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The main result of this paper is to prove the existence of a finite basis inthe description logic ${\cal ALC}$. We show that the set of General ConceptInclusions (GCIs) holding in a finite model has always a finite basis, i.e.these GCIs can be derived from finitely many of the GCIs. This result extends aprevious result from Baader and Distel, which showed the existence of a finitebasis for GCIs holding in a finite model but for the inexpressive descriptionlogics ${\cal EL}$ and ${\cal EL}_{gfp}$. We also provide an algorithm forcomputing this finite basis, and prove its correctness. As a byproduct, weextend our finite basis theorem to any finitely generated complete covariety(i.e. any class of models closed under morphism domain, coproduct and quotient,and generated from a finite set of finite models).

A finite basis theorem for the description logic ${\cal ALC}$