We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin' s monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations. (Implementations by themselves are not addressed here.) Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: all measures on the real line can be modeled by minimal valuations on the domain IR ⊥ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples in order to demonstrate that our semantics computes exactly as one would expect, and in order to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance.

We investigate the impact that general concept inclusions and role-value maps have on the complexity and decidability of reasoning in the description logic FL 0. On the one hand, we give a more direct proof for ExpTime-hardness of subsumption w.r.t. general concept inclusions in FL 0. On the other hand, we determine restrictions on role-value maps that ensure decidability of subsumption, but we also show undecidability for the cases where these restrictions are not satisfied.

We investigate the impact that general concept inclusions and role-value maps have on the complexity and decidability of reasoning in the Description Logic FL₀. On the one hand, we give a more direct proof for ExpTimehardness of subsumption w.r.t. general concept inclusions in FL₀. On the other hand, we determine restrictions on role-value maps that ensure decidability of subsumption, but we also show undecidability for the cases where these restrictions are not satisfied.