Inherent enumerability of strong jump-traceability

TL;DR

This paper demonstrates that strongly jump-traceable sets are inherently enumerable, obey all benign cost functions, and have several equivalent characterizations, revealing deep structural properties in computability theory.

Contribution

It establishes the inherent enumerability of strongly jump-traceable sets and generalizes their properties to all such sets, linking them to various randomness and lowness notions.

Findings

Strongly jump-traceable sets obey all benign cost functions.

Every strongly jump-traceable set is computable from a c.e. strongly jump-traceable set.

Strongly jump-traceable sets form an ideal in the Turing degrees.

Abstract

We show that every strongly jump-traceable set obeys every benign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise properties of c.e.\ strongly jump-traceable sets to all such sets. For example, the strongly jump-traceable sets induce an ideal in the Turing degrees; the strongly jump-traceable sets are precisely those that are computable from all superlow Martin-L\"{o}f random sets; the strongly jump-traceable sets are precisely those that are a base for $\text{Demuth}_{\text{BLR}}$-randomness; and strong jump-traceability is equivalent to strong superlowness.