It is shown that if A and C are sets of degrees uniformly recursive in 0' with $\mathbf{0} \nonin \mathscr{C}$ then there is a degree b with b' = 0', b ∪ c = 0' for every c ∈ C, and $\mathbf{a} \nleq \mathbf{b}$ for every a ∈ A ∼ {0}. The proof is given as an oracle construction recursive in 0'. It follows that any nonrecursive degree below 0' can be joined to 0' by a degree strictly below 0'. (...) Also, if $\mathbf{a ' and a" = 0" then there is a degree b such that a ∪ b = 0' and a ∩ b = 0. (shrink)

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of (...) learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs. (shrink)

Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$, then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results.We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic (...) one. While there have been full approximation constructions of ${\rm{\Delta }}_3^0$ minimal degrees before, our proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$. (shrink)

We show that at each type k ≥ 2, there exist c-irreducible functionals of c-r.e. degree, as defined in [Nor 1]. Our proofs are based on arguments due to Hinman, [Hin 1], and Dvornikov, [Dvo 1].