A Non-Well-Founded Approach to Terminological Cycles

Robert Dionne, Eric Mays, Frank J. Oles

In this paper, we propose a new approach to intensional semantics of term subsumption languages. We introduce concept algebras, whose signatures are given by sets of primitive concepts, roles, and the operations of the language. For a given set of variables, standard results give us free algebras. We next define, for a given set of concept definitions, a term algebra, as the quotient of the free algebra by a congruence generated by the definitions. The ordering on this algebra is called descriptive subsumption (la). We also construct a universal concept algebra, as a non-well-founded set given by the greatest fixed point of a certain equation. The ordering on this algebra is called structural subsumption (?a). We prove there are unique mappings from the free algebras, to each of these, and establish that our method for classifying cycles in a term subsumption language, K-REP, consists of constructing accessible pointed graphs, representing terms in the universal concept algebra, and checking a simulation relation between terms.

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