DFKI Research Report-90-13
by Franz Baader
Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Terminological Cycles
In Baader (1990a,1990b), we have considered different types of semantics for terminologicial cycles in the concept language 0 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 0 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 0 with cyclic definitions without running into the troubles caused by cycles in larger languages. Starting with the language of Schmidt-Schau"s&Smolka (1988) -- which allows negation, conjunction and disjunction of concepts as well as value restrictions and exists-in restrictions -- we shall disallow cyclic concept definitions, 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. Surprisingly, this algorithm can also be used to decide subsumption with respect to concept equations, i.e., arbitrary equational axioms of the form C = D where C and D are concept terms. This is so because concept terms of our extended language can be used to encode finite sets of concept equations.
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