  Deep Neural Networks are increasingly adopted in critical tasks that require
a high level of safety, e.g., autonomous driving. While state-of-the-art
verifiers can be employed to check whether a DNN is unsafe w.r.t. some given
property (i.e., whether there is at least one unsafe input configuration),
their yes/no output is not informative enough for other purposes, such as
shielding, model selection, or training improvements. In this paper, we
introduce the #DNN-Verification problem, which involves counting the number of
input configurations of a DNN that result in a violation of a particular safety
property. We analyze the complexity of this problem and propose a novel
approach that returns the exact count of violations. Due to the #P-completeness
of the problem, we also propose a randomized, approximate method that provides
a provable probabilistic bound of the correct count while significantly
reducing computational requirements. We present experimental results on a set
of safety-critical benchmarks that demonstrate the effectiveness of our
approximate method and evaluate the tightness of the bound.
