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：它是否能在给定有效输入的情况下计算出有效输出？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形式给出的输入/输出维度规范，该问题是NP完全的。我们重新梳理了该证明，并修复了原始上界和下界证明中的一些缺陷。基于这一一般性结果，我们证明NP难性已经适用于简单规范和神经网络的受限类别。仅需一个隐藏层和输出维度为1，以及仅包含一个负权重、零权重和一个正权重或偏置的神经网络，就足以确保NP难性。此外，我们对该神经网络验证研究方向的可能扩展进行了深入讨论与展望。