We present a new system, EEV, for verifying binarized neural networks (BNNs). We formulate BNN verification as a Boolean satisfiability problem (SAT) with reified cardinality constraints of the form $y = (x_1 + \cdots + x_n \le b)$, where $x_i$ and $y$ are Boolean variables possibly with negation and $b$ is an integer constant. We also identify two properties, specifically balanced weight sparsity and lower cardinality bounds, that reduce the verification complexity of BNNs. EEV contains both a SAT solver enhanced to handle reified cardinality constraints natively and novel training strategies designed to reduce verification complexity by delivering networks with improved sparsity properties and cardinality bounds. We demonstrate the effectiveness of EEV by presenting the first exact verification results for $\ell_{\infty}$-bounded adversarial robustness of nontrivial convolutional BNNs on the MNIST and CIFAR10 datasets. Our results also show that, depending on the dataset and network architecture, our techniques verify BNNs between a factor of ten to ten thousand times faster than the best previous exact verification techniques for either binarized or real-valued networks.

The computing industry is currently facing a major architectural shift. Extra computing power is not coming anymore from higher processor frequencies, but from a growing number of computing cores and processors. For AI, and constraint solving in particular, this raises the question of how to scale current solving techniques to massively parallel architectures. While prior work focusses mostly on small scale parallel constraint solving, we conduct the first study on scalability of constraint solving on 100 processors and beyond in this paper. We propose techniques that are simple to apply and show empirically that they scale surprisingly well. These techniques establish a performance baseline for parallel constraint solving technologies against which more sophisticated parallel algorithms need to  compete  in the future.