Neighbourhood Structures: Bisimilarity and Basic Model Theory

Abstract

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 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.

Dynamic Logics of Evidence-Based Beliefs

“…[4,Chapter 7] and [18,10], for modern introductions). For convenience, we will stick with finite models, though most of our results are easily generalized to infinite settings.…”

“…A natural generalization here is the "monotonic bisimulation" familiar from the literature on neighbourhood semantics [10] and game logics [19]. It is a standard fact that the sublanguage of L 0 without belief modalities is invariant under total bisimulations (totality is needed for the universal modality).…”

“…Nevertheless, the space of potential operations on neighborhood models is much larger, even if we impose conditions of bisimulation invariance as in [10]). Instead of exploring this wide realm, we show one new operation.…”

Strong Completeness for Iteration-Free Coalgebraic Dynamic Logics

“…N maps a set X to P(P(X)), and function f to the double-inverse-image map N (f ) = (f −1 ) −1 . An N -coalgebra ν : X → N (X) is known in modal logic as a neighbourhood frame, and N -coalgebra morphisms as bounded neighbourhood morphisms [3,10]. The neighbourhood modality is interpreted via the predicate lifting given by λ X (U ) = {N ∈ N (X) | U ∈ N }.…”

Reasoning about Strategies

“…For those under (b), see (14). The first law of (c) is the fixed-point recursion in the Zermelo argument, and the second an introduction law reminiscent of the axiom for the universal iteration modality in propositional dynamic logic.…”