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Neighbourhood structures : bisimilarity and basic model theory

Hansen, Helle Hvid and Kupke, Clemens and Pacuit, Eric (2009) Neighbourhood structures : bisimilarity and basic model theory. Logical Methods in Computer Science, 5 (2). ISSN 1860-5974

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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 2². We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2²-bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2² does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 2²-bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2²-bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem's characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.