A Structural Dichotomy in the Enumeration Degrees

Ganchev, Hristo; Kalimullin, I. Sh.; Miller, Joseph S.; Soskova, Mariya Ivanova

doi:10.1017/jsl.2019.72

Cited by 2 publications

(5 citation statements)

References 18 publications

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“…Proof. One direction in this theorem was already shown to be true in [11]: every continuous degree is PA bounded. Here we prove that if A is not of continuous degree, then A is not PA bounded.…”

Section: Continuous Is the Same As Pa Boundedmentioning

confidence: 85%

“…(2) There is a single Π 0 1 (A) class whose members are all PA relative to A. Enumeration degrees that preserve the first property are called PA bounded and those that preserve the second property have a universal class. Ganchev et al [11] introduced these classes and proved several relationships between them and the continuous, the cototal, and the self -PA degrees. We prove that the PA bounded enumeration degrees are exactly the continuous degrees.…”

Section: Deg T (A)) = Deg E (A ⊕ A)mentioning

confidence: 99%

“…We note that this definition is slightly more demanding than the one originally given in [11]. There we merely required that X ⊕ X is PA relative to A for every X ∈ P. Here we have decided to ask for some additional uniformity.…”

Section: Theorem 210 (Andrews Et Al [1]) a Has Cototal Degree If And ...mentioning

confidence: 99%

“…Ganchev et al [11] left the following questions open: Are there cototal degrees that are not PA bounded? Can a self -PA degree be PA bounded?…”

Section: Continuous Is the Same As Pa Boundedmentioning

confidence: 99%

“…A natural question is therefore: Recall that when we reintroduced the definition of a universal class, we added a little uniformity. Ganchev et al [11] gave a definition with no uniformity:…”

mentioning

confidence: 99%

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Pa Relative to an Enumeration Oracle

Goh¹,

Kalimullin²,

Miller³

et al. 2022

J. symb. log.

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Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question.

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Section: Continuous Is the Same As Pa Boundedmentioning

confidence: 85%

Section: Deg T (A)) = Deg E (A ⊕ A)mentioning

confidence: 99%

Section: Theorem 210 (Andrews Et Al [1]) a Has Cototal Degree If And ...mentioning

confidence: 99%

“…Ganchev et al [11] left the following questions open: Are there cototal degrees that are not PA bounded? Can a self -PA degree be PA bounded?…”

Section: Continuous Is the Same As Pa Boundedmentioning

confidence: 99%

mentioning

confidence: 99%