Boolean Operations, Joins, and the Extended Low Hierarchy
Lane A. Hemaspaandra, Zhigen Jiang, Joerg Rothe, Osamu Watanabe

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.
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Taxonomy
TopicsAdvanced Algebra and Logic · Complexity and Algorithms in Graphs · Advanced Graph Theory Research
