This is the first of two talks on a dualized view of Markov processes. The first part will describe how to view a probabilistic transition system as a transformer of functions rather than as a transformer of probability distributions. A Markov process is normally viewed as a Markov kernel i.e. a map from S x Σ → [0,1] where S is a state space and Σ is a σ-algebra on S. These Markov kernels are morphisms in the Kleisli category of the Giry monad. In recent work by Chaput, Danos, Panangaden and Plotkin, Markov processes were reinterpreted as linear maps on the space of positive L1 functions on S. This is analogous to taking the predicate transformer view of Markov processes. A number of dualities and isomorphisms emerge in this picture. Most interestingly conditional expectation can be understood functorially.
In the second part (by Panangaden) the theory of bisimulation and approximation will be presented from this dualized viewpoint.