Logic-Induced Bisimulations
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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.
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