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Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 507–521 | Cite as

Bounded low and high sets

  • Bernard A. Anderson
  • Barbara F. Csima
  • Karen M. LangeEmail author
Article

Abstract

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 \(A \le _{bT}B\) if and only if \(A^b \le _1 B^b\)? We show the forward direction holds but not the reverse.

Keywords

Bounded jump Bounded Turing degrees Weak truth table degrees High sets Low sets 

Mathematics Subject Classification

Primary 03D28 Secondary 03D30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceGordon State CollegeBarnesvilleUSA
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  3. 3.Department of MathematicsWellesley CollegeWellesleyUSA

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