Refinement Algebra with Explicit Probabilism

Refinement algebra provides axioms for the stepwise removal of abstraction, in the form of demonic nondeterminism, in a first-order system that supports reasoning about loops. It has been extended by Solin and Meinecke to computations involving implicit probabilistic choices: demonic nondeterminism then satisfies weaker properties. In this paper their axiom system is extended to capture explicit probabilistic choices. The first form is an unquantified probabilistic choice; the second is a partial quantified probabilistic choice (from which the usual binary probabilistic choice can be recovered). The new refinement algebra is sound with respect to 1-bounded expectation transformers, the premier model of probabilistic computations, but also with respect to a new model introduced here to capture more directly partial quantified computations. In this setting a `normal form' result of Kozen is proved, replacing multiple loops with a single loop that does the same job; and the extent to which the two forms of loop have the same expected number of steps to termination is considered. Being entirely first-order, the new refinement algebra is targeted to automation.