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Most modern formalisms used in Databases and Artificial Intelligence for describing an application domain are based on the notions of class (or concept) and relationship among classes. One interesting feature of such formalisms is the possibility of defining a class, i.e., providing a set of properties that precisely characterize the instances of the class. Many recent articles point out that there are several ways of assigning a meaning to a class definition containing some sort of recursion. In this paper, we argue that, instead of choosing a single style of semantics, we achieve better results by adopting a formalism that allows for different semantics to coexist. We demonstrate the feasibility of our argument, by presenting a knowledge representation formalism, the description logic µALCQ, with the above characteristics. In addition to the constructs for conjunction, disjunction, negation, quantifiers, and qualidied number restrictions, µALCQ includes special fixpoint constructs to express (suitably interpreted) recursive definitions. These constructs enable the usual frame-based descriptions to be combined with definitions of recursive data structures such as directed acyclic graphs, lists, streams, etc. We establish several properties of µALCQ, including the decidability and the computational complexity of reasoning, by formulating a correspondence with a particular modal logic of programs called the modal mu-calculus.

A Uniform Framework for Concept Definitions in Description Logics