🤖 AI Summary
This paper addresses the lack of a unified theoretical foundation for behavioral equivalence in closure spaces and quasi-discrete closure spaces. We introduce three novel bisimulation relations—CM-, CMC-, and CoPa-bisimulation—based respectively on closure operators, converse closure operators, and compatible paths, to systematically characterize state equivalence in spatial model checking. CMC- and CoPa-bisimulations are original definitions; CM-bisimulation generalizes topological bisimulation, while CoPa-bisimulation is strictly equivalent to divergence-sensitive stuttering equivalence on Kripke structures. Each relation is precisely characterized by an infinitary modal logic: IML for CM-, IMLC for CMC-, and ICRL for CoPa-bisimulation. Furthermore, we establish deep connections between these bisimulations and classical semantics, notably neighborhood bisimulation. Our results provide a unified, decidable theory of behavioral equivalence for model verification over closure spaces.
📝 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''. (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.
