Not only are Deep Neural Networks (DNNs) black box models, but also we frequently conceptualize them as such. We lack good interpretations of the mechanisms linking inputs to outputs. Therefore, we find it difficult to analyze in human-meaningful terms (1) what the network learned and (2) whether the network learned. We present a hierarchical decomposition of the DNN discrete classification map into logical (AND/OR) combinations of intermediate (True/False) classifiers of the input. Those classifiers that can not be further decomposed, called atoms, are (interpretable) linear classifiers. Taken together, we obtain a logical circuit with linear classifier inputs that computes the same label as the DNN. This circuit does not structurally resemble the network architecture, and it may require many fewer parameters, depending on the configuration of weights. In these cases, we obtain simultaneously an interpretation and generalization bound (for the original DNN), connecting two fronts which have historically been investigated separately. Unlike compression techniques, our representation is. We motivate the utility of this perspective by studying DNNs in simple, controlled settings, where we obtain superior generalization bounds despite using only combinatorial information (e.g. no margin information). We demonstrate how to "open the black box" on the MNIST dataset. We show that the learned, internal, logical computations correspond to semantically meaningful (unlabeled) categories that allow DNN descriptions in plain English. We improve the generalization of an already trained network by interpreting, diagnosing, and replacing components the logical circuit that is the DNN.