Ranking and Invariants for Lower-Bound Inference in Quantitative Verification of Probabilistic Programs

Quantitative properties of probabilistic programs are often characterised by the least fixed point of a monotone function $K$. Giving lower bounds of the least fixed point is crucial for quantitative verification. We propose a new method for obtaining lower bounds of the least fixed point. Drawing inspiration from the verification of non-probabilistic programs, we explore the relationship between the uniqueness of fixed points and program termination, and then develop a framework for lower-bound verification. We introduce a generalisation of ranking supermartingales, which serves as witnesses to the uniqueness of fixed points. Our method can be applied to a wide range of quantitative properties, including the weakest preexpectation, expected runtime, and higher moments of runtime. We provide a template-based algorithm for the automated verification of lower bounds. Our implementation demonstrates the effectiveness of the proposed method via an experiment.