Cupping and noncupping in the enumeration degrees of ∑20 sets

Cooper, S. Barry; Sorbi, Andrea; Yi, Xiaoding

doi:10.1016/s0168-0072(96)00009-7

Cited by 24 publications

(18 citation statements)

References 11 publications

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“…In the local structure of enumeration degrees, the noncuppable degrees form a subclass of the properly Σ 0 2 degrees. This follows from the result that every ∆ 0 2 degree cups [CSY96]. In fact every noncuppable degree a is downwards properly Σ 0 2 in the sense that every non zero degree below a is properly Σ 0 2 .…”

Section: Introductionmentioning

confidence: 77%

Non-cuppable enumeration degrees via finite injury

Harris¹

2011

Journal of Logic and Computation

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We exhibit finite injury constructions of a high Σ 0 2 enumeration degree incomparable with all intermediate ∆ 0 2 enumeration degrees, as also of both an upwards properly Σ 0 2 high and a low 2 noncuppable Σ 0 2 enumeration degree †. We also outline how to apply the same methods to prove that, for every Σ 0 2 enumeration degree b there exists a noncuppable degree a such that b ≤ a and a ≤ b , thus showing that there exist noncuppable Σ 0 2 enumeration degrees at every possible level of the high/low jump hierarchy.

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Section: Introductionmentioning

confidence: 77%

Non-cuppable enumeration degrees via finite injury

Harris¹

2011

Journal of Logic and Computation

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“…The existence of noncuppable 0 2 enumeration degrees was first shown by Cooper et al in [7]. However it was also shown in [7] that every 0 2 enumeration degree is cuppable.…”

Section: Introductionmentioning

confidence: 92%

“…such that y(l, s) ∈ T A (e, s)) and all 7 L m such that 6 Notice also that if e ≤ l < s and P(l, s) = 0 then y(l, s) / ∈ A s and P(l, s) = 1 which means that P l is already permanently satisfied. 7 Note that reinitialisation of L requirements in this case is applied only in order to simplify the verification of the construction. Indeed this is what forces the fact that L(e, s e ) = 0 as used in the first sentence of the proof of Claim 3. e < m < s and L(m, s) = 1 for reinitialisation during Step D. We say in this case that L e receives attention at stage s + 1.…”

Section: The Constructionmentioning

confidence: 99%

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Badness and jump inversion in the enumeration degrees

Harris

2012

Arch. Math. Logic

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This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a 0 2 set A whose enumeration degree a is bad-i.e. such that no set X ∈ a is good approximable-and whose complement A has lowest possible jump, in other words is low 2 . This also ensures that the degrees y ≤ a only contain 0 3 sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author's previous characterisation of the double jump of good approximable sets, the triple jump of a 0 2 set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith's jump interpolation technique to show that there exists a high quasiminimal 0 2 enumeration degree.

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“…Cooper et al [8] constructed below 0 ′ e an enumeration degree not cuppable to 0 ′ e , but showed that every non-zero ∆ 0 2 e-degree is cuppable to 0 ′ e . In particular, every non-zero low e-degree is so cuppable.…”

Section: Non-cupping and The Ershov Hierarchymentioning

confidence: 99%

Splitting and nonsplitting in the Σ20 enumeration degrees

Arslanov

Cooper

Kalimullin

et al. 2011

Theoretical Computer Science

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a b s t r a c tThis paper continues the project, initiated in (Arslanov, Cooper and Kalimullin 2003) [3], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting.The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′′ -priority construction of a Π 0 1 e-degree unsplittable over a 3-c.e. e-degree below it.

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Cupping and noncupping in the enumeration degrees of ∑20 sets

Cited by 24 publications

References 11 publications

Non-cuppable enumeration degrees via finite injury

Non-cuppable enumeration degrees via finite injury

Badness and jump inversion in the enumeration degrees

Splitting and nonsplitting in the Σ20 enumeration degrees

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