Bounded-low sets and the high/low hierarchy.

Anderson and Csima (Notre Dame J Form Log 55:245–264, 2014) defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. (Arch Math Logic 56:507–521, 2017) constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable (c.e. for short). Ng and Yu (Notre Dame J Form Log, to appear) proved that bounded-high c.e. sets are Turing complete, thus answered the conjecture positively. Wu and Wu (Lecture notes in computer science, vol 11436, 647–658. Springer, Cham, 2019) showed that bounded-low sets can be superhigh by constructing a Turing complete bounded-low c.e. set. In this paper, we continue the study of the comparison between the bounded-jump and Turing jump. We show that low c.e. sets are not all bounded-low and that incomplete superhigh c.e. sets can be bounded-low. [ABSTRACT FROM AUTHOR]