Reachability In Simple Neural Networks

TL;DR

This paper examines the computational complexity of the reachability problem in neural networks, establishing NP-hardness even for simple network architectures and specifications.

Contribution

It clarifies and corrects previous proofs of NP-completeness and demonstrates NP-hardness for restricted neural network classes.

Findings

NP-hardness holds for simple neural networks with one hidden layer and single output.

Even networks with minimal weights and biases are NP-hard to analyze.

The paper discusses potential extensions and future directions in neural network verification.

Abstract

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.