Extending $\mathcal {ALC}$ with the Power-Set Construct

Abstract

We continue our exploration of the relationships between Description Logics and Set Theory, which started with the definition of the description logic $\mathcal {ALC}^\varOmega$. We develop a set-theoretic translation of the description logic $\mathcal {ALC}^\varOmega$ in the set theory $\varOmega$, exploiting a technique originally proposed for translating normal modal and polymodal logics into $\varOmega$.

We first define a set-theoretic translation of $\mathcal {ALC}$ based on Schild’s correspondence with polymodal logics. Then we propose a translation of the fragment $\mathcal {LC}^{\varOmega }$ of $\mathcal {ALC}^\varOmega$ without roles and individual names. In this—simple—case the power-set concept is mapped, as expected, to the set-theoretic power-set, making clearer the real nature of the power-set concept in $\mathcal {ALC}^\varOmega$. Finally, we encode the whole language of $\mathcal {ALC}^\varOmega$ into its fragment without roles, showing that such a fragment is as expressive as $\mathcal {ALC}^\varOmega$. The encoding provides, as a by-product, a set-theoretic translation of $\mathcal {ALC}^\varOmega$ into the theory $\varOmega$, which can be used as basis for extending other, more expressive, DLs with the power-set construct.