Data-driven barrier certificate generation using deep learning and symbolic regression

Barrier certificate generation is an efficient and powerful technique for formally verifying the safety properties of cyber–physical systems. Neural networks are commonly used as the templates for barrier certificates, but the complex network structure makes it a challenge to verify the correctness of neural certificates. In this paper, we propose a novel data-driven framework that leverages deep learning and symbolic regression to synthesize barrier certificates in analytical form, with high efficiency and scalability. The framework is structured as an inductive loop with neural network training, distillation and verification. Specifically, a Learner leverages deep learning to train neural barrier candidates, which are then used as input for a Distiller to generate analytical barrier candidates via symbolic regression. Due to the simple analytical expressions, a Verifier then efficiently ensures the formal soundness of the analytical barrier candidates via an satisfiability modulo theories (SMT) solver, or generates counterexamples to further guide the Learner. We implement the tool SR4BC, and evaluate its performance over a set of benchmarks, which validates that SR4BC is much more efficient and effective than the state-of-the-art approaches.