Chain of Thought Empowers Transformers to Solve Inherently Serial
  Problems
Zhiyuan Li Hong Liu Denny Zhou Tengyu Ma
Instructing the model to generate a sequence of intermediate steps, a.k.a., a
chain of thought (CoT), is a highly effective method to improve the accuracy of
large language models (LLMs) on arithmetics and symbolic reasoning tasks.
However, the mechanism behind CoT remains unclear. This work provides a
theoretical understanding of the power of CoT for decoder-only transformers
through the lens of expressiveness. Conceptually, CoT empowers the model with
the ability to perform inherently serial computation, which is otherwise
lacking in transformers, especially when depth is low. Given input length $n$,
previous works have shown that constant-depth transformers with finite
precision $\mathsf{poly}(n)$ embedding size can only solve problems in
$\mathsf{TC}^0$ without CoT. We first show an even tighter expressiveness upper
bound for constant-depth transformers with constant-bit precision, which can
only solve problems in $\mathsf{AC}^0$, a proper subset of $ \mathsf{TC}^0$.
However, with $T$ steps of CoT, constant-depth transformers using constant-bit
precision and $O(\log n)$ embedding size can solve any problem solvable by
boolean circuits of size $T$. Empirically, enabling CoT dramatically improves
the accuracy for tasks that are hard for parallel computation, including the
composition of permutation groups, iterated squaring, and circuit value
problems, especially for low-depth transformers.