a random left-c.e. real (an Ω-number).

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A classification of low c.e. sets and the Ershov hierarchy

Faizrahmanov

2023

Mathematical Logic Qtrly

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In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ‐levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ‐level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ‐ and ‐bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ‐ or ‐bound of a c.e. set A is equivalent to the fact that . Finally, we consider the generalized truth‐table reducibilities and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, iff .

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TOTALLY Ω-Computably ENUMERABLE DEGREES AND BOUNDING CRITICAL TRIPLES

Abstract: We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ω-c.e. approximation.

Cited by 30 publications

References 24 publications

Π₁⁰ classes and strong degree spectra of relations

Π₁⁰ classes and strong degree spectra of relations

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

A classification of low c.e. sets and the Ershov hierarchy

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TOTALLY Ω-Computably ENUMERABLE DEGREES AND BOUNDING CRITICAL TRIPLES

Abstract: We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ω-c.e. approximation.

Cited by 30 publications

References 24 publications

Π10 classes and strong degree spectra of relations

Π10 classes and strong degree spectra of relations

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

A classification of low c.e. sets and the Ershov hierarchy

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Product

Resources

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Π₁⁰ classes and strong degree spectra of relations

Π₁⁰ classes and strong degree spectra of relations