Kantorovich Functors and Characteristic Logics for Behavioural Distances

Abstract Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest generic approaches to behavioural distances. In particular, coalgebraic methods capture variations system type (nondeterministic, probabilistic, game-based etc.), and notion of quantale abstracts over actual values take, thus covering, e.g., two-valued equivalences, (pseudo)metrics, (pseudo)metrics. Coalgebraic have been based either on liftings $$\textsf{Set}$$ Set -functors categories metric spaces, lax extensions relations. Every extension induces a functor lifting but not every comes from extension. It was shown recently that Kantorovich, i.e. induced by suitable choice monotone predicate liftings, implying via Hennessy-Milner theorem can be characterized modal logics. Here, we essentially show same more general setting liftings. lifting, indeed (quantale-valued) preserves isometries so distance (on systems suitably restricted branching degree) logic.