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VOLUME 5, ISSUE 2, PAPER 2


Neighbourhood Structures: Bisimilarity and Basic Model Theory

©Helle Hvid Hansen, Vrije Universiteit amsterdam
©Clemens Kupke, Centrum voor Wiskunde en Informatica
©Eric Pacuit, Stanford University

Abstract
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.

Publication date: April 9, 2009

Full Text: PDF | PostScript
DOI: 10.2168/LMCS-5(2:2)2009

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