Settling Time Reducibility Orderings

It is known that orderings can be formed with settling time domination and strong settling time domination as relations on c.e. sets. However, it has been shown that no such ordering can be formed when considering computation time domination as a relation on $n$-c.e. sets where $n \geq 3$. This will be extended to the case of $2$-c.e. sets, showing that no ordering can be derived from computation time domination on $n$-c.e. sets when $n\geq 2$. Additionally, we will observe properties of the orderings given by settling time domination and strong settling time domination on c.e. sets, respectively denoted as $\mathcal{E}_{st}$ and $\mathcal{E}_{sst}$. More specifically, it is already known that any countable partial ordering can be embedded into $\mathcal{E}_{st}$ and any linear ordering with no infinite ascending chains can be embedded into $\mathcal{E}_{sst}$. Continuing along this line, we will show that any finite partial ordering can be embedded into $\mathcal{E}_{sst}$

THE SETTLING TIME REDUCIBILITY ORDERING AND ∆ 0 2 SETS

Abstract. The Settling Time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The <st ordering is in fact an ordering on c.e. sets, since it is independent of the particular enumeration chosen. In this paper we show that it is not possible to extend this ordering in an approximation-independent way to ∆0 2 sets in general, or even to n-c.e. sets for any fixed n ≥ 3. 1