Semidecidable numberings in admissible sets

Kalimullin, I. Sh.; Puzarenko, V. G.; Faĭzrahmanov, M. Kh.

doi:10.33048/alglog.2020.59.308

Cited by 1 publication

(6 citation statements)

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“…In Section 4, we discuss a collection of classes of enumeration oracles that were introduced by Kalimullin and Puzarenko [15]. They grouped oracles into classes based on whether or not the principal ideal (with respect to enumeration reducibility) that an oracle defines possesses a certain property coming from descriptive set theory or from classical computability theory.…”

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

confidence: 99%

“…Kalimullin and Puzarenko [15] prove that deg e (X ) has the reduction property if and only if deg e (X ) has the uniformization property: if R ≤ e X is a binary relation then there is a function f with graph G f ≤ e X such that G f ⊆ R and dom(f) = dom(R) (i.e., the first projection of R).…”

Section: The Reduction Propertymentioning

confidence: 99%

“…We can also observe that not every continuous degree has the reduction property. Kalimullin and Puzarenko [15] prove that if A is nontotal and has the reduction property, then the set of total degrees bounded by A is a jump ideal, i.e., an ideal closed under the jump operator. The existence of low Δ 0 2 continuous degrees implies that the two classes are incomparable.…”

Section: The Reduction Propertymentioning

confidence: 99%

“…To see that (1) ⇒ (2), we note that if is a partial computable extension of a {0, 1}-valued function ϕ then the partial computable function * (x) = 0, if (x) ↓ = 0, 1, if (x) ↓ = 0 is {0, 1}-valued and extends ϕ. For ( 2) ⇒ (1), we use a proof from [15]. Suppose that G ϕ ≤ e X is the graph of a partial function.…”

Section: The Computable Extension Propertymentioning

confidence: 99%

“…In particular, since by [15] every oracle that has the computable extension property has a universal function, it follows that having a universal class strictly implies having a universal function.…”

Section: S (∃Xmentioning

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

confidence: 99%

Section: The Reduction Propertymentioning

confidence: 99%

Section: The Reduction Propertymentioning

confidence: 99%

Section: The Computable Extension Propertymentioning

confidence: 99%

Section: S (∃Xmentioning

confidence: 99%

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