Automatic Probabilistic Program Verification through Random Variable Abstraction

TL;DR

This paper introduces an automatic method using random variable abstraction within abstract interpretation to verify quantitative properties of probabilistic and nondeterministic programs on infinite state spaces.

Contribution

It presents the technique of random variable abstraction and a sufficient condition for exact fixed point computation, enabling automated analysis of complex probabilistic systems.

Findings

Successfully applied to compute expected running time

Determined expected gains in gambling strategies

Works on general probabilistic and nondeterministic transition systems

Abstract

The weakest pre-expectation calculus has been proved to be a mature theory to analyze quantitative 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 sufficient 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 instead of plain pGCL programs, allowing us to easily model a wide range of systems including distributed ones and unstructured programs. We present the operational and weakest precondition semantics for this programs and prove its equivalence.