Strong jump traceability versus LR reducibility — Problem 2.2

Computability Theory

Assume throughout that $A,B\subset\mathbb{N}$. One writes

if for every order function $h$ there exists a uniformly $B$-c.e. trace $(T_{n})$ satisfying that

and

Also recall that

We note that $A\leq_{SJT}\emptyset$ implies $A\leq_{LR}\emptyset$, namely, every strongly jump traceable set is $K$-trivial, equivalently, low for ML-random. For background and references, see [1]. The authors there give several characterisations of $\leq_{SJT}$ on the $K$-trivials; for instance, for $A,B$ $K$-trivial,

Let Rec denote the class of recursive sets.

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