We propose an approach for extending both the terminological and the assertional part of a Description Logic knowledge base by using information provided by the knowledge base and by a domain expert. The use of techniques from Formal Concept Analysis ensures that, on the one hand, the interaction with the expert is kept to a minimum, and, on the other hand, we can show that the extended knowledge base is complete in a certain, well-defined sense.

Given a finite set $C := C_1, C_n$ of description logic concepts, we are interested in computing the subsumption hierarchy of all least common subsumers of subsets of $C$ as well as the hierarchy of all conjunctions of subsets of $C$. These hierarchies can be used to support the bottom-up construction of description logic knowledge bases. The point is to compute the first hierarchy without having to compute the least common subsumer for all subsets of $C$, and the second hierarchy without having to check all possible pairs of such conjunctions explicitly for subsumption. We will show that methods from formal concept analysis developed for computing concept lattices can be employed for this purpose.