  Intensive algorithmic efforts have been made to enable the rapid improvements
of certificated robustness for complex ML models recently. However, current
robustness certification methods are only able to certify under a limited
perturbation radius. Given that existing pure data-driven statistical
approaches have reached a bottleneck, in this paper, we propose to integrate
statistical ML models with knowledge (expressed as logical rules) as a
reasoning component using Markov logic networks (MLN, so as to further improve
the overall certified robustness. This opens new research questions about
certifying the robustness of such a paradigm, especially the reasoning
component (e.g., MLN). As the first step towards understanding these questions,
we first prove that the computational complexity of certifying the robustness
of MLN is #P-hard. Guided by this hardness result, we then derive the first
certified robustness bound for MLN by carefully analyzing different model
regimes. Finally, we conduct extensive experiments on five datasets including
both high-dimensional images and natural language texts, and we show that the
certified robustness with knowledge-based logical reasoning indeed
significantly outperforms that of the state-of-the-arts.
