On Bisimilarity for Quasi-discrete Closure Spaces

• Main Authors: Ciancia, Vincenzo, Latella, Diego, Massink, Mieke, de Vink, Erik P
• Format: Journal Article
• Language: English
• Published: 27-01-2023
• Subjects: Computer Science - Logic in Computer Science
• DOI: 10.48550/arxiv.2301.11634

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.