### 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.

Original language | English |
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Article number | 2 |

Number of pages | 38 |

Journal | Logical Methods in Computer Science |

Volume | 5 |

Issue number | 2 |

DOIs | |

Publication status | Published - 9 Apr 2009 |

### Keywords

- Neighbourhood semantics
- non-normal modal logic
- bisimulation
- behavioural equivalence

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## Cite this

Hansen, H. H., Kupke, C., & Pacuit, E. (2009). Neighbourhood structures: bisimilarity and basic model theory.

*Logical Methods in Computer Science*,*5*(2), [2]. https://doi.org/10.2168/LMCS-5(2:2)2009