Coalgebraic logic for stochastic right coalgebras

Abstract

MSC: 03B45 03B70Keywords: Predicate liftings Behavioral equivalence Hennessy-Milner Theorem Stochastic relations Coalgebraic and modal logic Measurable selections a b s t r a c t 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.

Bisimilarity of Distributionally Equivalent Markov Transition Systems

“…Conjunction serves not only as a representative of the Boolean operations that are usually included in a logic (here disjunction would do as well), it is also -with a view towards Lemma 2.2 -decisive for the development from a technical vantage point, and here disjunction would not do; this was first observed in [3]. Disjunction may be added, however, through a natural transformation; this technique is discussed at length in [9,10]. The set Act of labels is assumed to be at most countably infinite.…”

“…Hence it invites a careful investigation of bisimilarity in the weak case; a look at the results in [6] shows that entering the discussion of Kleisli morphisms adds considerable color. Second, the treatment of coalgebraic stochastic logic [9,10] should be extended to Kleisli morphisms for the Giry monad. Here some non-trivial technical obstacles need to be overcome.…”

Stochastic coalgebraic logic: Bisimilarity and behavioral equivalence

“…Results similar to the ones obtained in the present paper could be obtained, provided the functor T has what we call the Hennessy-Milner property which means essentially that it admits a suitable selector. Details can be found in [13].…”

Coalgebraic logic over general measurable spaces – a survey

“…if we work in a Polish or analytic space, expressivity for this kind of coalgebraic logic has been investigated in Doberkat and Schubert (2009) and Doberkat (2008); the approach is unified and discussed in detail in Doberkat (2009b). if we work in a Polish or analytic space, expressivity for this kind of coalgebraic logic has been investigated in Doberkat and Schubert (2009) and Doberkat (2008); the approach is unified and discussed in detail in Doberkat (2009b).…”