On the Semantic Foundations of Probabilistic Synchronous Reactive Programs

Abstract

In this paper we consider synchronous parallel programs ▪ that are composed by sequential randomized processes ▪1,…, ▪k which communicate via shared variables. First, we give an operational semantics for the sequential components ▪i on the basis of a transition relation defined in the classical SOS-style à la Plotkin [Plo81] which we use to specify the behaviour of ▪ by a Markov chain whose transitions stand for the cumulative effect of the activities of the components ▪1,…▪k within one time step. Second, we provide a denotational semantics for ▪ that also models ▪ by a Markov chain. It is based on a (denotational) least fixed 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 a compiler might use. We establish a consistency result stating that the Markov chains induced by the operational and denotational semantics are bisimilar in the sense of [LS91].