A Scalable Approach to Probabilistic Neuro-Symbolic Verification

Neuro-Symbolic Artificial Intelligence (NeSy AI) has emerged as a promising direction for integrating neural learning with symbolic reasoning. In the probabilistic variant of such systems, a neural network first extracts a set of symbols from sub-symbolic input, which are then used by a symbolic component to reason in a probabilistic manner towards answering a query. In this work, we address the problem of formally verifying the robustness of such NeSy probabilistic reasoning systems, therefore paving the way for their safe deployment in critical domains. We analyze the complexity of solving this problem exactly, and show that it is $\mathrm{NP}^{\# \mathrm{P}}$-hard. To overcome this issue, we propose the first approach for approximate, relaxation-based verification of probabilistic NeSy systems. We demonstrate experimentally that the proposed method scales exponentially better than solver-based solutions and apply our technique to a real-world autonomous driving dataset, where we verify a safety property under large input dimensionalities and network sizes.