“…This relationship, illustrated in Figure 2, matches exactly the relationship that we have established between the oracles with continuous degrees, -PA oracles, oracles that have a universal class, and the low for PA oracles in Figure 1. We will see that there is a good explanation for this: all but one of the properties described above can be characterized in terms of the relation -PA restricted to separation classes. Figure 2 Summary of the results in [15]. …”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
confidence: 90%
“…Kalimullin and Puzarenko [15] isolate a series of enumeration oracles based on properties that are satisfied by the ideal of sets enumeration reducible to them. They study the oracles that have the reduction property , oracles that have the separation property , oracles that have a universal function , and oracles that have the computable extension property .…”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
confidence: 99%
“…Kalimullin and Puzarenko [15] prove that has the reduction property if and only if has the uniformization property : if is a binary relation then there is a function f with graph such that and (i.e., the first projection of R ).…”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
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: Forcing Separationsmentioning
confidence: 99%
“…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.…”
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.
“…This relationship, illustrated in Figure 2, matches exactly the relationship that we have established between the oracles with continuous degrees, -PA oracles, oracles that have a universal class, and the low for PA oracles in Figure 1. We will see that there is a good explanation for this: all but one of the properties described above can be characterized in terms of the relation -PA restricted to separation classes. Figure 2 Summary of the results in [15]. …”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
confidence: 90%
“…Kalimullin and Puzarenko [15] isolate a series of enumeration oracles based on properties that are satisfied by the ideal of sets enumeration reducible to them. They study the oracles that have the reduction property , oracles that have the separation property , oracles that have a universal function , and oracles that have the computable extension property .…”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
confidence: 99%
“…Kalimullin and Puzarenko [15] prove that has the reduction property if and only if has the uniformization property : if is a binary relation then there is a function f with graph such that and (i.e., the first projection of R ).…”
Section: Combinatorial Principles From Descriptive Set Theorymentioning
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: Forcing Separationsmentioning
confidence: 99%
“…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.…”
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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