Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Terminological Cycles

Abstract

In Baader (1990,1990a)1 , we · have considered different types of semantics for . terminologitial cycles in · the concept language ~Lo which allows only conjunction of concepts and value-restrictions. It turned out that greatest fixed-point semantics (gfp-semantics) seems to be most appropriate for cycles in this language. In 'the present paper we shall show that the concept defining facilities of ~Lo with cyclic definitions and gfp-semantics can also be obtained in a different way. One may replace cycles by role definitions involving union, composition, and transitive closure of roles. This proposes a way of retaining, in an extended language, the pleasant features of gfp-semantics for ~Lo with cyclic definitions without running into the troubles caused by cycles in larger languages. Starting with the language JILC of Schmidt-Schau8&Smolka (1988)--which allows negation, conjunction and disjunction of concepts as well as value-restrictions and exists-in-restrictions-we shall disallow cyclic concept defmitions, but instead shall add the possibility of role definitions involving union, composition, and transitive closure of roles. In contrast to other terminological KR-systems which incorporate the transitive closure operator for roles, we shall be able to give a sound and complete algorithm for concept subsumption.