Stochastic Relations Interpreting Modal Logic

Abstract

We propose an interpretation of modal logic through stochastic relations, providing a probabilistic complement to the usual nondeterministic interpretations using Kripke models. A simple temporal logic and a logic with a countable number of diamonds illustrate the approach. The main technical result of this paper is a probabilistic analogon to the well-known Hennessy-Milner Theorem characterizing models that have the same theories for their states and bisimilarity as equivalent properties. This requires the study of congruences for stochastic relations that underly the interpretation, for which a general bisimilarity result is also established. The results depend on the existence of semi-pullbacks for stochastic relations over analytic spaces.