Pa Relative to an Enumeration Oracle

Abstract

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.

Introenumerability, Autoreducibility, and Randomness

“…For proof of the first arrow, see [9]. The third arrow and the strictness of the first arrow are proved in [5] by Goh et al It is still unknown whether there is a set of introenumerable e-degree that does not have uniformly introenumerable e-degree. §2.…”

“…The definition of uniform introenumerability we give here is slightly different by using an enumeration operator instead of a c.e. operator, though the two definitions were shown to be equivalent in [6] by Greenberg et al Recently, Goh et al [5] also showed that Jockush's notion of (non-uniform) introenumerability is equivalent to the following notion:…”