The powerdomain of indexed valuations

This paper is about combining nondeterminism and probabilities. We study this phenomenon from a domain theoretic point of view. In domain theory, nondeterminism is modeled using the notion of powerdomain, while probability is modeled using the powerdomain of valuations. Those two functors do not combine well, as they are. We define the notion of powerdomain of indexed valuations, which can be combined nicely with the usual nondeterministic powerdomain. We show an equational characterization of our construction. Finally we discuss the computational meaning of indexed valuations, and we show how they can be used, by giving a denotational semantics of a simple imperative language.