Search

Closure spaces are a generalisation of topological spaces obtained by removing the idempotence requirement on the closure operator. We adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure models -- we call the resulting equivalence CM-bisimilarity -- and refine it for quasi-discrete closure models. We also define two additional notions of bisimilarity that are based on paths on space, namely Path-bisimilarity and Compatible Path-bisimilarity, CoPa-bisimilarity for short. The former expresses (unconditional) reachability, the latter refines it in a way that is reminishent of Stuttering Equivalence on transition systems. For each bisimilarity we provide a logical characterisation, using variants of the Spatial Logic for Closure Spaces (SLCS). We also address the issue of (space) minimisation via the three equivalences.

Topological Spatial Model Checking is a recent paradigm where model checking techniques are developed for the topological interpretation of Modal Logic. The Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability connectives that, in turn, can be used for expressing interesting spatial properties, such as "being near to" or "being surrounded by". SLCS constitutes the kernel of a solid logical framework for reasoning about discrete space, such as graphs and digital images, interpreted as quasi discrete closure spaces. Following a recently developed geometric semantics of Modal Logic, we propose an interpretation of SLCS in continuous space, admitting a geometric spatial model checking procedure, by resorting to models based on polyhedra. Such representations of space are increasingly relevant in many domains of application, due to recent developments of 3D scanning and visualisation techniques that exploit mesh processing. We introduce PolyLogicA, a geometric spatial model checker for SLCS formulas on polyhedra and demonstrate feasibility of our approach on two 3D polyhedral models of realistic size. Finally, we introduce a geometric definition of bisimilarity, proving that it characterises logical equivalence.