On Bisimilarity for Polyhedral Models and SLCS

The notion of bisimilarity plays an important role in concurrency theory. It provides formal support to the idea of processes having “equivalent behaviour” and is a powerful tool for model reduction. Furthermore, bisimilarity typically coincides with logical equivalence of an appropriate modal logic enabling model checking to be applied on reduced models. Recently, notions of bisimilarity have been proposed also for models of space, including those based on polyhedra. The latter are central in many domains of application that exploit mesh processing and typically consist of millions of cells, the basic components of face-poset models, discrete representations of polyhedral models. This paper builds on the polyhedral semantics of the Spatial Logic for Closure Spaces (SLCS) for which the geometric spatial model checker PolyLogicA has been developed, that is based on face-poset models. We propose a novel notion of spatial bisimilarity for face-poset models, called ±-bisimilarity. We show that it coincides with logical equivalence induced by SLCS on such models. The latter corresponds to logical equivalence with respect to SLCS on polyhedra which, in turn, coincides with simplicial bisimilarity, a notion of bisimilarity for continuous spaces.