Labelled transition systems as a Stone space

TL;DR

This paper demonstrates that the space of labelled transition systems, when viewed through a domain model, forms a Stone space with a topology that measures bisimilarity, unifying various semantics and enabling compactness results.

Contribution

It establishes a maximal-points space model for labelled transition systems as a Stone space, connecting bisimulation, modal transition systems, and domain theory in a novel way.

Findings

The quotient of labelled transition systems is a Stone space with a specific topology.

Image-finite labelled transition systems are dense in this space.

The set of systems refining a modal transition system is compact.

Abstract

A fully abstract and universal domain model for modal transition systems and refinement is shown to be a maximal-points space model for the bisimulation quotient of labelled transition systems over a finite set of events. In this domain model we prove that this quotient is a Stone space whose compact, zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree of bisimilarity such that image-finite labelled transition systems are dense. Using this compactness we show that the set of labelled transition systems that refine a modal transition system, its ''set of implementations'', is compact and derive a compactness theorem for Hennessy-Milner logic on such implementation sets. These results extend to systems that also have partially specified state propositions, unify existing denotational, operational, and metric semantics on partial processes, render robust consistency measures for modal transition systems, and yield an abstract interpretation of compact sets of labelled transition systems as Scott-closed sets of modal transition systems.