On Bisimilarity for Quasi-discrete Closure Spaces

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 neighbourhood 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 and to be an instantiation of neighbourhood bisimilarity, when CMs are seen as (augmented) neighbourhood models. CM-bisimilarity corresponds to equivalence with respect to the infinitary modal logic IML that includes the modality N for “being near to”. (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, (Formula Presented) and (Formula Presented), stemming from the binary relation underlying closure and its converse. CMC-bisimilarity, is captured by the infinitary modal logic IMLC including two modalities, (Formula Presented) and (Formula Presented), 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 (Formula Presented) and (Formula Presented) 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 models.