  We study comparisons between interpretations in description logics with
respect to "logical consequences" of the form of semi-positive concepts (like
semi-positive concept assertions). Such comparisons are characterized by
conditions similar to the ones of bisimulations. The simplest among the
considered logics is a variant of PDL (propositional dynamic logic). The others
extend that logic with inverse roles, nominals, quantified number restrictions,
the universal role, and/or the concept constructor for expressing the local
reflexivity of a role. The studied problems are: preservation of semi-positive
concepts with respect to comparisons, the Hennessy-Milner property for
comparisons, and minimization of interpretations that preserves semi-positive
concepts.
