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A Bounded Jump for the Bounded Turing Degrees

Bernard Anderson & Barbara Csima

Notre Dame Journal of Formal Logic 55 (2):245-264 (2014)

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A Hierarchy of Computably Enumerable Degrees.Rod Downey & Noam Greenberg - 2018 - Bulletin of Symbolic Logic 24 (1):53-89.details We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain. Download Export citation Bookmark 3 citations
Bounded-low sets and the high/low hierarchy.Huishan Wu - 2020 - Archive for Mathematical Logic 59 (7-8):925-938.details Anderson and Csima defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable. Ng and Yu proved that bounded-high c.e. sets are Turing complete, thus answered the conjecture positively. Wu and Wu 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 (...) Download Export citation Bookmark
Bounded low and high sets.Bernard A. Anderson, Barbara F. Csima & Karen M. Lange - 2017 - Archive for Mathematical Logic 56 (5-6):507-521.details Anderson and Csima :245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the Jump (...) Download Export citation Bookmark 2 citations
Effective Domination and the Bounded Jump.Keng Meng Ng & Hongyuan Yu - 2020 - Notre Dame Journal of Formal Logic 61 (2):203-225.details We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use (...) Download Export citation Bookmark 1 citation

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