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.

Anderson和Csima（Notre Dame J Form Log 55：245–264，2014）定义了有界Turning约简的有界跳跃算子，并研究了其基本性质。安德森等。（Arch Math Logic 56：507–521，2017）构建了一个低界高集合，并推测此类集合不可计算地枚举（简称ce）。Ng和Yu（Notre Dame J Form Log，即将出现）证明了有界高ce集是Turing完全的，因此肯定地回答了这一猜想。Wu和Wu（计算机科学讲义，第11436卷，第647-658页.Springer，Cham，2019年）表明，通过构建图灵完整的有界-低ce集，有界-低集可以是超高的。在本文中，我们继续研究有界跳跃和图灵跳跃之间的比较。我们显示出低ce集并不仅限于有界低和超高ce不完整