Unifying Qualitative and Quantitative Safety Verification of DNN-Controlled Systems

The rapid advance of deep reinforcement learning techniques enables the oversight of safety-critical systems through the utilization of Deep Neural Networks (DNNs). This underscores the pressing need to promptly establish certified safety guarantees for such DNN-controlled systems. Most of the existing verification approaches rely on qualitative approaches, predominantly employing reachability analysis. However, qualitative verification proves inadequate for DNN-controlled systems as their behaviors exhibit stochastic tendencies when operating in open and adversarial environments. In this paper, we propose a novel framework for unifying both qualitative and quantitative safety verification problems of DNN-controlled systems. This is achieved by formulating the verification tasks as the synthesis of valid neural barrier certificates (NBCs). Initially, the framework seeks to establish almost-sure safety guarantees through qualitative verification. In cases where qualitative verification fails, our quantitative verification method is invoked, yielding precise lower and upper bounds on probabilistic safety across both infinite and finite time horizons. To facilitate the synthesis of NBCs, we introduce their $k$-inductive variants. We also devise a simulation-guided approach for training NBCs, aiming to achieve tightness in computing precise certified lower and upper bounds. We prototype our approach into a tool called $\textsf{UniQQ}$ and showcase its efficacy on four classic DNN-controlled systems.