Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has $K \not\leq_{ss} B$ (respectively, $K \not\leq_{\overline{s}} B$): here $\leq_{\overline{s}}$ is the finite-branch version of s-reducibility, $\leq_{ss}$ is the computably bounded version of $\leq_{\overline{s}}$, and $\overline{K}$ is the complement of the halting set. Restriction to $\Sigma^0_2$ sets provides a similar characterization of the $\Sigma^0_2$ hyperhyperimmune sets in terms of s-reducibility. We also showthat no $A \geq_{\overline{s}} \overline{K}$ is hyperhyperimmune. As a consequence, $deg_s (\overline{K})$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed