### Abstract

Language | English |
---|---|

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 |

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### Keywords

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

### Cite this

*Logical Methods in Computer Science*,

*5*(2), [2]. https://doi.org/10.2168/LMCS-5(2:2)2009

}

*Logical Methods in Computer Science*, vol. 5, no. 2, 2. https://doi.org/10.2168/LMCS-5(2:2)2009

**Neighbourhood structures : bisimilarity and basic model theory.** / Hansen, Helle Hvid; Kupke, Clemens; Pacuit, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Neighbourhood structures

T2 - Logical Methods in Computer Science

AU - Hansen, Helle Hvid

AU - Kupke, Clemens

AU - Pacuit, Eric

PY - 2009/4/9

Y1 - 2009/4/9

N2 - 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.

AB - 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.

KW - Neighbourhood semantics

KW - non-normal modal logic

KW - bisimulation

KW - behavioural equivalence

U2 - 10.2168/LMCS-5(2:2)2009

DO - 10.2168/LMCS-5(2:2)2009

M3 - Article

VL - 5

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

IS - 2

M1 - 2

ER -