Neighbourhood Structures: Bisimilarity and Basic Model Theory

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2 . We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2 -bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose wit- nessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 2 2 -bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbour- hood structures, precocongruences are a better approximation of behavioural equivalence than 2 2 -bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with denability and image-niteness. We prove a Hennessy-Milner theorem for modally saturated and for image-nite neighbourhood mod- els. Our main results are an analogue of Van Benthem's characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.