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

Franz Baader
DFKI, DFKI Research Reports (RR), Vol. 90-13, 1990.


In Baader (1990a,1990b), we have considered different types of semantics for terminologicial cycles in the concept language FL0 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 FL0 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 FL0 with cyclic definitions without running into the troubles caused by cycles in larger languages. Starting with the language FL of Schmidt-Schauß & 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.