A Minkowski type duality mediating between state and predicate transformer semantics for a 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. We write cond(b, P,Q) for the conditional usually denoted as if b then P else Q fi and while(b, P ) for the while loop usually denoted as while b do P od The program P8Q nondeterministically executes either P or Q. The program P p⊕Q executes P with probability p and Q with probability 1−p.