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.

Verifying properties and interpreting the behaviour of deep neural networks (DNN) is an important task given their ubiquitous use in applications, including safety-critical ones, and their black-box nature. We propose an automata-theoric approach to tackling problems arising in DNN analysis. We show that the input-output behaviour of a DNN can be captured precisely by a (special) weak B\"uchi automaton and we show how these can be used to address common verification and interpretation tasks of DNN like adversarial robustness or minimum sufficient reasons.