Towards Spatial Bisimilarity for Closure Models: Logical and Coalgebraic Characterisations

Abstract

The topological interpretation of modal logics provides descriptive languages and proof systems for reasoning about points of topological spaces. Recent work has been devoted to model checking of spatial logics on discrete spatial structures, such as finite graphs and digital images, with applications in various case studies including medical image analysis. These recent developments required a generalization step, from topological spaces to closure spaces. In this work we initiate the study of bisimilarity and minimization algorithms that are consistent with the closure spaces semantics. For this purpose we employ coalgebraic models. We present a coalgebraic definition of bisimilarity for quasi-discrete models, which is adequate with respect to a spatial logic with reachability operators, complemented by a free and open-source minimization tool for finite models. We also discuss the non-quasi-discrete case, by providing a generalization of the well-known set-theoretical notion of topo-bisimilarity, and a categorical definition, in the same spirit as the coalgebraic rendition of neighbourhood frames, but employing the covariant power set functor, instead of the contravariant one. We prove its adequacy with respect to infinitary modal logic.

Geometric Model Checking of Continuous Space

“…Furthermore, bisimilarity hints at minimization in order to reduce the complexity of the analysis. The preliminary results presented in [CLMV20], including the tool MiniLogicA could be useful in this research direction. Note that the quotient mapping each simplex in P to a point in K in Definition 4.1 is an open map, thus it preserves and reflects logical equivalence of the modal fragment of our language; additionally, as shown in Section 6, it preserves and reflects logical equivalence of the full language, thus simplicial bisimilarity.…”

Analysing Spatial Properties on Neighbourhood Spaces

“…To alleviate this, we could introduce a converse modality to near, to distinguish points appropriately. Ciancia et al [11] achieved such a distinction by employing "strong" variants of the reachability modalities, which allows them to define such a converse modality as an abbreviation.…”

“…We intend to examine this type of modality in the future. Recently, Ciancia et al investigated SLCS with coalgebraic methods [11]. They provide several definitions of bisimulations, both with and without a coalgebraic flavour, on quasi-discrete models, show that they coincide, and present an algorithm and an implementation to minimise a given model with respect to these bisimulations.…”