Coalgebraic logic for stochastic right coalgebras

We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor F ⋅ S with F as an endofunctor on the category of analytic spaces, and S is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the set of predicate liftings. Consequently, behavioral and logical equivalence are really the same. From the cospan we construct a span. The central argument is a selection argument giving us the dynamics of a mediating coalgebra from the domains of the cospan. This construction is used to establish that behavioral equivalent coalgebras are bisimilar, yielding the equivalence of all three characterizations of a coalgebra’s behavior as in the case of Kripke models or Markov transition systems.