Geometric Model Checking of Continuous Space

Abstract

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.

Polyhedral Completeness of Intermediate Logics: The Nerve Criterion

“…Recently, this semantics has been applied to the field of model checking. The authors of [Bez+21] developed a geometric spatial model checker using polyhedral semantics, introducing the notion of bisimularity for polyhedra along the way.…”

Consistency-Checking 3D Geological Models

“…However, they are not yet applied to the evaluation of geological models, especially in combination with temporal relations, despite being applied to the evaluation of models in other domains (e.g. Van Oosterom, 1997;Gong and Mu, 2000;Arora et al, 2021;Nikoohemat et al, 2021;Bezhanishvili et al 2022).…”