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

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.