Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy [BBS86] than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL2, yet their join is in EL2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL2 and prove that EL2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL2 lower bounds for certain notions generalizing Selman’s P-selectivity [Sel79], which may be regarded as an interesting result in its own right.