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Deep Learning (DL) has achieved tremendous success in many cutting-edge applications. However, the state-of-the-art DL systems still suffer from quality issues. While some recent progress has been made on the analysis of feed-forward DL systems, little study has been done on the Recurrent Neural Network (RNN)-based stateful DL systems, which are widely used in audio, natural languages and video processing, etc. In this paper, we initiate the very first step towards the quantitative analysis of RNN-based DL systems. We model RNN as an abstract state transition system to characterize its internal behaviors. Based on the abstract model, we design two trace similarity metrics and five coverage criteria which enable the quantitative analysis of RNNs. We further propose two algorithms powered by the quantitative measures for adversarial sample detection and coverage-guided test generation. We evaluate …