On Bisimilarity for Quasi-discrete Closure Spaces

TL;DR

This paper introduces three new notions of bisimilarity for closure spaces and quasi-discrete closure spaces, along with their logical characterizations, enhancing the theoretical framework for spatial model checking.

Contribution

It defines and characterizes three bisimilarity notions for closure models, extending the understanding of spatial modal logics and their relation to neighborhood and path-based equivalences.

Findings

CM-bisimilarity generalizes topo-bisimilarity

CMC-bisimilarity refines CM-bisimilarity for quasi-discrete models

CoPa-bisimilarity relates to divergence-blind stuttering equivalence

Abstract

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 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 ${\cal N}$ 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, 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 models.