Hybrid Reasoning of Terminological Cycles in Description Logic εL

Description logic is a logical reconstruction of the frame-based knowledge representation languages, with the aim of providing a simple well-established declarative semantics to capture the meaning of structured representation of knowledge. Terminological cycles have been a difficult spot in the study of description logic for quite a few years. The basic problems, such as their semantics and reasoning mechanisms, have not been reasonably well settled. The current research progresses and the existing problems of terminological cycles in description logic are analyzed in this paper. Based on the works of Baader and Brandt, the hybrid reasoning of terminological cycles in description logic εL is further studied. The syntax and semantics (including fixpoint semantics and descriptive semantics) of hybrid knowledge bases in εL are given. Aiming at the requirement of hybrid reasoning of terminological cycles in εL, TBox-completion is presented, and the description graphs (including syntax description graph and semantics description graph) are redefined. The instance checking reasoning algorithms of hybrid knowledge bases in εL w.r.t. greatest fixpoint semantics and descriptive semantics are presented using TBox-completion and description graphs. The correctness of reasoning algorithms is proved, and the complexity property of reasoning algorithms is given. Theoretical foundation for the hybrid reasoning of terminological cycles in εL is provided through the instance checking reasoning algorithms.