Challenge: Existing studies have shown that RNNs can efficiently generate bounded hierarchical languages with high syntactic fidelity, but their success is not well-understood theoretically.
Approach: They propose a language of well-nested brackets and m-bounded nesting depth . they prove that an RNN with O(m log k) hidden units suffices, an exponential reduction in memory, by an explicit construction.
Outcome: The proposed language is well-nested brackets and has m-bounded nesting depth . it shows that an RNN with O(m log k) hidden units suffices, an exponential reduction in memory, by an explicit construction.

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Challenge: recurrent models have been effective in NLP tasks but performance on context-free languages (CFLs) is weak.
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Challenge: Existing models that can process formal languages with hierarchical structure are limited in their performance.
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Challenge: Existing theories of expressive power of RNNs are limited.
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Challenge: Recent work shows that recurrent neural networks can implicitly capture hierarchical information when trained to solve common natural language processing tasks.
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Challenge: A variety of hierarchical RNN models have been proposed to incorporate hierarchically-based hierarchic information in modeling languages in the literature.
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