Geometric Model Checking of Continuous Space
This work addresses the need for spatial reasoning in continuous domains like 3D scanning and visualization, representing an incremental advancement by adapting existing logical frameworks to new geometric models.
The authors tackled the problem of extending spatial model checking from discrete to continuous spaces by proposing an interpretation of the Spatial Logic of Closure Spaces (SLCS) on polyhedra, resulting in the development of PolyLogicA, a geometric model checker demonstrated on two realistic 3D polyhedral models.
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.