We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of rst-order logic. Semantically, one has the unpleasant e ect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only nitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are only applied to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a nite terminological default theory, which means that this type of default reasoning is decidable.

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Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be es timated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possi bility degree or a necessity degree that ex presses to what extent the fo rmula is pos sibly or necessarily true. Possibilistic reso lution yields a calculus for possibilistic logic which respects the semantics developed for possibilistic logic. A drawback, which possibilistic resolution in herits from classical resolution, is that it may not terminate if applied to fo rmulas belong ing to decidable fragments of first-order log ic. Therefore we propose an alternative proof method for possibilistic logic. The main fea ture of this method is that it completely ab stracts from a concrete calculus but uses as basic operation a test for classical entailment. We then instantiate possibilistic logic with a terminological logic, which is a decidable subclass of first-order logic but nevertheless much more expressive than propositional log ic. This yields an extension of terminological logics towards the representation of uncer tain knowledge which is satisfactory from a semantic as well as algorithmic point of view.

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Much of the research on concept languages, also called terminological languages, has focused on t he computational complexity of subsumption. The intractability results can be divided into two groups. First, it has been shown that extending the basic language F £with constructs containing some form of logical disjunction leads to co-NPhard subsumption problems. Second, adding negation to F £makes subsumption PSPACE-complete.The main result of this paper is that extending F £with unrestricted existential quantification makes subsumption NP-complete. This is the first proof of intractability for a concept language containing no construct expressing disjunction-whether explicitly or implicitly. Unrestricted existential quantification is therefore, alongside disjunction, a source of computational complexity in concept languages.

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