Abstract:
We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove that countable Aumann algebras and countably-generated continuous-space Markov processes defined on a certain class of Hausdorff spaces are dual in the sense of Stone-type duality. The first idea might be to define Markov processes on top of Stone spaces and just leverage ordinary Stone duality. However, there are some twists in the tale. One has to remove certain bad ultrafilters in order to get the duality. In doing so, one loses compactness and one has to use other ideas. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes. This is joint work with Dexter Kozen, Kim Larsen and Radu Mardare.