Proof rules for probabilistic loops

Probabilistic predicate transformers provide a semantics for imperative programs containing both demonic and probabilistic nondeterminism. Like the (standard) predicate transformers popularised by Dijkstra, they model programs as functions from final results to the initial conditions sufficient to achieve them. This paper presents practical proof rules, using the probabilistic transformers, for reasoning about iterations when probability is present. They are thoroughly illustrated by example: probabilistic binary chop, faulty factorial, the martingale gambling strategy and Herman’s probabilistic self-stabilisation. Just as for traditional programs, weakest-precondition based proof rules for program derivation are an important step on the way to designing more general refinement techniques, or even a refinement calculus, for imperative probabilistic programming.