Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the expressivity of LLMs with CoT in solving fundamental mathematical and decision-making problems. By using circuit complexity theory, we first give impossibility results showing that bounded-depth Transformers are unable to directly produce correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of constant size suffice to solve both tasks by generating CoT derivations using a commonly used math language format. Moreover, we show LLMs with CoT can handle a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, an extensive set of experiments show that, while Transformers always fail to directly predict the answers, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.

翻译：近期研究发现，思维链提示（CoT）能显著提升大型语言模型（LLMs）的性能，尤其在处理涉及数学或推理的复杂任务时。尽管取得了巨大的实证成功，但CoT背后的内在机制以及它如何激发LLMs的潜力仍不明确。本文首次尝试从理论上回答这些问题。具体而言，我们考察了结合CoT的LLMs在解决基础数学和决策问题时的表达能力。通过运用电路复杂性理论，我们首先给出了不可能性结果：对于基本算术/方程任务，有界深度Transformer无法直接生成正确答案，除非模型规模随输入长度超多项式增长。相反，我们通过构造证明：固定大小的自回归Transformer在采用常用数学语言格式生成CoT推导步骤后，足以解决这两类任务。此外，我们证明具有CoT的LLMs能处理一类称为动态规划的通用决策问题，从而论证了其在解决复杂现实任务中的能力。最后，大量实验表明：虽然Transformer始终无法直接预测答案，但在获得足够CoT示范后，它们能逐步学习生成正确的解题步骤。