  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.
