On Bisimilarity for Quasi-discrete Closure Spaces
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Closure spaces, a generalisation of topological spaces, have shown to be a convenient theoretical framework for spatial model checking. The closure operator of closure spaces and quasi-discrete closure spaces induces a notion of neighborhood akin to that of topological spaces that build on open sets. For closure models and quasi-discrete closure models, in this paper we present three notions of bisimilarity that are logically characterised by corresponding modal logics with spatial modalities: (i) CM-bisimilarity for closure models (CMs) is shown to generalise Topo-bisimilarity for topological models. CM-bisimilarity corresponds to equivalence with respect to the infinitary modal logic IML that includes the modality N for “being near”. (ii) CMC-bisimilarity, with `CMC' standing for CM-bisimilarity with converse, refines CM-bisimilarity for quasi-discrete closure spaces, carriers of quasi-discrete closure models. Quasi-discrete closure models come equipped with two closure operators, Direct C and Converse C, stemming from the binary relation underlying closure and its converse. CMC-bisimilarity, is captured by the infinitary modal logic IMLC including two modalities, Direct N and Converse N, corresponding to the two closure operators. (iii) CoPa-bisimilarity on quasi-discrete closure models, which is weaker than CMC-bisimilarity, is based on the notion of compatible paths. The logical counterpart of CoPa-bisimilarity is the infinitary modal logic ICRL with modalities Direct ζ and Converse ζ, whose semantics relies on forward and backward paths, respectively. It is shown that CoPa-bisimilarity for quasi-discrete closure models relates to divergence-blind stuttering equivalence for Kripke structures.
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