# The join can lower complexity

Session 7

First Online:

## Abstract

We prove that the join of sets may actually be simpler than the sets themselves: there exist sets that are not in the second level of the extended low hierarchy [BBS86], EL_{2}, yet their join *is* in EL_{2}. That is, in terms of extended lowness, the join operator can lower complexity. We study the closure properties of EL_{2} 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 [Sel79], which may be regarded as an interesting result in its own right.

## Keywords

Polynomial Hierarchy Boolean Connective Extended Lowness Infinite Hierarchy Approximate Polynomial Time
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© Springer-Verlag Berlin Heidelberg 1996