Boolean Operations, Joins, and the Extended Low Hierarchy

TL;DR

This paper investigates the extended low hierarchy in computational complexity, revealing that the join operation can reduce complexity levels and that EL_2 is not closed under certain Boolean operations, challenging its naturalness as a measure.

Contribution

It proves that join can lower the extended low hierarchy level and establishes EL_2's non-closure under specific Boolean operations, with new lower bounds for generalized Selman's P-selectivity.

Findings

Join of two sets can be in a lower EL_2 level than either set.

EL_2 is not closed under certain Boolean operations.

Established optimal EL_2 lower bounds for generalized P-selectivity.

Abstract

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. 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 EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.