Title: LIMOPro: Reasoning Refinement for Efficient and Effective Test-time Scaling
Venue: NeurIPS 2025
Accepted: yes
Keywords: LLM

reasoning

Test-time Scaling
Abstract: Large language models (LLMs) have demonstrated remarkable reasoning capabilities through test-time scaling approaches, particularly when fine-tuned with chain-of-thought (CoT) data distilled from more powerful large reasoning models (LRMs). However, these reasoning chains often contain verbose elements that mirror human problem-solving, categorized as progressive reasoning (the essential solution development path) and functional elements (verification processes, alternative solution approaches, and error corrections). While progressive reasoning is crucial, the functional elements significantly increase computational demands during test-time inference. We introduce PIR (Perplexity-based Importance Refinement), a principled framework that quantitatively evaluates the importance of each reasoning step based on its impact on answer prediction confidence. PIR systematically identifies and selectively prunes only low-importance functional steps while preserving all progressive reasoning components, creating optimized training data that maintains the integrity of the core solution path while reducing verbosity. Models fine-tuned on PIR-optimized data exhibit superior test-time scaling properties, generating more concise reasoning chains while achieving improved accuracy (+0.9\% to +6.6\%) with significantly reduced token usage (-3\% to -41\%) across challenging reasoning benchmarks (AIME, AMC, and GPQA Diamond). Our approach demonstrates strong generalizability across different model sizes, data sources, and token budgets, offering a practical solution for deploying reasoning-capable LLMs in scenarios where efficient test-time scaling, response time, and computational efficiency are valuable constraints. Code and dataset are available at the [LIMOPro GitHub repository.](https://github.com/GAIR-NLP/LIMOPro)

LIMOPro: Reasoning Refinement for Efficient and Effective Test-time Scaling

Abstract

Large language models (LLMs) have demonstrated remarkable reasoning capabilities through test-time scaling approaches, particularly when fine-tuned with chain-of-thought (CoT) data distilled from more powerful large reasoning models (LRMs). However, these reasoning chains often contain verbose elements that mirror human problem-solving, categorized as progressive reasoning (the essential solution development path) and functional elements (verification processes, alternative solution approaches, and error corrections). While progressive reasoning is crucial, the functional elements significantly increase computational demands during test-time inference. We introduce PIR (Perplexity-based Importance Refinement) , a principled framework that quantitatively evaluates the importance of each reasoning step based on its impact on answer prediction confidence. PIR systematically identifies and selectively prunes only low-importance functional steps while preserving progressive reasoning components, creating optimized training data that maintains the integrity of the core solution path while reducing verbosity. Models fine-tuned on PIR-optimized data exhibit superior test-time scaling properties, generating more concise reasoning chains while achieving improved accuracy (+0.9% to +6.6%) with significantly reduced token usage (-3% to -41%) across challenging reasoning benchmarks (AIME, AMC, and GPQA Diamond). Our approach demonstrates strong generalizability across different model sizes, data sources, and token budgets, offering a practical solution for deploying reasoningcapable LLMs in scenarios where efficient test-time scaling, response time, and computational efficiency are valuable constraints. Code and dataset are available at the LIMOPro.

1 Introduction

Large language models (LLMs) have demonstrated remarkable capabilities in complex reasoning tasks through chain-of-thought (CoT) [33], where models generate step-by-step solutions to problems. Recent advances of test-time scaling [12, 26] can significantly enhance LLMs' reasoning abilities by increasing the compute at test time. One approach to the test-time scaling involves fine-tuning LLMs on high-quality reasoning data distilled from more powerful large reasoning models (LRMs) [20, 37]. LRMs like DeepSeek-R1 [10], OpenAI o1 [12], and QwQ [29] represent the state of the art in this paradigm, producing reasoning chains that lead to accurate solutions.

However, this approach faces a significant challenge: reasoning chains distilled from LRMs often contain numerous functional elements that, while reflecting human problem-solving processes, possibly produce unnecessarily verbose outputs [7, 32, 4]. In a typical mathematical scenario, an LRMmight solve a problem by establishing an initial solution path, verifying calculations, identifying

SII

AIME

LIMO-P

50.0

52.5

55.0

57.5

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65.0

67.5

70.0

Pass@1

56.7

63.3

LIMO-P

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Avg. Response Tokens

12497

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Pass@1

91.9

93.8

LIMO-P

5000

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5300

5400

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5800

Avg. Response Tokens

5516

5235

LIMO-P

1.60

1.65

1.70

1.75

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1.85

Efficiency

1e

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1.67e-04

1.79e-04

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LIMO-P

66.0

67.0

68.0

69.0

70.0

71.0

72.0

73.0

74.0

Pass@1

67.2

71.2

LIMO-P

6900

6950

7000

7050

7100

7150

7200

Avg. Response Tokens

7173

6969

LIMO-P

0.85

0.90

0.95

1.00

1.05

1.10

Efficiency

1e

4

9.36e-05

1.02e-04

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Pass@1

LIMO-P

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-15%

+32%

+2%

-5%

+7%

+6%

-3%

+9%

errors, revising the approach, and ultimately confirming the final answer. This thorough process generates lengthy reasoning chains with redundant or marginally valuable steps. When these verbose chains are used to fine-tune target models, the resulting models inevitably adopt similar behaviors, producing equally elaborate reasoning sequences despite many steps contributing minimally to solution accuracy. Consequently, inference time increases substantially, along with computational demands and response latency. This inefficiency poses a considerable obstacle to implementing reasoning-enhanced LLMs in practical applications where timely, precise responses are essential.

To address this challenge, we introduce PIR (Perplexity-based Importance Refinement) , a novel framework that systematically refines reasoning chains to optimize the efficiency-effectiveness balance. Our refinement approach builds upon three key innovations that work in concert: (1) We develop a systematic methodology to classify functional patterns in complex reasoning chains, identifying four distinct modes-progressive reasoning and three types of functional steps: verification, multimethod validation, and error correction. (2) Through comprehensive analysis across diverse problem domains, our analysis indicates that progressive reasoning forms the essential logical backbone, directly advancing solution derivation, and must be preserved intact. In contrast, functional steps frequently introduce computational overhead with redundancies that can be strategically pruned without compromising solution integrity. This differential treatment-preserving progressive reasoning while selectively optimizing functional elements-maintains the core problem-solving logic while significantly improving computational efficiency. (3) Building on (1) and (2), we propose the PIR metric, which quantitatively measures each functional step's contribution to the final solution by comparing answer perplexity with and without specific steps. This perplexity-based evaluation provides a principled mechanism to refine reasoning chains by identifying and selectively removing low-importance functional steps while preserving the progressive reasoning chain. By selectively targeting only non-essential functional components, our refinement approach maintains the logical coherence of the solution process while significantly reducing verbosity.

By applying our refinement framework to datasets distilled from different foundation models (LIMO [37] from Deepseek-R1, S1 [20] from Gemini Flash Thinking [2], and LIMO-V2 [37] from QwQ), we create PIR-optimized training datasets. Models fine-tuned on these refined datasets maintain or enhance accuracy while significantly reducing response length compared to models trained on the original unrefined data. Our experiments across challenging reasoning benchmarks demonstrate that PIR-refined models consistently outperform their counterparts in both effectiveness and efficiency, achieving up to 71% efficiency improvement.

The contributions of our work include: 1. A novel perplexity-based refinement framework (PIR) for quantifying the importance of reasoning steps and optimizing reasoning chains, balancing efficiency and effectiveness. 2. A systematic analysis of reasoning patterns in reasoning problem-solving, providing insights into the structure and function of different reasoning elements. 3. Comprehensive empirical validation showing that PIR-refined models achieve improved accuracy (+0.9% to +6.6%) with significantly reduced token usage (-3% to -41%) across diverse benchmarks. 4. Demonstration of the framework's generalizability across different model sizes, data sources, and token budgets.

Our work addresses a critical gap in current approaches to LLM reasoning enhancement, offering a practical solution for more efficient reasoning without sacrificing solution quality. By systematically refining training data to preserve essential reasoning while eliminating redundant functional steps (verification, validation, and error correction processes), we enable LLMs to produce concise yet equally effective reasoning chains, advancing the practical deployment of reasoning-capable LLMs across scenarios where response time and computational resources are valuable constraints.

2 Reasoning Refinement: Reasoning Optimization Framework

Reasoning chains produced by LRMs typically contain numerous functional steps-including verification processes, multiple solution approaches, and error corrections-that mirror human problemsolving but significantly increase computational overhead without proportionally enhancing solution accuracy. This section presents our Perplexity-based Importance Refinement (PIR) framework, which quantitatively evaluates reasoning step importance and systematically refines reasoning chains by preserving essential solution elements while removing less valuable components.

2.1 Problem Formulation

In this paper, we address the challenge of optimizing reasoning chains for complex reasoning tasks. Formally, we consider a dataset D containing question-reasoning-answer triplets ( q, r, a ) , where q ∈ Q represents a reasoning problem, r ∈ R is the reasoning chain, and a ∈ A is the answer. We define a reasoning chain r as a sequence of intermediate steps { s 1 , s 2 , ..., s n } , where each step s i represents a logical deduction that bridges the gap between the question and the final answer.

Our goal is to refine each reasoning chain r into an optimized version r ′ such that: (1) The answer accuracy is preserved: f ( q, r ′ ) = f ( q, r ) = a ; (2) The token length is reduced: | r ′ | < | r | ; (3) The essential reasoning logic is maintained without harming the quality of the dataset.

2.2 Theoretical Foundations

Cognitive Reasoning Patterns Through extensive analysis of reasoning patterns, we identify four representative distinct modes [9, 8] that characterize problem-solving processes: (1) Progressive Reasoning , characterized by forward-chaining inference that follows a deductive logical progression from premises to conclusion, forming the essential backbone of solution development; (2) Verification , which represents metacognitive monitoring processes where previous calculations are systematically validated for accuracy, often using phrases like 'Let me check'; (3) Multi-method Validation , demonstrating convergent thinking by applying diverse methodological approaches to reinforce conclusions, potentially introducing redundancy; and (4) Error Correction embodies a self-regulatory mechanism through which logical inconsistencies, computational errors, or potential mistakes are identified and remediated. This pattern captures the process of recognizing when a path of reasoning may be flawed, reassessing assumptions, and correcting course. While progressive reasoning constitutes the critical path to solution derivation, the other three functional patterns, though valuable in human-like reasoning, often contain redundancies that can be optimized without

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compromising solution integrity. Details about the four reasoning patterns and corresponding cases can be found in Appendix A.

PIR: Perplexity-Based Importance Refinement of Reasoning Steps PIR quantifies reasoning step importance by measuring perplexity changes when specific steps are removed. The indicator compares answer perplexity with and without a particular reasoning step:

Where PPL θ ( R ) and PPL θ ( R \ { x i } ) represent perplexities calculated by model θ with and without step i :

Here, m represents the number of answer tokens, a j is the j -th answer token, x 1: n represents all reasoning steps, and x 1: i -1 , x i +1: n represents all steps except step i . The perplexity calculations are performed using a model θ . A higher PIR value indicates greater step importance: when a critical reasoning step is eliminated, the model becomes significantly less confident in generating the correct answer, lacking essential information for solution derivation, resulting in higher perplexity and thus a higher PIR score.

2.3 Analysis and Optimization of Reasoning Chains

Our approach first employs a hierarchical decomposition process where reasoning chains are segmented into logical steps by Claude 3.7 Sonnet [1], with each step typically comprising multiple coherent sentences that form a cohesive reasoning unit. For classification, we implement a two-phase system: initially, a rule-based pattern matching component identifies steps containing characteristic linguistic markers (such as 'Let me check' for verification or 'I made a mistake' for error correction). For steps lacking explicit markers, we again apply Claude 3.7 Sonnet to perform contextual analysis, capturing more nuanced reasoning structures and assigning appropriate pattern classifications.

To validate our classification methodology, we randomly select 5% of classified steps across our datasets for human evaluation. Four postgraduates independently assess whether each step is correctly classified according to our defined patterns. Steps are considered correctly classified only when all four annotators unanimously agree with the system's classification. This rigorous evaluation protocol revealed that 93.4% of the steps are unanimously verified as correctly classified, demonstrating the robust performance of our hybrid classification approach for reasoning step categorization.

Building on this classification system and the PIR metric, we implemented a targeted pruning approach that selectively identifies and removes low-importance functional steps while maintaining data quality and preserving reasoning integrity. This process is shown in Figure 2. Unlike approaches that might indiscriminately compress reasoning chains, our method specifically targets functional steps (verification, multi-method validation, and error correction) while preserving all progressive reasoning steps, which constitute the essential deductive core of the solution process. For each identified functional step, we compute its PIR value to quantify its importance based on the impact its removal has on answer prediction confidence. The process then selectively removes functional steps with the lowest PIR values according to a predefined ratio threshold, resulting in more concise reasoning chains that maintain effectiveness while significantly reducing verbosity.

This selective pruning approach creates more efficient reasoning chains that maintain the integrity of core problem-solving logic while eliminating redundant verification processes. The resulting optimized chains are both more efficient to process and more effective as training exemplars for downstream models, without sacrificing the quality and completeness of the essential reasoning. The Algorithm 1 in the appendix demonstrates this whole process. In our experiments, we apply Qwen2.5-32B-Instruct to calculate PPL.

3 Experimental Results and Analysis

3.1 Experimental Setup

Training Dataset To empirically validate our proposed PIR reasoning refinement framework, we conduct experiments using established datasets from prior work. As demonstrated in Table 1, our approach leverages three representative reasoning datasets: LIMO (distilled from DeepSeek R1), LIMO-V2 (distilled from QwQ), and S1K (distilled from Gemini Thinking). These datasets, distilled from different foundation models, represent diverse reasoning patterns and problem-solving approaches. Applying our framework across these varied sources allows us to validate the generalizability of our method across different data sources, ensuring that the PIR optimization is robust and not dependent on specific characteristics of any single source model. We apply our PIR framework to these datasets to create optimized versions: LIMO-P, LIMO-V2-P, and S1K-P. Additionally, we construct several variant datasets by implementing different refinement ratios to investigate the optimal balance between conciseness and effectiveness. Detailed information regarding different refinement ratios and the specific token counts of the optimized datasets is provided in the Appendix B.1.

Benchmark Datasets To rigorously assess our methodology, we utilized three representivate reasoning-intensive benchmarks: (1) AIME24 evaluation set encompasses 30 challenging problems from the American Invitational Mathematics Examination administered in early 2024, requiring sophisticated reasoning across mathematical domains; (2) GPQA Diamond corpus incorporates 198 doctoral-level scientific inquiries across biological, chemical, and physical disciplines, presenting formidable challenges that even subject matter experts struggle to master fully; (3) AMC23 includs 40 problems from the AIMO progress prize competition. Using data after 2023.

Performance metrics: Reasoning Effectiveness and Efficiency To evaluate reasoning effectiveness (ACC) , we employ the pass@1 accuracy as our primary performance indicator. For each

problem in our benchmark, we sample eight responses from the model and calculate ACC under the Zero-shot Chain-of-Thought (CoT) setting with the instruction of: 'Please reason step by step, and put your final answer within boxed.' We utilize Qwen2.5-Math evaluators [35] to systematically assess solution correctness across all solutions, with each sampling conducted at a temperature setting of 0.7 to balance deterministic reasoning with exploration of solution paths. Building on our effectiveness measure, we quantify reasoning efficiency (EFF) as the ratio between model performance and resource utilization: EFF = ACC / TOK, where TOK represents the average number of response tokens across all benchmark problems. This efficiency metric captures the utility produced per unit of test-time resource consumption, effectively highlighting the critical trade-off between performance and computational cost.

Evaluation of PIR Framework Across Multiple Training Datasets We establish baseline models (LIMO, LIMO-V2, and S1-32B) that were trained on the original unmodified datasets using Qwen2.5-32B-Instruct [35]. These baseline models were obtained from their official Hugging Face repositories. 1 To test our PIR framework, we fine-tune the same Qwen2.5-32B-Instruct base model on the pruned datasets: LIMO-P, LIMO-V2-P, and S1K-P, using identical training scripts as described in the original papers and yielding our PIR-optimized models: LIMO-P, LIMO-V2-P, and S1-32B-P. All evaluations follow consistent testing protocols across methods to maintain comparative validity.

Comparison Against Alternative Reasoning Optimization Methods We compare our PIR framework with other competitive reasoning optimization methods. First, we establish S1-32B as our primary baseline trained on the original unmodified S1K dataset without any reasoning optimization. We then compare against two leading competing methods from prior work: Prompt Reduction [6], denoted as S1-PROMPT , which develops innovative prompting strategies that encourage LLMs to use shortcuts to quickly exploit reasoning clues and bypass detailed procedural steps. We apply this method for model S1-32B as a baseline; and SPIRIT [5], denoted as S1-SPIRIT , which applies a non-discriminative pruning approach for all reasoning steps based solely on perplexity values. We apply this method to the S1K dataset to get a pruned dataset, and fine-tuning Qwen2.5-32B-Instruct with the pruned dataset. Unlike our PIR method that selectively preserves essential progressive reasoning steps, S1-SPIRIT's uniform filtering risks removing critical components necessary for accurate solutions. Additionally, we include an ablation of our method, Rule-based Filtering (denoted as S1-RULE ), which identifies functional step categories but randomly removes steps from these categories without using PIR metrics to determine their importance. 2

3.2 Main Results

Performance Across Benchmarks Table 2 demonstrates the consistent effectiveness of our PIR optimization framework across multiple challenging reasoning benchmarks. The PIR-optimized variants achieve superior efficiency-accuracy trade-offs across all model families. S1-32B-P shows remarkable improvements on AIME with a 4.2 percentage point accuracy increase alongside a 29% token reduction, yielding a 56% efficiency improvement. Similarly, LIMO-P demonstrates consistent gains with enhanced accuracy (+6.6, +1.9, and +4.0 percentage points across AIME, AMC, and GPQA Diamond, respectively) while reducing token consumption by up to 15%, achieving efficiency improvements of 32%, 7%, and 9% across the three benchmarks. LIMO-V2-P exhibits comparable enhancements with token reductions of 12%, 7%, and 13%, paired with accuracy improvements of 4.9, 2.2, and 3.0 percentage points. These consistent improvements across diverse benchmarks confirm that our PIR framework effectively identifies and preserves high-value reasoning steps while eliminating low-importance functional components.

Generalizability to Different Data Sources The substantial performance improvements across models trained on data distilled from diverse foundation models (S1 from Gemini Flash Thinking, LIMO from DeepSeek-R1, and LIMO-V2 from QwQ) suggest PIR's strong generalizability potential. While individual gains vary in magnitude, the consistent pattern of simultaneous accuracy increases and token reductions across these different data sources indicates that our framework successfully identifies important reasoning patterns independent of the original distillation source. This suggests that PIR captures fundamental aspects of reasoning quality rather than exploiting source-specific characteristics.

Comparison with Alternative Optimization Methods Table 3 demonstrates our PIR framework's superior performance compared to other reasoning optimization approaches. PIR (S1-32B-P) consistently outperforms all alternatives across benchmarks, achieving significant improvements over the baseline S1-32B model with accuracy gains of 4.2, 2.2, and 0.9 percentage points alongside token reductions of 29%, 16%, and 41% on AIME, AMC, and GPQA Diamond respectively. Notably, our approach of selectively pruning only functional steps while preserving all progressive reasoning components proves superior to whole-sentence pruning methods (S1-SPIRIT), which apply filtering indiscriminately across all reasoning steps. The superior accuracy of S1-32B-P over S1-SPIRIT, particularly on AIME (5.0 percentage point advantage), empirically validates our hypothesis that progressive reasoning steps constitute the essential solution backbone and should remain intact.

3.3 Analysis

3.3.1 Generalizability to Test-Time Scaling

Test-time scaling represents a critical dimension of LLM reasoning deployment, where more computational resources are allocated during inference. To evaluate whether our PIR optimization framework maintains its effectiveness across varying inference-time token budgets, we conducted experiments measuring accuracy as a function of token budget constraints. Figure 1 presents the test-time scaling curves across three benchmarks of LIMO and LIMO-P. The results demonstrate that PIR-optimized models consistently outperform their non-optimized counterparts across most token budget levels. The superior performance of PIR-optimized models across different token budgets demonstrates the generalizability of our approach to different resource constraints. This makes our PIR framework particularly valuable for real-world applications where inference efficiency and response latency are crucial considerations. The ability to maintain performance advantages highlights the robustness of our optimization methodology to different deployment scenarios.

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3.3.2 Impact of ratios

We investigated the impact of various pruning ratios on model performance by creating multiple LIMO dataset variants with different proportions of functional reasoning steps removed. As shown in Figure 3, our experiments revealed clear performance trade-offs across metrics. For the AIME benchmark, lower pruning ratios (0.2-0.3) yielded optimal accuracy improvements over the baseline, while higher ratios (0.8) achieved the greatest response length reduction. For AMC, a moderate pruning ratio of 0.3 delivered peak accuracy while maintaining efficiency gains. Test time efficiency consistently improved with pruning across both benchmarks, with particularly strong gains on AIME. These results demonstrate the existence of an optimal refinement threshold that balances the removal of redundant functional steps with the preservation of critical reasoning components. Excessive pruning beyond this threshold leads to declining accuracy as valuable reasoning elements are removed, even when categorized as functional steps. This pattern validates our PIR framework's approach of selectively identifying and preserving high-value reasoning steps based on their quantitative importance scores.

3.3.3 Generalizability to Model Size

To evaluate PIR's generalizability across parameter scales, we conducted experiments using Qwen2.5 models ranging from 3B to 32B parameters, comparing the performance of models trained with optimized LIMO-P versus original LIMO datasets. As shown in Figure 4, our method demonstrates robust scalability with performance improvements across most model sizes. The benefits of PIR refinement become increasingly pronounced as model size increases, particularly for the AIME benchmark where the 32B model shows impressive gains across all metrics (11.8% accuracy improvement, 15.3% response length reduction, and 32.0% efficiency increase). For AMC, mid-sized models (7B-14B) yield the strongest efficiency improvements (up to 23.0%). The consistent pattern of enhancement for most model sizes suggests that our method's scalability and practical utility across various model sizes.

4 Related Work

4.1 Test-Time Scaling of LLMs

Test-time scaling [12, 26, 29, 10] enhances LLM reasoning by increasing inference-time computation. Approaches include non-training methods, which optimize existing model inference strategies, and training-based methods, which modify model parameters. Non-training techniques encompass Best-of-N sampling [18, 27], majority voting [31, 3], and tree search [36, 30] for exploring multiple reasoning paths. Training-based approaches divide into supervised fine-tuning (SFT) and reinforcement learning (RL). SFT methods train on high-quality reasoning traces [20, 37, 16, 15], with S1 and LIMO demonstrating improved performance through careful sample selection. RL approaches [24, 10, 12, 29] have yielded exceptional results, with DeepSeek-R1 using GRPO and OpenAI's o1 and QwQ enabling autonomous development of reasoning chains that adaptively allocate computation based on problem complexity. Our work focuses on optimizing SFT-based approaches through perplexity-based refinement to improve efficiency while preserving accuracy.

4.2 Efficient Reasoning

Research on efficient reasoning has gained significant traction as LLMs face challenges with computational overhead and verbosity. At inference time, various optimization approaches have emerged without requiring parameter updates. Length budgeting techniques like Token-Budget-Aware LLM Reasoning [11] enforce token limits via prompting, while S1 [20] appends end-of-thinking delimiters. Dual-process inspired system switching methods alternate between fast intuitive and deliberative reasoning; Dualformer [28] selectively drops reasoning traces during training, while System 1.x [23] employs a controller to assess task difficulty. Model switching approaches such as BiLD [14] and EAGLE [17] leverage speculative decoding with smaller models for initial predictions, while RouteLLM [21] routes queries based on complexity. For supervised fine-tuning, C3ot [13] preserves essential information while reducing redundancies, and TokenSkip [34] omits less important tokens. SPIRIT [5] prunes low-importance reasoning steps. RL-based methods either incorporate explicit length penalties [19, 25] or balance exploitation with exploration [22]. Most closely related to our work, SPIRIT [5] calculates perplexity for all reasoning steps and filters them based on a predetermined ratio. Our approach fundamentally differs by distinguishing between reasoning step types rather than treating all equally. We preserve all progressive reasoning steps-the essential solution backbone-while only pruning less critical functional components (verification, multi-method validation, and error correction), ensuring core reasoning integrity and avoiding the risk of removing critical solution elements.

5 Conclusion

Contributions This paper introduces PIR (Perplexity-based Importance Refinement), a novel framework that optimizes reasoning chains by quantitatively assessing step importance and selectively pruning low-value functional elements while preserving essential reasoning paths. Our comprehensive evaluation demonstrates that models fine-tuned on PIR-optimized datasets achieve both improved accuracy and significantly reduced token usage. By strategically balancing thorough problemsolving with computational efficiency, PIR establishes a principled approach for deploying advanced reasoning capabilities in latency-sensitive applications, opening new avenues for research on efficient reasoning in foundation models.

Limitations While our approach demonstrates significant improvements, several limitations warrant further investigation. First, our evaluation primarily focuses on mathematical reasoning tasks and science tasks; future work should validate PIR's effectiveness across broader reasoning domains including logical, commonsense, and causal reasoning. Second, our refinement strategy relies on perplexity as the primary importance indicator, which may not fully capture the semantic contribution of certain reasoning steps. Alternative metrics incorporating semantic relevance could enhance refinement precision. Third, the optimal pruning ratio may vary across different reasoning tasks and model architectures, suggesting the need for adaptive refinement strategies. Finally, our approach currently requires access to model perplexity outputs, which may limit applicability with closedsource models. Addressing these limitations could further advance efficient reasoning frameworks for real-world applications.

References

Claude 3.7 Sonnet and Claude Code - anthropic.com. https://www.anthropic.com/ news/claude-3-7-sonnet . [Accessed 30-04-2025].

Thinking | Generative AI on Vertex AI | Google Cloud - cloud.google.com. https: //cloud.google.com/vertex-ai/generative-ai/docs/thinking . [Accessed 30-042025].

Lingjiao Chen, Jared Quincy Davis, Boris Hanin, Peter Bailis, Ion Stoica, Matei A Zaharia, and James Y Zou. Are more llm calls all you need? towards the scaling properties of compound ai systems. Advances in Neural Information Processing Systems , 37:45767-45790, 2024.

Xingyu Chen, Jiahao Xu, Tian Liang, Zhiwei He, Jianhui Pang, Dian Yu, Linfeng Song, Qiuzhi Liu, Mengfei Zhou, Zhuosheng Zhang, et al. Do not think that much for 2+ 3=? on the overthinking of o1-like llms. arXiv preprint arXiv:2412.21187 , 2024.

Yingqian Cui, Pengfei He, Jingying Zeng, Hui Liu, Xianfeng Tang, Zhenwei Dai, Yan Han, Chen Luo, Jing Huang, Zhen Li, et al. Stepwise perplexity-guided refinement for efficient chain-of-thought reasoning in large language models. arXiv preprint arXiv:2502.13260 , 2025.

Mengru Ding, Hanmeng Liu, Zhizhang Fu, Jian Song, Wenbo Xie, and Yue Zhang. Break the chain: Large language models can be shortcut reasoners. arXiv preprint arXiv:2406.06580 , 2024.

Chenrui Fan, Ming Li, Lichao Sun, and Tianyi Zhou. Missing premise exacerbates overthinking: Are reasoning models losing critical thinking skill? arXiv preprint arXiv:2504.06514 , 2025.

Yichao Fu, Junda Chen, Siqi Zhu, Zheyu Fu, Zhongdongming Dai, Aurick Qiao, and Hao Zhang. Efficiently serving llm reasoning programs with certaindex. arXiv preprint arXiv:2412.20993 , 2024.

Kanishk Gandhi, Ayush Chakravarthy, Anikait Singh, Nathan Lile, and Noah D Goodman. Cognitive behaviors that enable self-improving reasoners, or, four habits of highly effective stars. arXiv preprint arXiv:2503.01307 , 2025.

Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948 , 2025.

Tingxu Han, Zhenting Wang, Chunrong Fang, Shiyu Zhao, Shiqing Ma, and Zhenyu Chen. Token-budget-aware llm reasoning. arXiv preprint arXiv:2412.18547 , 2024.

Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed El-Kishky, Aiden Low, Alec Helyar, Aleksander Madry, Alex Beutel, Alex Carney, et al. Openai o1 system card. arXiv preprint arXiv:2412.16720 , 2024.

Yu Kang, Xianghui Sun, Liangyu Chen, and Wei Zou. C3ot: Generating shorter chain-ofthought without compromising effectiveness. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 39, pages 24312-24320, 2025.

Sehoon Kim, Karttikeya Mangalam, Suhong Moon, Jitendra Malik, Michael W Mahoney, Amir Gholami, and Kurt Keutzer. Speculative decoding with big little decoder. Advances in Neural Information Processing Systems , 36:39236-39256, 2023.

Bespoke Labs. Bespoke-stratos: The unreasonable effectiveness of reasoning distillation. www.bespokelabs.ai/blog/bespoke-stratos-the-unreasonable-effectiveness-of-reasoningdistillation, 2025. Accessed: 2025-01-22.

Dacheng Li, Shiyi Cao, Tyler Griggs, Shu Liu, Xiangxi Mo, Eric Tang, Sumanth Hegde, Kourosh Hakhamaneshi, Shishir G Patil, Matei Zaharia, et al. Llms can easily learn to reason from demonstrations structure, not content, is what matters! arXiv preprint arXiv:2502.07374 , 2025.

Yuhui Li, Fangyun Wei, Chao Zhang, and Hongyang Zhang. Eagle: Speculative sampling requires rethinking feature uncertainty. arXiv preprint arXiv:2401.15077 , 2024.

Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let's verify step by step. In The Twelfth International Conference on Learning Representations , 2023.

Haotian Luo, Li Shen, Haiying He, Yibo Wang, Shiwei Liu, Wei Li, Naiqiang Tan, Xiaochun Cao, and Dacheng Tao. O1-pruner: Length-harmonizing fine-tuning for o1-like reasoning pruning. arXiv preprint arXiv:2501.12570 , 2025.

Niklas Muennighoff, Zitong Yang, Weijia Shi, Xiang Lisa Li, Li Fei-Fei, Hannaneh Hajishirzi, Luke Zettlemoyer, Percy Liang, Emmanuel Candès, and Tatsunori Hashimoto. s1: Simple test-time scaling. arXiv preprint arXiv:2501.19393 , 2025.

Isaac Ong, Amjad Almahairi, Vincent Wu, Wei-Lin Chiang, Tianhao Wu, Joseph E Gonzalez, MWaleed Kadous, and Ion Stoica. Routellm: Learning to route llms with preference data, 2024. URL https://arxiv. org/abs/2406.18665 .

Yuxiao Qu, Matthew YR Yang, Amrith Setlur, Lewis Tunstall, Edward Emanuel Beeching, Ruslan Salakhutdinov, and Aviral Kumar. Optimizing test-time compute via meta reinforcement fine-tuning. arXiv preprint arXiv:2503.07572 , 2025.

Swarnadeep Saha, Archiki Prasad, Justin Chih-Yao Chen, Peter Hase, Elias Stengel-Eskin, and Mohit Bansal. System-1. x: Learning to balance fast and slow planning with language models. arXiv preprint arXiv:2407.14414 , 2024.

John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347 , 2017.

Yi Shen, Jian Zhang, Jieyun Huang, Shuming Shi, Wenjing Zhang, Jiangze Yan, Ning Wang, Kai Wang, and Shiguo Lian. Dast: Difficulty-adaptive slow-thinking for large reasoning models. arXiv preprint arXiv:2503.04472 , 2025.

Charlie Snell, Jaehoon Lee, Kelvin Xu, and Aviral Kumar. Scaling llm test-time compute optimally can be more effective than scaling model parameters. arXiv preprint arXiv:2408.03314 , 2024.

Yifan Song, Guoyin Wang, Sujian Li, and Bill Yuchen Lin. The good, the bad, and the greedy: Evaluation of llms should not ignore non-determinism. arXiv preprint arXiv:2407.10457 , 2024.

DiJia Su, Sainbayar Sukhbaatar, Michael Rabbat, Yuandong Tian, and Qinqing Zheng. Dualformer: Controllable fast and slow thinking by learning with randomized reasoning traces. In The Thirteenth International Conference on Learning Representations , 2024.

Qwen Team. Qwq-32b: Embracing the power of reinforcement learning, March 2025.

Ye Tian, Baolin Peng, Linfeng Song, Lifeng Jin, Dian Yu, Lei Han, Haitao Mi, and Dong Yu. Toward self-improvement of llms via imagination, searching, and criticizing. Advances in Neural Information Processing Systems , 37:52723-52748, 2024.

Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain of thought reasoning in language models. arXiv preprint arXiv:2203.11171 , 2022.

Yue Wang, Qiuzhi Liu, Jiahao Xu, Tian Liang, Xingyu Chen, Zhiwei He, Linfeng Song, Dian Yu, Juntao Li, Zhuosheng Zhang, et al. Thoughts are all over the place: On the underthinking of o1-like llms. arXiv preprint arXiv:2501.18585 , 2025.

Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. Chain-of-thought prompting elicits reasoning in large language models. Advances in neural information processing systems , 35:24824-24837, 2022.

Heming Xia, Yongqi Li, Chak Tou Leong, Wenjie Wang, and Wenjie Li. Tokenskip: Controllable chain-of-thought compression in llms. arXiv preprint arXiv:2502.12067 , 2025.

An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, et al. Qwen2. 5 technical report. arXiv preprint arXiv:2412.15115 , 2024.

Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. Advances in neural information processing systems , 36:11809-11822, 2023.

Yixin Ye, Zhen Huang, Yang Xiao, Ethan Chern, Shijie Xia, and Pengfei Liu. Limo: Less is more for reasoning. arXiv preprint arXiv:2502.03387 , 2025.

A Reasoning Refinement: Reasoning Optimization Framework

A.1 Phrases Used for Pattern Matching

In our rule-based pattern matching system for reasoning step classification, we identify four distinct cognitive reasoning patterns characterized by specific linguistic markers. Table 4 presents these patterns alongside their characteristic phrases. Figure 5 demonstrates one sample from S1K containing the four patterns.

Progressive Reasoning constitutes the critical path to solution derivation, characterized by forwardchaining inference that follows a deductive logical progression from premises to conclusion. This pattern forms the essential backbone of solution development and is indicated by phrases that signal logical advancement, such as 'Let's solve,' 'First/Then/Next,' and 'Therefore.'

Verification represents metacognitive monitoring processes where previous calculations are systematically validated for accuracy. This pattern is identified through phrases like 'Let me check,' 'Let me verify,' and 'Double-check,' which signal when the model reviews its prior work.

Multi-method Validation demonstrates convergent thinking through the application of diverse methodological approaches to reinforce conclusions. This pattern is recognized through expressions indicating alternative approach consideration, including 'Alternatively,' 'Another way,' and 'Let's try a different approach,' potentially introducing redundancy.

Error Correction embodies a self-regulatory mechanism through which logical inconsistencies or computational errors are identified and remediated. This pattern is captured by phrases acknowledging mistakes, such as 'This is wrong,' 'The mistake was,' and 'This contradicts.'

A.2 Prompt used for Pattern Recognition

Our hybrid classification system combines rule-based pattern matching with contextual analysis using Claude 3.7 Sonnet for more nuanced reasoning pattern recognition. In Figure 6 and 7, we present the prompts used for the sophisticated pattern recognition of Claude 3.7 Sonnet.

Solution:

Okay, so I need to find the last three digits of the product of the positive

roots of the equation √(1995) * x^(log_{1995} x) = x². Hmm, let's see. First, let me

try to understand the equation and figure out how to solve it. \n\nThe equation is

given as √(1995) * x^(log_{1995} x) = x². That looks a bit complicated with the

logarithm in the exponent. Maybe I can take logarithms on both sides to simplify it?

Or perhaps rewrite the equation using properties of exponents and logarithms. \n\n

Let me recall that log_b a = (ln a)/(ln b), so log_{1995} x is equal to (ln x)/(ln 1995).

Also, x raised to log_{1995} x can be rewritten using exponentials. Wait, maybe I can express x^(log_{1995} x) as

e^{(ln x) * (log_{1995} x)}. But that might not be helpful. Let me think differently.\n\nAlternatively, since the equation

has x in the exponent and also x squared, maybe substituting t = log_{1995} x would help. Let me try that. Let's set t =

log_{1995} x, which means that x = 1995^t. Then substituting back into the equation:\n\n√(1995) * (1995^t)^{t} =

(1995^t)^2\n\nSimplify both sides:\n\nLeft side: √(1995) * 1995^{t^2}\n\nRight side: 1995^{2t}\n\nSince 1995 is a

positive real number, we can equate the exponents on both sides, but first, let's express √(1995) as 1995^{1/2}. So the

left side becomes 1995^{1/2} * 1995^{t^2} = 1995^{t^2 + 1/2}\n\nThe right side is 1995^{2t}\n\nTherefore, since the

bases are the same and positive, we can set the exponents equal:\n\nt^2 + 1/2 = 2t\n\nThat simplifies to t^2 - 2t + 1/2

= 0. Hmm, quadratic equation. Let's solve for t:\n\nt = [2

±

sqrt(4 - 2)] / 2 = [2

±

sqrt(2)] / 2 = 1

±

(sqrt(2)/2)\n\nSo t = 1

+ (√2)/2 or t = 1 - (√2)/2\n\nBut t was defined as log_{1995} x. Therefore, x = 1995^t. So the two solutions for x

are:\n\nx1 = 1995^{1 + (√2)/2} and x2 = 1995^{1 - (√2)/2}\n\nTherefore, the positive roots are x1 and x2 as above. The

problem asks for the product of the positive roots, which is x1 * x2.\n\nLet's compute that product:\n\nx1 * x2 =

1995^{1 + (√2)/2} * 1995^{1 - (√2)/2} = 1995^{(1 + (√2)/2 + 1 - (√2)/2)} = 1995^{2}\n\nBecause the exponents add up to

1 + 1 + (√2)/2 - (√2)/2 = 2. So the product is 1995 squared. Therefore, the product is 1995². Then we need to find the

last three digits of this product.\n\nSo the problem reduces to computing 1995² mod 1000. That is, find the last three

digits of 1995 squared.\n\nTo compute 1995² mod 1000, note that 1995 mod 1000 is 995. So 1995 = 1000 + 995.

Therefore, 1995² = (1000 + 995)² = 1000² + 2*1000*995 + 995².\n\nCompute each term modulo 1000:\n\n1000² mod

1000 = 0\n\n2*1000*995 mod 1000 = (2000*995) mod 1000. Since 2000 mod 1000 = 0, this term is also 0.\n\nSo

1995² mod 1000 ≡ 995² mod 1000.\n\nTherefore, compute 995² mod 1000.\n\nCompute 995²: 995*995. Let's compute

this.\n\nNote that 995 = 1000 - 5. So (1000 - 5)^2 = 1000² - 2*1000*5 + 5² = 1,000,000 - 10,000 + 25 =

990,025.\n\nTherefore, 995² = 990,025. Then 990,025 mod 1000 is 025. So the last three digits are 025.\n\nBut wait,

let me verify that. 990,025 divided by 1000 is 990 with a remainder of 25, so yes, the last three digits are 025.

Therefore, 1995² mod 1000 is 025.\n\nTherefore, the last three digits of the product of the positive roots are

025.\n\nBut wait, let me double-check. Maybe I made a mistake in calculating 995 squared. Let's do it step by

step:\n\n995 * 995:\n\nFirst, compute 1000 * 1000 = 1,000,000\n\nSubtract 5 * 1000 = 5,000\n\nSubtract 5 * 1000 =

5,000\n\nAdd 5 * 5 = 25\n\nWait, that's using the formula (a - b)^2 = a² - 2ab + b². So (1000 - 5)^2 = 1000² - 2*1000*5

+ 5² = 1,000,000 - 10,000 + 25 = 990,025. Correct. So 990,025. The last three digits are indeed 025. So 025.

Therefore, the answer should be 025. But leading zeros are usually not written, but since the question asks for the last

three digits, we need to include the leading zero. So the last three digits are 025.\n\nBut let me confirm once again.

Suppose I compute 995 * 995 directly:\n\n995 * 995:\n\nBreak it down as (1000 - 5)(1000 - 5) = 1000*1000 - 5*1000 -

5*1000 + 5*5 = 1,000,000 - 5,000 - 5,000 + 25 = 1,000,000 - 10,000 + 25 = 990,000 + 25 = 990,025. Yep, that's

correct.\n\nTherefore, 1995 squared is 1995*1995. But since 1995 = 2000 - 5, maybe compute (2000 - 5)^2 = 2000² -

2*2000*5 + 5² = 4,000,000 - 20,000 + 25 = 3,980,025. Wait, that can't be. Wait, 2000² is 4,000,000, minus 2*2000*5 =

20,000, so 4,000,000 - 20,000 = 3,980,000, plus 25 is 3,980,025. So 1995² is 3,980,025. Then the last three digits are

025. So that's correct. Therefore, the last three digits are 025.\n\nTherefore, the answer is 025, which in boxed form is

\\boxed{025}.\n\nBut wait, hold on. Let me check if the problem is indeed asking for the product of the positive roots. In

the original equation, we found two positive roots, x1 and x2, whose product is 1995². Therefore, their product is 1995²,

whose last three digits are 025, as we found. So the answer is 025. But in some contexts, when asked for the last

three digits, leading zeros are included, so 025 is correct. However, sometimes they might write it as 25, but since it's

three digits, 025 is appropriate. \n\nWait, but let me check once again the problem statement: \"Find the last three

digits of the product of the positive roots...\" So yes, the last three digits. Therefore, 025 is the correct answer.

Therefore, the final answer is \\boxed{025}.\n\n**Final Answer**\n\\boxed{025}

Progressive Reasoning

Verification

Multi-method Validation

Error Correction

Question

: Find the last three digits of the product of the positive roots of $\\sqrt{1995}x^{\\log_{1995}x}=x^2.$

Answer

: 25

A.3 The Pseudocode of the Pipeline of PIR

Algorithm 1 outlines the complete pipeline of our Perplexity-based Importance Refinement (PIR) framework. Our PIR framework implements a streamlined three-phase process for optimizing reasoning chains. First, Claude 3.7 Sonnet performs hierarchical decomposition of reasoning into coherent steps, which are classified using our hybrid system combining rule-based pattern matching and contextual analysis. Second, we quantify each functional step's importance by calculating its PIR value-the logarithmic ratio between the perplexity of generating the answer with and without that step using Qwen2.5-32B-Instruct. Finally, we apply pattern-specific selective pruning, preserving all progressive reasoning while removing only low-value functional components (verification, multimethod validation, and error correction) based on their PIR scores, thereby maintaining solution integrity while significantly reducing verbosity.

"# Mathematical Reasoning Analysis JSON Prompt\n\nAnalyze the provided mathematical reasoning solution by identifying the thinking patterns into a structured JSON format.\n\n## Pattern Definitions and Examples\n\n### 1. Progressive Reasoning\n\n**Definition**: The standard forward-moving problem-solving process that follows logical order.\n\n**Key Indicators**: \"Let's solve\", \"First/Then/Next\", \"Therefore\", \"We need to\", \"Given that\"\n\n**Example**:\n\n```\n\"Let's solve this step by step:\"\n\"First, we need to calculate n = 2^31 * 3^19\"\n\"So, we start by calculating this value.\"\n\"Now, let's compute the value of n.\"\n\"Therefore, let's find n = 2^31 * 3^19.\"\n\n```\n\n### 2. Verification\n\n**Definition**: Process of returning to check previous steps for accuracy.\n\n**Key Indicators**: \"Wait\", \"Let me check\", \"Let me verify\", \"Double-check\", \"Going back to\"\n\n**Example**:\n\n```\n\"Wait, let me check the multiplication:\"\n\"32 * 20 = 640 ✓ \"\n\"Let me verify this calculation:\"\n\"32 * 20 = 640 ✓ \"\n\"Just to be extra sure:\"\n\"32 * 20 = 640 ✓ \"\n\"We got 640, which looks right.\"\n\n```\n\n### 3. Multi-method Validation\n\n**Definition**: Using different methods or perspectives to verify a conclusion.\n\n**Key Indicators**: \"Alternatively\", \"Another way\", \"Let's try a different approach\", \"Using another method\", \"We can also verify\"\n\n**Example**:\n\n```\n\"Let's verify with n = 6:\"\n\"[calculation steps with n = 6]\"\n\"Alternatively, let's try n = 12:\"\n\"[calculation steps with n = 12]\"\n\"Let's also check n = 24:\"\n\"[calculation steps with n = 24]\"\n\"And verify with n = 36:\"\n\"[calculation steps with n = 36]\"\n\n```\n\n### 4. Error Correction Pattern\n\n**Definition**: Process of identifying and correcting mistakes in reasoning.\n\n**Key Indicators**: \"This is wrong\", \"The mistake was\", \"That's impossible\", \"This contradicts\", \"The error is\"\n\n**Example**:\n\n```\n\"This can't be right because the total is odd.\"\n\"The mistake is in assuming all pairs are non-divisors.\"\n\"Wait, I made a mistake in the pairing assumption.\"\n\"Let me correct this pairing error.\"\n\"The problem was in how we thought about pairs.\"\n\"We need to fix our understanding of pairs.\"\n\n```\n\n## Analysis Requirements\n\n1. **Group Related Sentences**: Identify related sentences that together form coherent sub-thinking processes.\n2. **Assign IDs**: Each sub-thinking process should have its own unique ID.\n3. **Structure Output as JSON**: Return your analysis in JSON format with:\n    - A dictionary with all sentences separated into sub-groups\n4. **Group Size Limitation**: Each sentence group should contain a reasonable number of sentences (typically 3-4) that represent one coherent step or thought. Avoid creating overly large groups that combine multiple distinct steps or thoughts.\n\n## Expected JSON Format\n\n```json\n{\n  \"sentence_groups\": {\n    \"group1\": [\"sentence1\", \"sentence2\", \"...\"],\n    \"group2\": [\"sentence1\", \"sentence2\", \"...\"],\n    \"...\": \"...\"\n  }}\n\n```\n\nFocus on identifying complete thinking processes rather than labeling individual sentences. Each group should represent a coherent step or thought in the mathematical reasoning. Do not delete or modify any sentences!!! Please just group the specified reasoning process."

B Experiments

B.1 Dataset Statistics

This section provides comprehensive statistics for the datasets used in our experiments before and after applying the PIR optimization framework at different thresholds. Table 5 presents the number of examples and total token counts for each dataset variant. The original datasets (S1, LIMO, and LIMO-V2) were derived from three different Large Reasoning Models: Gemini Flash Thinking, DeepSeek-R1, and QwQ, respectively. Each dataset was then processed using our PIR framework at varying optimization ratios (from 0.2 to 0.8), where higher values indicate more aggressive pruning of functional reasoning steps.

B.2 Main Results with Training Tokens

Table 6 presents a comprehensive evaluation of our PIR optimization approach across three challenging benchmarks: AIME (American Invitational Mathematics Examination), AMC (American Mathematics Competition), and GPQA Diamond. We include training token counts alongside accuracy, response length, and test-time efficiency to highlight the relationship between data efficiency and model performance. The baseline models-R1-Distill-Qwen-32B, Qwen2.5-32B-Instruct, and QWQ-establish performance references across the benchmarks. QWQ demonstrates superior accuracy but requires substantially longer responses, which impacts its test-time efficiency. Qwen2.532B-Instruct offers the highest efficiency due to its concise responses, albeit with lower accuracy. For our optimized models, we observe consistent patterns across all three dataset families (S1, LIMO, and LIMO-V2): S1 datasets: The PIR-optimized variant (S1-32B-P) achieves higher accuracy on AIME (+4.2%) and AMC (+2.2%) while using 4.4% fewer training tokens. The average response length decreases by 29% for AIME and 16% for AMC, resulting in efficiency improvements of 56% and 22%, respectively. LIMO datasets: The PIR-optimized model (LIMO-P) demonstrates accuracy improvements across all benchmarks (+6.6% on AIME, +1.9% on AMC, +4% on GPQA) while requiring 8.8% fewer training tokens. Response length reductions of 15% for AIME and 5% for AMC translate to efficiency gains of 32% and 7%, respectively. LIMO-V2 datasets: PIR optimization (LIMO-V2-P) achieves the consistent accuracy improvements (+4.9% on AIME, +2.2% on AMC, +3% on GPQA) while using 5.3% fewer training tokens. Response lengths decrease

"# Mathematical Reasoning Analysis JSON Prompt\n\nAnalyze the provided mathematical reasoning solution by categorizing the subprocess of the reasoning solution to thinking patterns into a structured JSON format.\n\n## Pattern Definitions and Examples\n\n### 1. Progressive Reasoning\n\n**Definition**: The standard forward-moving problemsolving process that follows logical order.\n\n**Key Indicators**: \"Let's solve\", \"First/Then/Next\", \"Therefore\", \"We need to\", \"Given that\"\n\n**Example**:\n\n```\n\"Let's solve this step by step:\"\n\"First, we need to calculate n = 2^31 * 3^19\"\n\"So, we start by calculating this value.\"\n\"Now, let's compute the value of n.\"\n\"Therefore, let's find n = 2^31 * 3^19.\"\n\n```\n\n### 2. Verification\n\n**Definition**: Process of returning to check previous steps for accuracy.\n\n**Key Indicators**: \"Wait\", \"Let me check\", \"Let me verify\", \"Double-check\", \"Going back to\"\n\n**Example**:\n\n```\n\"Wait, let me check the multiplication:\"\n\"32 * 20 = 640 ✓ \"\n\"Let me verify this calculation:\"\n\"32 * 20 = 640 ✓ \"\n\"Just to be extra sure:\"\n\"32 * 20 = 640 ✓ \"\n\"We got 640, which looks right.\"\n\n```\n\n### 3. Multi-method Validation\n\n**Definition**: Using different methods or perspectives to verify a conclusion.\n\n**Key Indicators**: \"Alternatively\", \"Another way\", \"Let's try a different approach\", \"Using another method\", \"We can also verify\"\n\n**Example**:\n\n```\n\"Let's verify with n = 6:\"\n\"[calculation steps with n = 6]\"\n\"Alternatively, let's try n = 12:\"\n\"[calculation steps with n = 12]\"\n\"Let's also check n = 24:\"\n\"[calculation steps with n = 24]\"\n\"And verify with n = 36:\"\n\"[calculation steps with n = 36]\"\n\n```\n\n### 4. Error Correction Pattern\n\n**Definition**: Process of identifying and correcting mistakes in reasoning.\n\n**Key Indicators**: \"This is wrong\", \"The mistake was\", \"That's impossible\", \"This contradicts\", \"The error is\"\n\n**Example**:\n\n```\n\"This can't be right because the total is odd.\"\n\"The mistake is in assuming all pairs are non-divisors.\"\n\"Wait, I made a mistake in the pairing assumption.\"\n\"Let me correct this pairing error.\"\n\"The problem was in how we thought about pairs.\"\n\"We need to fix our understanding of pairs.\"\n\n```\n\n## Analysis Requirements\n\n1. **Structure Output as JSON**: Return your analysis in JSON format with:\n- A dictionary mapping pattern types to lists of sub-thinking process IDs\n\n2. Each sentence group should be categorized as either Regular Reasoning OR one of the other patterns, but never both. Each sentence group should be categorized as one of the four patterns.\n3. **Pattern Identification**:\n    **Progressive Reasoning**: One continuous pattern from start to finish with logical steps\n    - **Verification**: Moments where solver checks previous work (reference which Regular Reasoning ID is being verified)\n    - **Multi-method Validation**: Separate different verification methods, each with its own ID (reference which Regular Reasoning ID is being checked with multi methods)\n    - **Error Correction Pattern**: Instances where mistakes are caught and fixed (reference which Regular Reasoning ID is being corrected)\n\n## Expected JSON Format\n\n```json\n{\n  \"patterns\": {\n    \"regular_reasoning\": [\"group1\", \"group3\", \"...\"],\n    \"backtracking_verification\": [\n      {\n        \"id\": \"group2\",\n        \"relates_to\": \"group1\"\n      }\n    ],\n    \"multiple_verification\": [\n      {\n        \"id\": \"group4\",\n \"relates_to\": \"group1\",\n        \"method\": \"Method Description\"\n      },\n      {\n        \"id\": \"group5\",\n \"relates_to\": \"group1\",\n        \"method\": \"Method Description\"\n      }\n    ],\n    \"error_correction\": [\n      {\n \"id\": \"group6\",\n        \"relates_to\": \"group1\"\n      }\n    ]\n  }\n}\n\n```\n\n"

consistently (12% for AIME, 7% for AMC, 13% for GPQA), yielding efficiency improvements of 23%, 10%, and 22%, respectively. These results demonstrate that our PIR framework effectively reduces training token requirements while simultaneously improving both accuracy and efficiency across diverse reasoning tasks. The consistent performance improvements across different model families validate the generalizability of our approach. Notably, the LIMO-V2-P model achieves state-of-the-art performance on all benchmarks while maintaining competitive efficiency, highlighting the effectiveness of optimizing reasoning chains by preserving essential progressive reasoning while removing less valuable functional steps.

B.3 Impact of Cognitive Reasoning Patterns

Our empirical analysis of cognitive reasoning patterns reveals distinct performance characteristics across the evaluated benchmarks. As shown in Figure 8, the four identified reasoning patterns exhibit different trade-offs between accuracy and computational efficiency. Progressive Reasoning (PR) provides a solid baseline, while PR+Error Correction demonstrates the most balanced performance improvement on several datasets, delivering notable accuracy gains with competitive or even improved efficiency. These findings validate our PIR framework's approach of quantitatively evaluating reasoning step importance to identify and selectively preserve high-value steps while pruning those with minimal contribution, creating optimized reasoning chains that balance accuracy gains with computational demands. The observed performance variations highlight that different reasoning patterns contribute differentially to model performance, with certain patterns delivering more substantial benefits in specific contexts. This suggests that selectively preserving the most valuable functional reasoning components while removing redundant steps can effectively optimize the efficiency-accuracy trade-off in reasoning chains.

Algorithm 1: Reasoning Chain Optimization via PIR

0.8

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1.2

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GPQA Diamond

Reasoning Pattern

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PR+MV

B.4 Case Study

To illustrate the effectiveness of our Perplexity-based Importance Refinement (PIR) framework, we present a detailed case study examining how models trained on refined reasoning chains differ in their inference behavior. Figure 9 provides a side-by-side comparison of responses generated by two models: one trained on the original LIMO dataset (left) and another trained on our PIRoptimized LIMO-P dataset (right), when presented with an identical mathematical problem. The model trained on original LIMO data exhibits characteristically verbose reasoning patterns inherited from its training data (left panel, 3,234 tokens). Despite reaching the correct answer, this model produces extensive verification steps, redundant calculations, and multiple self-checking procedures. The response includes numerous instances of recalculation, approach reassessment, and duplicate validations-reflecting the verbose nature of the LRM-distilled training data it was fine-tuned on. In striking contrast, the model trained on PIR-optimized data produces a significantly more concise response while maintaining solution accuracy (right panel, 1,612 tokens). This model has learned to focus on essential progressive reasoning pathways while minimizing unnecessary verification steps. The 50% reduction in output token count demonstrates that models trained on PIR-refined data effectively internalize more efficient reasoning strategies without compromising problem-solving

capabilities. Qualitative analysis of both responses reveals that while the token count differs substantially, both models arrive at the correct solution (204 minutes). However, the PIR-trained model achieves this with greater efficiency, focusing on core mathematical operations and direct solution paths rather than extensive self-verification. This confirms that models inherit the reasoning patterns present in their training data-when trained on strategically refined examples, they learn to generate more streamlined responses. This case study highlights a critical advantage of our approach: by optimizing training data through selective removal of low-importance functional steps (based on our PIR metric), we influence downstream model behavior to favor efficiency without sacrificing effectiveness. The PIR-trained model demonstrates that the extensive verification processes prevalent in LRM reasoning-while mimicking human problem-solving-can be substantially reduced in practical applications without compromising solution quality.

LIMO Okay, let's see. I need to solve this problem where Aya is walking different distances at different speeds, and we have to find out how long her walk takes when she walks at s + 1/2 km/h, including the time t she spends in the coffee shop. <Omitted>n\nWait, 11.25 is equal to 22.5 / 2? Wait, no. Wait, 11.25 is 45/4. Hmm, better to keep it as fractions perhaps.\n\nAlternatively, maybe work with decimals here. Let me see.\n\nSo, quadratic equation: s² + 2s - 11.25 = 0\n\nLet me multiply both sides by 4 to eliminate decimals:\n\n4s² + 8s - 45 = 0\n\nNow, quadratic equation: 4s² + 8s - 45 = 0\n\nUsing quadratic formula:\n\ns = [ -b ± sqrt(b² - 4ac) ]/(2a)\n\nHere, a=4, b=8, c=-45\n\nDiscriminant D = b² -4ac = (8)^2 - 4*4*(-45) = 64 + 720 = 784\n\nsqrt(784)=28\n\nTherefore,\n\ns = [ -8 ± 28 ] / (2*4) = (-8 ± 28)/8\n\nTaking the positive root since speed can't be negative:\n\ns = (-8 +28)/8 = 20/8 = 2.5 km/h\n\nSo, s = 2.5 km/h\n\nNow, let's verify this value. Let me check if substituting s=2.5 into the equations gives consistent results.\n\nFirst equation: 9/s + t/60 =4\n\n9/2.5 = 3.6\n\nSo 3.6 + t/60 =4 → t/60 =0.4 → t=24 minutes\n\nSecond equation: 9/(s+2) + t/60 =2.4\n\ns+2=4.5 km/h\n\n9/4.5 =2 hours\n\nSo 2 + t/60 =2.4 → t/60=0.4 → t=24 minutes, which matches.\n\nSo, s=2.5 km/h and t=24 minutes.\n\nNow, the question asks: If Aya walks at s + 1/2 km/h, find the number of minutes the walk takes her, including the t minutes spent in the coffee shop.\n\nFirst, compute s + 1/2:\n\ns =2.5 km/h, so s + 0.5 = 3 km/h\n\nWalking speed is 3 km/h. The distance is still 9 km, so walking time is 9 /3 = 3 hours.\n\nIncluding the coffee time t=24 minutes, total time is 3 hours +24 minutes.\n\nConvert 3 hours to minutes: 3*60=180 minutes, plus 24 minutes gives 204 minutes.\n\nWait,  but  the  question  says  \"the walk  takes  her\",  does  that  include  the  coffee  time  or  not?  Wait,  the problem says: \"Find the number of minutes the walk takes her, including the  t  minutes  spent  in  the  coffee  shop.\"\n\nAh, so the total time is the walking time plus t minutes. So yes, exactly as above.\n\nSo total time is 3 hours walking +24 minutes coffee, which is 3*60 +24 = 180 +24=204 minutes.\n\nWait, but let me double-check my steps to make sure I didn't make  any  miscalculations.\n\nFirst,  solving  for  s  and  t:\n\nFrom  the  first scenario:\n\nWalking at 2.5 km/h, walking time is 9/2.5 = 3.6 hours. Then total time is 3.6 + t/60 =4, so t/60=0.4 → t=24 minutes. Correct.\n\nSecond scenario:  walking  at  2.5+2=4.5  km/h,  walking  time  9/4.5=2  hours.  Total time 2 + t/60 =2.4 → t=24 minutes. Correct.\n\nThen, walking at s + 1/2 =2.5 +0.5=3 km/h. Walking time is 9/3=3 hours. Total time is 3 hours +24 minutes,  which  is  3*60  +24=180+24=204  minutes.\n\nYes,  that  seems correct.\n\nBut just to be thorough, let me check if there was another way someone  might  misinterpret  the  problem.\n\nFor  example,  maybe  the problem says \"the walk takes her 4 hours\" including the coffee shop time, so  perhaps  the  walking  time  is  4  hours  minus  t  minutes.  Wait,  actually, that's exactly how we modeled it. The walking time is distance over speed, then add  the coffee time to get the total time. So  our model  is correct.\n\nAlternatively, maybe someone might think that the \"walk takes her\"  refers  only  to  the  walking  time,  but  the  problem  specifically  says \"including t minutes spent in the coffee shop\". So the total time is walking time plus t. Therefore, the answer is indeed 204 minutes.\n\nWait, but let me confirm once more. Let me check units again. The problem states that in the first scenario, when she walks at s km/h, the walk takes her 4 hours, including  t  minutes.  So  total  time  is  4  hours,  which  is  240  minutes.  So walking time plus t minutes equals 240 minutes. So if walking time is 3.6 hours,  which  is  216  minutes,  then  t=24  minutes,  which  adds  up  to  240. Correct.\n\nSimilarly, second scenario: 2 hours 24 minutes is 144 minutes. Walking time at 4.5 km/h is 2 hours, which is 120 minutes, plus 24 minutes gives 144. Correct.\n\nTherefore, when walking at 3 km/h, walking time is 3 hours (180 minutes), plus 24 minutes, totaling 204 minutes. Therefore, the answer is 204 minutes.\n\nBut wait, the problem says \"the number of minutes\",  so  204  minutes  is  the  answer.  But  let  me  check  if  I  did everything correctly. Let me re-calculate the quadratic solution.\n\nOriginal equations:\n\nEquation  1:  9/s  +  t/60  =4\n\nEquation  2:  9/(s+2)  +  t/60 =2.4\n\nSubtracting equation 2 from equation 1:\n\n9/s - 9/(s+2)=1.6\n\nWhich  led  to:\n\n18/(s(s+2))=1.6\n\nMultiply  both  sides  by s(s+2):\n\n18=1.6*s(s+2)\n\nDivide both sides by 1.6:\n\ns(s+2)=18/1.6=11.25\n\nThus, s² +2s -11.25=0\n\nMultiply by 4:\n\n4s² +8s -45=0\n\nQuadratic formula:\n\ns = [-8 ± sqrt(64 + 720)] /8 = [-8 ± sqrt(784)] /8 = [-8 ± 28]/8\n\nPositive solution: (20)/8=2.5. Correct.\n\nTherefore,  all  steps  are  correct.  Thus,  the  answer  is  204 minutes.\n\nWait,  but  the  problem  says  \"put  your  final  answer  within \\boxed{}\". Since the question asks for the number of minutes, so boxed 204.\n\ <omitted> Token:3234

LIMO-P Okay, let's see. Aya goes for a 9-kilometer walk every morning and then stops at a coffee shop. The problem is about figuring out how long her entire outing takes when she walks at a speed of s plus a half kilometers per hour, including the time t spent in the coffee shop. Hmm. <Omitted> \n\nWait, so her original speed s is  2.5  km/h?  That  seems  slow  for  walking,  but  maybe it's  okay.  Let's  check.\n\nIf  s  is  2.5  km/h,  then walking 9 km would take 9 / 2.5 = 3.6 hours, which is 3  hours  and  36  minutes.  Then,  adding  t  minutes,  the total  time  is  4  hours.  So  3  hours  36  minutes  plus  t minutes  equals  4  hours.  Therefore,  t  must  be  24 minutes.\n\nSimilarly, when she walks at s + 2 = 4.5 km/h, the walking time is 9 / 4.5 = 2 hours. Adding t = 24 minutes gives 2 hours + 24 minutes = 2 hours 24 minutes,  which  matches  the  second  scenario.  Okay,  so that  checks  out.\n\nNow, the question is asking, if she walks at s + 1/2 km/h, so that's 2.5 + 0.5 = 3 km/h, how many minutes does the entire outing take, including t minutes in the coffee shop?\n\nSo, walking at 3 km/h, the walking time T_w is 9 / 3 = 3 hours. Converting that to minutes, 3 hours is 180 minutes. Adding t, which is 24 minutes, so total time is 180 + 24 = 204 minutes.\n\nWait,  but  let  me  confirm  again. The problem says \"the walk takes her 4 hours, including t minutes...\" So the total time is 4 hours, which  is  240  minutes.  If  walking  time  is  3  hours  36 minutes  (which  is  216  minutes)  plus  t  minutes  equals 240,  then  t  is  indeed  24  minutes.  Similarly,  in  the second  case,  walking  time  is  2  hours  (120  minutes) plus  24  minutes  is  144  minutes,  which  is  2  hours  24 minutes. So that's consistent.\n\nTherefore, when walking at 3 km/h, walking time is 3 hours (180 minutes) plus 24 minutes in the coffee shop, totaling 204 minutes. So the answer should be 204 minutes. \n\nBut  just  to  double-check,  let  me  go  through  the steps again.\n\nOriginal speed s = 2.5 km/h, so walking 9 km takes 9/2.5 = 3.6 hours. 0.6 hours is 36 minutes, so 3 hours 36 minutes. Adding t minutes gives 4 hours. So 3h36m + t = 4h => t = 4h - 3h36m = 24m. Correct.\n\nAt  s  +  2  =  4.5  km/h,  walking  time  is  2 hours.  2h  +  24m  =  2h24m.  Correct.\n\nNow,  s  +  1/2  = 2.5  +  0.5  =  3  km/h.  Walking  time  is  9/3  =  3  hours. Total time is 3h + 24m = 3h24m, which is 3*60 + 24 = 180  +  24  =  204  minutes.  \n\nYes,  that  seems  right. Therefore, the number of minutes is 204.\n\n**Final Answer**\nThe number of minutes the walk takes her, including the \\( t \\) minutes spent in the coffee shop, is \\boxed{204}." Progressive Reasoning Token:1612

Verification

Multi-method Validation

Error Correction

Question : "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, including $t$ minutes spent in the coffee shop. Suppose Aya walks at $s+\\frac{1}{2}$ kilometers per hour. Find the number of minutes the walk takes her, including the $t$ minutes spent in the coffee shop.' Answer : 204

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Peer reviews:
comments: SUMMARY:
This paper introduces a novel framework called PIR (Perplexity-based Importance Refinement) to optimize the reasoning chains of large language models for efficient and effective test-time scaling. The PIR framework classifies reasoning steps into four distinct modes: progressive reasoning (essential solution development path) and three types of functional elements: verification, multi-method validation, and error correction. PIR quantitatively evaluates the importance of each reasoning step based on its impact on answer prediction confidence, using the perplexity metric. It selectively prunes low-importance functional steps while preserving progressive reasoning components, creating optimized training data.

STRENGTHS AND WEAKNESSES:
- The quality of this paper is fair since the data selection method is quite straightforward. 
- The paper is well-structured and clear.
- The paper contributes to the ongoing research on efficient reasoning in foundation models, while this area has many similar works such as AdaptThink and ThinkLess.
- This work is based on LIMO and incrementally prunes some extra reasoning steps, which is not very original.

QUESTIONS:
-  The paper focuses primarily on mathematical and science reasoning tasks.  Can your method generalize to other domains like instruction following?
- Will shorten the reasoning steps hurt the model's exploration ability? 
- The paper relies on perplexity as the primary metric for evaluating the importance of reasoning steps. Are there alternative metrics that could be used to assess the semantic contribution of steps and potentially provide more nuanced insights into the reasoning process?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-19

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an innovative framework named PIR (Perplexity-based Importance Refinement) to optimize the reasoning process of large language models (LLMs). The core idea is to decompose verbose reasoning chains into “progressive reasoning” (the core problem-solving path) and “functional steps” (e.g., verification, correction). Using a perplexity-based metric to quantify the importance of each functional step, PIR systematically identifies and removes low-importance steps that contribute little to the final answer while preserving the core progressive logic. Experimental results across several challenging benchmarks (AIME, AMC, GPQA) demonstrate that PIR-optimized models not only significantly reduce token usage during inference but also improve answer accuracy , thus achieving a better trade-off between efficiency and effectiveness.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The highlight of this work is its elegant and principled methodology. Instead of pruning all reasoning steps indiscriminately like some previous methods, it innovatively distinguishes between “progressive” and “functional” reasoning steps. This categorization aligns well with human cognitive processes when solving complex problems and targets the redundancy issue in modern LLMs. The use of PIR—a perplexity-based metric measuring answer confidence change—as a step-importance signal provides a clear and interpretable criterion for pruning, making the optimization both theoretically grounded and practically reasonable.\n2. The experimental design is thorough. The authors validate the method on datasets from multiple sources (LIMO, S1K) and across challenging benchmarks (AIME, GPQA Diamond), demonstrating broad applicability. Comparisons with related methods such as SPIRIT clearly highlight PIR’s superiority, particularly its strategy of preserving the progressive reasoning path. Additional ablation studies on pruning ratios and model scales reinforce the scalability of the proposed method.\n3. The paper is well written, with each part detailed enough for understanding. Besides, the authors give clear explanations for their motivation, methodology, and experiments. Therefore, this paper is easy to reproduce and implement.\n\nWeaknesses:\n1. The initial step of the PIR pipeline—reasoning step segmentation and classification—relies heavily on Claude 3.7 Sonnet. This introduces an external dependency, making the framework’s success partially contingent on the capabilities of a strong proprietary model. Two concerns arise: (a) reproducibility and generality—how would it perform if a weaker open-source model were used instead? and (b) conceptual coherence—the framework’s effectiveness may partially derive from the classifier’s intelligence rather than the PIR logic itself.\n2. The evaluation is primarily focused on mathematical and scientific reasoning (AIME, AMC, GPQA), where functional steps like verification and correction are clearer and more common. However, it remains unclear how the framework performs on other types of reasoning tasks such as commonsense reasoning, logic puzzles, code generation\n3. The current method uses a predefined fixed ratio to decide how many functional steps to prune. As shown in Figure 3, the optimal pruning ratio varies across benchmarks, indicating that a fixed global setting may not be ideal. A more adaptive strategy—e.g., based on problem difficulty, initial reasoning chain length, or dynamic importance thresholds—could potentially yield better performance.\n\nQUESTIONS:\n1. The classification of reasoning steps is foundational to the PIR framework. Could the authors comment on the feasibility of replacing Claude 3.7 Sonnet with a smaller, more accessible open-source model (e.g., LLaMA-3 8B or Qwen-2.5-7B)? What minimum classification accuracy do you believe is required for PIR to remain effective?\n2.The PIR score quantifies a step’s importance based on the perplexity change when it’s removed. Is there a risk that a low-importance functional step (e.g., a verification) may actually be critical for enabling a downstream progressive step? Has the framework considered the possibility that pruning such steps could disrupt local reasoning dependencies?\n3. Given that the optimal pruning ratio varies across tasks and model sizes, have the authors considered designing a dynamic or adaptive pruning mechanism?\n\nLIMITATIONS:\nYes, the content is discussed in Conlusion.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nReasoning datasets for training language models often contain long reasoning chains, which may include unnecessary logical steps that do not make meaningful contributions to deriving the final answer. In this work, the authors tackle this challenge by proposing Perplexity-based Importance Refinement (PIR). Intuitively, they break down a reasoning chain into multiple reasoning steps and classify them into four categories: progressive reasoning, verification, multi-method validation, and error correction. They deem the first category, progressive reasoning, as an essential type of reasoning that cannot be excluded from the chain, whereas the remaining three, called functional elements, become the candidates for elimination. Specifically, using a specific language model (Qwen 2.5 32B Instruct in this work), they compute and compare the perplexities of the answer sequence generation with and without each functional element to derive their Perplexity-based Importance Refinement (PIR). The authors exclude a pre-specified portion (a hyperparameter) of functional elements with the lowest PIR values in each reasoning chain. Empirically, the authors demonstrate that the proposed approach can meaningfully reduce the number of output tokens while retaining higher accuracies on AIME, AMC, and GPQA Diamond.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- Decomposing each reasoning chain and filtering out reasoning steps based on how the exclusion of each step affects the perplexity of answer generation can be a viable approach for making the reasoning chains more compact with less decrease in accuracy.\n- The presented empirical results demonstrate that this approach may improve both the accuracy and efficiency of the fine-tuned model on the reasoning benchmarks.\n- The authors provide various empirical analyses, which provide helpful insights. This spans from how different portions of functional elements removed affect the accuracy and efficiency (in terms of number of tokens) of the resulting, fine-tuned models to how the proposed method works with models of different sizes and different benchmarks.\n\nWeaknesses\n- One of my biggest concerns regarding this work is about the decision to use the pruning ratio as a hyperparameter. Intuitively, it should be possible that depending on the sample, all the functional steps are essential or most of the functional steps are not needed i.e., I'm not sure if a constant pruning ratio would be an ideal choice of hyperparameter to apply to various samples (even though each dataset is generated by a single teacher model). It may be useful for things like the analysis in Figure 3, but it doesn't sound like a generally usable hyperparameter in practice. For instance, if the source datasets are generated differently or have already gone through similar processes of removing unnecessary reasoning steps, the current pruning ratios would not be directly usable. Given the definition of PIR (Eq.(1)), pruning all the functional steps with negative PIRs or PIRs smaller than some threshold value makes more sense to me.\n- On a related topic, I'm also primarily concerned about the use of different pruning ratios for different target benchmarks. Although I couldn't find the exact pruning ratios used for the main empirical results, Table 6 (combined with Table 5) provides some clues; it looks that the pruning ratios for the different target benchmarks (AIME, AMC, and GPQA Diamond) differ to some degree. This could indicate the possibility of overfitting of this parameter to each target benchmark (not sure by how much, though).\n- For the empirical validation of how each reasoning step is classified by the LLM, the authors employed human annotators to check the accuracy of the classifications and concluded that 93.4% of the classification results had unanimous agreement with the 4 annotators, which sounds great. On the other hand, I wonder how well each reasoning chain is decomposed into *cohesive* reasoning steps (units), because each reasoning step becomes a target of the exclusion and mis-decomposition may lead to having incomplete logical reasoning steps. This kind of granularity aspect may be more challenging to evaluate, but at least some qualitative justifications would be great to have.\n- This may be a minor thing, but the charts in Figure 1 have various scales of the y-axis, which could be somewhat misleading.\n\nQUESTIONS:\n- What do the empirical results (especially accuracies) look like if you use an absolute value of 0 (zero) as a threshold for PIR (i.e., pruning the functional steps with negative PIRs)? Could you demonstrate that with some constant thresholding of PIR values themselves, you can achieve similar empirical results?\n\nBesides this question, please take a look at the weaknesses I wrote above. I am open to updating my score based on the author's response.\n\nLIMITATIONS:\nYes, I think the current limitations section is fair.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
The paper introduces PIR, a method for refining LLM reasoning traces by classifying every step, scoring its importance with a perplexity delta, and pruning only low-value functional steps. Training on these leaner traces supports LLMs that answer more accurately and concisely. There is a sound experimental design across 3 contemporary reasoning benchmarks and a sample of model sizes, demonstrating a practical argument to cost-effective test-time scaling.

STRENGTHS AND WEAKNESSES:
Strengths

- Principled pruning metric grounded in answer-perplexity, not heuristics.

- Clear argument, supported by a well designed empirical analysis.

Weaknesses

- Lack of a qualitative characterization of the analysis. For which reasoning types the approach performs best and why?

- This can help to make sense of the limits of using perplexity as a proxy.

QUESTIONS:
My main questions are on the characterization of the qualitative aspects of the framework.

- For which reasoning types the approach performs best and why?

- In which scenarios does it have a negative impact?

- I believe it is essential to provide a systematic characterization of the qualitative points (i.e. non-anecdotal).

- What is the formal selection criteria (inclusion and exclusion) for the base models? How does it affect the universality of the claims.

- What is the formal selection criteria for the benchmarks?

- How do they limit the universality of the claims?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-05