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Analyzing the Power of Chain of Thought through Memorization Capabilities

Lijia Yu, Xiao-Shan Gao, Lijun Zhang

TL;DR

This work investigates whether chain-of-thought (CoT) enhances transformer reasoning across all tasks by analyzing memorization capacities under fixed precision. It provides complete necessary and sufficient conditions for memorization with and without CoT on finite reasoning datasets, showing tight parameter bounds of order $ar{Θ}(N)$ and revealing that CoT does not universally increase reasoning power. The authors also extend the analysis to infinite languages, proving that some infinite datasets cannot be memorized by either CoT or no-CoT transformers, and they offer insights into the roles of position encoding and symbol augmentation. Collectively, the results clarify the limits of CoT, offering practical guidance on when CoT is beneficial and highlighting fundamental limits in algorithmic memorization for autoregressive transformers.

Abstract

It has been shown that the chain of thought (CoT) can enhance the power of large language models (LLMs) to solve certain mathematical reasoning problems. However, the capacity of CoT is still not fully explored. As an important instance, the following basic question has not yet been answered: Does CoT expand the capability of transformers across all reasoning tasks? We demonstrate that reasoning with transformers is essentially a memorization problem for reasoning datasets. Thus, examining the power of CoT across all reasoning tasks amounts to analyzing the memorization capabilities of CoT transformers. In this paper, we give a complete description of the memorization capabilities of fixed-precision transformers with or without CoT and give a negative answer to the above-mentioned question. Precisely, we first give necessary and sufficient conditions for fixed-precision transformers with and without CoT to memorize a finite reasoning dataset and show that these two conditions do not imply each other. Then, we give lower and upper bounds for the number of parameters needed for transformers with or without CoT to memorize a finite reasoning dataset with $N$ elements, which are $\overlineΘ(N)$ in all cases. This implies that there exist reasoning tasks for which CoT does not enhance the reasoning power of transformers, leading to a negative answer to the above-mentioned question. Finally, we give the first results on memorizing infinite reasoning datasets by CoT transformers and show that some simple infinite datasets cannot be memorized by transformers with or without CoT.

Analyzing the Power of Chain of Thought through Memorization Capabilities

TL;DR

This work investigates whether chain-of-thought (CoT) enhances transformer reasoning across all tasks by analyzing memorization capacities under fixed precision. It provides complete necessary and sufficient conditions for memorization with and without CoT on finite reasoning datasets, showing tight parameter bounds of order and revealing that CoT does not universally increase reasoning power. The authors also extend the analysis to infinite languages, proving that some infinite datasets cannot be memorized by either CoT or no-CoT transformers, and they offer insights into the roles of position encoding and symbol augmentation. Collectively, the results clarify the limits of CoT, offering practical guidance on when CoT is beneficial and highlighting fundamental limits in algorithmic memorization for autoregressive transformers.

Abstract

It has been shown that the chain of thought (CoT) can enhance the power of large language models (LLMs) to solve certain mathematical reasoning problems. However, the capacity of CoT is still not fully explored. As an important instance, the following basic question has not yet been answered: Does CoT expand the capability of transformers across all reasoning tasks? We demonstrate that reasoning with transformers is essentially a memorization problem for reasoning datasets. Thus, examining the power of CoT across all reasoning tasks amounts to analyzing the memorization capabilities of CoT transformers. In this paper, we give a complete description of the memorization capabilities of fixed-precision transformers with or without CoT and give a negative answer to the above-mentioned question. Precisely, we first give necessary and sufficient conditions for fixed-precision transformers with and without CoT to memorize a finite reasoning dataset and show that these two conditions do not imply each other. Then, we give lower and upper bounds for the number of parameters needed for transformers with or without CoT to memorize a finite reasoning dataset with elements, which are in all cases. This implies that there exist reasoning tasks for which CoT does not enhance the reasoning power of transformers, leading to a negative answer to the above-mentioned question. Finally, we give the first results on memorizing infinite reasoning datasets by CoT transformers and show that some simple infinite datasets cannot be memorized by transformers with or without CoT.

Paper Structure

This paper contains 51 sections, 28 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Prerequisite
  4. Data.
  5. Autoregressive Transformer.
  6. Classification result for transformer.
  7. Transformers with or without CoT.
  8. Memorization expressive ability of transformers
  9. Memorization using no-CoT-transformer
  10. Memorization using CoT-transformer
  11. CoT and no-CoT-transformers: comparison and more results
  12. The memorization powers for no-CoT-transformer and CoT-transformer are different.
  13. Position embedding is important for no-CoT-transformer, but not for CoT-transformer.
  14. More basic symbols are important for CoT-transformer, but not for no-CoT-transformer.
  15. More basic symbols are important for CoT-transformer, but not for no-CoT-transformer.
  16. ...and 36 more sections

Key Result

Theorem 4.1

Let $S$ be a finite language of basic symbols $\Gamma=\{\gamma_{i}\}_{i=1}^T$, $N=|S|$, $L=\max_{(x,y)\in S}\{{\rm{len}}(x)\}$, and $q\in{\mathbb{Z}}_+$. Then $S$ can be memorized by a no-CoT-transformer if and only if $(x_1,y_1),(x_2,y_2)\in S$ and ${\rm{typ}}(x_1)={\rm{typ}}(x_2)=\{\gamma_k\}$ for

Theorems & Definitions (69)

  • Definition 3.1
  • Example 3.2
  • Remark 3.3
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 4.1
  • Corollary 4.2
  • ...and 59 more