Stochastic Coalgebraic Logic - Preamble

Preface Motivation Modal logics are usually interpreted through Kripke models, branching logics find their interpretation through models which deal with infinite paths. These seemingly structurally different interpretations can be unified by considering coalgebras which model the underlying worlds suitably; the predicates through which the formulas are represented in their interpretation are modelled using natural transformations between functors, to which the functor that underlies the coalgebra contributes. The basic functor is usually based on the power set functor. Adopting this general approach, we see that a fairly general and uniform way of interpreting modal logics and their step twins arises through coalgebras and the generalization of predicates into suitable natural transformations. We will show in this treatise that coalgebras based on the subprobability functor are amenable to these ideas as well. Having arrived at this general approach of interpreting a rather broad family of logics, the question of comparing different models relative to a given logic presents itself. So we ask (and answer) the question about the conditions under which the well investigated relationship of logical equivalence, bisimilarity, and behavioral equivalence holds in this generalized, uniform scenario. This is one of two driving topics in this book. The other one notes that stochastic interpretations — such as the ones indicated above — rest on stochastic relations, which in turn are the Kleisli morphisms for the Giry monad. The morphisms considered so far are based on measurable maps which are the morphisms of the base category. But at least bisimilarity and behavioral equivalence are formulated through morphisms, viz., through the existence of a span or a cospan, respectively. These formulations apply verbatim to Kleisli morphisms as well. Generalizing logical equivalence to distribu-tional equivalence lifts the entire stage to the level of the Kleisli category, and, again, the problem of the relationship of the various behavioral descriptions vii viii Preface presents itself. Since morphisms and congruences are very closely related, a study of these issues needs to be accompanied by a careful investigation of congruences on the Kleisli category. A brief survey of the contents of the individual chapters is in order. Stochastic relations form the mathematical basis for a probabilistic interpretation of coalgebraic logics. They provide also a foundation for Markov transition systems. These relations in turn are based on transition probabilities, and because we do not confine ourselves to the probabilistic case but rather accept models in which mass vanishes, we …