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 ${\cal N}$ 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 ${\cal C}$ and Converse ${\cal C}$, stemming from the binary relation underlying closure and its converse. CMC-bisimilarity, is captured by the infinitary modal logic IMLC including two modalities, Direct ${\cal N}$ and Converse ${\cal N}$, 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 $\zeta$ and Converse $\zeta$, 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.
翻译:关闭模型和准分解封闭空间的封闭操作器引发了类似于以开放机组为基础的表面空间的概念。对于封闭模型和准分解封闭模型模型模型,我们在本文件中提出了三个具有相应的模式逻辑特征的两样概念:(一)关闭模型(CM-CM)的CM-两样概念被显示为具有空间模式的相应模式检查。(一)关闭模型(CM)的CM-两样概念被显示为具有表层模型的泛化 Topo-双样性。关闭空间和准分解封闭空间模型的CM-BO。C-BIM(C-C)的关闭模型与不固定模式的ILML(美元)相匹配。CM(C-C-Cal-Cal-Cal-Conal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-Cal-CM)的两种关闭操作和C-Cal-C-C-Cal-C-C-C-CMIL-C-C-C-C-C-C-C-C-C-C-C-C-C-IFIL-IL-IL-和C-IL-IL-C-C-C-C-IL)的基)和基-C-C-C-C-C-C-和C-C-IL-IL-C-C-C-C-ID-ID-和C-和C-I-和C-I-I-I)的基关系,它以基、C-和C-和C-C-C-C-C-C-C-C-C-C-C-C-C-和C-和C-C-C-C-C-C-C-I-I-C-C-和C-C-C-C-C-C-C-C-I-I-IF-I-I-I