Locality and subsumption testing in EL and some of its extensions

In this paper we show that subsumption problems in many lightweight description logics (including ${\cal EL}$) 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 ${\cal EL}$ and extensions thereof. These locality considerations allow us to present a new family of logics which extend ${\cal EL}$ with $n$-ary roles and numerical domains and also a new extension of ${\cal EL}^+$