On bisimulations for description logics

Abstract

Abstract. We study bisimulations for useful description logics. The simplest among the considered logics is ALCreg (a variant of PDL). 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. They also allow role axioms. We give results about invariance of concepts, TBoxes and ABoxes, preservation of RBoxes and knowledge bases, and the Hennessy-Milner property w.r.t. bisimulations in the considered description logics. Using the invariance results we compare the expressiveness of the considered description logics w.r.t. concepts, TBoxes and ABoxes. Our results about separating the expressiveness of description logics are naturally extended to the case when instead of ALCreg we have any sublogic of ALCreg that extends ALC. We also provide results on the largest auto-bisimulations and quotient interpretations w.r.t. such equivalence relations. Such results are useful for minimizing interpretations and concept learning in description logics. To deal with minimizing interpretations for the case when the considered logic allows quantified number restrictions and/or the constructor for the local reflexivity of a role, we introduce a new notion called QS-interpretation, which is needed for obtaining expected results. 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.

Bisimulation-based concept learning for information systems in description logics

“…As DLs are variants of modal logics, bisimulation can be used to characterize indiscernibility of objects [9,10] and hence for concept learning in DLs. The earlier discussed work by Nguyen and Szałas [22] is devoted to Setting 3.…”

“…The idea is to use models of the given knowledge base and bisimulation in those models to guide the search for the concept to be learnt. Based on the BBCL method, Divroodi et al [8] studied C-learnability in DLs for Setting 3. Tran et al [29] developed the BBCL2 method for concept learning in DLs using Setting 2.…”

“…Divroodi and Nguyen [9,10] studied bisimulations for a number of DLs. Nguyen and Szałas [22] generalized that notion to model indiscernibility of objects and study concept learning.…”

“…Nguyen and Szałas [22] generalized that notion to model indiscernibility of objects and study concept learning. In this section, we generalize the notion of bisimulation further for the class of DLs studied in this paper, which is larger and more general than the one studied in [9,10,22]. …”

Computing Minimal Models Modulo Subset-Simulation for Modal Logics

“…This includes logics such as modal logics, description logics, and temporal logics (even those that are not translatable to fragments of first-order logic). It is known from the literature, for example [6], that bisimulation needs to be extended depending on the expressivity of the logic. This is because bisimulation, like simulation and subset-simulation, is a local definition.…”