Geometric Model Checking of Continuous Space

TL;DR

This paper extends topological spatial model checking to continuous space using polyhedra, enabling reasoning about 3D models with applications in visualization and mesh processing.

Contribution

It introduces a geometric interpretation of SLCS for continuous space and presents PolyLogicA, a model checker for polyhedral models, with proven logical equivalence via bisimilarity.

Findings

Feasibility demonstrated on large 3D polyhedral models

PolyLogicA effectively checks SLCS formulas on continuous space

Bisimilarity characterizes logical equivalence in this setting

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.