A characterization of the Δ20 hyperhyperimmune sets

Let A be an infinite \Delta^0_2 set and let K be creative: we show that K \leq_Q A if and only if K \leq_{Q_1} A. (Here \leq_Q denotes Q-reducibility, and \leq_{Q_1} is the subreducibility of \leq_Q obtained by requesting that Q-reducibility be provided by a computable function f such that W_{f(x)} \cap W_{f(y)}=\emptyset, if x \ne y.) Using this result we prove that A is hyperhyperimmune if and only if no \Delta^0_2 subset B of A is s-complete, i.e. there is no \Delta^0_2 subset B of A such that \overline{K} \leq_s B, where \le_s denotes s-reducibility, and \overline{K} denotes the complement of K.