Bounded low and high sets

Bounded low and high sets Anderson and Csima (Notre Dame J Form Log 55(2):245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing (or weak truth table) 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 Theorem holds for the bounded jump: do we have $$A \le _{bT}B$$ A ≤ b T B if and only if $$A^b \le _1 B^b$$ A b ≤ 1 B b ? We show the forward direction holds but not the reverse. Archive for Mathematical Logic Springer Journals

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Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
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