On the semantic foundations of Probabilistic VERUS

In this paper we explain the semantic foundations of the tool Probabilistic VERUS Har98]. A Probabilistic VERUS program P consists of sequential randomized processes S 1 ; : : : ; S k that are executed in parallel and that communicate via shared variables. First we give an operational semantics for the sequential components on the basis of a transition relation deened in the classical SOS-style a la Plotkin Plo81] which we use to specify the behaviour of the P by a Markov chain whose transitions stand for the cumulative eeect of the activities of the components S 1 ; : : : S k within one time step. Second, we provide a denotational semantics for P that is based on a (denota-tional) least xed point semantics for the sequential components which formalizes the input/output behaviour of the sequential components within one time step. While the operational (declarative) semantics might be the one that a designer (who provides the input for the tool) has in mind the denotational (procedural) semantics is the one that the compiler of Probabilistic VERUS uses. We establish a consistency result stating that the Markov chains induced by the operational and denotational semantics are bisimilar in the sense of LS91].