We define a new logic-induced notion of bisimulation (called ρ-bisimulation) for coalgebraic modal logics given by a logical connection, and investigate its properties. We show that it is structural in the sense that it is defined only in terms of the coalgebra structure and the one-step modal semantics and, moreover, can be characterised by a form of relation lifting. Furthermore we compare ρ-bisimulations to several well-known equivalence notions, and we prove that the collection of bisimulations between two models often forms a complete lattice. The main technical result is a Hennessy-Milner type theorem which states that, under certain conditions, logical equivalence implies ρ-bisimilarity. In particular, the latter does not rely on a duality between functors T (the type of the coalgebras) and L (which gives the logic), nor on properties of the logical connection ρ.
de Groot, J., H. H. Hansen and A. Kurz, Logic-induced bisimulations, in: Advances in Modal Logic (AiML’14) (2020), pp. 289-308.