Notes on degrees of relative computable categoricity

We are studying the degrees in which a computable structure is relatively computably categoricity, i.e., computably categorcial among all non-computable copies of the structure. Unlike the degrees of computable categoricity we can bound the possible degrees of relative computable categoricity by the oracle 0". In the case of rigid structures the bound is in fact 0'. These estimations are precise, in particular we can build a computable structure which is relatively computably categorical only in the degrees above 0"

Degrees of selector functions and relative computable categoricity

We study the degrees of selector functions related to the degrees in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such degrees can be represented as the unions of upper cones of c.e. degrees. In addition we show that there are non-c.e. upper cones realized as the degrees in which some computable structure is relatively computably categorical

Density results in the Delta-0-2 e-degrees

We show that th Delta-0-2 enumeration degrees are dense. We also show that for everynonzero n-c. e. e-degree a, with n≥3, one can always find a nonzero 3-c. e. e-degree b such that b < a; on the other hand there is a nonzero omega-c. e. e-degree which bounds no nonzero n-c. e. e-degree

Total Degrees and Nonsplitting Properties of Σ 0 2 Enumeration Degrees

This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees. The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, and a construction of a Π 0 1 e-degree unsplittable over a ∆2 e-degree below it. In [ACK] it was shown that using semirecursive sets one can construct minimal pairs of e-degrees by both effective and uniform ways, following which new results concerning the local distribution of total e-degrees and of the degrees of semirecursive sets enabled one to proceed, via the natural embedding of the Turing degrees in the enumeration degrees, to results concerning embeddings of the diamond lattice in the e-degrees. A particularly striking application of these techniques was a relatively simple derivation of a strong generalisation of the Ahmad Diamond Theorem. This paper extends the known constraints on further progress in this direction, such as the result of Ahmad and Lachlan [AL98] showing the existence of a nonsplitting ∆ 0 2 e-degree> 0e, and the recent result of Soskova [Sos07] showing that 0 ′ e is unsplittable in the Σ 0 2 e-degrees above some Σ 0 2 e-degree < 0 ′ e. This work also relates to results (e.g. Cooper and Copestake [CC88]) limiting the local distribution of total e-degrees. For further background concerning enumeration reducibility and its degree structure, the reader is referred to Cooper [Co90], Sorbi [Sor97] or Cooper [Co04], chapter 11. Theorem 1 If a < h ≤ 0 ′ , a is low and h is total and high then there is a low total e-degree b such that a ≤ b < h