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Deep Neural Networks are increasingly adopted in critical tasks that requirea high level of safety, e.g., autonomous driving. While state-of-the-artverifiers can be employed to check whether a DNN is unsafe w.r.t. some givenproperty (i.e., whether there is at least one unsafe input configuration),their yes/no output is not informative enough for other purposes, such asshielding, model selection, or training improvements. In this paper, weintroduce the #DNN-Verification problem, which involves counting the number ofinput configurations of a DNN that result in a violation of a particular safetyproperty. We analyze the complexity of this problem and propose a novelapproach that returns the exact count of violations. Due to the #P-completenessof the problem, we also propose a randomized, approximate method that providesa provable probabilistic bound of the correct count while significantlyreducing computational requirements. We present experimental results on a setof safety-critical benchmarks that demonstrate the effectiveness of ourapproximate method and evaluate the tightness of the bound.

The #DNN-Verification Problem: Counting Unsafe Inputs for Deep Neural Networks