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

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