Title: Analyzing the Power of Chain of Thought through Memorization Capabilities
Venue: NeurIPS 2025
Accepted: yes
Keywords: Theory

Chain of Thought

Transformer

Memorization
Abstract: It has been shown that the chain of thought (CoT) can enhance the power of LLMs to simulate a Turing machine or an algorithm, and in particular its mathematical reasoning ability. The memorization capability of LLMs is an important aspect of their expressive ability, which offers valuable insight into designing models with enhanced generalization potential. Currently, the optimal memorization capacities of transformers have been established for both the general dataset and the dataset that satisfies a specific separability condition. However, the question of whether the CoT can improve the memorization capability of LLMs remains unexamined. To fill this gap, we establish the memorization capability for fixed-precision autoregressive transformers with or without CoT. Precisely, we first give the necessary and sufficient conditions for transformers to memorize a finite language and then provide the upper and lower bounds for the number of parameters of the memorization transformers. Our result indicates that the classes of languages that can be memorized by transformers with or without CoT do not contain each other, and the same number of parameters is needed for transformers with or without CoT to memorize, implying that CoT does not enhance a transformer’s memorization power significantly. We further show that CoT can not help transformers to memory certain infinite languages.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies whether CoT prompting can enhance the expressive power of Transformers in the context of memorization. Focusing on fixed-precision Transformers and finite datasets, the authors claim that CoT does not significantly improve a transformer's ability to memorize general datasets. They show that Transformers with CoT do not strictly dominate those without CoT in terms of memorization capacity. Additionally, the paper demonstrates that CoT does not enable Transformers to memorize certain infinite languages.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The paper presents an interesting and somewhat counter-intuitive result that CoT may not enhance the expressive power of Transformers from the perspective of memorization. This aspect seems to be largely overlooked in prior literature.\n2. The theoretical analysis appears solid, covering both necessary and sufficient conditions as well as providing matching upper and lower bounds.\n\nWeaknesses\n1. In my understanding, CoT is primarily intended to to improve the capability of LLMs for reasoning tasks whose solving typically involves multiple algorithmic steps. From this perspective, the significance of analyzing CoT's impact on memorization is somewhat unclear.\n2. The paper considers fixed-precision Transformers with flexible width and depth, in contrast to prior works that focus on log-precision, constant-depth Transformers when analyzing CoT’s effect on expressivity. In those earlier studies, constant depth is a critical constraint, where CoT serves to increase the model’s effective depth. While it is reasonable to explore a different setting, additional justification may be necessary to explain its relevance (e.g., better alignment with practical models) and to clarify how the differences in setup affect the interpretation and comparability of results with prior work.\n\nQUESTIONS:\nSee Weaknesses.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This work studies whether CoT can enhance a model's memorization capability from a theoretical perspective. The authors first identified the necessary and sufficient conditions for Transformers to memorize a language. Then, the authors presented lower and upper bound on the number of model parameters required to memorize a given language with and without CoT. These bounds show that (1) CoT affects the languages that can be memorized by the Transformer (2) CoT does not increase the memorization capability of Transformers significantly.

STRENGTHS AND WEAKNESSES:
(Disclaimer: I do not work on formal language or learning theory, hence might not be the right person to judge the significance of this work)

**Strengths**
* This work studies an interesting problem: whether CoT enhance the memorization capacity of Transformers.
* This work includes a detailed analysis comparing memorization capacities of CoT and non-CoT Transformers under different language classes, position embeddings setup, etc.
* The paper is generally well written. The settings and assumptions are clearly stated in the paper.

**Weaknesses**
* This work studies one particular implementation of CoT Transformer (as defined on L154-158). Is it possible that another implementation of CoT Transformers could potentially lead to different memorization capabilities, especially for different languages?

QUESTIONS:
See weaknesses.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the expressiveness of transformers without positional encoding, both with and without chain-of-thought (COT). It establishes a simplified theoretical framework, derives necessary and sufficient conditions for the languages expressible by these models, and compares their expressive power, illustrated through the LCP problem. Additionally, it provides bounds on parameter size estimates for valid languages. While the problem is of significant interest to the community, the paper is difficult to follow due to unclear notations and insufficient intuitive explanations of theorems.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper addresses a compelling problem and provides theoretical insights into the expressiveness of transformers with and without COT. The analysis of expressive power differences and the application to the LCP problem are notable strengths. However, the paper suffers from insufficient explanations. Theorems lack intuitive interpretations, and key concepts are not adequately clarified, hindering readability.\n\nQUESTIONS:\nMajor Problems:\n1. What is the meaning of {K_k} in Theorem 4.1 and κ_j in Theorem 4.4? Is it a one-element set of κ_k or a k-element subset of κ?\n2. How does a COT transformer memorize LCP^{=1}_{n}? By Theorem 4.1, a non-COT transformer only considers the set of distinct input symbols, not their positions or counts. For inputs x = A and x = A ···A, the COT process should be identical since they share the same typ() at each step, leading to the same output. How does the COT transformer produce different outputs based on input length?\n3. Does Theorem 4.1 simply state that a non-COT transformer is a universal approximator for functions dependent only on typ(x), i.e., it can express all functions in UAT(typ(x))?\n4. What mechanism allows a COT transformer to leverage input information beyond typ(x)?\n\nMinor Problems:\n1. The notation 2^κ, where κ is a set, is confusing. A more conventional notation would be κ^k, representing a length-k array with elements from κ.\n2. What is the relationship between embedding dimension d and width W in Section 3?\n\nThese questions are adequately addressed by the authors in the responses.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nCoT-enabled and standard fixed-precision autoregressive Transformers are formally compared, treating CoT as a scratch-pad workspace added at inference time.\nBoth models memorize any finite dataset with N parameters, though each obeys distinct necessary conditions; CoT does not lower the parameter count required for memorization.\nThus CoT alters which languages can be memorized but leaves raw memorization capacity unchanged, implying its gains come from reasoning structure rather than extra memory.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. Gives exact necessary and sufficient conditions for memorization with and without CoT.\n2. Proves tight N parameter bounds, showing CoT does not cut model size.\n3. Identifies finite languages each model can memorize exclusively, clarifying expressive differences.\n\nWeaknesses\n1. Limits analysis to single-token labels, leaving sequence outputs unaddressed.\n2. Omits positional encodings and LayerNorm, diverging from real-world Transformers.\n3. Provides no empirical experiments, so theoretical claims are not validated in practice.\n\nQUESTIONS:\nQ1. Could you report empirical results on real LLMs to confirm the theoretical predictions?\n\nQ2. Section 4 shows CoT and no-CoT memorize disjoint finite-language classes. Can you further characterize which natural NLP tasks map onto each class, or give examples where this distinction has concrete practical consequences?\n\nQ3. Could you clarify what concrete practical or scientific insights are gained by measuring CoT’s memorization limits, and why those insights cannot be obtained more directly by studying CoT’s reasoning performance instead?\nGiven that Chain-of-Thought is primarily a reasoning aid, not a storage mechanism, the motivation for analysing its memorisation capacity feels under-justified.\n\nQ4. As in Q3, your study focuses solely on whether finite datasets can be memorized, leaving untouched the core advantages of Chain-of-Thought—namely algorithmic reasoning difficulty and accuracy.\nCould you justify why these reasoning metrics were excluded and provide evidence (theoretical or empirical) that the reported memorization bounds translate into meaningful gains—or at least no regressions—in actual CoT reasoning performance?\n\nLIMITATIONS:\nSince no empirical experiments validate the theoretical results on real LLMs, their practical reliability remains untested.\nThe work considers only whether data can be memorized, ignoring CoT’s core strengths—algorithmic reasoning difficulty and accuracy.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}