Bisimulation Equivalence in Description Logics and Its Applications Author's Declaration Supervisor's Declaration

In this dissertation, we study bisimulations and bisimulation-based comparisons in a uniform way for a large class of description logics that extend ALCreg (a variant of propositional dynamic logic) with an arbitrary set of features among I (inverse roles), O (nominals), Q (qualified number restrictions), U (the universal role), Self (local reflexivity of a role) as well as role axioms. We give results on invariance of concepts, TBoxes and ABoxes, preservation of RBoxes and knowledge bases, the Hennessy-Milner property w.r.t. bisimulations, the largest auto-bisimulations and quotient interpretations w.r.t. such equivalence relations. By adapting Hopcroft’s automaton minimization algorithm and the Paige-Tarjan algorithm, we give efficient algorithms for computing the partition corresponding to the largest auto-bisimulation of a finite interpretation. We provide results on preservation of semi-positive concepts, the Hennessy-Milner property w.r.t. bisimulation-based comparisons, characterizing bisimulation for tidy interpretations by semi-positive concepts, and minimization of interpretations that preserves semi-positive concepts. We separate the expressiveness of the description logics that extend L, where ALC ≤ L ≤ ALCreg, with any combination of the features I, O, Q, U , Self. Our separation results are w.r.t. concepts, positive concepts, TBoxes and ABoxes. We prove that any concept in any description logic that extends ALC with some features amongst I, Self, Qk (qualified number restrictions with numbers bounded by a constant k) can be learned if the training information system (specified as an interpretation) is good enough.