Autoregressive + Chain of Thought (CoT) $\simeq$ Recurrent: Recurrence's Role in Language Models and a Revist of Recurrent Transformer

The Transformer architecture excels in a variety of language modeling tasks, outperforming traditional neural architectures such as RNN and LSTM. This is partially due to its elimination of recurrent connections, which allows for parallel training and a smoother flow of gradients. However, this move away from recurrent structures places the Transformer model at the lower end of Chomsky's computational hierarchy, imposing limitations on its computational abilities. Consequently, even advanced Transformer-based models face considerable difficulties in tasks like counting, string reversal, bracket pairing, and multiplication. These tasks, though seemingly elementary, require a level of computational complexity that exceeds the capabilities of the Transformer architecture. Concurrently, the emergence of ``Chain of Thought" (CoT) prompting has enabled Transformer-based language models to tackle tasks that were previously impossible or poorly executed. Despite some previous research primarily interpreting CoT from a psychological perspective, a comprehensive understanding of \textit{why} CoT proves so effective in the reasoning process remains elusive. In this work, we thoroughly investigate the influence of recurrent structures in language models on their reasoning abilities, shedding light on how the CoT approach can mimic recurrent computation and act as a bridge between autoregression and recurrence. It is this approximated recurrence that notably improves the model's performance and computational capacity. Moreover, we revisit recent recurrent-based Transformer model designs, focusing on their computational abilities through our proposed concept of ``recurrence-completeness" and identify key theoretical limitations in models like Linear Transformer and RWKV. Through this, we aim to provide insight into the neural model architectures and prompt better model design.