Problem 9: Capacity of Recurrent Neural Networks (RNNs) for Sequence Modeling Statement: Prove that recurrent neural networks (RNNs) with a certain architecture and activation functions can represent any computable sequence function. Specifically, show that RNNs are Turing complete under appropriate conditions. Key Points for the Proof: • Define the computational capabilities of RNNs and relate them to models of computation like Turing machines. Construct an RNN architecture that can simulate the operations of a Turing machine. • Address the role of activation functions and network depth in achieving computational universality. Provide a formal argument demonstrating the equivalence of RNNs to Turing-complete systems.

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