Automatic Probabilistic Program Verification through Random Variable Abstraction

The weakest pre-expectation calculus [20] has been proved to be a mature theory to analyze quan-titative properties of probabilistic and nondeterministic programs. We present an automatic method for proving quantitative linear properties on any denumerable state space using iterative backwards fixed point calculation in the general framework of abstract interpretation. In order to accomplish this task we present the technique of random variable abstraction (RVA) and we also postulate a suf-ficient condition to achieve exact fixed point computation in the abstract domain. The feasibility of our approach is shown with two examples, one obtaining the expected running time of a probabilistic program, and the other the expected gain of a gambling strategy. Our method works on general guarded probabilistic and nondeterministic transition systems in-stead of plain pGCL programs, allowing us to easily model a wide range of systems including dis-tributed ones and unstructured programs. We present the operational and wp semantics for this pro-grams and prove its equivalence.