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We provide a generic algorithm for constructing formulae that distinguishbehaviourally inequivalent states in systems of various transition types suchas nondeterministic, probabilistic or weighted; genericity over the transitiontype is achieved by working with coalgebras for a set functor in the paradigmof universal coalgebra. For every behavioural equivalence class in a givensystem, we construct a formula which holds precisely at the states in thatclass. The algorithm instantiates to deterministic finite automata, transitionsystems, labelled Markov chains, and systems of many other types. The ambientlogic is a modal logic featuring modalities that are generically extracted fromthe functor; these modalities can be systematically translated into custom setsof modalities in a postprocessing step. The new algorithm builds on an existingcoalgebraic partition refinement algorithm. It runs in time $\mathcal{O}((m+n)\log n)$ on systems with $n$ states and $m$ transitions, and the sameasymptotic bound applies to the dag size of the formulae it constructs. Thisimproves the bounds on run time and formula size compared to previousalgorithms even for previously known specific instances, viz. transitionsystems and Markov chains; in particular, the best previous bound fortransition systems was $\mathcal{O}(m n)$.

Explaining Behavioural Inequivalence Generically in Quasilinear Time