## Notre Dame Journal of Formal Logic

### Effective Domination and the Bounded Jump

#### Abstract

We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: (1) We prove that the analogue of Sacks jump inversion fails for the bounded jump and the $\mathrm{wtt}$-reducibility. (2) We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use bounded by an $\omega$-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 61, Number 2 (2020), 203-225.

Dates
Accepted: 25 April 2019
First available in Project Euclid: 7 April 2020

https://projecteuclid.org/euclid.ndjfl/1586224879

Digital Object Identifier
doi:10.1215/00294527-2020-0005

Mathematical Reviews number (MathSciNet)
MR4092531

Zentralblatt MATH identifier
07222687

Subjects
Primary: 03D30: Other degrees and reducibilities
Secondary: 03D28: Other Turing degree structures

#### Citation

Ng, Keng Meng; Yu, Hongyuan. Effective Domination and the Bounded Jump. Notre Dame J. Formal Logic 61 (2020), no. 2, 203--225. doi:10.1215/00294527-2020-0005. https://projecteuclid.org/euclid.ndjfl/1586224879

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