Locality and applications to subsumption testing and interpolation in $\mathcal{EL}$ and some of its extensions

In this paper we show that subsumption problems in lightweight description logics (such as $\mathcal{EL}$ and $\mathcal{EL}^+$) can be expressed as uniform word problems in classes of semilattices with monotone operators. We use possibilities of efficient local reasoning in such classes of algebras, to obtain uniform PTIME decision procedures for CBox subsumption in $\mathcal{EL}$, $\mathcal{EL}^+$ and extensions thereof. These locality considerations allow us to present a new family of (possibly many-sorted) logics which extend $\mathcal{EL}$ and $\mathcal{EL}^+$ with $n$-ary roles and/or numerical domains. As a by-product, this allows us to show that the algebraic models of ${\cal EL}$ and ${\cal EL}^+$ have ground interpolation and thus that ${\cal EL}$, ${\cal EL}^+$, and their extensions studied in this paper have interpolation. We also show how these ideas can be used for the description logic $\mathcal{EL}^{++}$.