Weak Simplicial Bisimilarity for Polyhedral Models and SLCS_eta -- Extended Version

In the context of spatial logics and spatial model checking for polyhedral models -- mathematical basis for visualisations in continuous space -- we propose a weakening of simplicial bisimilarity. We additionally propose a corresponding weak notion of $\pm$-bisimilarity on cell-poset models, a discrete representation of polyhedral models. We show that two points are weakly simplicial bisimilar iff their repesentations are weakly $\pm$-bisimilar. The advantage of this weaker notion is that it leads to a stronger reduction of models than its counterpart that was introduced in our previous work. This is important, since real-world polyhedral models, such as those found in domains exploiting mesh processing, typically consist of large numbers of cells. We also propose SLCS_eta, a weaker version of the Spatial Logic for Closure Spaces (SLCS) on polyhedral models, and we show that the proposed bisimilarities enjoy the Hennessy-Milner property: two points are weakly simplicial bisimilar iff they are logically equivalent for SLCS_eta. Similarly, two cells are weakly $\pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS_eta. This work is performed in the context of the geometric spatial model checker PolyLogicA and the polyhedral semantics of SLCS.