# Bounded Jump and the High/Low Hierarchy

• Guohua Wu
• Huishan Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

## 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 AB, 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. (1)

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

2. (2)

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

3. (3)

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

4. (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.

## Keywords

Bounded jump High/low hierarchy Bounded high/low sets Pseudo-jump inversion

## References

1. 1.
Anderson, B., Csima, B.: A bounded jump for the bounded turing degrees. Notre Dame J. Formal Logic 55, 245–264 (2014)
2. 2.
Anderson, B., Csima, B., Lange, K.: Bounded low and high sets. Arch. Math. Logic 56, 523–539 (2017)
3. 3.
Jockusch Jr., C.G., Shore, R.A.: Pseudo-jump operators. I: the r.e. case. Trans. Am. Math. Soc. 275, 599–609 (1983)
4. 4.
Mohrherr, J.: A refinement of $$low_{n}$$ and $$high_{n}$$ for r.e. degrees. Z. Math. Logik Grundlag. Math. 32, 5–12 (1986)
5. 5.
Bickford, M., Mills, C.: Lowness properties of r.e. sets (1982, typewritten unpublished manuscript)Google Scholar
6. 6.
Nies, A.: Computability and Randomness. Oxford University Press, Inc., New York (2009)

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## Authors and Affiliations

1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
2. 2.School of Information ScienceBeijing Language and Culture UniversityBeijingChina