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.