Compositional Analysis for Almost-Sure Termination of Probabilistic Programs

In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on compositional analysis as their theoretical basis. In non-probabilistic programs, the classical variant rule (V-rule) of Floyd-Hoare logic is the foundation for compositional analysis. Extension of this rule to almost-sure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in [15]. While the proposed probabilistic variant cautiously addresses the key issue of integrability, we show that the proposed compositional rule is still not sound for almost-sure termination of probabilistic programs. Besides establishing unsoundness of the previous rule, our contributions are as follows: First, we present a sound compositional rule for almost-sure termination of probabilistic programs. Our approach is based on a novel notion of descent supermartingales. Second, for algorithmic approaches, we consider descent supermartingales that are linear and show that they can be synthesized in polynomial time. Finally, we present experimental results on several natural examples that model various types of nested while loops in probabilistic programs and demonstrate that our approach is able to efficiently prove their almost-sure termination property.