Helle Hvid Hansen ; Clemens Kupke ; Eric Pacuit - Neighbourhood Structures: Bisimilarity and Basic Model Theory

lmcs:1167 - Logical Methods in Computer Science, April 9, 2009, Volume 5, Issue 2 - https://doi.org/10.2168/LMCS-5(2:2)2009
Neighbourhood Structures: Bisimilarity and Basic Model Theory

Authors: Helle Hvid Hansen ; Clemens Kupke ; Eric Pacuit

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 witnessing 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 neighbourhood 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 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.

Volume: Volume 5, Issue 2
Published on: April 9, 2009
Accepted on: June 25, 2015
Submitted on: December 19, 2007
Keywords: Computer Science - Logic in Computer Science,F.1.1,F.3.2,F.4.1,I.2.4


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