The Equivalence of Model-Theoretic and Structural Subsumption in Description Logics

A new approach to the semantics of description logics, using concept algebras, was introduced in [Dionne et al., 1992b]. In that approach, the terms of a description logic, i.e., concept descriptions, were viewed as elements of a free algebra. In the context of a given knowledge base possibly involving cycles, an intensional semantics was given by mapping every concept description to a possibly non-well-founded set that embodied the abstract structure of the concept description. This is in contrast to an extensional semantics that assigns to a concept description a specific set it describes. These sets that embody the abstract structures of concept descriptions are the elements of the universal concept algebra. The novelty of this approach is that one can define an ordering on the terms in these algebras that corresponds directly to the structural subsumption algorithms that most of these logics employ in their implementations. In this paper we prove that structural subsumption in the universal concept algebra is the most abstract wi th respect to all the greatest fixed point models, i.e., that subsumption defined structurally is equivalent to subsumption defined model-theoretically. This result provides the l ink between our intensional semantics based on concept algebras and the usual extensional models.