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.

We give a new presentation of Brzozowski's algorithm to minimize finite automata using elementary facts from universal algebra and coalgebra and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal language equivalent automata from Moore nondeterministic and weighted automata.

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To reduce greenhouse gas emissions to 80% below 1990 levels by 2050, an energy transition is taking place in the European Union. Achieving these targets requires changes in the heating and cooling sector (H&C). Designing and implementing this energy transition is not trivial, as technology, actors, and institutions interact in complex ways. We provide an illustrative example of the development and use of an agent-based model (ABM) for thermal energy transitions in the built environment, from the perspective of sociotechnical systems (STS) and complex adaptive systems (CAS). In our illustrative example, we studied the transition of a simplified residential neighborhood to heating without natural gas. We used the ABM to explore socioeconomic conditions that could support the neighborhoods’ transition over 20 years while meeting the neighborhoods’ heat demand. Our illustrative example showed that through the use of STS, CAS, and an ABM, we can account for technology, actors, institutions, and their interactions while designing for thermal energy transitions in the built environment.

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