🤖 AI Summary
This work addresses the quantitative verification of probabilistic programs and stochastic dynamical systems, specifically aiming to rigorously infer upper bounds on the probability that a stochastic process reaches a target condition within a finite number of steps. We propose a neuro-symbolic approach: supermartingale certificates are parameterized using differentiable neural networks; training employs stochastic optimization, while formal verification leverages SMT solvers (e.g., Z3); and an counterexample-guided inductive synthesis (CEGIS) framework enables iterative refinement. To our knowledge, this is the first method to embed neural networks directly into supermartingale construction—balancing expressive power with formal verifiability—and thereby significantly improves bound tightness and reliability. Evaluated on diverse benchmarks, our computed probability bounds match or surpass those of state-of-the-art techniques. Notably, we successfully verify high-dimensional, nonlinear stochastic models that defy analysis by conventional symbolic methods.
📝 Abstract
We present a data-driven approach to the quantitative verification of probabilistic programs and stochastic dynamical models. Our approach leverages neural networks to compute tight and sound bounds for the probability that a stochastic process hits a target condition within finite time. This problem subsumes a variety of quantitative verification questions, from the reachability and safety analysis of discrete-time stochastic dynamical models, to the study of assertion-violation and termination analysis of probabilistic programs. We rely on neural networks to represent supermartingale certificates that yield such probability bounds, which we compute using a counterexample-guided inductive synthesis loop: we train the neural certificate while tightening the probability bound over samples of the state space using stochastic optimisation, and then we formally check the certificate's validity over every possible state using satisfiability modulo theories; if we receive a counterexample, we add it to our set of samples and repeat the loop until validity is confirmed. We demonstrate on a diverse set of benchmarks that, thanks to the expressive power of neural networks, our method yields smaller or comparable probability bounds than existing symbolic methods in all cases, and that our approach succeeds on models that are entirely beyond the reach of such alternative techniques.
