Topological Cones: Foundations for a Domain Theoretical Semantics Combining Probability and Nondeterminism

There are more and more papers dealing with situations where probabilistic features occur together with ordinary nondeterminism. Based on the PhD thesis by R. Tix [18], domain theoretical tools for combining probability with nondeterminism were developped in [19]. A motivating situation was the semantics of an imperative language with both probabilistic and nondeterministic choice as considered by McIver and Morgan [9, 10] for discrete state spaces. In [19] dicrete state spaces are replaced by arbitrary continuous domains in the sense of, e.g., [4]. For a denotational model within domain theory of concurrent systems, similar ideas have been developped by Mislove, Ouaknine and Worrell in [11, 12] for a probabilistic process algebra also both with probabilistic and purely nondeterministic choice. D. Varacca [20, 21] has undertaken a subtle investigations of a semantics for imperative languages taking in account schedulers that lead him to slightly weaker notions than those considered in the other papers just mentioned. In this extended abstract we intend to show that the theory can be extended to larger classes of spaces including in particular stably locally compact state spaces. We are indebted to G.Plotkin who has developped similar ideas (see e.g. [13]). It turned out that, dealing with domain theoretical versions of probabilities and spaces of probabilities (and, more generally, spaces of measures), a domain theoretical variant of functional analytic concepts and tools like topological vector spaces, their topological duals and Hahn-Banach type separation theorems had to be developped. In [19] the underlying setting was domain theoretical in a strict sense: All spaces arose from directed complete partially ordered sets (dcpos, for short) and continuous domains with their Scott topology. In order to include more general state spaces and in particular arbitrary stably locally compact spaces one has to extend the results to more general topological setting including topologies that correspond to weak*topologies in duals of topological vector spaces. Thus, our approach is basically topological and not an order theoretical one. Nevertheles, often proofs