Semantics of Probabilistic Programs: A Weak Limit Approach

For a simple probabilistic language we present a semantics based on linear operators on infinite dimensional Hilbert spaces.We show the equivalence of this semantics with a standard operational one and we discuss its relationship with the well-known denotational semantics introduced by Kozen. For probabilistic programs, it is typical to use Banach spaces and their norm topology to model the properties to be analysed (observables). We discuss the advantages in considering instead Hilbert spaces as denotational domains, and we present a weak limit construction of the semantics of probabilistic programs which is based on the inner product structure of this space, i.e. the duality between states and observables.