Patterns Over Principles: The Fragility of Inductive Reasoning in LLMs under Noisy Observations
Authors: Chunyang Li, Weiqi Wang, Tianshi Zheng, Yangqiu Song
Accepted at: acl25

Patterns Over Principles: The Fragility of Inductive Reasoning in LLMs under Noisy Observations

Chunyang Li, Weiqi Wang, Tianshi Zheng, Yangqiu Song
Department of Computer Science and Engineering, HKUST, Hong Kong SAR, China
{cliei, wwangbw, tzhengad, yqsong}@cse.ust.hk

Abstract
Inductive reasoning, a cornerstone of human cognition, enables generalization from limited data but hasn’t yet been fully achieved by large language models (LLMs). While modern LLMs excel at reasoning tasks, their ability to maintain stable and consistent rule abstraction under imperfect observations remains underexplored. To fill this gap, in this work, we introduce Robust Rule Induction, a task that evaluates LLMs’ capability in inferring rules from data that are fused with noisy examples. To address this task, we further propose Sample-steered Rule Refinement (SRR), a method enhancing reasoning stability via observation diversification and execution-guided feedback. Experiments across arithmetic, cryptography, and list functions reveal: (1) SRR outperforms other methods with minimal performance degradation under noise; (2) Despite slight accuracy variation, LLMs exhibit instability under noise (e.g., 0% accuracy change with only 70% consistent score); (3) Counterfactual task gaps highlight LLMs’ reliance on memorized patterns over genuine abstraction. Our findings challenge LLMs’ reasoning robustness, revealing susceptibility to hypothesis drift and pattern overfitting, while providing empirical evidence critical for developing human-like inductive systems. Code and data are available at https://github.com/HKUST-KnowComp/Robust-Rule-Induction.

1 Introduction
Inductive reasoning—the cognitive capacity to generalize from specific instances to universal principles, is fundamental to human intelligence (Lake et al., 2017). This ability enables humans to abstract latent patterns from sparse data while maintaining robustness against conflicting evidence (Feldman, 1997), as exemplified by children mastering linguistic rules despite exposure to occasional grammatical errors. The robustness of inductive reasoning manifests in resistance to pattern interference (preserving learned rules when encountering contradictory examples) and tolerance to observational imperfections (maintaining stable performance under noisy learning conditions). Understanding and quantifying this robustness becomes increasingly crucial as artificial systems approach complex real-world applications where clean data remains elusive.

Recent advances in large language models (LLMs) have demonstrated remarkable performance across various reasoning tasks (OpenAI, 2024b; DeepSeek-AI et al., 2024), reigniting interest in comparing machine and human reasoning paradigms (Collins et al., 2024). While contemporary studies showcase LLMs’ proficiency in inductive reasoning, their capacity for robustness remains questionable. Unlike humans who can rapidly converge on correct rules through Bayesian hypothesis updating (Tenenbaum et al., 2011; Lake et al., 2015), current models often exhibit unstable reasoning trajectories when noise disrupts their thinking process (Zhou et al., 2024). A growing body of evidence suggests that LLMs primarily engage in pattern matching rather than genuine reasoning (Mirzadeh et al., 2024; Wu et al., 2024).

Despite increasing attention to LLM inductive reasoning capabilities, current works exhibit limitations that obscure true inductive reasoning robustness. First, in prevailing and concurrent evaluations (Alet et al., 2021; Gendron et al., 2024; Zheng et al., 2025b), models just predict outputs for novel inputs given exemplars, bypassing explicit rule verification. It fails to diagnose where and why rule induction fails. Existing studies that do generate intermediate rules either disregard noisy learning conditions (Wang et al., 2024; Liu et al., 2024) or assess performance solely through aggregate metrics like task accuracy gap (Qiu et al., 2024). This neglects instance-level reasoning consistency—whether models maintain stable rule interpretations when exposed to conflicting patterns,
which is a key indicator of human-like robust induction. Besides, prior methods often repurpose existing benchmarks (e.g., SCAN; Lake and Baroni 2018, ARC; Chollet 2019) without modification, where potential data contamination may undermine validity as models could use memorized solutions.

To address these gaps, we introduce a novel task of robust rule induction, which challenges models to identify underlying functions from input-output exemplars containing controlled noise injections. Our approach features three innovations: (1) an evaluation pipeline from data synthesis to automatic evaluation, requiring models to output rules explicitly and enabling direct validation through programmatic execution. (2) A multi-dimensional robustness assessment with conventional accuracy and instance-level metrics. (3) A sample-steered refinement (SRR) methodology that improves both the capability and stability of LLMs’ inductive reasoning through diverse sampling and iterative diagnostic refinement based on rule execution.

We curate three datasets spanning arithmetic, cryptography, and list operations—domains requiring progressively abstract rule formalization. The proposed SRR substantially improves performance across most tasks and maintains stability compared to other methods. However, comprehensive experiments reveal a critical dichotomy: while the accuracy variance achieves less than 5%, the consistency metric exposes fundamental instability. Furthermore, LLMs exhibit significant performance inconsistency with the same task category, performing considerably better on more familiar tasks than others, despite comparable complexity. Correct predictions may stem from recited patterns rather than stable rule abstraction. These findings challenge the “real” reasoning capabilities of LLMs.

2 Preliminary

2.1 Problem Definitions

We formulate robust rule induction as the problem of identifying latent mapping functions from input-output pairs while maintaining consistent reasoning performance under noisy conditions. While prior works (Wang et al., 2024; Liu et al., 2024) focus on basic rule discovery capabilities, we specifically investigate the robustness of inductive reasoning when the observations contain conflicting patterns. Some examples are shown in Figure 1. The observations may contain both normal and noisy examples, and the model is required to infer the correct rule regardless of the noise.

![Example instances with noise and rules from Arithmetic_{base-7}, Cryptography_{Caesar} and List Functions.](page_682_370_682_180.png)

Figure 1: Example instances with noise and rules from Arithmetic_{base-7}, Cryptography_{Caesar} and List Functions.

Formally, given a dataset \( \mathcal{D} = \{(x_i, y_i)\}_{i=1}^N \) where \( x \in \mathcal{X} \) denotes inputs and \( y \in \mathcal{Y} \) represents outputs generated by an underlying function \( f : \mathcal{X} \to \mathcal{Y} \), the objective is to induce an approximation \( \hat{f} \) from observed examples \( \mathcal{D}_{\text{seen}} \subseteq \mathcal{D} \). Robust rule induction should satisfy: \( \hat{f}(x) = f(x) \) for \( (x, y) \in \mathcal{D}_{\text{unseen}} \) despite \( \mathcal{D}_{\text{seen}} \) containing noisy samples \( \mathcal{D}_{\text{noise}} \) where \( y \neq f(x) \).

2.2 Evaluation Pipeline

As illustrated in figure 2, to evaluate the robustness of inductive reasoning in large language models, we propose a novel evaluation pipeline consisting of three main stages: data synthesis, model inference, and performance evaluation. In the data synthesis stage, due to the characteristics of our data, normal examples, noisy examples, and test examples for each instance in the dataset can be automatically generated by programs given the rule and noise definitions. The first two are mixed to form seen examples, while the latter is used to evaluate the model’s reasoning ability. In the model inference stage, we prompt the language model with seen examples in standard input-output (IO) format, and the induced rules are restricted to Python function form for automatic evaluation. Finally, we evaluate the rule by executing it on the test examples. The instance is considered solved if all the test examples are correctly solved. More details about the evaluation pipeline, like the data synthesis process, can be found in appendix A.

We use task accuracy, which is the proportion of solved instances over the total number of instances, and its change under different conditions as the evaluation metrics. While task accuracy is a comprehensive metric to evaluate the model’s reasoning ability, it may not fully capture the robustness of the model at the instance level. To address this, we introduce the consistency score, which is defined
Figure 2: Evaluation pipeline exemplified by base-9 addition, consisting of three stages: (1) Data Synthesis, generating normal, noisy and test examples; (2) Model Inference, prompting models with seen examples to induce rules in Python function form; (3) Performance Evaluation, executing induced rules on test examples to assess correctness and robustness under noise.

as follows:

\[
\text{Consistency Score} := \frac{1}{N} \sum_{i=1}^{N} \mathbb{I}[Sol_i^c = Sol_i^n]
\]

where \( N \) is the total number of instances in the dataset, \( \mathbb{I}[\cdot] \) is the indicator function, \( Sol_i^c \) is whether the \( i \)-th instance is solved without noise, 1 if solved and 0 otherwise, and \( Sol_i^n \) is whether the \( i \)-th instance is solved with noise. This metric quantifies the model’s ability to maintain stable reasoning conclusions under noise and offers a more granular view of the model’s robustness.

3 Sample-steered Rule Refinement

Motivated by previous methods (Qiu et al., 2024; Wang et al., 2024) using iterative hypothesis refinement, we propose Sample-steered Rule Refinement (SRR), a framework to improve inductive reasoning in both clean and noisy conditions. This method conducts inductive reasoning through three-phase optimization: (1) contrastive hypothesis generation to bootstrap diverse rule candidates, (2) diagnostic feedback construction through evidence-aware sampling, and (3) iterative rule refinement using failure-driven corrections. The full algorithm is detailed in Algorithm 1.

Diversity-aware Hypothesis Generation Different from previous methods based on the whole seen examples, we first generate \( K \) hypotheses from random subsets of the seen examples, plus one hypothesis using the full observations. This step ensures coverage of both local patterns and global consistency. The number of examples in the subset is less than the full seen set, thus the noise in the subset has a greater impact. The sampled subset may also contain completely noise-free examples, which can better distinguish the rules.

Diagnostic Feedback Construction After generating the initial hypotheses, the induced rule is applied to the seen examples through code interpreter. We collect the correct and incorrect cases and sample these cases as feedback. We focus more on the incorrect cases, as they provide more information about the rule refinement, while the correct

Algorithm 1 SRR Framework

Require: Seen examples \( \mathcal{D}_{\text{seen}} \), LLM \( M \), max iterations \( T \), max subsets \( K \), threshold \( \tau \)
Ensure: Best hypothesis \( \hat{f}^* \) (in Python function form)
\( \mathcal{H}_0 \leftarrow \emptyset \)
for \( k = 1 \) to \( K \) do
    \( \mathcal{D}_k \leftarrow \text{SampleSubset}(\mathcal{D}_{\text{seen}}) \)
    \( \mathcal{H}_0 \leftarrow \mathcal{H}_0 \cup \{M(\mathcal{D}_k, "Generate\ rule\ function")\} \)
end for
\( \mathcal{H}_0 \leftarrow \mathcal{H}_0 \cup \{M(\mathcal{D}_{\text{seen}}, "Generate\ rule\ function")\} \)
\( \hat{f}_0 \leftarrow \arg\max_{h \in \mathcal{H}_0} \text{Acc}(h, \mathcal{D}_{\text{seen}}) \)
for \( t = 1 \) to \( T \) do
    if \( \text{Acc}(\hat{f}_{t-1}, \mathcal{D}_{\text{seen}}) \geq \tau \) then
        return \( \hat{f}_{t-1} \)
    end if
    \( \mathcal{C}_t \leftarrow \text{CorrectExamples}(\hat{f}_{t-1}, \mathcal{D}_{\text{seen}}) \)
    \( \mathcal{E}_t \leftarrow \text{WrongExamples}(\hat{f}_{t-1}, \mathcal{D}_{\text{seen}}) \)
    Feedback \( \leftarrow \{\mathcal{C}_t[:n], \mathcal{E}_t[:m]\}\ \text{/* Sample n,m cases */} \)
    \( h_{\text{new}} \leftarrow M(\hat{f}_{t-1}, \text{Feedback}, "Revise\ rule") \)
    if \( \text{Acc}(h_{\text{new}}, \mathcal{D}_{\text{seen}}) > \text{Acc}(\hat{f}_{t-1}, \mathcal{D}_{\text{seen}}) \) then
        \( \hat{f}_t \leftarrow h_{\text{new}} \)
    else
        \( \hat{f}_t \leftarrow \hat{f}_{t-1} \)
    end if
end for
return \( \hat{f}^* = \arg\max_t \text{Acc}(\hat{f}_t, \mathcal{D}_{\text{seen}}) \)
cases are used as positive feedback to reinforce the rule. The hypothesis with the highest accuracy on the seen examples is selected as the initial rule.

Iterative Rule Refinement At each iteration, the model receives the current selected rule with sampled feedback to generate a new rule. The new hypothesis is compared to the previous one, and the more accurate one is selected. This iterative process continues until the accuracy of the rule on the seen examples exceeds a predefined threshold or the maximum number of iterations is reached.

4 Experiments

In this section, we evaluate the robustness of inductive reasoning in language models on different tasks. We also compare the performance of the models under different noise levels and analyze the effectiveness of the different methods in enhancing the robustness of inductive reasoning.

4.1 Experimental Setup

Consistent with previous studies (Wang et al., 2024; Qiu et al., 2024), we adopt few-shot prompting to assess the models’ inductive capabilities. Each instance contains 10 normal examples, 5 noisy examples, and 10 test examples. The normal and noisy examples are mixed to form 10 seen examples, which serve as prompts. This approach preserves task semantics while introducing controlled perturbations, thereby simulating real-world scenarios where observational data often contains inherent imperfections. LLMs are explicitly informed that examples may contain some noise. We formulate the output of LLMs as Python functions and execute them on the test examples to automatically evaluate the inferred rules. More details about the experiments can be found in the appendix B.

4.2 Datasets

We use three datasets with their subsets: Arithmetic, Cryptography, and List Functions. These datasets offer different rule induction challenges, including mathematical calculations, symbol representation, and list operations. To ensure that the noise reflects plausible perturbations encountered in real-world scenarios, we balance realism with experimental control while minimizing artificiality in designing the noise. Comprehensive description and statistics are shown in Table 1.

<table>
  <tr>
    <th>Dataset</th>
    <th>Noise Construction</th>
    <th>Subset</th>
    <th># Instance</th>
  </tr>
  <tr>
    <td rowspan="3">Arith.</td>
    <td>Decimal Addition</td>
    <td>7-base<br>8-base<br>9-base</td>
    <td>100<br>100<br>100</td>
  </tr>
  <tr>
    <td rowspan="3">Crypto.</td>
    <td>Random Replacement</td>
    <td>Caesar<br>Atbash<br>Keyboard</td>
    <td>100<br>100<br>100</td>
  </tr>
  <tr>
    <td>List Func.</td>
    <td>Random Replacement</td>
    <td>N/A</td>
    <td>250</td>
  </tr>
</table>

Table 1: Overview of datasets, including noise construction methods, subsets, and the number of instances per dataset. Arith., Crypto., List Func. denote Arithmetic, Cryptography and List Functions respectively.

Arithmetic The arithmetic task, introduced by Wu et al. (2024), involves counterfactual two-digit addition in non-decimal bases. To modulate difficulty and prioritize reasoning over memorization, we focus on the base-7, 8, 9 systems, and the noisy example is the common 10-based equations. The input is two two-digit numbers in the corresponding base, connected by a plus sign, and the output is the sum of the two numbers. To ensure the model can reason correctly, we guarantee that there must be a carry-over in the addition process so that the model must reason about the carry-over instead of simply adding the numbers.

Cryptography We use three types of substitution ciphers: Caesar, Atbash, and Keyboard. The Caesar cipher is a cipher that shifts the alphabet by a fixed number of positions. The Atbash cipher is a cipher that replaces each letter with the letter symmetrically opposite in the alphabet (e.g., A to Z). The Keyboard cipher replaces letters according to their positions in the alphabet with the corresponding positions on the keyboard (e.g., A to Q, B to W). The input is a word, and the output is the encrypted text. We randomly replace the characters in the output with other characters to generate noisy examples.

List Functions The list functions dataset (Rule, 2020) evaluates the concept learning ability in the domain of cognitive science. The task is to induce a function that maps a list of numbers to another list of numbers. Each instance in the dataset corresponds to a list manipulation operation, such as sorting, reversing, or filtering. We generate the examples of each instance under controlled conditions to ensure the rule can be induced. The noisy examples are generated by randomly replacing the numbers in the output list with other numbers.
Figure 3: Consistency score(%) with clean data of different models on the Cryptography and List Functions datasets under different noise levels.

<table>
  <tr>
    <th rowspan="2">Model</th>
    <th colspan="2">Cryptography</th>
    <th colspan="2">List Functions</th>
  </tr>
  <tr>
    <th>Caesar</th>
    <th>Atbash</th>
    <th>Keyboard</th>
    <th>List Functions</th>
  </tr>
  <tr>
    <td>GPT-4o-mini</td>
    <td>28.7<sub>±1.2</sub></td>
    <td>6.3<sub>±1.2</sub></td>
    <td>0.0<sub>±0.0</sub></td>
    <td>31.3<sub>±0.2</sub></td>
  </tr>
  <tr>
    <td>10% noise</td>
    <td>32.0<sub>±2.4</sub></td>
    <td>12.7<sub>±2.5</sub></td>
    <td>3.7<sub>±0.5</sub></td>
    <td>28.4<sub>±1.4</sub></td>
  </tr>
  <tr>
    <td>20% noise</td>
    <td>29.0<sub>±0.8</sub></td>
    <td>13.3<sub>±2.6</sub></td>
    <td>1.3<sub>±1.2</sub></td>
    <td>23.5<sub>±1.0</sub></td>
  </tr>
  <tr>
    <td>30% noise</td>
    <td>21.3<sub>±0.9</sub></td>
    <td>16.0<sub>±0.0</sub></td>
    <td>3.3<sub>±0.5</sub></td>
    <td>19.7<sub>±0.2</sub></td>
  </tr>
  <tr>
    <td>GPT-4o</td>
    <td>68.3<sub>±0.5</sub></td>
    <td>48.0<sub>±2.2</sub></td>
    <td>4.7<sub>±0.5</sub></td>
    <td>39.9<sub>±0.5</sub></td>
  </tr>
  <tr>
    <td>10% noise</td>
    <td>75.3<sub>±0.5</sub></td>
    <td>34.7<sub>±0.9</sub></td>
    <td>3.7<sub>±0.5</sub></td>
    <td>36.9<sub>±1.0</sub></td>
  </tr>
  <tr>
    <td>20% noise</td>
    <td>75.3<sub>±2.1</sub></td>
    <td>27.3<sub>±0.5</sub></td>
    <td>1.7<sub>±0.9</sub></td>
    <td>31.9<sub>±0.8</sub></td>
  </tr>
  <tr>
    <td>30% noise</td>
    <td>64.7<sub>±4.7</sub></td>
    <td>22.7<sub>±2.4</sub></td>
    <td>2.3<sub>±1.2</sub></td>
    <td>26.1<sub>±0.5</sub></td>
  </tr>
  <tr>
    <td>DeepSeek-V3</td>
    <td>50.0<sub>±1.6</sub></td>
    <td>21.7<sub>±2.5</sub></td>
    <td>25.0<sub>±0.8</sub></td>
    <td>43.1<sub>±0.4</sub></td>
  </tr>
  <tr>
    <td>10% noise</td>
    <td>24.3<sub>±1.7</sub></td>
    <td>15.3<sub>±2.4</sub></td>
    <td>15.7<sub>±1.2</sub></td>
    <td>40.8<sub>±0.9</sub></td>
  </tr>
  <tr>
    <td>20% noise</td>
    <td>15.3<sub>±3.7</sub></td>
    <td>21.3<sub>±2.5</sub></td>
    <td>11.3<sub>±1.2</sub></td>
    <td>35.3<sub>±1.9</sub></td>
  </tr>
  <tr>
    <td>30% noise</td>
    <td>11.0<sub>±0.8</sub></td>
    <td>16.7<sub>±0.5</sub></td>
    <td>11.3<sub>±3.1</sub></td>
    <td>30.5<sub>±0.7</sub></td>
  </tr>
</table>

Table 2: Task accuracy (%) on Cryptography (including Caesar, Atbash, and Keyboard subtasks) and List Functions under different noise levels (proportion of noise in seen examples). Results are shown in mean ± standard deviation over 3 independent runs.

4.3 Robustness Under Different Noise Levels

We first evaluate the robustness of inductive reasoning in language models under different noise injection ratios. Noise levels are defined as the proportion of noisy examples in the seen examples. We test three representative models: GPT-4o-mini (OpenAI, 2024a), GPT-4o (OpenAI, 2024b), and DeepSeek-V3 (DeepSeek-AI et al., 2024) and ask the models to directly infer the rules from the seen examples without any additional output. Since the models cannot solve the Arithmetic task under the Direct Output (DO) setting, we present the results for the Cryptography and List Functions. The results are shown in Table 2. To better evaluate the robustness of the models, we further investigate the consistency score with clean data under different noise levels, as shown in Figure 3. According to the results, we have the following observations.

First, contrary to conventional assumptions, noise introduction does not universally degrade performance, instead, the models exhibit performance fluctuations, demonstrating their inherent sensitivity to conflicting patterns. For example, GPT-4o achieves improved accuracy on Caesar cipher tasks at 10% noise (7.0% absolute improvement over clean data), while List Functions exhibit monotonic performance decay with increasing noise levels. In some cases, moderate noise even improves performance.

With the exception of tasks where models fundamentally struggle, experimental results demonstrate a decline in consistency scores as noise levels escalate, and it declines more sharply than its task accuracy variation. This discrepancy indicates noise introduces bidirectional reasoning instability: models not only fail on previously solvable instances (noise interference) but also succeed on originally challenging cases (incorrect generalization), leading to different performance changes.

4.4 Method-wise Effectiveness Comparison

Direct Output may limit the model’s reasoning ability, as the model must output the rule directly without any intermediate steps. To assess the robustness of different reasoning paradigms, we compare our Sample-steered Rule Refinement (SRR) method with four reasoning methods: (1) Chain of Thought (CoT; Wei et al. 2022), which decomposes reasoning into step-by-step rationales; (2) Self-Consistency (SC; Wang et al. 2023), which aggregates multiple CoT trajectories through majority voting; (3) Self-Refine (SR; Madaan et al. 2023), which iteratively improves hypotheses using self-generated feedback; and (4) Iterative Hypothesis Refinement (HR; Qiu et al. 2024), which iteratively generates and ranks multiple hypotheses at each step, then refines the best hypothesis with feedback from incorrect predictions. Table 3 presents the task accuracy under 10% noise and the deviation from clean data for GPT-4o and DeepSeek-V3.
<table>
  <tr>
    <th rowspan="2">Model</th>
    <th rowspan="2">Method</th>
    <th colspan="3">Arithmetic</th>
    <th colspan="3">Cryptography</th>
    <th rowspan="2">List Functions</th>
  </tr>
  <tr>
    <th>7-base</th>
    <th>8-base</th>
    <th>9-base</th>
    <th>Caesar</th>
    <th>Atbash</th>
    <th>Keyboard</th>
  </tr>
  <tr>
    <td rowspan="6">GPT-4o</td>
    <td>DO</td>
    <td>0.0</td>
    <td>0.0</td>
    <td>0.0</td>
    <td>75.3(± 7.0)</td>
    <td>34.7(± 13.3)</td>
    <td>3.7(± 1.0)</td>
    <td>36.9(± 3.0)</td>
  </tr>
  <tr>
    <td>Cot</td>
    <td>3.0(± 0.0)</td>
    <td>8.0(± 14.0)</td>
    <td>8.0(± 5.0)</td>
    <td><b>85.0(± 1.0)</b></td>
    <td>20.0(± 7.0)</td>
    <td>4.0(± 0.0)</td>
    <td>40.4(± 5.2)</td>
  </tr>
  <tr>
    <td>SC</td>
    <td>0.0(± 1.0)</td>
    <td>6.0(± 7.0)</td>
    <td>0.0(± 3.0)</td>
    <td><b>85.0(± 0.0)</b></td>
    <td>29.0(± 11.0)</td>
    <td>5.0(± 2.0)</td>
    <td>42.4(± 3.2)</td>
  </tr>
  <tr>
    <td>SR</td>
    <td>1.0(± 4.0)</td>
    <td>18.0(± 4.0)</td>
    <td>4.0(± 2.0)</td>
    <td>81.0(± 2.0)</td>
    <td>23.0(± 8.0)</td>
    <td>5.0(± 0.0)</td>
    <td>39.6(± 4.4)</td>
  </tr>
  <tr>
    <td>HR</td>
    <td>1.0(± 4.0)</td>
    <td>28.0(± 0.0)</td>
    <td><b>19.0(± 1.0)</b></td>
    <td>84.0(± 0.0)</td>
    <td><b>56.0(± 1.0)</b></td>
    <td><b>22.0(± 2.0)</b></td>
    <td>55.6(± 2.0)</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>5.0(± 1.0)</td>
    <td><b>51.0(± 6.0)</b></td>
    <td><b>19.0(± 2.0)</b></td>
    <td><b>85.0(± 3.0)</b></td>
    <td>52.0(± 3.0)</td>
    <td>9.0(± 1.0)</td>
    <td><b>57.2(± 1.6)</b></td>
  </tr>
  <tr>
    <td rowspan="6">Deepseek-V3</td>
    <td>DO</td>
    <td>0.0</td>
    <td>0.0</td>
    <td>0.0</td>
    <td>24.3(± 25.7)</td>
    <td>15.3(± 6.4)</td>
    <td>15.7(± 9.3)</td>
    <td>40.8(± 2.3)</td>
  </tr>
  <tr>
    <td>Cot</td>
    <td>77.5(± 6.0)</td>
    <td>83.0(± 13.0)</td>
    <td>67.5(± 14.0)</td>
    <td>80.5(± 4.0)</td>
    <td>26.0(± 5.5)</td>
    <td>5.0(± 8.5)</td>
    <td>52.0(± 6.4)</td>
  </tr>
  <tr>
    <td>SC</td>
    <td>83.0(± 3.0)</td>
    <td>93.0(± 6.0)</td>
    <td>81.0(± 3.0)</td>
    <td><b>86.0(± 1.0)</b></td>
    <td>40.0(± 3.0)</td>
    <td>7.0(± 4.0)</td>
    <td>56.0(± 3.2)</td>
  </tr>
  <tr>
    <td>SR</td>
    <td>70.0(± 10.0)</td>
    <td>74.0(± 9.0)</td>
    <td>68.0(± 6.0)</td>
    <td>72.0(± 10.0)</td>
    <td>19.0(± 1.0)</td>
    <td>8.0(± 3.0)</td>
    <td>47.2(± 10.0)</td>
  </tr>
  <tr>
    <td>HR</td>
    <td>82.0(± 4.0)</td>
    <td>89.0(± 2.0)</td>
    <td>78.0(± 0.0)</td>
    <td><b>86.0(± 0.0)</b></td>
    <td><b>56.0(± 0.0)</b></td>
    <td><b>17.0(± 7.0)</b></td>
    <td>63.6(± 6.0)</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td><b>96.0(± 1.0)</b></td>
    <td><b>95.0(± 4.0)</b></td>
    <td><b>94.0(± 0.0)</b></td>
    <td><b>86.0(± 1.0)</b></td>
    <td>52.0(± 1.0)</td>
    <td>11.0(± 2.0)</td>
    <td><b>64.8(± 2.8)</b></td>
  </tr>
</table>

Table 3: Task accuracy (%) on different datasets under 10% noise. The numbers in parentheses are the change compared to the clean data, and the arrows indicate the direction of the change. Bold indicates the best performance, and underline indicates the smallest change.

<table>
  <tr>
    <th rowspan="2">Dataset</th>
    <th colspan="3">GPT-4o</th>
    <th colspan="3">DeepSeek-V3</th>
  </tr>
  <tr>
    <th>SC</th>
    <th>HR-0</th>
    <th>SRR-0</th>
    <th>SC</th>
    <th>HR-0</th>
    <th>SRR-0</th>
  </tr>
  <tr>
    <td>Arithmetic<sub>c</sub></td>
    <td>0.0</td>
    <td>0.0</td>
    <td>4.0</td>
    <td>83.0</td>
    <td>81.0</td>
    <td>95.0</td>
  </tr>
  <tr>
    <td>Arithmetics<sub>c</sub></td>
    <td>6.0</td>
    <td>11.0</td>
    <td>40.0</td>
    <td>93.0</td>
    <td>85.0</td>
    <td>95.0</td>
  </tr>
  <tr>
    <td>Arithmetics<sub>r</sub></td>
    <td>0.0</td>
    <td>4.0</td>
    <td>15.0</td>
    <td>81.0</td>
    <td>68.0</td>
    <td>91.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>Caesar</sub></td>
    <td>85.0</td>
    <td>84.0</td>
    <td>85.0</td>
    <td>86.0</td>
    <td>86.0</td>
    <td>86.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>Atbash</sub></td>
    <td>29.0</td>
    <td>42.0</td>
    <td>51.0</td>
    <td>40.0</td>
    <td>52.0</td>
    <td>48.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>Keyboard</sub></td>
    <td>5.0</td>
    <td>14.0</td>
    <td>8.0</td>
    <td>7.0</td>
    <td>10.0</td>
    <td>9.0</td>
  </tr>
  <tr>
    <td>List Functions</td>
    <td>42.4</td>
    <td>48.8</td>
    <td>54.0</td>
    <td>56.0</td>
    <td>59.2</td>
    <td>62.8</td>
  </tr>
  <tr>
    <td>Average</td>
    <td>23.9</td>
    <td>29.1</td>
    <td>36.7</td>
    <td>63.7</td>
    <td>63.0</td>
    <td>69.5</td>
  </tr>
</table>

Table 4: Task accuracy (%) under 10% noise of SC and the initial rule in Hypothesis Refinement (HR-0) and SRR (SRR-0) on different datasets.

Superior Performance of SRR  As shown in Table 3, our SRR framework achieves state-of-the-art performance across 10 out of 14 task-model combinations while exhibiting relatively low performance degradation (2.1% average gap). This dual advantage stems from two mechanisms: (1) Diversity-aware hypothesis generation explores broader solution spaces through subset sampling, outperforming SC’s majority voting and the Hypothesis Refinement (Qiu et al., 2024) that amplifies the similar patterns at the rule initialization stage (Table 4); (2) Execution-guided feedback leverages code interpreter for objective error detection, circumventing LLMs’ inherent deductive limitations (Chen et al., 2023b; Cheng et al., 2024; Zheng et al., 2025a).

Consistency Scores Reveal Hidden Instability  While task accuracy provides comprehensive insights, consistency scores uncover fundamental reasoning fragility. As depicted in Figure 4, the Atbash cipher task exhibits particularly low consistency despite modest accuracy changes. The slight variation in accuracy may merely be an illusion created by the combined effects of noise interference (solved→unsolved cases) and incorrect generalization (unsolved→solved cases). Prior and concurrent works’ singular focus on accuracy fluctuations (Qiu et al., 2024; Zhou et al., 2024; Mirzadeh et al., 2024; Huang et al., 2025a) overlooks this critical duality in reasoning robustness.

![Consistency score (%) between clean data and data with 10% noise of DeepSeek-V3.](page_1092_1342_340_340.png)

Figure 4: Consistency score (%) between clean data and data with 10% noise of DeepSeek-V3.

Counterfactual Challenges Expose Knowledge Reliance  We observe dramatic performance gaps on counterfactual tasks like base-7 and base-9 arithmetic (90%+ vs. 10%– for DeepSeek-V3 vs. GPT-4o) and Keyboard ciphers. Diagnostic analysis of responses from LLMs shows models default to familiar rules rather than induction. GPT-4o persistently misinterprets base-7 and base-9 addition as “base-8” or “decimal sum with constant”, while Atbash and Keyboard ciphers get erroneously classified as Caesar shifts in the reasoning process or self-generated feedback. Although Atbash and Caesar ciphers exhibit similar levels of complex-
ity in transformation, their performance gaps are significant. These failures reveal that there is a pattern overfitting in the reasoning process. Current models’ inductive reasoning essentially operates through pattern matching rather than abstract induction. When the scarcity of counterfactual tasks in training data forces models to rely on “real” rule induction with unseen rules rather than recitation, model performance plummets dramatically.

4.5 Extended Explorations

To further investigate the robustness of LLMs’ inductive reasoning, we conduct a comparative evaluation against DeepSeek-R1 (DeepSeek-AI et al., 2025), one of the state-of-the-art reasoning models, and human reasoning patterns.

Comparison with DeepSeek-R1 We focus on the Atbash cipher and List Functions. Atbash provides unified task semantics yet challenges models with unfamiliar transformation logic, while List Functions captures diverse rule abstraction scenarios. Table 5 compares task accuracy and consistency scores under clean and 10% noise conditions. Notably, DeepSeek-R1 achieves higher task accuracy on both tasks than DeepSeek-V3. However, its consistency scores are still not competitive enough, indicating unresolved instability. Detailed breakdowns of consistency scores are shown in 6. For Atbash, the comparable number of right-to-wrong and wrong-to-right suggests instability and randomness in its reasoning process. Manual inspection shows DeepSeek-R1 still interprets Atbash cipher as character shifts in failed cases, mirroring previous pattern-overfitting behavior. For List Functions, the elevated right-to-wrong rate indicates its sensitivity to input perturbations.

<table>
  <tr>
    <th>Dataset</th>
    <th>Method</th>
    <th>Clean Acc</th>
    <th>Noise Acc</th>
    <th>Consistency</th>
  </tr>
  <tr>
    <td rowspan="2">Atbash</td>
    <td>CoT</td>
    <td>65.0</td>
    <td>65.0(±0.0)</td>
    <td>70.0</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>82.0</td>
    <td>79.0(−3.0)</td>
    <td>93.0</td>
  </tr>
  <tr>
    <td rowspan="2">List Func.</td>
    <td>CoT</td>
    <td>76.0</td>
    <td>68.4(−7.6)</td>
    <td>87.6</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>83.6</td>
    <td>78.0(−5.6)</td>
    <td>89.6</td>
  </tr>
</table>

Table 5: Task accuracy (%) and consistency score (%) between DeepSeek-R1 with CoT and SRR under clean and noisy conditions.

Human Reasoning Comparison Rule (2020) reports human performance on List Functions. We divide the tasks into three difficulty levels in a 5 : 3 : 2 ratio based on the sorted mean human performance. To contrast with human-like thinking patterns, we analyze consistency across 12 trials (4 noise levels × 3 runs) in Section 4.3. Figure 5 visualizes the distribution of consistency and performance per task in List Functions, stratified by difficulty. LLMs show macro-level alignment yet micro-level divergence compared to human. While LLMs broadly mirror human-like stability, showing higher consistency on both simple and hard tasks, with moderate instability on medium-difficulty tasks, their internal patterns reveal critical deviations. For hard tasks, models display unpredictability (e.g., sporadic success) rather than systematic incapacity. Simple tasks, despite their low complexity, exhibit in consistent performance unrelated to intrinsic difficulty. Notably, GPT-4o and DeepSeek-V3 demonstrate similar behavioral trends, suggesting a shared inductive bias.

<table>
  <tr>
    <th>Dataset</th>
    <th>Method</th>
    <th># BothR</th>
    <th># BothW</th>
    <th># RtoW</th>
    <th># WtoR</th>
  </tr>
  <tr>
    <td rowspan="2">Atbash</td>
    <td>CoT</td>
    <td>50</td>
    <td>20</td>
    <td>15</td>
    <td>15</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>77</td>
    <td>16</td>
    <td>5</td>
    <td>2</td>
  </tr>
  <tr>
    <td rowspan="2">List Func.</td>
    <td>CoT</td>
    <td>165</td>
    <td>54</td>
    <td>25</td>
    <td>6</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>189</td>
    <td>35</td>
    <td>20</td>
    <td>6</td>
  </tr>
</table>

Table 6: Detailed breakdowns of consistency score. BothR, BothW, RtoW, WtoR represent both right, both wrong, right to wrong from clean condition to noisy condition and wrong to right respectively.

![Task-solving consistency of DeepSeek-V3 and GPT-4o on List Functions. Each cell represents a task, arranged by ascending difficulty (top-to-bottom, left-to-right). Colors denote correctness patterns: 12R (all correct) to 12W (all wrong), with intermediate states (e.g., 1W11R: 1 wrong, 11 correct).](page_1012_1342_388_246.png)

Figure 5: Task-solving consistency of DeepSeek-V3 and GPT-4o on List Functions. Each cell represents a task, arranged by ascending difficulty (top-to-bottom, left-to-right). Colors denote correctness patterns: 12R (all correct) to 12W (all wrong), with intermediate states (e.g., 1W11R: 1 wrong, 11 correct).

5 Discussion

Our experimental results reveal limitations in LLMs’ inductive reasoning. We try to analyze the results in this section.
Process Analysis Inductive reasoning can be conceptualized through a Bayesian paradigm (Tenenbaum et al., 2011), where models update posterior distributions over a hypothesis space based on observations. The introduction of noise disrupts the reasoning through dual mechanisms: Noise-Induced Hypothesis Drift, where the initial hypothesis space becomes misaligned with true rules when noise introduces conflicting patterns. This drift particularly impacts Direct Output (DO) method, which lacks intermediate reasoning steps and relies heavily on the initial hypothesis space. This is more evident for tasks with similar internal patterns, the model’s performance fluctuates significantly (Cryptography) or even fails (Arithmetic) compared to other methods. Evidence Ambiguity Amplification, where noise reduces the effective signal-to-noise ratio during posterior optimization. Methods with iterative or step-by-step reasoning suffer error accumulation cascades—each reasoning step propagates uncertainty. As shown in Table 3, CoT and SR exhibit larger average accuracy drops and worse performance. SRR avoids it by objectively identifying errors through execution.

Implications for Robust Inductive Reasoning LLMs heavily rely on prior knowledge rather than pure induction. They blend memorized pattern matching with shallow reasoning. When observations align with priors (e.g., Caesar cipher), they demonstrate pseudo-robustness. However, conflicting patterns and counterfactual scenarios expose this fragility—models either default to familiar templates or enter unstable hypothesis oscillations, as shown in Table 7. Even the state-of-the-art reasoning model exhibits unstable performance in unified tasks. This contrasts sharply with human few-shot learning, where true rule abstraction enables stable generalization (Lake et al., 2015). Achieving human-level robust reasoning and few-shot induction requires disentangling knowledge recitation from inductive rule formation.

<table>
  <tr>
    <th>Unstable Hypothesis Oscillation</th>
  </tr>
  <tr>
    <td><b>Seen Examples</b> (base-9 arithmetic): I: 68+68 O: 147; I: 80+42 O: 132; I: 76+17 O: 104; I: 57+56 O: 113; I: 38+14 O: 53; I: 43+84 O: 137; I: 48+55 O: 114; ...</td>
  </tr>
  <tr>
    <td><b>Answer:</b> To solve this problem, let’s analyze the examples to identify a potential pattern: 1. \( 68 + 68 = 147 \): The expected output of 68 + 68 should be 136, but the output is 147. This suggests there is a transformation applied to the arithmetic result. 2. \( 80 + 42 = 132 \): Normally, 80 + 42 equals 122, but the output is 132. ... 4. \( 57 + 56 = 113 \): The expected result is 113, which matches the output. ... The examples seem to suggest a pattern of adding an extra 10 to the arithmetic sum of the inputs, except in the case where the sum already matches the desired output (as in the case of 57 + 56 = 113). With these observations, we can propose a rule: If the sum of the numbers is not equal to the expected output, add 10 to the arithmetic sum.</td>
  </tr>
  <tr>
    <th>Default to Familiar Templates</th>
  </tr>
  <tr>
    <td><b>Seen Examples</b> (Atbash cipher): I: cuissard O: xfrfhziw; I: tripisis* O: girkhrh; I: Maccabaeus O: Nzxxzyzvfh; ...</td>
  </tr>
  <tr>
    <td><b>Answer:</b> To solve this problem, let’s analyze the given examples to find a pattern: 1. cuissard->xfrfhziw 2. tripisis->girkhrh 3. Maccabaeus->Nzxxzyzvfh ... Observations: 1. Character Shifts: The transformation seems to involve shifting characters. For example, 'c' in "cuissar" becomes 'x' in "xfrfhziw". This suggests a possible Caesar cipher or similar character shift. ... 4. Pattern Recognition: The shifts are not consistent across all examples, suggesting that some examples might be incorrect or that multiple rules are being applied. However, a common shift that appears frequently is 21 positions backward (e.g., 'c' to 'x', 'P' to 'K', 'C' to 'X', 'W' to 'D'). Hypothesis: Given the frequent appearance of a 21-position backward shift, let’s hypothesize that the rule involves shifting each character 21 positions backward in the alphabet.</td>
  </tr>
</table>

Table 7: The failure cases of the models on the Arithmetic and Cryptography tasks. The red text indicates the noise introduced, and the orange text indicates the wrong hypothesis of the model.

tributions. Comparative studies (Lake et al., 2015, 2017) have highlighted the contrast between human learners and machine intelligence. With the advent of pre-trained language models (Brown et al., 2020), research on inductive reasoning has shifted from domain-specific and neural formulation (Tian et al., 2020; Odena et al., 2021; Sablé-Meyer et al., 2022) to natural language. Initial approaches (Alet et al., 2021; Gendron et al., 2024; Mirchandani et al., 2023; Yang et al., 2024) predominantly relied on input-output (IO) prompting, which evaluates model performance on unseen examples without explicit rule articulation. However, this paradigm overlooks the rule-inference process, as it conflates rule induction with rule execution capabilities. Recent efforts (Wang et al., 2024; Qiu et al., 2024) explicitly evaluate intermediate rules, and our set-
tings and methods are similar to theirs, despite the sampling part in initialization and iteration. Specifically, Qiu et al. (2024) also proposed an evaluation on noisy conditions, but their analysis is confined to a single dataset and prioritized accuracy metrics. We further explore the consistency and draw a different conclusion that LLMs’ inductive reasoning capabilities are not robust and generalizable.

Robustness of Reasoning in LLMs In this work, robustness refers to the ability to maintain consistent performance under imperfections or counterfactual scenarios (Elazar et al., 2021), which is intrinsically linked to the diversity and unpredictability of the generation process (Zhang et al., 2023; Huang et al., 2025b). Prior works on LLM behavioral variability exhibit inconsistent performance across time (Tu et al., 2024; Chen et al., 2023a). Recent works focus on understanding their internal reasoning mechanisms by altering conditions, such as changing numerical values in mathematical tasks (Mirzadeh et al., 2024; Huang et al., 2025a) or examining performance on counterfactual tasks (Wu et al., 2024). Zhou et al. (2024) investigates the impact of noisy rationales on model performance. These studies primarily measure robustness through global accuracy changes, overlooking inconsistencies at the instance level. Our work addresses this gap by introducing a consistency score that quantifies intra-task stability, providing a more granular view of model robustness.

7 Conclusion

In this paper, we explore the robustness of inductive reasoning in large language models under imperfect observations. Through introducing the Robust Rule Induction task with metrics of both holistic and individual levels, we systematically evaluate the ability of LLMs to maintain stable and consistent rule abstraction. The Sample-steered Rule Refinement outperforms other reasoning paradigms by effectively leveraging diversity-aware hypothesis generation and execution-guided feedback. Our findings reveal that while LLMs can demonstrate impressive reasoning capabilities, they are inherently sensitive to noise and prone to hypothesis drifting and pattern overfitting.

Limitations

We discuss the limitations of our work as follows. First, the general goal of our research is to investigate the relationship between a model’s reasoning and its understanding, specifically examining whether correctly solving a problem indicates true comprehension and reasoning of the underlying issue. To this end, our evaluation primarily focuses on highly formalized and symbolic tasks that are easily verifiable. However, real-world inductive reasoning often involves ambiguous rules like social norms from text or visual patterns, which are not captured by our formalism. Second, the tasks in Arithmetic and Cryptography may be relatively simple and unified, potentially limiting insights into the capabilities of large language models. Expanding task diversity could uncover deeper limitations in LLMs’ reasoning abilities. Finally, conducting a large-scale assessment of human performance across all tasks under varying levels of noise was beyond the scope of this study, which restricts our ability to compare human and model reasoning patterns comprehensively.

Ethics Statement

We discuss the ethical considerations of our work in the context of the following aspects: (1) Data collection and use. We use publicly available datasets and self-generated data for evaluation. We ensure that the data is only used for academic research purposes and no personal data is involved. The List Functions dataset is released with Apache-2.0 license1 in Big-bench (bench authors, 2023). We strictly follow the license terms and conditions. (2) LLMs API. We comply with the terms of service of the LLMs API providers strictly, maintaining fair use. (3) Transparency and reproducibility. We provide detailed descriptions of our methods, datasets, and evaluation metrics to ensure transparency. Our code and evaluation results are publicly available. (4) AI assistance. We use ChatGPT to paraphrase some sentences.

Acknowledgement

We thank all the anonymous reviewers and meta reviewers for their valuable comments. The authors of this paper were supported by the ITSP Platform Research Project (ITS/189/23FP) from ITC of Hong Kong, SAR, China, and the AoE (AoE/E-601/24-N), the RIF (R6021-20) and the GRF (16205322) from RGC of Hong Kong, SAR, China.

1https://www.apache.org/licenses/LICENSE-2.0
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Appendices

A  Details on Evaluation Pipeline

In this section, we provide detailed information on the evaluation pipeline, including the data construction and the performance assessment.

For Arithmetic, we generate the base-7, base-8, and base-9 tasks by randomly sampling two two-digit numbers in the corresponding base, and then we check whether there is a carry-over in the addition process. If there is no carry-over, we regenerate the numbers. The noisy examples are generated in base-10. This choice reflects a common human-like calculation error—defaulting to the familiar decimal system—making it a “meaningful” and realistic form of noise in a mathematical context.

For Cryptography, we randomly select words of appropriate lengths from the NLTK Word Lists2, and then we encrypt the words using the Caesar, Atbash, and Keyboard ciphers. The noisy examples are generated by randomly replacing the letters in the output with other letters. This mimics common errors in text processing or encryption, such as typographical mistakes or transmission noise, which are naturally occurring in symbolic domains.

For List Functions, we first write the corresponding rule functions for each task, and then we automatically generate the input data with appropriate lengths and ranges. The inputs are generated by randomly sampling numbers from a specific range with some constraints. Noise is introduced by randomly replacing some numbers in the output list with other values. This simulates natural errors, such as data corruption or misrecording, that might occur in sequence-based tasks.

During the data synthesis process, we attach manual supervision to ensure that the generated data can correctly induce the rules. During the evaluation process, we use the inputs in the test set as the input for rule execution. We evaluate the model’s performance using exact match. If the model’s output is correct on all the test set examples, we consider the model to have successfully induced the rule. If the model fails to output a valid programmatic rule or the program contains an infinite loop or errors, we consider it a failure.

2https://www.nltk.org/nltk_data/

B  Experimental Details

B.1  Experimental Settings

For robustness under different noise levels, we run each experiment three times and report the mean and standard deviation of the results to avoid randomness. For chain-of-thought (CoT), self refine (SR) and direct output (DO), we set the temperature to 0.0 for all models to ensure reproducibility. For self-consistency (SC), hypothesis refinement (HR) and sample-steered rule refinement (SRR), the temperature is set to 0.7, consistent with the original work of SC (Wang et al., 2023). The positions of noise in seen examples are random to avoid positional bias (Lu et al., 2022). In implementation, we choose 2 subsets for SRR by splitting the seen examples into two subsets, and we generate 3 hypotheses at each iteration of HR. For SC, we generate 5 trajectories at the same time. The number of iterations is set to 3 (including the initial round) for SR, HR and SRR.

For the GPT-4o-mini and GPT-4o models, the experiments are conducted through the official OpenAI3 API platform. For the DeepSeek-V3 and DeepSeek-R1 models, the experiments are conducted through the official DeepSeek4 API platform and the Volcengine5 platform. We spent about 700 USD in total for the experiments.

<table>
  <tr>
    <th>Method</th>
    <th>Acc<sub>clean</sub></th>
    <th>Acc<sub>10\% noise</sub></th>
    <th>Consistency</th>
  </tr>
  <tr>
    <td>CoT</td>
    <td>66.0</td>
    <td>56.0</td>
    <td>64.5</td>
  </tr>
  <tr>
    <td>SC</td>
    <td>79.0</td>
    <td>72.0</td>
    <td>63.0</td>
  </tr>
  <tr>
    <td>SR</td>
    <td>67.0</td>
    <td>60.0</td>
    <td>65.0</td>
  </tr>
  <tr>
    <td>HR</td>
    <td>89.0</td>
    <td>76.0</td>
    <td>79.0</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>100.0</td>
    <td>97.0</td>
    <td>97.0</td>
  </tr>
</table>

Table 8: Task accuracy (%) and consistency score (%) of DeepSeek-V3 on the base-7 Arithmetic task with distribution noise.

B.2  Additional Results

Different Types of Noise  To ensure the noise reflects perturbations encountered in real-world scenarios, the noise of Arithmetic is different from that of Cryptography and List Functions. To explore the consistency of the results under a more uniform noise strategy, we conducted additional experiments on the base-7 Arithmetic task using

3https://openai.com/api/
4https://platform.deepseek.com/usage
5https://www.volcengine.com/
<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="4">Arithmetic<sub>c</sub></th>
    <th colspan="4">Arithmetic<sub>sc</sub></th>
    <th colspan="4">Crypto<sub>sc</sub></th>
    <th colspan="4">Crypto<sub>ab</sub></th>
    <th colspan="4">List Func.</th>
  </tr>
  <tr>
    <th>BR</th><th>BW</th><th>RW</th><th>MR</th>
    <th>BR</th><th>BW</th><th>RW</th><th>MR</th>
    <th>BR</th><th>BW</th><th>RW</th><th>WR</th>
    <th>BR</th><th>BW</th><th>RW</th><th>MR</th>
    <th>BR</th><th>BW</th><th>RW</th><th>MR</th>
    <th>BR</th><th>BW</th><th>RW</th><th>MR</th>
  </tr>
  <tr>
    <th colspan="21">GPT-4o-mini</th>
  </tr>
  <tr>
    <td>DO<sub>10%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>64</td><td>182</td><td>22</td><td>32</td><td>15</td><td>258</td><td>4</td><td>23</td><td>0</td><td>289</td><td>0</td><td>11</td><td>190</td><td>492</td><td>45</td><td>23</td>
  </tr>
  <tr>
    <td>DO<sub>20%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>51</td><td>178</td><td>35</td><td>36</td><td>9</td><td>250</td><td>10</td><td>31</td><td>0</td><td>296</td><td>0</td><td>4</td><td>160</td><td>499</td><td>75</td><td>16</td>
  </tr>
  <tr>
    <td>DO<sub>30%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>38</td><td>188</td><td>48</td><td>26</td><td>14</td><td>247</td><td>5</td><td>34</td><td>0</td><td>290</td><td>0</td><td>10</td><td>126</td><td>493</td><td>109</td><td>22</td>
  </tr>
  <tr>
    <th colspan="21">GPT-4o</th>
  </tr>
  <tr>
    <td>DO<sub>10%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>195</td><td>64</td><td>10</td><td>31</td><td>85</td><td>137</td><td>59</td><td>19</td><td>7</td><td>282</td><td>7</td><td>4</td><td>259</td><td>433</td><td>40</td><td>18</td>
  </tr>
  <tr>
    <td>DO<sub>20%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>180</td><td>49</td><td>25</td><td>46</td><td>66</td><td>140</td><td>78</td><td>16</td><td>28</td><td>12</td><td>2</td><td>178</td><td>432</td><td>41</td><td>19</td>
  </tr>
  <tr>
    <td>DO<sub>30%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>155</td><td>56</td><td>39</td><td>47</td><td>105</td><td>195</td><td>55</td><td>16</td><td>5</td><td>182</td><td>12</td><td>4</td><td>178</td><td>431</td><td>121</td><td>18</td>
  </tr>
  <tr>
    <td>CoT</td><td>0</td><td>94</td><td>3</td><td>2</td><td>72</td><td>20</td><td>6</td><td>0</td><td>89</td><td>3</td><td>8</td><td>82</td><td>13</td><td>2</td><td>1</td><td>60</td><td>20</td><td>13</td><td>1</td><td>93</td><td>3</td><td>2</td><td>96</td><td>131</td><td>18</td><td>5</td>
  </tr>
  <tr>
    <td>SC</td><td>0</td><td>99</td><td>1</td><td>0</td><td>82</td><td>12</td><td>5</td><td>0</td><td>97</td><td>3</td><td>0</td><td>83</td><td>13</td><td>2</td><td>2</td><td>19</td><td>50</td><td>21</td><td>10</td><td>0</td><td>88</td><td>7</td><td>5</td><td>100</td><td>130</td><td>14</td><td>6</td>
  </tr>
  <tr>
    <td>SR</td><td>0</td><td>94</td><td>5</td><td>1</td><td>4</td><td>64</td><td>18</td><td>14</td><td>0</td><td>90</td><td>6</td><td>4</td><td>78</td><td>14</td><td>5</td><td>3</td><td>10</td><td>56</td><td>21</td><td>13</td><td>2</td><td>92</td><td>3</td><td>3</td><td>88</td><td>129</td><td>22</td><td>11</td>
  </tr>
  <tr>
    <td>HR</td><td>0</td><td>94</td><td>5</td><td>1</td><td>12</td><td>56</td><td>16</td><td>16</td><td>5</td><td>66</td><td>15</td><td>14</td><td>82</td><td>14</td><td>2</td><td>2</td><td>40</td><td>29</td><td>15</td><td>16</td><td>6</td><td>64</td><td>14</td><td>16</td><td>124</td><td>91</td><td>20</td><td>15</td>
  </tr>
  <tr>
    <td>SRR</td><td>0</td><td>89</td><td>6</td><td>5</td><td>23</td><td>27</td><td>22</td><td>28</td><td>4</td><td>64</td><td>17</td><td>15</td><td>84</td><td>14</td><td>4</td><td>59</td><td>32</td><td>16</td><td>13</td><td>0</td><td>83</td><td>8</td><td>9</td><td>136</td><td>96</td><td>11</td><td>7</td>
  </tr>
  <tr>
    <th colspan="21">DeepSeek-V3</th>
  </tr>
  <tr>
    <td>DO<sub>10%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>65</td><td>142</td><td>65</td><td>8</td><td>35</td><td>224</td><td>30</td><td>11</td><td>15</td><td>213</td><td>40</td><td>12</td><td>282</td><td>403</td><td>41</td><td>24</td>
  </tr>
  <tr>
    <td>DO<sub>20%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>141</td><td>214</td><td>113</td><td>9</td><td>37</td><td>208</td><td>28</td><td>27</td><td>24</td><td>215</td><td>10</td><td>73</td><td>308</td><td>405</td><td>85</td><td>27</td>
  </tr>
  <tr>
    <td>DO<sub>30%</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>31</td><td>148</td><td>119</td><td>2</td><td>28</td><td>213</td><td>37</td><td>22</td><td>19</td><td>210</td><td>56</td><td>15</td><td>198</td><td>396</td><td>125</td><td>31</td>
  </tr>
  <tr>
    <td>CoT</td><td>141</td><td>19</td><td>26</td><td>14</td><td>160</td><td>2</td><td>32</td><td>6</td><td>110</td><td>12</td><td>53</td><td>25</td><td>156</td><td>26</td><td>13</td><td>5</td><td>29</td><td>114</td><td>34</td><td>23</td><td>5</td><td>168</td><td>22</td><td>5</td><td>236</td><td>184</td><td>56</td><td>24</td>
  </tr>
  <tr>
    <td>SC</td><td>76</td><td>7</td><td>10</td><td>7</td><td>92</td><td>0</td><td>7</td><td>1</td><td>69</td><td>4</td><td>15</td><td>12</td><td>84</td><td>11</td><td>3</td><td>2</td><td>25</td><td>42</td><td>18</td><td>15</td><td>1</td><td>83</td><td>10</td><td>6</td><td>129</td><td>91</td><td>19</td><td>11</td>
  </tr>
  <tr>
    <td>SR</td><td>58</td><td>8</td><td>22</td><td>12</td><td>66</td><td>9</td><td>17</td><td>8</td><td>45</td><td>15</td><td>7</td><td>23</td><td>64</td><td>10</td><td>18</td><td>8</td><td>56</td><td>66</td><td>15</td><td>14</td><td>2</td><td>83</td><td>9</td><td>6</td><td>105</td><td>94</td><td>38</td><td>13</td>
  </tr>
  <tr>
    <td>HR</td><td>78</td><td>8</td><td>12</td><td>8</td><td>82</td><td>22</td><td>7</td><td>9</td><td>82</td><td>11</td><td>11</td><td>18</td><td>84</td><td>12</td><td>2</td><td>2</td><td>46</td><td>46</td><td>16</td><td>13</td><td>2</td><td>103</td><td>21</td><td>20</td><td>6</td><td>103</td><td>21</td><td>20</td><td>6</td>
  </tr>
  <tr>
    <td>SRR</td><td>93</td><td>0</td><td>4</td><td>3</td><td>94</td><td>0</td><td>5</td><td>1</td><td>88</td><td>0</td><td>6</td><td>6</td><td>84</td><td>11</td><td>3</td><td>2</td><td>37</td><td>32</td><td>16</td><td>15</td><td>2</td><td>78</td><td>11</td><td>9</td><td>154</td><td>73</td><td>15</td><td>8</td>
  </tr>
</table>

Table 9: Detailed breakdown of the consistency score of different methods or different noise levels. The notation BR, BW, RW, and WR represent both right, both wrong, right to wrong from clean condition to noisy condition, and wrong to right from clean condition to noisy condition, respectively.

DeepSeek-V3. Instead of base-10 outputs, we introduce disturbance noise by adding a random number between 1 and 6 (base_num−1) to the correct results, aligning this noise more closely with the random corruption used in other tasks. The results are presented in Table 8.

From the results, we find that the performance exhibits the same trend as in the original experiments, with a consistent drop in accuracy when noise is introduced. The consistency score also shows a similar trend, indicating the robustness fragility of inductive reasoning in the presence of noise. The consistency drop is more evident than the accuracy drop, suggesting the accuracy drop is not solely due to the failure of previous solvable cases. This further supports the need to consider a more detailed granularity except for the accuracy.

Breakdown of Consistency Score For a more comprehensive understanding of the consistency score, we provide a detailed breakdown of the consistency score in Table 9. From the results, we observe that the introduction of noise not only leads to a performance drop in the original solvable cases but also results in some previously unsolved cases being solved. This reveals the fragility and inconsistency in the rule induction process.

Ablation We conduct ablation studies to investigate the impact of sampling right and wrong examples during the refinement process of SRR. The results are shown in Table 10. From the results, we observe that sampling right and wrong examples during the refinement process of SRR has a positive impact on the performance when noise is introduced, while it has a slightly negative impact on the performance in the clean condition. This suggests that sampling can help the model to better adapt to the noise, the performance gap between the clean and noisy conditions is reduced.

<table>
  <tr>
    <th rowspan="2">Dataset</th>
    <th colspan="2">Accuracy<sub>clean</sub></th>
    <th colspan="2">Accuracy<sub>10% noise</sub></th>
  </tr>
  <tr>
    <th>w/sample</th><th>w/o sample</th>
    <th>w/sample</th><th>w/o sample</th>
  </tr>
  <tr>
    <td>Arithmetic<sub>c</sub></td><td>97.0</td><td>99.0</td><td>96.0</td><td>91.0</td>
  </tr>
  <tr>
    <td>Arithmetic<sub>sc</sub></td><td>99.0</td><td>98.0</td><td>95.0</td><td>96.0</td>
  </tr>
  <tr>
    <td>Arithmetic<sub>sc</sub></td><td>94.0</td><td>94.0</td><td>94.0</td><td>88.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>sc</sub></td><td>87.0</td><td>86.0</td><td>86.0</td><td>85.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>ab</sub></td><td>53.0</td><td>51.0</td><td>52.0</td><td>44.0</td>
  </tr>
  <tr>
    <td>Crypto<sub>sc</sub></td><td>13.0</td><td>18.0</td><td>11.0</td><td>19.0</td>
  </tr>
  <tr>
    <td>List Func.</td><td>67.6</td><td>66.8</td><td>64.8</td><td>64.4</td>
  </tr>
  <tr>
    <th>Average</th><th>72.9</th><th>73.3</th><th>71.3</th><th>69.6</th>
  </tr>
</table>

Table 10: The impact of sampling right and wrong examples during the refinement process of SRR using DeepSeek-V3. The results are reported in terms of task accuracy (%). w/sample and w/o sample represent with and without sampling, respectively.

We also investigate the performance of SR, HR, and SRR across different iterations. The results are shown in Figure 6. Compared to the performance of SRR and HR, the performance of SR is not stable across different refinement iterations, and even shows a performance drop in some cases. This indicates that the LLM-generated feedback in SR is not always helpful for the model to refine the rule induction process. In contrast, both HR and SRR show a more stable performance improvement across iterations, and the improvement is more pro-
Figure 6: The performance of SR, HR and SRR across different iterations with different models and noise levels. The results are reported in terms of task accuracy (%).

nounced in HR. This is because HR generates different refined hypotheses at the same iteration and selects the best one, while SRR only refines the existing hypothesis. Specifically, HR also shows a performance drop in the DeepSeek-V3 model on the List Functions task with 10% noise, which is likely due to HR only selecting the best hypothesis from the generated ones at each iteration. At the same time, SRR chooses the best one from all the generated rules.

Computational Cost  The computational cost of different methods is an important factor to consider the effectiveness, which is measured by the number of output tokens in the whole process. In general, methods that require more samples or generate more responses tend to have higher computational costs. We summarize the computational cost of different methods in Table 11.

<table>
  <tr>
    <th>Method</th>
    <th>DeepSeek-V3<sub>clean</sub></th>
    <th>DeepSeek-V3<sub>noise</sub></th>
    <th>GPT-4<sub>clean</sub></th>
    <th>GPT-4<sub>noise</sub></th>
  </tr>
  <tr>
    <td>Cot</td>
    <td>1,120</td>
    <td>1,194</td>
    <td>828</td>
    <td>800</td>
  </tr>
  <tr>
    <td>SC</td>
    <td>5,271</td>
    <td>5,620</td>
    <td>3,719</td>
    <td>3,709</td>
  </tr>
  <tr>
    <td>SR</td>
    <td>2,333</td>
    <td>2,673</td>
    <td>1,289</td>
    <td>1,192</td>
  </tr>
  <tr>
    <td>HR</td>
    <td>4,261</td>
    <td>4,884</td>
    <td>4,089</td>
    <td>4,211</td>
  </tr>
  <tr>
    <td>SRR</td>
    <td>3,388</td>
    <td>3,574</td>
    <td>2,791</td>
    <td>2,879</td>
  </tr>
</table>

Table 11: The computational cost of different methods. The results are reported in terms of the average number of output tokens per instance per dataset.

More Reasoning Models  Except for the reasoning model used in 4.5, we also conduct experiments with QwQ-32B, another reasoning model implementing a reinforcement learning scaling approach (Team, 2025). Our goal is to explore whether the introduced noise can impact the performance of the reasoning models. The results based on Chain-of-Thought are shown in Table 12. Even the performance of reasoning models is more powerful than the LLMs, they still suffer from the noise and counterfactual tasks.

<table>
  <tr>
    <th rowspan="2">Dataset</th>
    <th colspan="3">DeepSeek-R1</th>
    <th colspan="3">QwQ-32B</th>
  </tr>
  <tr>
    <th>Acc<sub>clean</sub></th>
    <th>Acc<sub>noise</sub></th>
    <th>Consistency</th>
    <th>Acc<sub>clean</sub></th>
    <th>Acc<sub>noise</sub></th>
    <th>Consistency</th>
  </tr>
  <tr>
    <td>Arithmetic<sub>c</sub></td>
    <td>99.0</td>
    <td>96.0</td>
    <td>95.0</td>
    <td>100.0</td>
    <td>99.0</td>
    <td>99.0</td>
  </tr>
  <tr>
    <td>Arithmetic<sub>n</sub></td>
    <td>100.0</td>
    <td>99.0</td>
    <td>99.0</td>
    <td>100.0</td>
    <td>100.0</td>
    <td>100.0</td>
  </tr>
  <tr>
    <td>Arithmetic<sub>h</sub></td>
    <td>98.0</td>
    <td>98.0</td>
    <td>96.0</td>
    <td>100.0</td>
    <td>99.0</td>
    <td>99.0</td>
  </tr>
  <tr>
    <td>Cryptoc<sub>c</sub></td>
    <td>84.0</td>
    <td>81.0</td>
    <td>93.0</td>
    <td>77.0</td>
    <td>75.0</td>
    <td>84.0</td>
  </tr>
  <tr>
    <td>Cryptob<sub>n</sub></td>
    <td>65.0</td>
    <td>65.0</td>
    <td>70.0</td>
    <td>46.0</td>
    <td>41.0</td>
    <td>69.0</td>
  </tr>
  <tr>
    <td>Cryptob<sub>h</sub></td>
    <td>2.0</td>
    <td>0.0</td>
    <td>98.0</td>
    <td>0.0</td>
    <td>0.0</td>
    <td>100.0</td>
  </tr>
  <tr>
    <td>List Func.</td>
    <td>76.0</td>
    <td>68.4</td>
    <td>87.6</td>
    <td>72.8</td>
    <td>68.0</td>
    <td>85.6</td>
  </tr>
</table>

Table 12: Task accuracy (%) and consistency score of DeepSeek-R1 and QwQ-32B with Chain-of-Thought prompting on different datasets.

C Prompt Templates

We provide the prompts we used in this section. For the self-refine, hypothesis refinement, and sample-steered rule refinement setting, we use the chain-of-thought prompt as the initial prompt; the iterative prompts are different due to the type of feedback.
PROMPT FOR DIRECT OUTPUT

Please generate a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}. Note that some examples may be wrong, and you should take this into account when proposing the rule.

{examples}

Please format your Python function as follows:
```python
def fn(x):
# Your code here
```
Your response should only include the function definition, not the function call or any other information.

PROMPT FOR CHAIN-OF-THOUGHT

Please generate a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}. Note that some examples may be wrong, and you should take this into account when proposing the rule.

{examples}

Please format your Python function as follows:
```python
def fn(x):
# Your code here
```
Think step-by-step and explain your reasoning. Your response should include your thought process and the function definition without the function call.
Prompt for Self Refine

Feedback Generation
You have generated a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}.

{examples}

In the last step, your rule is:

```python
{rule}
```

Give some feedback on the rule you have generated, like how it can be improved, what is wrong with it, etc.
Your response should only include the feedback. If you think the rule is good enough, your response should be “NO FEEDBACK” without other information. Note that some examples may be wrong, and you should take this into account when proposing the feedback.

Iteration
You have generated a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}. Note that some examples may be wrong, and you should take this into account when proposing the rule.

{examples}

In the last step, your rule is:

```python
{rule}
```

The feedback you have given is:

{feedback}

Generate a new rule that maps the given inputs to their corresponding outputs using a Python function. Please format your rule as follows:
```python
def fn(x):
# Your code here
```
Think step-by-step and explain your reasoning. Your response should include your thought process and the function definition without the function call.
Prompt for Iteration in Hypothesis Refinement

You have generated a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}. Note that some examples may be noisy, and you should take this into account when proposing the rule.

{examples}

In the last step, your rule is

```python
{rule}
```

It does not work for the following examples:

{wrong examples}

Generate a new rule that maps the given inputs to their corresponding outputs using a Python function. Please format your rule as follows:
```python
def fn(x):
# Your code here
```

Think step-by-step and explain your reasoning. Your response should include your thought process and the function definition without the function call. You can either modify the existing rule or propose a new one.
PROMPT FOR ITERATION IN SAMPLE-STEERED RULE REFINEMENT

You have generated a rule that maps the following inputs to their corresponding outputs using a Python function. The input is {Input Description}. The output is {Output Description}. Note that some examples may be noisy, and you should take this into account when proposing the rule. In the last step, your rule is

```python
{rule}
```

But this rule is not correct. It works for the following examples:

{sampled right examples}

However, it does not work for the following examples:

{sampled wrong examples}

Generate a new rule that maps the given inputs to their corresponding outputs using a Python function. Please format your rule as follows:
```python
def fn(x):
# Your code here
```

Think step-by-step and explain your reasoning. Your response should include your thought process and the function definition without the function call. You can either modify the existing rule or propose a new one.

Reviewer paper summaries: This paper evaluates robust rule induction tasks under controlled noise and proposes a sample-steered refinement methodology that enhances both the accuracy and stability of LLMs’ inductive reasoning. Furthermore, it verifies the previously demonstrated claim that LLMs are inherently sensitive.

This paper introduces a new benchmark task called Robust Rule Induction to evaluate the robustness of inductive reasoning in large language models (LLMs) under noisy and conflicting observations. The authors design a systematic evaluation pipeline and propose a novel method called Sample-steered Rule Refinement (SRR), which enhances reasoning stability through hypothesis diversity and execution-guided feedback. Extensive experiments across arithmetic, cryptography, and list operations demonstrate the effectiveness of SRR and reveal that current LLMs are fragile under noise, often relying on memorized patterns rather than genuine abstraction. The paper further proposes a consistency score to expose model instability at the instance level, providing a more fine-grained understanding of reasoning robustness.
Summary of strengths: 1. The writing is clear and well-structued.
2. The analysis experiments are comprehensive, as described in Section 4.5 and 5.

Well-motivated task: The paper targets an important gap in LLM evaluation—robust inductive reasoning—especially under noisy conditions, which is both practical and underexplored.
    Novel methodology: The proposed SRR framework is conceptually clean and effectively leverages diversity-aware hypothesis sampling and execution-based feedback, outperforming strong baselines (CoT, SC, SR).
    Thorough evaluation: The authors provide extensive experiments on diverse domains with varying noise levels, comparing multiple models and reasoning paradigms, including human-level analysis.
    Consistency metric: Introducing a new metric (consistency score) offers valuable insights into instance-level stability that task accuracy alone fails to capture.
Summary of weaknesses: 1. The proposed SRR method naturally performs well on tasks with controlled noise, as the sampled subsets may contain fewer noisy examples, and the selected rule is the one that yields the highest accuracy. Therefore, it is not surprising (and arguably even unfair) that SRR achieves better accuracy and stability in such settings. However, it remains inferior to large reasoning models like DeepSeek-R1. Given that SRR also requires multiple iterations to refine hypotheses, its overall advantage is relatively limited.
2. Based on the results in Table 2 and Figure 3, the introduction of noise can, in some cases, improve accuracy. In such scenarios, achieving higher accuracy requires changes in the consistency score. As a result, it is questionable to consider a high consistency score as a universally positive metric.

Therefore, while the proposed task has some merit, I find the overall contribution of the paper to be unconvincing.

Limited domain coverage: The benchmark focuses heavily on symbolic/formalized tasks (e.g., base-n arithmetic, ciphers), which may not generalize to real-world inductive reasoning scenarios. Some discussion or qualitative analysis on broader applicability would strengthen the work.
    Model-specific behavior underexplored: While the paper compares GPT-4o and DeepSeek, it could benefit from a deeper analysis of why different models behave differently (e.g., pretraining biases, CoT usage, instruction tuning effects).
    Limited model coverage: The current evaluation only includes GPT-4o and DeepSeek series, lacking comparisons with a broader range of mainstream models—particularly the latest reasoning-optimized models such as the ChatGPT-o1 series, Qwen series, and DeepSeek-R1. Including these models would significantly strengthen the generality and representativeness of the findings.
Comments, suggestions and typos: NA

Limited domain coverage: The benchmark focuses heavily on symbolic/formalized tasks (e.g., base-n arithmetic, ciphers), which may not generalize to real-world inductive reasoning scenarios. Some discussion or qualitative analysis on broader applicability would strengthen the work.
    Model-specific behavior underexplored: While the paper compares GPT-4o and DeepSeek, it could benefit from a deeper analysis of why different models behave differently (e.g., pretraining biases, CoT usage, instruction tuning effects).
    Limited model coverage: The current evaluation only includes GPT-4o and DeepSeek series, lacking comparisons with a broader range of mainstream models—particularly the latest reasoning-optimized models such as the ChatGPT-o1 series, Qwen series, and DeepSeek-R1. Including these models would significantly strengthen the generality and representativeness of the findings.
Overall assessment scores: 2.5

2.5