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On the one hand, class ical terminological knowledge representation excludes the possi bility of handling uncertain concept descrip tions involving, e.g., "usually true" concept properties, generalized quantifiers, or excep tions. On the other hand, purely numer ical approaches for handling uncertainty in general are unable to consider terminologi cal knowledge. This paper presents the lan guage At:CP which is a probabilistic extension of terminological logics and aims at closing the gap between the two areas of research. We present the formal semantics underlying the language At:CP and introduce the prob abilistic formalism that is based on class es of probabilities and is realized by means of probabilistic constraints. Besides infer ing implicitly existent probabilistic relation ships, the constraints guarantee terminologi cal and probabilistic consistency. Altogether, the new language .AirP applies to domains where both term descriptions and uncertain ty have to be handled.

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This paper proposes a probabilistic extension of terminological logics. The extension maintains the original performance of drawing inferences in a hierarchy of terminological definitions. It enlarges the range of applicability to real world domains determined not only by definitional but also by uncertain knowledge. First, we introduce th e propositionally complete terminological language ACe. On the basis of the language construct "probabilistic implication" it is shown how statistical information on concept dep endencies can be represented. To guarantee (terminological and probabilistic) consistency, several requirements have to be met. More.over , these requirements allow one to infer implicitly existent probabilistic relationships and their quantitative computation. By explicitly introducing restrictions for the ranges derived by instantiating the consistency requirements, exceptions can also be handled. In the categorical cases this corresponds to the overriding of properties in nonmonotonic inheritance networks. Consequently, our model applies to domains wh ere both term descriptions and non-categorical relations between term extensions have to be represented. "This work has been carried out in the WIP project which is supported by the German Ministry for Research and Technology BMFT under contract ITW 8901 8. I would like to thank Fahiem Bacchus, Bernhard Neb el, Bernd Owsnicki-Klewe , Hans-Jiirgen Profitlich, Alessandro Saffiotti, and the members of the Berlin BACK group for valuable comments on earlier versions of this paper. 1 Contents 1 Introduction 2 The Terminological Language .ACe 4 3 The Probabilistic Extension 4 Probabilistic Consistency and Inferences 4.1 Triangular Cases-Concept Specializations 9 4.2 Triangular Cases-Concept Definitions 5 Related Work 17

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