Relating direct and predicate transformer partial correctness semantics for an imperative probabilistic-nondeterministic language

where b ranges over a set BExp of boolean expressions, a ranges over a set Act of basic actions and p is a real number with 0 < p < 1. The aim of the current paper is to perform this task for the partial correctness case where the direct semantics of a program P is given by a function from the set S of states to PLV(S), the convex lower powerdomain of valuations on S. Here V(S) is the set of all subprobability distributions on S which we identify with functions μ : S → [0, 1] such that ∑ s∈S μ(s) ≤ 1. Thus V(S) is a subdomain of I where I is the unit interval [0, 1] considered as a domain with the usual order ≤. The domain PLV(S) consist of all nonempty convex closed lower subsets of V(S) ordered by ⊆. With every f : S → PLV(S) one may associate the function Wp(f) : I → I defined as