Semantic Importance Sampling for Statistical Model Checking

Statistical Model Checking SMC is a technique, based on Monte-Carlo simulations, for computing the bounded probability that a specific event occurs during a stochastic systems execution. Estimating the probability of a rare event accurately with SMC requires many simulations. To this end, Importance Sampling IS is used to reduce the simulation effort. Commonly, IS involves tilting the parameters of the original input distribution, which is ineffective if the set of inputs causing the event i.e., input-event region is disjoint. In this paper, we propose a technique called Semantic Importance Sampling SIS to address this challenge. Using an SMT solver, SIS recursively constructs an abstract indicator function that over-approximates the input-event region, and then uses this abstract indicator function to perform SMC with IS. By using abstraction and SMT solving, SIS thus exposes a new connection between the verification of non-deterministic and stochastic systems. We also propose two optimizations that reduce the SMT solving cost of SIS significantly. Finally, we implement SIS and validate it on several problems. Our results indicate that SIS reduces simulation effort by multiple orders of magnitude even in systems with disjoint input-event regions.