Bounded Jump and the High/Low Hierarchy

Abstract

The notion of bounded-jump operator, $A^\dag$, was proposed by Anderson and Csima in paper [1], where they tried to find an appropriate jump operator on weak-truth-table (wtt for short) degrees. For a set A, the bounded-jump of A is defined as the set $A^{\dag } =\{e\in \mathbb {N}: \exists i\le e[\varphi _{i}(e)\downarrow \ \& \ ~\varPhi ^{A\upharpoonright _{\varphi _{i}(e)}}_{e}(e)\downarrow ]\}$. In [1], Anderson and Csima pointed out this bounded-jump operator $^\dag$ behaves likes Turing jump $'$, like (1) $\emptyset ^\dag$ and $\emptyset '$ are 1-equivalent, (2) for any set A, $A<_{wtt}A^\dag$, and (3) for any sets A, B, if $A\le _{wtt}B$, then $A^\dag \le _{wtt}B^\dag$. A set A is bounded-low, if $A^{\dag }\le _{wtt}\emptyset ^{\dag }$, and a set $B\le _{wtt}\emptyset ^{\dag }$ is bounded-high if $\emptyset ^{\dag \dag }\le _{wtt}B^{\dag }$. Anderson, Csima and Lange constructed in [2] a high bounded-low set and a low bounded-high set, showing that the bounded jump and Turing jump can behave very different. In this paper, we will answer several questions raised by Anderson, Csima and Lange in their paper [2] and show that:

• (1) there is a bounded-low c.e. set which is low, but not superlow;

• (2) $\mathbf{0}'$ contains a bounded-low c.e. set;

• (3) there are bounded-low c.e. sets which are high, but not superhigh;

• (4) there are bounded-high sets which are high, but not superhigh.

In particular, we will develop new pseudo-jump inversion theorems via bounded-low sets and bounded-high sets respectively.

Guohua Wu is partially supported by M4020333 (MOE2016-T2-1-083(S)), M4011672 (RG32/16) and M4011274 (RG29/14) from Ministry of Education of Singapore. Huishan Wu is supported by Science Foundation of Beijing Language and Culture University (by “the Fundamental Research Funds for the Central Universities”) (Grant No. 18YBB19).