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

Abstract

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 [R. Tix. Some results on Hahn-Banach type theorems for continuous d-cones. Theoretical Computer Science, 264:205–218, 2001], domain theoretical tools for combining probability with nondeterminism were developed in [R. Tix, K. Keimel, G.D. Plotkin. Semantic Domains Combining Probabilty and Nondeterminism. Electronic Notes in Theoretical Computer Science, 129:1–104, 2005]. A motivating situation was the semantics of an imperative language with both probabilistic and nondeterministic choice as considered by McIver and Morgan [A. McIver and C. Morgan. Partial correctness for probablistic demonic programs. Theoretical Computer Science, 266:513–541, 2001; A. McIver, and C. Morgan, Specification and Refinement of Probabilistic Systems. Monographs in Computer Science, Springer Verlag, 2004, 402 pages] for discrete state spaces. In [R. Tix, K. Keimel, G.D. Plotkin. Semantic Domains Combining Probabilty and Nondeterminism. Electronic Notes in Theoretical Computer Science, 129:1–104, 2005] discrete state spaces are replaced by arbitrary continuous domains. 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. Proofs are mostly omitted. In dealing with domain theoretical versions of probabilities and spaces of probabilities (and, more generally, spaces of measures), domain theoretical variants of functional analytic concepts and tools like topological vector spaces, their topological duals and Hahn-Banach type separation theorems have to be developed.