CHARACTERISING SJT REDUCIBILITY

SJT reducibility between sets A , B ⊆ N $A,B \subseteq \mathbb N$ upper A comma upper B subset of or equal to double struck upper N is defined by A ≤ S J T B $A \le _{SJT} B$ upper A less than or equals Subscript upper S upper J upper T Baseline upper B if for each computable function h that is unbounded and nondecreasing, there is an h -bounded uniformly B -c.e. trace ( T n ) n ∈ N $(T_n)_{n \in \mathbb N} $ left parenthesis upper T Subscript n Baseline right parenthesis Subscript n element of double struck upper N such that for each n , the value J A ( n ) $J^A(n)$ upper J Superscript upper A Baseline left parenthesis n right parenthesis of the jump is in T n $T_n$ upper T Subscript n , if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the K -trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.