A Local Version of the Slaman–Wehner Theorem and Families Closed Under Finite Differences

Abstract

The main question of this article is whether there is a family closed under finite differences (i.e., if A belongs to the family and $B{=}^{\ast }A$, then B also belongs to the family) that can be enumerated by any noncomputable c.e. degree, but which cannot be enumerated computably. This question was formulated by Greenberg et al. (2020) in their recent work in which families that are closed under finite differences, close to the Slaman–Wehner families, are deeply studied.