On directed simulations in description logics

We study directed simulations in description logics, which are comparisons between interpretations with respect to logical consequences of the form of semi-positive concepts (like semi-positive concept assertions). Such relations 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 a set of features among inverse roles, nominals, qualified number restrictions, the universal role and the concept constructor for expressing the local reflexivity of a role. The studied problems are: preservation of semi-positive concepts by directed simulations; separating the expressiveness of description logics w.r.t. semi-positive concepts and positive concepts; the Hennessy-Milner property w.r.t. semi-positive concepts and positive concepts; the relationship between bisimilarity and directed similarity; interpretation minimization that preserves semi-positive concepts.