Finite Model Reasoning in DL-Lite with Cardinality Constraints

The relationship of description logics (DLs) and conceptual modelling has been extensively studied in the literature [5, 4, 1]. One of the advantages of using description logics as modelling languages is that along with their capability of representing knowledge they provide also reasoning services. More precisely, a conceptual model can be represented by a DL ontology (TBox), and standard reasoning services (e.g., satisfiability and subsumption) allow to verify some properties of the conceptual model (e.g., consistency) and infer relations between concepts (IS-A relationships between classes or entities) that are not explicitly expressed. In order to use DLs effectively for conceptual modelling we need to ensure (1) that the chosen DL language is expressive enough to capture faithfully the intended semantics of traditional modelling languages (e.g., UML class diagrams, ER schema), and (2) that the complexity of reasoning in the chosen DL is acceptable (e.g., tractable). Regarding (1), it is worth noticing that the domain of interest in most applications is finite, therefore, reasoning on conceptual models should be understood as reasoning w.r.t. finite models. The latter is not the usual assumption in DLs mainly because traditional description logics enjoy the finite model property (FMP), and hence there is no need to distinguish between reasoning w.r.t. arbitrary models, and w.r.t. finite ones. Notably, ALC (one of the traditional DLs) is not expressive enough for capturing cardinality constraints. In DLs cardinality constraints are expressed by (qualified) number restrictions. ALCQI –which extends ALC with qualified number restrictions and inverse roles– captures the semantics of UML class diagrams [4]. However, this extension of ALC does not enjoy the FMP any more. One drawback for the use of ALCQI is the complexity of reasoning: finite satisfiability of ALCQI knowledge bases is ExpTime-complete [11]. The high complexity of reasoning makes ALCQI not very attractive for the conceptual modelling task; specially because no optimized algorithms for finite model reasoning exist. As an alternative, members of the DL-Lite-family of description logics including unqualified