We investigate the complexity of the reachability problem for (deep) neural networks: does it compute valid output given some valid input? It was recently claimed that the problem is NP-complete for general neural networks and specifications over the input/output dimension given by conjunctions of linear inequalities. We recapitulate the proof and repair some flaws in the original upper and lower bound proofs. Motivated by the general result, we show that NP-hardness already holds for restricted classes of simple specifications and neural networks. Allowing for a single hidden layer and an output dimension of one as well as neural networks with just one negative, zero and one positive weight or bias is sufficient to ensure NP-hardness. Additionally, we give a thorough discussion and outlook of possible extensions for this direction of research on neural network verification.

翻译：我们研究了（深度）神经网络中可达性问题的复杂性：给定有效输入，它是否能计算出有效输出？最近有研究声称，对于一般神经网络以及由线性不等式合取给出的输入/输出维度规范，该问题是NP完全的。我们重新阐述了证明过程，并修正了原上下界证明中的若干缺陷。基于这一一般性结果，我们证明NP困难性在简单规范和神经网络的限制类别中已经成立。仅需考虑具有单隐藏层、输出维度为1的神经网络，以及仅包含一个负、零和正权重或偏置的神经网络，就足以确保NP困难性。此外，我们对神经网络验证这一研究方向的可能扩展进行了深入讨论和展望。