Abstract
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.
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The author appreciates the referees for useful comments and helpful suggestions. This research project is supported by Science Foundation of Beijing Language and Culture University (supported by “the Fundamental Research Funds for the Central Universities”) (Grant No. 20YJ040004).
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Wu, H. Bounded-low sets and the high/low hierarchy. Arch. Math. Logic 59, 925–938 (2020). https://doi.org/10.1007/s00153-020-00726-7
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DOI: https://doi.org/10.1007/s00153-020-00726-7