Extending CL-reducibility on array noncomputable degrees

Abstract

Given a function f, f-bounded-Turing (f-bT-) reducibility is the Turing reducibility with use function bounded by f. In the special case where $f=\mathrm{id}+c$ (with id being the identity function and c a constant), this is referred to as cl-reducibility. In a work by Barmpalias, Fang, and Lewis-Pye, it was proven that there exist two left-c.e. reals such that no left-c.e. real $(\mathrm{id}+g)$-bT-computes both of them whenever g is computable, nondecreasing, and satisfies ${\sum }_n{2}^{-g\left(n\right)}=\infty$. Moreover, such maximal pairs exist precisely within every array noncomputable degree. This result generalizes a prior result on cl-reducibility, which states that there exist two left-c.e. reals such that no left-c.e. real cl-computes both of them. An open question remained as to whether a similar extension could apply to another result on cl-reducibility, which asserts that there exists a left-c.e. real not cl-reducible to any random left-c.e. real. We answer this question affirmatively, providing a simpler proof compared to previous works. Additionally, we streamline the proof of the initial extension.