Modular Verification for AlmostSure Termination of Probabilistic Programs. In this work, we consider the almostsure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on modularity as their theoretical basis. In nonprobabilistic programs, the classical variant rule (Vrule) of FloydHoare logic provides the foundation for modular analysis. Extension of this rule to almostsure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in Fioriti and Hermanns 2015. While the proposed probabilistic variant cautiously addresses the key issue of integrability, we show that the proposed modular rule is still not sound for almostsure termination of probabilistic programs.  Besides establishing unsoundness of the previous rule, our contributions are as follows: First, we present a sound modular rule for almostsure 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 a variety of benchmarks and 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 almostsure termination property.

Proving Expected Sensitivity of Probabilistic Programs with Randomized VariableDependent Termination Time. The notion of program sensitivity (aka Lipschitz continuity) specifies that changes in the program input result in proportional changes to the program output. For probabilistic programs the notion is naturally extended to expected sensitivity. A previous approach develops a relational program logic framework for proving expected sensitivity of probabilistic while loops, where the number of iterations is fixed and bounded. In this work, we consider probabilistic while loops where the number of iterations is not fixed, but randomized and depends on the initial input values. We present a sound approach for proving expected sensitivity of such programs. Our sound approach is martingalebased and can be automated through existing martingalesynthesis algorithms. Furthermore, our approach is compositional for sequential composition of while loops under a mild side condition. We demonstrate the effectiveness of our approach on several classical examples from Gambleru0027s Ruin, stochastic hybrid systems and stochastic gradient descent. We also present experimental results showing that our automated approach can handle various probabilistic programs in the literature.