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=∗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.

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher (...) random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)