Chain-of-Thought Data

• Chain-of-thought data are structured representations of a model’s intermediate reasoning steps, providing a clear, stepwise breakdown of complex problem-solving processes.
• They exist in natural language and programmatic forms, where natural language CoTs enhance interpretability and programmatic CoTs enable executable verification.
• Utilizing CoT data has proven to significantly boost accuracy and robustness in symbolic and mathematical reasoning tasks, as demonstrated by improved performance on benchmarks like GSM8K.

Chain-of-thought (CoT) data are structured representations of the intermediate reasoning steps taken by a model or solver when addressing complex problems, most notably in mathematical, logical, or code-based tasks. Rather than mapping input directly to output, CoT data provide an explicit, stepwise decomposition of the solution process, either in natural language or as structured, executable code. Over the past several years, research has established that CoT data not only improve the interpretability of LLMs’ outputs but also enhance accuracy and robustness in many reasoning-intensive domains.

1. Forms and Representations of CoT Data

There are two primary modalities for CoT data in reasoning tasks:

1. Natural Language CoTs These present the problem-solving process as a series of explanatory steps articulated in plain text. Each step narrates the logical progression from problem statement to solution, facilitating interpretability, but these explanations are not machine-executable (Jie et al., 2023).
2. Programmatic CoTs These encode the reasoning chain as executable code, enabling direct verification of each intermediate step. Programmatic CoTs further subdivide into:
   • Self-Describing Program (SDP): Uses semantically meaningful variable names closely tied to the question (e.g., “total_cost”).
   • Comment-Describing Program (CDP): Standardizes variable names (e.g., v1, v2) but augments steps with natural language comments.
   • Non-Describing Program (NDP): Eschews both descriptive names and comments, reducing context but increasing determinism (Jie et al., 2023).

The design choice between these forms profoundly impacts model performance, diversity of solutions, and the ability to verify results automatically.

2. Effectiveness across Reasoning Domains

The impact of CoT data varies markedly by task type:

• Mathematical and Symbolic Reasoning:

Empirical and meta-analytic research consistently finds that CoT data give the largest performance gains in domains requiring explicit multi-step logic or symbolic manipulation. In benchmarks such as GSM8K, MathQA, and SVAMP, implementation of Python-based SDP CoTs raises accuracy remarkably (e.g., achieving 80.9% on GSM8K, well above GPT-3.5-turbo's 75.3%) (Jie et al., 2023). A comprehensive meta-analysis over 110 papers and 1,200+ experiments confirmed that almost all measurable improvements from CoT prompting stem from mathematically or symbolically grounded tasks. The presence of specific tokens, such as an equals sign “=”, is a strong predictor of where CoT excels (Sprague et al., 18 Sep 2024).

• Commonsense, Open-Domain, and Non-Symbolic Tasks:

For problems relying on general knowledge, language understanding, or where stepwise decomposition is inapplicable, CoT data confer little additional benefit and can sometimes hurt performance. The selective application of CoT—wherein a classifier detects when formal reasoning is appropriate—enables resource savings without loss of accuracy (Sprague et al., 18 Sep 2024).

• Code Generation:

The logical structure of code naturally aligns with CoT-based decomposition, making programmatic CoT design particularly impactful for these tasks (Yu et al., 2023).

3. Design Principles and Performance Metrics

The structure and prompt design of CoT examples are pivotal. Key principles include:

• Demonstration Quality:

Demonstrations should balance complexity (springboarding richer reasoning), relevance, and diversity to avoid overfitting to a specific chain pattern (Yu et al., 2023).

• Prompt Construction:

Prompts generally bundle the problem statement, a stepwise reasoning trace (CoT), and a final answer as a triple $(x, \mathbf{r}, y)$. The addition of textual instructions (e.g., “Let’s think step by step”) further primes the model to decompose tasks (Yu et al., 2023).

• Ensemble and Extension Strategies:

Variants like majority voting across diverse CoT generations or reward-model reranking can be used to increase robustness. Programmatic CoT outputs lend themselves to execution-based validation, ensuring syntactic and semantic correctness (Jie et al., 2023).

Performance is measured by:

• Accuracy: The proportion of correct final answers, often determined either directly or by executing generated code.
• Execution Rate: The fraction of CoTs that can be run without error (programmatic CoTs).
• Precision: The percentage of executable outputs that yield the correct result.
• Diversity Metrics: Such as “correct@100”—the likelihood that at least one of 100 generated CoTs is valid (Jie et al., 2023).

4. Programming Language and Coding Style

Programmatic CoT efficacy depends on the programming language and the stylistic conventions embedded in CoT data:

• Language Choice:

Python is empirically favored over alternatives like Wolfram, owing to its prevalence in model pretraining data and its rich scientific ecosystem. Models produce more syntactically valid and semantically useful outputs in Python, which in turn boosts code execution rates and precision (Jie et al., 2023).

• Diversity vs. Determinism:

Self-describing programs introduce more diversity, increasing the chance that ensemble strategies will yield a correct solution, but may also lead to lower execution rates due to syntactic errors in novel code paths. Conversely, non-describing, deterministic code is easier to parse and execute, but generally less effective in modeling linguistic context and abstraction (Jie et al., 2023).

5. Implementation Resources and Best Practices

The development of CoT data and its integration into reasoning systems involves several pragmatic considerations:

• Dataset Construction:

Public resources now exist for benchmarking and experimentation—including datasets annotated with natural language, programmatic (SDP, CDP, NDP), and multiple programming language CoTs. Notable repositories, such as https://github.com/lqtrung1998/mwp_cot_design, provide templates for further research and system development (Jie et al., 2023).

• Model and Task Matching:

CoT-based improvements are most pronounced in large models (typically over 10B parameters). Smaller models, even when supplied with CoT-augmented data, may not benefit unless additional methods (e.g., fine-tuning, data distillation) are implemented (Yu et al., 2023).

• Future Design Guidelines:

Research suggests integrating natural language descriptions alongside executable code, using Python for program CoTs, and exploring hybrids—where multiple CoT types are deployed and ensembled. The selective use of CoT, triggered by task-type or symbolic cues, is advised for balancing efficiency with accuracy (Jie et al., 2023, Sprague et al., 18 Sep 2024).

6. Limitations and Open Research Directions

While CoT data have markedly advanced model interpretability and accuracy on symbolic tasks, several limitations remain:

• Faithfulness and Error Propagation:

Current models may produce CoT rationales that seem plausible but do not reflect underlying correct reasoning. Faithfulness, explicitly ensuring that narrated steps cause the output, is an ongoing research challenge (Yu et al., 2023).

• Applicability to Non-Symbolic or Ambiguous Tasks:

The benefit of CoT is clearly delimited; for many open-ended or non-symbolic problems, direct answering outperforms explicit stepwise decomposition. There is also the risk that verbose CoT sequences may dilute relevant signal (Sprague et al., 18 Sep 2024).

• Computational Considerations:

Programmatic CoT methods can increase inference costs due to code execution and ensemble reranking. Selective triggering of CoT reduces unnecessary overhead (Sprague et al., 18 Sep 2024).

• Hybrid and Ensemble Paradigms:

The field is shifting toward new methodologies that couple prompt-based CoT with external symbolic tools, search-based (tree-of-thought) frameworks, or modular systems that separate planning from execution. This is motivated by the observation that, even for math tasks, specialized symbolic solvers outperform LLMs when provided with a CoT plan (Sprague et al., 18 Sep 2024).

7. Summary Table: CoT Method Comparison in Math Problem Solving (Python, 30B Model, Reward Rerank) (Jie et al., 2023)

Method	GSM8K	MathQA	SVAMP	NLP vs Programmatic
NL CoT	66.6%	72.6%	77.9%	Linguistically clear, unverifiable
Python SDP	80.9%	78.1%	87.0%	High diversity, executable
Python CDP	80.7%	78.6%	85.3%	Moderately diverse, clear comments
Python NDP	77.3%	73.7%	78.8%	Low diversity, highly deterministic

• "Design of Chain-of-Thought in Math Problem Solving" (Jie et al., 2023)
• "To CoT or not to CoT? Chain-of-thought helps mainly on math and symbolic reasoning" (Sprague et al., 18 Sep 2024)
• "Towards Better Chain-of-Thought Prompting Strategies: A Survey" (Yu et al., 2023)

This synthesis reflects both the effectiveness and the current limits of CoT data in state-of-the-art reasoning systems, highlighting avenues for continued investigation and refinement.