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The paper develops a methodology for the online built-in self-testing of deep neural network (DNN) accelerators to validate the correct operation with respect to their functional specifications. The DNN of interest is realized in the hardware to perform in-memory computing using non-volatile memory cells as computational units. Assuming a functional fault model, we develop methods to generate pseudorandom and structured test patterns to detect hardware faults. We also develop a test-sequencing strategy that combines these different classes of tests to achieve high fault coverage. The testing methodology is applied to a broad class of DNNs trained to classify images from the MNIST, Fashion-MNIST, and CIFAR-10 datasets. The goal is to expose hardware faults which may lead to the incorrect classification of images. We achieve an average fault coverage of 94% for these different architectures, some of which are large and complex.