Finite Basis for gfp-Model of Description Logic FLε

Description logics (DLs) are a class of knowledge representation formalisms in the tradition of semantic networks and frames, which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. The finite basis problems in DLs, especially the fundamentality and the current research progresses of the finite basis problems in DLs are analyzed in this paper. The attribute implication and the Duguenne-Guigues basis in the formal concept analysis (FCA) are studied. Based on the existence of the Duguenne-Guigues basis in FCA and the works of F Baader, a new context, namely description context and the implications of description logic formulas are defined. It is proved that there exists a unique greatest fixed-points(gfp) model in the terminological cycles of the description logic system FLε. Based on the gfp-model, the existence of the finite basis (a finite set of implications) in the terminological cycles in the description logic system FLε is proved. Moreover, the soundness and the completeness of the finite basis are proved, too. Such a finite basis provides the knowledge engineers with interesting information on the application domain described by the description context. The knowledge engineers can use these implications as starting point for building an ontology describing this application domain.