Weak Simplicial Bisimilarity and Minimisation for Polyhedral Model Checking
Nick Bezhanishvili, Laura Bussi, Vincenzo Ciancia, David Gabelaia, Mamuka Jibladze, Diego Latella, Mieke Massink, Erik P. de Vink
TL;DR
This work advances scalable spatial model checking over polyhedral spaces by introducing SLCS_η, a weaker variant of SLCS_γ, and by establishing weak simplicial bisimilarity in polyhedral models and weak ±-bisimilarity in their cell poset counterparts. It proves that η-equivalence aligns with these bisimulation notions (the Hennessy–Milner Property), enabling robust model reduction through a three-step minimisation: encode poset models as LTSs, minimise via branching bisimulation, and reconstruct a minimal Kripke model that preserves η. The authors provide a rigorous correctness framework and demonstrate practical gains through an experimental toolchain that couples PolyLogicA, mCRL2, and PolyVisualizer, yielding dramatic reductions in model size and substantial speedups in η-based reachability checks. This combination of theory and tooling promises scalable verification for complex polyhedral geometries in 3D graphics and related domains, with future work aimed at deeper integration and extension to additional operators such as distance-aware modalities.
Abstract
The work described in this paper builds on the polyhedral semantics of the Spatial Logic for Closure Spaces (SLCS) and the geometric spatial model checker PolyLogicA. Polyhedral models are central in domains that exploit mesh processing, such as 3D computer graphics. A discrete representation of polyhedral models is given by cell poset models, which are amenable to geometric spatial model checking on polyhedral models using the logical language SLCS$η$, a weaker version of SLCS. In this work we show that the mapping from polyhedral models to cell poset models preserves and reflects SLCS$η$. We also propose weak simplicial bisimilarity on polyhedral models and weak $\pm$-bisimilarity on cell poset models, where by ``weak'' we mean that the relevant equivalence is coarser than the corresponding one for SLCS, leading to a greater reduction of the size of models and thus to more efficient model checking. We show that the proposed bisimilarities enjoy the Hennessy-Milner property, i.e. two points are weakly simplicial bisimilar iff they are logically equivalent for SLCS$η$. Similarly, two cells are weakly $\pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS$η$. Furthermore we present a model minimisation procedure and prove that it correctly computes the minimal model with respect to weak $\pm$-bisimilarity, i.e. with respect to logical equivalence of SLCS$η$. The procedure works via an encoding into LTSs and then exploits branching bisimilarity on those LTSs, exploiting the minimisation capabilities as included in the mCRL2 toolset. Various examples show the effectiveness of the approach.