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This paper proposes a novel neural architecture — Attentive Tensor Product Learning (ATPL) — to represent grammatical structures of natural language in deep learning models. ATPL exploits Tensor Product Representations (TPR), a structured neural-symbolic model developed in cognitive science, to integrate deep learning with explicit natural language structures and rules. The key ideas of ATPL are: 1) unsupervised learning of role-unbinding vectors of words via the TPR-based deep neural network; 2) the use of attention modules to compute TPR; and 3) the integration of TPR with typical deep learning architectures including long short-term memory and feedforward neural networks. The novelty of our approach lies in its ability to extract the grammatical structure of a sentence by using role-unbinding vectors, which are obtained in an unsupervised manner. Our ATPL approach is applied to 1) image captioning, 2) part of speech (POS) tagging, and 3) constituency parsing of a natural language sentence. The experimental results demonstrate the effectiveness of the proposed approach in all these three natural language processing tasks.