# 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

# Bounded low and high sets

, Volume 56 (6) – Apr 28, 2017
15 pages

/lp/springer_journal/bounded-low-and-high-sets-ARvUs9xmtK
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0537-8
Publisher site
See Article on Publisher Site

### 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\$\$ 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.

### Journal

Archive for Mathematical LogicSpringer Journals

Published: Apr 28, 2017

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