Computing reachability probabilities is a fundamental problem in the analysis of randomized programs. This article aims at a comprehensive and comparative account of various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points—a classic topic in computer science—in the analysis of randomized programs. This leads us to two new martingale-based techniques, too. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales.

Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account of various martingale-based methods for over-and underapproximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points-a classic topic in computer science-in the analysis of probabilistic programs. This leads us to two new martingale-based techniques, too. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales.

We introduce a novel sampling algorithm for Bayesian inference on imperative probabilistic programs. It features a hierarchical architecture that separates control flows from data: the top-level samples a control flow, and the bottom level samples data values along the control flow picked by the top level. This separation allows us to plug various language-based analysis techniques in probabilistic program sampling; specifically, we use logical backward propagation of observations for sampling efficiency. We implemented our algorithm on top of Anglican. The experimental results demonstrate our algorithm's efficiency, especially for programs with while loops and rare observations.