Characterizing the continuous degrees

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

“…For a partial computable function f :⊆ X → Y, the direction (2)⇒(1) always holds. Moreover, if X is a computable compactum, the converse direction (1)⇒(2) holds as well, that is, the following (1) and (2) are equivalent:…”

“…If x ∈ R n is weakly 2-generic, then dim P (x) = n.Proof. For any k ∈ , the set S k of all x such that K r+1 (x)/r ≥ n(1 − 1/k) for some r ≥ k is dense, since if p is random then K r (p) ≥ nr − O(1). Again consider the c.e.…”

“…In particular, the theory of generalized Turing degrees has achieved great success. The researchers have found a number of unexpected applications of the notion of metric/topological Turing degrees: Day-Miller [7] explained the behavior of Levin's neutral measures in algorithmic randomness theory; Gregoriades-Kihara-Ng [10] gave a partial answer to the open problem on generalizing the Jayne-Rogers theorem in descriptive set theory; Kihara-Pauly [22] proved a result related to descriptive set theory, infinite dimensional topology, and Banach space theory; Andrews-Igusa-Miller-Soskova [1] showed that the PA degrees are first-order definable in the enumeration degrees; Kihara-Lempp-Ng-Pauly [19] established a classification theory of the enumeration degrees; et cetera.…”

Enumeration Degrees and Topology

“…Miller had observed in [30] that all continuous degrees share a peculiar property, namely being almost-total (the name was coined later, in [2]). The name is explained by noting that an enumeration degree is called total, if it is in the range of the embedding of the Turing degrees.…”

“…That the existence of non-total almost-total degrees is proven via a seeming detour through the continuous degrees is not accident: Andrews, Igusa, Miller and Soskova proved that the almost-total degrees are exactly the continuous degrees [2]. Their proof proceeds via a number of characterizations, essentially showing that every almost-total degree has a certain representative, and that the collection of these representatives forms an effective regular topological space.…”