Bisimulation for Neighbourhood Structures

Abstract

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2 . In our paper, we investigate the coalgebraic equivalence notions of 2 2 -bisimulation, behavioural equivalence and neighbourhood bisimulation (a notion based on pushouts), with the aim of finding the logically correct notion of equivalence on neighbourhood structures. Our results include relational characterisations for 2 2 -bisimulation and neighbourhood bisimulation, and an analogue of Van Benthem's characterisation theorem for all three equivalence notions. We also show that behavioural equivalence gives rise to a Hennessy-Milner theorem, and that this is not the case for the other two equivalence notions.

Structures for Epistemic Logic

“…This semantics is characterised by an axiomatisation given by [78], summarised in Table 4. Common Knowledge for neighbourhood models Table 4 The axioms and rules above are added to the propositional Taut and MP It is also possible to generalise the notion of bisimulation (to behavioural equivalence) to neighbourhood models, as well as to have a suitable notion of standard translation to a two-sorted first-order language, where the crucial clause for the translation is ST x (2ϕ) = ∃T (xNT ∧ ∀y(T Ey ↔ ST y (ϕ))), where xNT iff T ∈ N(x) and T Ey iff y ∈ T . With such an apparatus in place, [60] has been able to prove a 'van Benthem-style' characterisation theorem for modal logic using a neighbourhood semantics. For completeness of modal logics wrt neighbourhood semantics we refer to e.g., [37] and [59].…”

6 Algebras and coalgebras

“…For instance, the definition of bisimilarity presented in HBML for monotone neighborhood frames really corresponds to behavioral equivalence for the functor UP . One can give a fairly simple example of such a relation between two structures that is not a UP -bisimulation in the sense of Definition 11.4, see HANSEN & KUPKE [55] for the details. In order to guarantee that the two notions do coincide, consider the following constraint on the functor.…”

Modelling simultaneous games in dynamic logic

“…[214] Synthese (2008) 165:247-268 251 deeper logical study of non-monotonic versions of powers in a general neighbourhood setting, we refer to Hansen et al (2007).…”