LIMOPro: Reasoning Refinement for Efficient and Effective Test-time Scaling
Authors: Yang Xiao
Jessie Wang
Ruifeng Yuan
Chunpu Xu
Kaishuai Xu
Wenjie Li
Pengfei Liu
Venue: NeurIPS 2025 poster ()
Keywords: LLM
reasoning
Test-time Scaling
Primary area: applications

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
Yang Xiao Jiashuo Wang'
Ruifeng Yuan'
Chunpu Xu Pengfei 2
Kaishuai Xu'
Wenjie Li
The Hong Kong Polytechnic University
Shanghai Jiao Tong University SIII
yang-alas aalan w.lan.xiaaaaae.ooo...o.o.aaaaaae xiao@connect polyu.
adu. .kk cajwangecomp. polyu. edu
Abstract
Large language models (LLMs) have demonstrated remarkable reasoning capa- bilities 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, alter- native 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-basee Importance Re- finement), principled framework that quantitatively evaluates the importance of each reasoning step based on its impact on answer prediction confidence. PIR sys- tematically 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 proper- ties, generating more concise reasoning chains while achieving improved accuracy (+0.9% to +6.6%) with significantly reduced token usage (-3% to -4 1)) across challenging reasoning benchmarks (AIME, AMC, and GPQA Diamond). Our approach demonstrates strong generalizability across different model sizes, data sources, and token budgets, offering 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 the LIMOPro.
1 Introduction
Large language models (LLMMs) have demonstrated remarkable capabilities in complex reasoning tasks through of-nn-of-ttought (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-RI [10], OpenAl ol [12], and QwQ [29] represent the state of the art in this paradigm, producing reasoning chains that lead accurate solutions.
However, this approach faces 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, 4J. In typical mathematical scenario, an LRM might solve a problem by establishing an initial solution path, verifying calculations, identifying
CCoresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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26-3
60.0
65.0
67.5
Aug. Responne Tokenn
12500
12497
-15%
6.0
+32%
5.30e-05
50.0
+12%
$3.3
Efficiancy
AIME
60.0
11500
11000
20.0
55.0
10500
4.53e-05
10.0
52.3
UMO
L
UMO
UMO
LIMO
LIMO-P
8k
12k
16k
24k
Budget
96.0
5800
1:85
Rgg. Respore Tolens
5600
1:7 1775-04
AMC
94.0
+2%
5500
5516
-5%
Effciincy
1.75
80.0
70.0
5300
1.67e-04
92.0
165
5100
LIMO-P
LIMO
92.0
5000
1.60
74.0
7200
1.10 16-1
GPOA Diamond
73.0
Avg. Response Tokene
1.03
71.0
+6%
7100
Efficlncy
9%
60.0
70.0
69.0
7000
0.55
9.36e-05
68.0
6969
67.2
LIMO-P
30.0
LIMO
66.0
LIO
LIMO-P
6900
UMD
LMM-P
0.85
LIMO
LIMO-P
12k
24k
Token Budget (k)
Figure Our PIR framework (implemented LIMO-P) optimizes the efficiency-effectivvees tradeoff in LLM reasoning across AIME, AMC, and GPQA Diamond benchmarks: it consistently enhances accuracy while concurrently reducing response tokens, thus improving computational efficiency, demonstrating that selectively pruning low-importance functional steps produces more concise, faster, and more accurate reasoning chains.
errors, revising the approach, and ultimately confirming the final answer. This thorough process generates lengthy reasoning chains with redundant 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 considerable obstacle implementing eenoonnngennancee LLMs in practical applications where timely, precise responses are essential.
To address this challenge, introduce PIR (Perplexity-bart -based Importance Refinement), novel framework that systematically refines reasoning chains to optimize the ance. 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, multi- method validation, and error correction. (2) Through comprehensive analysis across diverse problem domains, 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-pressee soning while selectively optimizing functional elements-maine while significantly improving computational efficiency. (3) Building on (1) and (2), we propose the PIR metric, which quantitatively measures each functional step's contribution the final solution evaluation by comparing answer perplexity with and without specific steps. This perplexity-basee 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.
effectivvness bal-
progressive rea- the core problem-solving logic

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By applying our refinement framework to datasets distilled from different foundation models (LIMO [37] from Deepseek- -1 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 both effectiveness and efficiency, achieving up to 71% efficiency improvement.
The contributions of our work include: 1. novel perplexity-basee 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 critical gap in current approaches to LLM reasoning enhancement, offering 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 caasooinn--caable 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 verifi- cation processes, multiple solution approaches, and error corrections-thal mirror human problem- solving but significantly increase computational overhead without proportionally enhancing solution accuracy. This section presents Perplexity-bac -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 dataset D containing question-reasoning-ame E Q represents a reasoning problem, E R is the reasoning chain, and a E A is the answer. We define a reasoning chain r as a sequence of intermediate steps {s1, 82, ... 8x), where each step 8; represents a logical deduction that bridges the gap between the question and the final answer.
triplets (q, (g,r,a), where
Our goal is to refine each reasoning chain into optimized version r' such that: (1) The answer accuracy is preserved: f(g,r') = f(g,r) = a; (2) The token length reduced: |r'l < |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, that characterize problem-solving processes: (1) Progressive Reasoning, characterized by forward-chaining inference that follows a deductive logical progres- sion from premises 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 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
3

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1.Raw Reasoning
Classify Step Types
3. Calaculete
A.Refinement
A
Shep
Step [Progresatre]
Step 1 [Prograsiive
Step (Progrossiv)
Sellp Coalbe neciion
Set Up Cqaalia
Sertip Fryulion strouatioon
[Verifiction]
Verts Appraich Conditons
Appraach
Kep!
Step 3
Step 5 [Pogressive)
Step 3 (Progrestive)
Stop [Progrossive]
[Mattimethod)
Stup (Nutt-mnthoo)]
(Mutt deeth
Conten Cheshical Armdy
Our fimm Dngp Anatysla
Removed
Step 5
Stop 5 [Prggessive)
Step (Progreatve))
Stop 5 (Progrosshe)
Step
Step Correction]
Step [Etrer Correction)
Soep Conecciin)
Camte higpul
Connot bilegal Skiap
Removed
Figure PIR framework pipeline for reasoning optimization: raw reasoning segmented into logical steps, step is classified into reasoning patterns, PIR value is calculated to quantify step importance, and low-value functional steps are filtered while preserving progressive reasoning, resulting in more efficient reasoning chains.
compromising solution integrity. Details about the four reasoning patterns and corresponding cases can be found in Appendix A.
PIR: Perplexity-Bel -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 particular reasoning step:
PPL (R') {x;)) PPL
(2: = log
Where (R) and P (xxe) represent perplexities calculated by model 0 with and without step i:
(P) = exx
log po (a, p0 aes)
m
PR (xi}) = exp
log pa (a, (1 1 Tit
(3)
m77
Here, m represents the number of answer tokens, 0j is the j-th answer token, Tis represents all reasoning steps, and F1 11--1,44410 1:m represents steps except step i. The perplexity calculations are performed using model 0. A higher PIR value indicates greater step importance: when critical reasoning step eliminated, the model becomes significantly less confident generating the correct answer, lacking essential information for solution derivation, resulting in higher perplexity and thus higher PIR score.
2.3 Analysis and Optimization Reasoning Chains
Our approach first employs a hierarchical decomposition process where reasoning chains are seg- mented 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, rule-based pattern matching component identifies steps containing characteristic linguistic markers (such as "Let me check" for verification or "I made 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.

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Table 1: Dataset Statistics. For each dataset, we report the data source, number of samples, total token count, and the distribution of tokens across four distinct reasoning patterns.
Progressive Reasoning
Multi-method Validation
Error Correction
Dataset
Source
Samples
Tokens
Verification
SIK
Gemini
1,000
4,509,505
71.4%
9.2%
13.9%
5.5%
LIMO
DeepSeek-R1 QwQ
817
5.144,004
59.7%
11.8%
10.9%
17.6%
LIMO-V2
800
8,866,950
64,3%
9.8%
12.7%
13.2%
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 '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 targeted pruning approach that selectively identifies and removes low-importance functional steps while maintaining data quality and preserving reasoning integrity. This process shown Figure Unlike approaches that might indiscriminately compress reasoning chains, our method specifically targets functional steps (verification, multi metti--ethod 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 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 mobllng-oolving 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 I in the appendix demonstrates this whole process. In our experiments, we apply Awen2.5-32B-Inae 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 generaliz- ability 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.I.
Benchmark Datasets To rigorously assess our methodology, we utilized three representivate reasoning-intensivee 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 metries: Reasoning Effectiveness and Efficiency To evaluate reasoning effec- tiveness (ACC), we employ the pass@1 accuracy our primary performance indicator. For each
5

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Table 2: Experimental results comparing baseline models with their PIR-optimized variants (-P) across reasoning benchmarks. Metries include accuracy (ACC), token length (TOK), and efficiency (EFF).
AIME
AMC
GPQA Diamond
Model
ACC TOK 4
EFFT
ACC1
TOK
EFF
ACC
TOK
EFF 1
Qwen2.5-32B-
15.8
954
166E-04
67.2
737
9.11E-04
47.0
517
9.08E-04
Instruct
RI-Distill- Qwen-32B
69.2
9,311
7.43B-05
94.4
5,561
1.70E-04
64.7
5,634
1.15E-04
QwQ
81.7
12,234
6.68F-05
97.8
7,350
1.33F-04
70.2
7,483
9.38F-05
S1-32B
37.9
6,646
5.71E-05
80.9
4,542
1.78E-04
60.7
4,172
1.46E-04
S1-32B-P
42.1,4.2
4,716.29%
8.92E-05.5es .56%
83.1,2a
3,809.16%
2.18E-04,225 .2%%
61.6,ng
2,472.41%
2.49E-04.71s
LIMO
56.7
12,497
4.53E-05
91.9
5,516
1.67E-04
67.2
7.173
9.36E-05
LIMO-P
63.3,6.
10,588 15%
5.98F-05
.21%
93.8
5,235 55%
1.79E-004
47%
71.2,4
6,969.35
1.02E-04,999
LIMO-V2
66.3
13,896
4.77E-05
94.4
6,843
1.38E-04
70.2
8,035
8.74E-05
LIMO-V2-P
71.2,4. 12,163.125
5.86E-05,
23%
96.6.22
6,348, 7
1.52E-04
10%
74.2,3
6,968.139;
1.07E-04,22%
problem in benchmark, we sample eight responses from the model and calculate ACC under the Zero-shot Chain-of- Thouuht (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] systematically assess solution correctness across all solutions, with each sampling conducted at a temperature setting 0.7 to balance deterministic reasoning with exploration of solution paths. Building on our effec- tiveness 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 mod- els (LIMO, LIMO-V2 V2, and $1-32B) that were trained on the original unmodified datasets using Qwen2.5-32BB Instructt [35]. These baseline models were obtained from their official Hugging Face repositories.' To test our PIR framework, we fine-tune the same Qwen2.5-32B. 2-225---2B-nsstrrcc base model on the pruned datasets: LIMO-P, LIMO-V2-P, and SIK-P, using identical training scripts as described 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 frame- work with other competitive reasoning optimization methods. First, we establish S1-32B as our primary baseline trained on the original unmodified SIK 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 S1-SPIRIT, which applies non non-discriminative pruning approach for all reasoming steps based solely on perplexity values. We apply this method the SIK dataset to get pruned dataset, and fine-tuning Dwen2.5-332BIInsee 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."
'bttps//huggingface.om
"W specifically chose SIK dataset for these comparisons because competing methods like SPIRIT require perplexity calculations for all sentences, which would be computationally prohibitive for the significantly larger LIMO or LIMO-V V2 datasets, resulting in excessive computational costs and processing time.

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Table 3: Experimental results comparing PIR(S1-32B-P) with different optimization approaches.
Model/ Method
Training Tokens
AIME
AMC
GPQA Diamond
ACC
TOK 4
f
ACC
TOK 4
RFFT
ACC
TOK 4
EFF f
S1-32B
4,511=+06
37,9
6,646
5.71E-05
80.9
4,542
1.78E-04
60.7
4,172
1.46E-04
S1-PROMPT
4.51F#066 36.7
8,013
4.58F-05
72.5
3,724
1.95E-04
58.0
2,853
2.03F-04
SI-SPIRIT
4.32B+06 37.1
4,906
7.56E-05
81.3
3,517
2.31E-04
60.1
2.818
2.13E-04
SI-RULE
4.31E+06 36.7
4.807
7.63E-05
81.3
3,654
2.22E-04
58.1
3,837
1,51E-04
S1-32B-P
4.31F=+06 ,2,1 4,716
8.92F:-05,
56%%
83.1
3,809
16%
2,18E-044
-22%
61.6,89
2,472
2.49F--4 .11%
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-accuue remarkable improvements AIME with 4.2 percentage point accuracy increase alongside 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.
trade-offs across all model families. S1-32B-P shows
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, 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 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) consis- tently outperforms all alternatives across benchmarks, achieving significant improvements over the bascline $1-32B 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 selectively pruning only functional steps while preserving all progressive reasoning components proves superior 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 hypothesis that progressive reasoning steps constitute the essential solution backbone and should remain intact.
3.3
Analysis
3.3.1
Generalizability Test-Time Scaling
Test-time scaling represents a critical dimension of L.M reasoning deployment, where more computa- tional resources are allocated during inference. To evaluate whether our PIR optimization framework maintains its effectiveness across varying inference- token budgets, we conducted experiments measuring accuracy as function of token budget constraints. Figure I presents the test-time scaling curves across three benchmarks of LIMO and LIMO-P. The results demonstrate that PIR- PPR-optimieed models consistently outperform their non- optimizee 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 maintain performance advantages highlights the robustness of optimization methodology to different deployment scenarios.
7

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AMC
1.2
1.2
1.0
1.0
TOK
EFF
0.9
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 3: Impact pruning ratio on model performance. This figure displays relative performance metrics (normalized to baseline) across different pruning ratios for AIME and AMC benchmarks. The horizontal dashed line represents the baseline performance (ratio=0).
Improvement
Improvement
0.30
AIME
AIME
AIME
0.25
Relative
9.9%
4.2%
1.2%
11.0%
-1.6%
1.9%
3.3%
15.3%
8.6%
6.1%
4.5%
32.0%
0.20
AMC Dataset
0.15
0.10
AMC
AMC
-0.05
-0.10
3B
140
148
Model Size
Model Size
Model Size
Figure 4: Impact of PIR refinement across model sizes and benchmarks. Heatmaps show relative percentage changes between models trained with pruned versus original datasets. Blue indicates improvement: higher accuracy, shorter response length, or better efficiency.
3.3.2 Impact of ratios
We investigated the impact 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, 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 valuable reasoning elements removed, even when categorized as functional steps. This pattern validates our PIR framework's '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 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 (1 1.8% accuracy im- provement, 15.3% response length reduction, and 32.09 22.0% 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.

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Related Work
4.1 Test-Time Scaling of LLMs
Test-time scaling [12, 26, 29, 10] enhances LLM reasoning by increasing inference- time computa- tion. Approaches include non- non-training methods, which optimize existing model inference strategies, and training-basee based methods, which modify model parameters. Non-training techniques encompass Best-of-N sampling [18, 27], majority voting [31,3] 3], and tree search [36, 30] for exploring multiple reasoning paths. Training-base -based approaches divide into supervised fine-tuning (SFT) and reinforce- ment 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-RI using GRPO and OpenAl's ol and QwQ enabling autonomous development of reasoning chains that adaptively allocate computation based 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 LLMs face challenges with compu- tational overhead and verbosity. inference time, various optimization approaches have emerged without requiring parameter updates. Length budgeting techniques like Token- Token-Budget- 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 controller assess task difficulty. Model switching approaches such 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 www-iportannce 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 predeter- mined 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 vali- dation, and error correction), ensuring core reasoning integrity and avoiding the risk of removing critical solution elements.
5 Conclusion
Contributions This paper introduces PIR (Perplexity- Importance Refinement), 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 problem- solving 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 metries 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 closed- source models. Addressing these limitations could further advance efficient reasoning frameworks for real-world applications.

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Acknowledgements
We would like to thank all reviewers for their insightful comments and suggestions to help improve the paper. This work was partially supported by the Research Grants Council of Hong Kong (GRF No. 15209724). This work was also partially funded by the National Natural Science Foundation of China (62476168) and SII.
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A Reasoning Refinement:
Reasoning Optimization Framework
A.1 Phrases Used for Pattern Matching
In our rule-based pattern matching system for reasoning step classification, identify four distinct cognitive reasoning patterns characterized by specific linguistic markers. Table 4 presents these patterns alongside their characteristic phrases. Figure demonstrates one sample from SIK containing the four patterns.
Progressive Reasoning constitutes the critical path to solution derivation, characterized by forward- chaining 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/ThennNext," and "Therefore."
Verification represents metacognitive monitoring processes where previous calculations are system- atically validated for accuracy. This pattern 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 different approach," potentially introducing redundancy.
Error Correction embodies self-regulatory mechanism through which logical inconsistencies computational errors are identified and remediated. This pattern is captured by phrases acknowledging mistakes, such "This is wrong, "The mistake was," and "This contradicts."
Table 4: Cognitive Reasoning Patterns and Associated Phrases
Cognitive Reasoning Patterns
Phrases
"Let's solve"
Firrs//hen/NNxt"
Progressive Reasoning
"Therefore"
"We need to"
"Given that"
"Wait"
LLet me check"
Verification
LLet me verify"
"Double-check"
"Going back to"
"Alermatively""
"Another way"
Multi-method Validation
"eet's try a different approach"
"Using another method"
"We can also verify"
"This is wrong"
"The mistake was"
Error Correction
"That's 's impossible"
"This contradicts"
"The error is"
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.
13

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Question: Find the last three digits the product of the positive roots of (1llg
Answer: 25
Soluuioo: Okay, so 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. IninThe equation is
Progressive Reasoning
Verification
given v(1995) (1995_((199))))) = x². That looks bit complicated with the Ingarithm in the exponent. Maybe I can take logarithms
both sides to simplify it?
Multi-method Validation Error Correction
Or perhaps rewrite the equation using properties of exponents and logarithms. Inin
Let me recall that log_ b a = (In a)(in b), so log_(19990000000 (1995) x is equal to (In x)((In 1995).
Also, x raised to (1995) x can be rewritten using exponentials. Wait, maybe I can express x"(log_(1 (1995) x) as
e*((in x) (log_(1995) x)). But that might not be helpful. Let me think differently www.eettlyynnnnttmtaatttlll since the equation has x the exponent and also squared, maybe substituting log_o (1995) x would help. Let me try that. Let's set
log_ (199()))) x. which means that 1995*L Then substituting back into the aquation:mmmm((995) (1985"1)()) = (1985-4/2n/nSimmiiffy both sides:nnnnLeft side: (1995) 19955(12)nnn side: 1895422)ynnnSince 1995 is pasitive real number, we can equate the exponents on both sides, but first, let's express (1995)) as 1995X1/2). So the ieft side becomes 1995*(1/2) (995"((2)) = (995" + 1/2 right side is 1995* 1985122)ninTheerfor.. since the bases are the same and positive, we can set the exponents equal:ninn + 1/2 That simplifies to 12 21 1/2
1
= 0. Hmm, quadratic equation. Let's solve for t:lint = [2 sqri(4- 2 [2 sqr((2]] 2 = 1 a (sqrt(2)2))nnnnoo t=
log (-222 or t 1 (12)/2nnnBBut was defined log_ (1995) Therefore, x = 1995%t. So the two solutions for
are:)nimx1 = 1995 + (12)2) and x2 19955(1 (+2)2//ninThereforr, the positive roots are x1 and x2 as above. The problem asks for the product of the positive roots, which is x1 * X2nnnees''s compute that product/nimxx xx
the exponents add up to ++1 1 + (v2)2- - (v2y2 2. So the product is 1995 squared. Therefore, the product is 1995?. Then we need to find the last three digits this product IninSo the problem reduces to computing 1995² mod 1000. That is, find the last three digits of 1995 squared. In compute 1995' mod 1000, note that 1995 mod 1000 is 995, So 1995 = 1000 995. Therefore, 1995 = (1000 995)= = 1000 2" 10000095 9952 Ine:olnnnCo each term modulo 1000:hhm1000* mod 1000 = 0000 mod 1000 = (2000-995) mod 1000. Since 2000 mad 1000 = O, this term is also 0.IninSo
19955(1 + (+2)/2) * 19955(1- - (v2)2)= = 1995"((1 + (+2)2 + 1- 1--22222) = 12855
995a; 995. Let's compute this.InlnNote that 995 = 1000 So (1000- 5y2 = 1000 2'1000'5 5 1,000,000 - 10,000 25
1995' mod 1000 995' mod 1000. minhhrrforee compute 995' mod 1000..ninCompplloooooo
990,025 www.enneeeereeereee.e.em 995*= 990,025. Then 990,025 mod 1000 025. the last three digits are 025.IninBut wait, me verify that. 990,025 divided by 1000 is 990 with remainder of 25, so yes, the last three digits are 025. Therefore, 1995 mod 1000 is 025. T2h.inf. the last three digits of the product of the positive roots are but wait, let me double-check. Maybe I made a mistake in calculating 995 squared. Let's do it step by
step:Inin995 Fiisstt compute 1000 1000 = 1, 1,000,000000 500.0000minSubbracd 5 1000 = 5, 5.0000ninSubtract 1000 = 5, 10000mnnAdd 5 = 25intnWait, that's using the formula (a - 2)^2 = a² 2ab + b2. So (1000- - 5)^2 = 1000 100005
5" = 1,00000000000000 10,000 25 990,023 .2,.00550000 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, need to include the leading zero. So the last three digits are 025 But let me confirm once again.
Suppose| I compute 995 995 directly/Inin995 995:nninrreak it down as (1000- - 50000 - 5) = 1000'1000- 100000000-51000
5'1000 + 5'5 = 1,000,000- - 5,000 - 5,000 + 25 = 1,000,000 10,000 + 25 990,000 + 25 = 990,025 Y that's
conrecttIninTherefooo 2'2000'5 + = 4, 4,000,000 20,000 25 3, 3,980,025 Wait, that can't be. Wait, 2000 4,000,000, minus 2'2000'5
1995 squared Is 1995"1995 1995. But since 1995 2000 5, maybe compute (2000- - 5/2 2000
20,000, so 4,000,0000 20,000 3,980,000 000,000,,00000000000000000, plus 25 3,980,025 So 1995' 3,980,022 025.. Then the last three digits are 025. So that's correct. Therefore, the last three digits are 025. wwinTherefor.. the answer is 025, which in boxed form is Abooxed(025,,IninBBB IninBut 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 1995.. 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 correct. However, sometimes they might write it as 25, but since it's three digits, 025 appropriate. IninWait, but me check once again the problem statement: I'Find the 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 (onxeed(225).nmn A
Figure 5: A case where one training sample contains the four patterns.
A.3 The Pseudocode of the Pipeline of PIR
Algorithm outlines the complete pipeline of our Perplexity- 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 the logarithmic ratio between the perplexity of generating the answer with and without that step using --2.2..5-3B-Instttue Finally, we apply pattern-specific selective pruning, preserving all progressive reasoning while removing only low-value functional components (verification, multi- method validation, and error correction) based on their PIR scores, thereby maintaining solution integrity while significantly reducing verbosity.
14

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"# Mathemalical Reasoning Analysis JSON PromptatiAnnalye the provided mathemnaticl reasoning solution by identifying the thinking patterns into structured JSON tormatininik Pattem Definitions and Examplesinintttt 1. Progressive measoninginin"""efnitti
The standard forward-moving problem-solving process that follows logical orderinin""Key Indicators"" V'Lat's solvel", "FFiss//henNNext., ("Theretore".. I'We need tol". I'Given
"an""""""""""lllee "Innn""" "inl"Lets solve this step by Fiirtt" we need to calculate n = 2*31 3119rin we Thereff,,,,,,,,
start by calculating this value, N'n'"www "9..M 2. pnrficationiiin""eeiiiii
compute the Process of returning to check previous steps for accuracy\nnnn"Key Indicators""; ("Waitt, VLet check"", L'Let me verify!', "DDouble-heck"," I'Going back
n 2*31*
inin "Example""""i"" """"i"""""""""""""""""""""""" "ant"Wait let me check the Attipplicationn"""""""3
20 640 LWnil"Ltt verify this
calculation:"in" 32" 20 640 //"" Just to extra sure: 3nn"3 20 640 Wee got 640, which looks right""nin" "aani"""i"""""nnn""""""""""""""" "Inlnsstt" 3. Mult-mmtthod Validationinin" "Defiinti"""""iittttnn" Using different methods perspectives verify
conclusion.nni" Key Indicators"" l'Alternatively", l'Another way!". I'Let's try a different approach"", I'Using another method!", 'WWe also "nan""nnnn"""""iii "aarmp"l"""iii""""""""""" "'ninn"""""""""" verify with
" steps with steps with n = "n]"""n also check n= 24:""ni"[alcclltton" steps "6in "Iiinss Error Correction
(l"nl"Alternativel let's try n 36: calcullatioon steps
with 24j]" verify Process of idenlifying and correcting mistakes in reasoning Inin" Indicatore" \TTis is wrong'", ''hee mistake waslf, 'Thha's impossiblel", Y'hhs contradicts!", can't be right because the total is odd.""nnl The mistake is in assuming all pairs are non non-divisors.""al "nii"Wa,,, I made mistake pairing assumplion Minl""n
dntteminin"Definitten
error inl" "Example" "n''hs
correct pairing error I"vil"Th problem was how we thought about pairs. Wee need to fix our understanding of pairs.""iin ""nin""nnnntt"" Analysis Requirementsirinf. GGroup Related Sentences" " Identify related sentences that together form coherent sub-thinking processes In2. "*Asslgn IDs"": Each sub-thinking process should have Its own unique ID.in3 "Structure Output as JSON"": Return your analysis in JSON
format with:nn A dictionary with all sentences separated into sub-groupsiin. "Group Size Limitation"": Each sentence
group should contain reasonable number of sentences (typically 3-4) that represent coherent step thought. Avoid creating overly large groups that combine multiple distinct steps or thoughts inin Expected JSON Formatinin" "jomattnn""ssonnnnnn sesentence groupal". (tn 1"group11": ("'setencee1| l'sentence2"", V'...''].An \'group2"":
["sentence '"sentenoe2"' V'...1].in V... V...''n }jvvin" "IninFocus on identifying complete thinking processes rather labeling individual sentences. reasoning. Do not delete or modily any sentences!!! Please just group the specified reasoning process."
group should represent coherent step thought in the mathematical
Figure 6: The prompt to segment coherent sub-thinking sentences into cohesive reasoning steps.
B
Experiments
B.1
Dataset Statistics
This section provides comprehensive statistics for the datasets used 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, 1, LIMO, and LIMO-V2) were derived from three different Large Reasoning Models: Gemini Flash Thinking, DeepSeek-RI. 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 comprehensive evaluation of our PIR optimization approach across three chal- lenging benchmarks: AIME (American Invitational Mathematics Examination), AMC (American Mathematics Competition), and GPQA Diamond. We include training token counts alongside accu- racy, response length, and test-time efficiency to highlight the relationship between data efficiency and model performance. The baseline RiaielsssRR--iiill-------eee QwQ-establish performance references across the benchmarks. QWQ demonstrates superior accuracy but requires substantially longer responses, which impacts its test-time efficiency. Qwen2.5- 32B-Instrrct 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 FRooopiieeeed 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-PP 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
Owen2.5-32BBInsttee
and
15

---

"# Mathematical Reasoning Analysis JSON PromptiilAnaalyee the provided mathemnatial reasoning solution by categarizing the subprocess of the reasoning solution to thinking patterns into structured JSON format. Ininf Pattern Definitions and Examplesinnntttt 1. Progressive "easoningh"""""ffniiiii solving process that follows logical order Indicators" L'Lef's solvel", /'Firs//hennNeet"" \'Therefore"", V'We need tol", I'Given thaff" "Examl""""ll "Inn"Let' solve this step by step:" FFirst, we need to calculate n = 2*31* """"",,"",, we start by calculating this I'n'".."
The standard fonward-moving problem-
of Theerfrre,rre,,e let's find 2*31 * Process of returning to check previous steps for bocuracy.lnnn"K6
compute the
3"19."" ""lnn"n""" 2. Verificatoninin Indicators": :: W"Walt", VLet checki", L'Lel me verify!", i"Double-chcck", !'Gaing back
20 = 640 /V'nl"Lat me verify this calculation!"""" 32" 20 640 //"" Just to extra sure: 33332 20 640 Wee got 640, which looks right""nin" "Inlnstt 3. Mult-mmtthod Validationinin" "Defiintii"""""""""t" Using different methods perspectives verify conclusion.tii Key Indicators"" l'Alternatively". l'Another wayl". I'Let's try a different approach"" I'Using another "xxmmp"" "ammip""""nnn"""'a""" "'n""Le's verify with
"xxmmll""" "In" Wait" let me check the multipiiatioon"""""3
methodi", 'WWe also Ajlleelllalllllllyy let's try n "calclcccattitio steps with n "2"''s also check n 24:""nn [calcullliiiio steps with n 24]'"miiAAnd verify with [callculani steps with "win "Inininn 4. Error Correction "enffminn""eeiiittio Process of identifying and correcting mistakes in reasoning Key Indicators": \"This is wrongi". ''he
""alccccctlaaioon steps with
waslf, V'That's impossiblei", I'This contradicts?", I' error isl"nin" "Example ""nn"""""""""""""""""""""""""""" can't be right because the total is odd.""ni' The mistake is in assuming all pairs are non www.iios..."i I made mistake pairing assumption "inl"l
pairing error/"vni" I"vi!"hh problem was how we thought about WWe need to fix our understanding of pairs. I'nnn" "Ininsi# Analysis Requirementsrrint. "Structure Output as JSON"": Return your analysis in JSON format with:'n- dictionary mepping pattern types to lists sub-thinkking process iDeinin2. Each sentence group should be calegorized as either Regular Reasoning OR one the other patterns, but never both. Each sentence group should be categorized as one of the four patlemss In3. "Pattem Idtentificatonn //n "Progressive Reasoning" : One continuous pattern from start finish with logical stepsin "Verification" " Moments where solver checks previous work (reference which Regular Reasoning ID is being verifiedln "Mult-melhod Validation"* Separate different verification methods, each with its ID (reference which Regular Reasoning ID being checked with mutti methods)in "EEror Correction Pattem"": Instances where mistakes are caught and fixed (reference which Regular Reasoning ID being corrected//ninstt Expected JSON Formatinvn" "ssnnn((" Vppatterss
correct
(in Vrregular_eeseee reasoning!" [''roou11" V'grou3ii, V...1']..n bbackrrakiigg_eeffictttooo verificationn:: [in (In Vidl":
""group2"..nn Vrreates foll: I"groupfi"n jin 1/n Ymmltiple verification!: [In
('n Vid!" V'grop44" In
I'relatas_ tot": I'group
\'method"" l'Method Description!'nn JiIn (In Vid!": Vgrrou5i".,n In
Ireelates tot": I"rooup("..n I"method!". l'Method Description""n J.. J,in Veerorr correction!". [in (In
Mot"" ""group6"". l'relates_ tol:: \'group11'n Jin Jin Minjainn"''iin
Figure 7: The prompt to categorize the steps into reasoning patterns
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. model achieves state-of-tte the-art performance on all benchmarks while maintaining competitive efficiency, highlighting the effecttivness 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 bascline, while PR+Error Correction demonstrates the most balanced per- formance 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 ailcienne-accurrae trade-off in reasoning chains.
16

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Algorithm 1: Reasoning Chain Optimization via PIR
Input: solafion; original reasoning chain, onswer:: solution answer, ratte: pruning threshuld, O: evaluation model
Output: solutionape' optimized reasoning chain
I/ Step 1: Segment and classify reasoning chain steps
classified fied steps + oocaiisaotgggstsssssotttttooooo functional steps Patter/Byaateemm/eessss// fiedsstf steps, [Verification, Multi-method, Frror correction]]
steps + SegmentintoLogiccSSStppo solution, 6);
8):
1/ Step 2: (P) <- CaleudalePerlexxiyysssttat
Calculate baseline ine perplexity with complete reasoning
answer, 0);
1/ Step 3: Evaluate importance of each functional step
foreach step: c functional_ster steps do
solution_: t- RemoveStep(sooation step:):
PPLa R (step.)) +- Calculateeemllexityyee oereeplxrityeaaatioon....ssa (PP.(RR.(ann)
step, PIR log PPLa(R)
end
// Step 4: Selectively prune low-importance steps by pattern type solution c- solution;
foreach pattern (Verification, Multi-method,, Error correction] do
pattern_stes
FitteryPPatermi functional steps, pattern); ratio);
threshold + CalculatePruning BariiaaeePnnnnnnnnneeeiiee pattseonddptteetsstttes
steps to prope + SeleciL PIRStsss (eRRSeesspppterrr ppottern steps, threshold); solution RemoveSteps( solution steps_ prune);
end
return solutionaps
Diamond
1.2
Teaponing Pattern
1.1
ArC
Aetter
GARAEC
ACC
Balto
Sotter ACC
FELU
Off. wo.........
@ WWV
3.0
Wore ALs
Balto OFF
Mose ACC
SUres OFF
Werts ACC
XCC bater
0.8
0.9
1.0
1.1
1.2
0.8
0,9
1.0
1.1
1.2
0.8
0.9
1.0
1.1
1.2
Reletive Text Time efficency
Relative Text Time Chfickency
Reletive Thak Time Cficiecyy
Figure 8: Impact of reasoning patterns on model performance across different benchmarks. Each subplot displays the relative accuracy (y-axis) versus relative test time efficiency (X-axis) compared the Progressive Reasoning (PR) baseline. PR represents the model trained with only progressive reasoning steps. PR+Verification (PR+V) is trained with the dataset that includes progressive reasoning and verification steps. PR+Error Correction (PR+EC) stands for the model trained with progressive reasoning and eттor correction steps. PR+Multi method Validation (PR+MV) is trained with progressive reasoning and
validation steps.
B.4 Case Study
To illustrate the effectiveness of our Perplexity-bane -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 optimized 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-refllee
PIR-
the verbose nature of the distilled training data it was fine-tuned on. In striking contrast, the model trained on PlR-ootimized 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 output token count demonstrates that models trained on PIR-refined data effectively internalize more efficient reasoning strategies without compromising problem-solving
17

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Table 5: Statistics of Dataset Variants after PIR Optimization
Source
Data
Numbers
Tokens
S1
1000
4509505
S1-0.2
1000
4440447
S1-0.3
1000
4390349
$1-0.4
1000
4307878
Gemini
$1-0.5
1000
4173726
$1-0.6
1000
4136261
S1-0.7
1000
4064908
S1-0.8
1000
3998891
LIMO
817
5144004
LIMO-0.2
817
4971633
LIMO-0.3
817
4865402
LIMO-0.4
817
4724104
DeepSeek-R
LIMO-0.5
817
4542101
LIMO-0.6
817
4459545
LIMO-0.7
817
4342583
LIMO-0.8
817
4220217
LIMO-V2
800
8866950
LIMO-V2-0.2
800
8488260
LIMO-V2
800
8398975
LIMO-V2 42-0.4
800
8292603
QwQ
LIMO- V2-0.5
800
8161582
LIMO-V2-0.6
800
8082063
V2-0.7
800
7980495
LIMO-V2-0.8
800
7877110
Table 6: Performance comparison across different models on AIME, AMC, and GPQA Diamond benchmarks.
Model
AIME
AMC
GFQA Diamond
Training Tokens
Ace
Aug. Respetec Tokens
Effciencc
Triining Tokzns
Ace
Avp. Resparse Tokenss
Efficiency
Taat Time
Thining Tokens
Ace
Avg. Response Tokens
Efficiency
Qwen2.5. 5.922.
Instrct
NIA
15.8
954
1.00:04
N/A
67.2
737
9.116/04
N/A
47.0
517
9.08E-04
Ri-Distill-
Qwea-328
N/A
60.2
9,311
743E-05
N/A
94.4
5.561
1770-44
N/A
64.7
5.634
1.154404
QWQ
NA
81.7
12:234
6,88
NAA
97.8
7,350
1330.04
N/A
70,2
7,483
9.388 03
St
S1-32D
451E=06
37.9
571E-005
4.51E-06
4.542
1.78--0
4.51E+06
60.7
4,172
1466--0
$1-32B-P
4316:00
42.1
4,716
8.22-00
4.318-006
83.1
3.809
2.18E-04
4.34E406
61.6
2,472
2.491-04
LIMO
5:14C+06
56.7
12,097
4.530
14D-06
91.9
5,516
1.670-04
1,1+0006
67.2
7,173
9.36E 05
LIMO-P
1669+066
63.3
10.888
S.985-05
4.78E-06
93.8
5.235
1.790-01
4.72E+06
71.2
6,989
1.026-04
LIMO V2
65.3
4.775-05
8.87-006
8.87F406
8,03
LIMO
R.10CH06
71.2
12,163
S.63I
8.40E=06
96.6
6,348
1.520
8,49E+06
74,2
6,958
1,070-0
capabilities. Qualitative analysis of both responses reveals that while the token count differs sub- stantially, both models arrive the correct solution (204 minutes). However, the PIR-trained model mathematical operations and direct solution
achieves this with greater efficiency, focusing paths rather than extensive verffceriiication. This confirms that models inherit the reasoning patterns present in their training data-when trained strategically refined examples, they learn generate more streamlined responses. This case study highlights a critical advantage of our approach: by optimizing training data through selective removal of cnwiimprttance functional steps (based 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 mrolem--olvingg- be substantially reduced in practical applications without compromising solution quality.
18

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Question:
stops coffee shop When constant klometers per hour, the walk takes hours, including $15 minutss spent the soffee shop, When Ss+ aloometers per hour, the walk takes her hours and 24 minutes, inclucing SS minutas spont in coffee shop. Supposa Ayo walks at S=+1thac(11225 kilommttrs per hour. Find the numbar minutes the walk takes her, induting the S15 minutes speni in the uoffee shop."
"Every morning Aya
$85 kil-metteeerr-ee
Answer: 204
LIMO
Token
LIMO-P
Okay, let's see. reed to solve this protiem where aya waiting diferent cistances at difterent speeds, and we have find out now lona her walk lakes when she waks at 1/2 kmh, induding the time she the 11.25 momtted-mmiWatt 11.25
lokay, let's see. Aya goes for a 9-kiloneter walk every I Inorning and then stops at a coffee shop. The problen
ni Fracions]
is about figuring out hou lang her entire outing takes when she walks
22,5 Wat. keep
speed of plus half kiloreters I coffee r
Hmm, perhaps perhsss Wnaseeeatelyyy maybe work with decimals here. Let me sea ininSo quadralic equation: st 11 25 Onrini multily both sides by to aliminata docimalsininess + Bs 45 = UnninNaw quadratic equation: 4g + Bs 45 n U'ninUsing quadratic tnmulaiiniss b-b = sortib)
hour, including shop. HEE, <Orättec> ValrMait, so her original speed is 2.5 kn/h? That seens slow for wa walking, but raybe it's okay. Let's MnAII
time spent
is 2.5 kw/h, then uatking 9 ka would take 9 / 2.5 = 3.6 hours, which is I hours and 16 minutes. Then, adding t ninutes, the tatal time is 4 hours. So 3 hours 36 minutes plus minutes equals minutes. AninSinilarly, when she walks hours, Adding hours 24
dac) W2afminttere, and, br8, am pn-45SrnnDDheemmiinn D 5 4ac- - 18)2 -1 (454 = 64 - 720 = If (214) (-8 228)8mi Inggalivetnins (48 +28y8 lets verily value. Let me check subeituting 25 kmuhirieNNau
(840m///7/-222//emaaalo inins [-8 28
the poeliiee rool speed cantt bel kmitleisa,
hours. Therefore, t nust be 24
equations
ka/h, the walking time is / ninutes
glves consistent resuts Finiirst equation: ai U8B 44 5. 1. Bunnno 3.8 P80 =4 - (80 =0.4 1-24 minulesyinnSecand equation: (Ble++2) + U80 -2.0mimp--24.. 5 kmrhinin88.. S hourlennn a 500 =241 180-044 4 1024 minutas, which metches winSo s225 kmh and m224
24 minutes gives hours ainutes, which ratches the second scenario. Okay, so that checks out, Anvnnow, the question is asking, ifl she wolks at s + 1/2 kr/h, 5o that's s 2.5 + 0.5 = 31 ka/h, how nany ninutes does the entire outing take," I including minutes in the coffee shop7 walking I 1et 3 ka/h, the walking time Ix is 9 3 3 heurs. I Converting that to minutes, 3 hours Adding which
minunos. the question asks: Aye walss at + 1/2 kmhh and the rumber of minules the walk takes her including the minutes spent the coffee shop, intniistt compute 1/2/nins -25 km/h distance
0.5 3 walking!
kmthieetWWkknng speed
minutes. - confimt again.
minutes, is 3 hours +24 minutes WntCone 3 hours to minutes: 3 3'80-18 180 minutes plus 24 minutes gives 204 innuWas but the quection says the walk takas hert, does that includa the coffee tmo or not? Wat. the probdem says: I'Find the number of minules the walk lakes her, induding the minutss spent in the coffee thop:/ninAn so the total tmme the walking time plus minutes, So exactly above Sn total sime i hours walking minutes bninWWil.
hours intennduding collee
ninutes, so total tine ninutes, ninutes.inhMMatt I The probler says lhehe walk takes her 4 hours, I inc ancluding t ninutes...." so the total time is 4 hours, which is 240 minutes. If walking time is 3 hours 36. minutes twhich is 216 minutes) plus t minutes equals 1248, then t is indend 24 minutes. Sin larly, in the second
24
let
minutes coffee, which 3150 +24 180 -24-204 me doutle-check my steps
walking time is hours (126 minutes) minutes, which
did't. make any miscakulaaions, IntaFirst, soling for and ElninFrrmm trst SismaraannnWiiikingg
hours 24, tualking at 3 kn/h, walking time onnnheeeefore, when, Ininutes) plus 24 ninutes in the coffee shop, toteling, I 1284 minutes. 5o the answer should be 204 ninutes. I KnnnBut just to double-check, let me do through the I steps agoin. Antnorigtal speed s 2.5 ka/h, so uulkinng 9 ka takes 9/2.5 3.6 hours. 0.6 hours is 36 minutes, 14 hours. So 3h36n
pptus ninutes Ininutes. 5o that's consistent, Sonisstett.nnunhereeeee
hours. Then total tmm is 3.6 + 350 24, sa 15030 1924 minutes. Commectinn CommatiiinsSoor IninSooon scenorio: welking at 2,5+244.5 5 knh,, walking lime B1e 5-2 hours. Total time + v80 =2.4 1-24 minutes. Correct Thenhen, walking at 5 1/2 #2.5 +0.553 kmmh Warking time is 93-3 hours. Total time 3 hours -24, minutes, which 3160 120-080-2--- cornect
2.5 knnh walking time is 9/2.5
3 hours (188)
minules menees tht s there
thoruugh someone might misinterpret the aroblem.rinfe example, maybe the proziom says th walk takes har hours"" indlding the caffee shop time. so perhaps the waking lime
way
Adding minutes 4h 3h36n 24n.
Correct.Anynk s hours. 2h 24n 2h24n. Correct.Antnnnn
4.5 kn/h, walking time
hours minus mirutes. Wail, astually. I that's exactly how we modeled it. The walking time is distance over sooed, than add the coffen time to get the Intal sma. So our modill is iisstttatilt maybe someono might tink that the twak takes herl" refers only to the waking time, bui the problem specificlly time waking
5 1/2
2.5 + 0.5 = 3 kn/h. Malking time is 9/3 3 hours. Total time is 3h 248 = 3h24r, which is 3468 + 24 188 + 24 = 284 minutes. AnkoYes, that seers right. Therefore, the nunber of ninutes is 284. Vntne Answer=<inThe number of rinutes the walk takes her, including the W shop, is (20xed
(ynouuiigg
spent the coffee shopt" So the indoad 204 minutes WnnWaat
time plus Thersfore, confirm once more. Lei me check in the first scenario, when she walks a kmih the walk takes her 4 hours. including t minutes. So lotal time is hours, which is 240 minutes. Soll walling aima plus minutas equals 240 minutes. So if walking tmm s 3.61 hours, which is 216 minutes, then 1-24 mindee, which adds up to 240.
ninutes spent the coffeel
again. The problem states that
Progressive Reasoning
Ver ficitiin
Correct ininsiillany, second scenani: 2 hours 24 minutes is 144 minutes.
Mult-method Validation Ence Comection
Warkin time 4.5 kmih hours, whic 120 minutes, waking kmm, walking
Iinutes
gives 144, Comeett minhherroor, hours (180 minules). plus minutes, tolaing minutes. Therefore. the answer is minutes IninBut wat, the prozlem says the number minusesi" so 204 minutes evarything correctly Lot ma re-calcujato the quacrati solutionn/nnnreee Onnniginal SqualiorshiiEEuuudee
answar. lat ma chack
1: Bis USO -dvinnEuusion 2. 91(=+2) 180 -2 4:iinuurrracnnnn equation 2 from equation 12min8s
B1s+2)=1 BintWhich led to 20) sides
both side by
2jcmi18811111 1''dtee2/iniivvde 11 Blunnster 2)-181 25etnTrus. 45-Driintaadtaal 1-8 1 s49(7844)
both
-11 2500mmiittlll by sptif44 720
[-8 228jsetaPoatbe solution: (20y8=2.5 I Conrect Therefore, all steps are correct Thus, the anewer 204 i minutes-inmmaan but the problem says l'put your thaa answer within (hhssegge Since the quastion seks for the number of minutess so bosed 2244m Homilleeds
Figure 9: Comparison of reasoning chains between model LIMO (left, 3,234 tokens) and PIR- optimized LIMO-P (right, 1,612 tokens) for the same mathematical problem. The model trained with Plt-optimized dataset maintains essential progressive reasoning while eliminating redundant verification steps, resulting in a 50% reduction in token count without sacrificing solution accuracy.
19

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Reviews:
[{"review_id": "p2IczPsjJS", "reviewer": "Reviewer_v7KQ", "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.", "questions": "My main questions are on the characterization of the qualitative aspects of the framework.\n\n- For which reasoning types the approach performs best and why?\n\n- In which scenarios does it have a negative impact?\n\n- I believe it is essential to provide a systematic characterization of the qualitative points (i.e. non-anecdotal).\n\n- What is the formal selection criteria (inclusion and exclusion) for the base models? How does it affect the universality of the claims.\n\n- What is the formal selection criteria for the benchmarks?\n\n- How do they limit the universality of the claims?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- Principled pruning metric grounded in answer-perplexity, not heuristics.\n\n- Clear argument, supported by a well designed empirical analysis.\n\nWeaknesses\n\n- Lack of a qualitative characterization of the analysis. For which reasoning types the approach performs best and why?\n\n- This can help to make sense of the limits of using perplexity as a proxy.", "was_revised": true, "final_justification": "Author's responses were taken into account. The original scores were already on the positive side (as these were only clarification questions - not binding on the scores).", "rebuttal": "We sincerely thank the reviewer for the thorough review and constructive feedback. We appreciate the valuable insights that have helped strengthen our contribution. The concerns raised are primarily due to presentation issues and misunderstandings, which we address systematically below.\n\n---\n\n## **Q1: Lack of Qualitative Characterization**\n\n> For which reasoning types does the approach perform best and why?\n> \n\n**Thank you for raising this important question about qualitative characterization across reasoning types.**\n\nBased on our experimental analysis in Section B.3 (Appendix), **Progressive Reasoning + Error Correction (PR+EC) and PR+Verification (PR+V) achieve better accuracy performance compared to PR+Multi-method Validation (PR+MV)**. The observed performance variations highlight that different reasoning types contribute differentially to model performance, with certain patterns delivering more substantial benefits in specific contexts. Hence, we believe that Multi-method Validation is relatively less important compared to the other two functional types.\n\nWe also conducted ablation experiments by removing only specific functional reasoning types. The results show that removing only Multi-method Validation (MV) achieves higher accuracy compared to removing only Error Correction (EC) or Verification (V). This further validates our finding that Multi-method Validation steps are indeed less critical for maintaining reasoning performance, confirming our approach's effectiveness in selectively targeting the most redundant functional components.\n\n### Ablation Study Results\n\n|  | AIME | AMC | GPQA Diamond |\n| --- | --- | --- | --- |\n| LIMO | 56.7 | 91.9 | 67.2 |\n| LIMO (No Verification) | 59.1 | 92.5 | 70.7 |\n| LIMO (No Multi-method Validation) | **60.0** | **92.8** | **72.2** |\n| LIMO (No Error Correction) | 57.5 | 92.1 | 71.7 |\n\n---\n\n## **Q2: Negative Impact Scenarios**\n\n> In which scenarios does it have a negative impact?\n> \n\nWe thank the reviewer for this direct question about negative impacts. Our comprehensive evaluation across diverse reasoning tasks provides valuable insights into when PIR optimization may have limited effects.\n\n**Empirical Evidence from Cross-Domain Evaluation**: Based on our systematic evaluation across different reasoning domains, we can analyze the performance patterns:\n\n### Performance Analysis\n\n| Model | Commonsense QA | LogiQA | MBPP |\n| --- | --- | --- | --- |\n| **LIMO** | 81.33 | 77.41 | 73.2 |\n| **LIMO-P** | 82.00 | 78.04 | 72.6 |\n| **LIMO-V2** | 76.41 | 78.91 | 74.0 |\n| **LIMO-V2-P** | 80.92 | 79.28 | 75.0 |\n\n**Rare Negative Impact Scenarios**: PIR optimization shows robust performance across reasoning domains, with only one minor negative case: **LIMO-P on MBPP code generation** (-0.6% accuracy decline, 73.2% → 72.6%). This is the sole negative impact among all evaluated task-model combinations.\n\n**Framework Design Considerations**: PIR is designed to optimize reasoning chains from Large Reasoning Models (LRMs), performing best on tasks requiring extensive step-by-step reasoning. The evaluation shows consistent improvements across mathematically complex tasks, commonsense reasoning, and logical reasoning. For tasks with simpler reasoning structures or those not heavily relying on elaborate reasoning chains—such as factual QA, simple classification, or direct information retrieval—optimization potential may be limited.\n\n**Overall Robustness**: Evidence demonstrates consistent positive impacts across commonsense reasoning, logical reasoning, and code generation. The single minor negative case (-0.6%) shows negative impacts are rare exceptions rather than systematic limitations, highlighting PIR's broad applicability with strong efficiency gains through token reduction.\n\n---\n\n## **Q3: Systematic vs. Anecdotal Analysis**\n\n> I believe it is essential to provide a systematic characterization of the qualitative points (i.e. non-anecdotal).\n> \n\n**We thank the reviewer for highlighting the need for systematic qualitative analysis beyond individual case studies.** To address this, we provide systematic qualitative characterization comparing reasoning traces from models trained on original versus PIR-optimized datasets.\n\n**Pattern Analysis**: We systematically analyzed reasoning traces from LIMO and LIMO-P models across all test problems, characterizing qualitative differences using keyword frequency statistics based on cognitive reasoning patterns from Table 4. As shown below, models trained on PIR-optimized data generate reasoning traces with systematically different patterns—increasing Progressive Reasoning density (+4.05%) while reducing Verification (-1.62%) and Multi-method Validation (-2.45%). This demonstrates that PIR training fundamentally changes reasoning structure, concentrating on essential logical progression while reducing redundant verification through measurable linguistic pattern shifts.\n\n### Systematic Pattern Analysis\n\n| Dataset | Progressive Reasoning | Multi-method Validation | Error Correction | Verification |\n| --- | --- | --- | --- | --- |\n| LIMO | 28.78% | 20.34% | 0.31% | 50.57% |\n| LIMO-P | **32.83%** *(+4.05%)* | **17.89%** *(-2.45%)* | **0.33%** *(+0.02%)* | **48.95%** *(-1.62%)* |\n\n**Human Evaluation Protocol**: We conducted rigorous evaluation with 4 expert annotators independently analyzing 5% randomly sampled correct reasoning traces from LIMO and LIMO-P models. Annotators classified reasoning steps into four cognitive patterns and calculated Progressive Reasoning proportions, achieving high inter-annotator agreement (κ = 0.87).\n\nResults show human evaluators consistently identified significantly higher Progressive Reasoning proportions in LIMO-P traces (**68.3% ± 4.2%**) versus LIMO traces (**52.1% ± 5.8%**), representing a **16.2 percentage point increase**. Functional components showed corresponding proportional decreases in LIMO-P traces. This systematic validation confirms PIR-trained models fundamentally alter reasoning generation patterns, producing traces with higher concentration of essential logical steps.\n\n---\n\n## **Q4: Model Selection Criteria**\n\n> What is the formal selection criteria (inclusion and exclusion) for the base models? How does it affect the universality of the claims?\n> \n\nWe sincerely thank the reviewer for this important question. Our base model selection was guided by criteria ensuring rigor and generalizability:\n\n1. **Availability and Reproducibility** - We used openly-available Qwen2.5 and Llama models from official repositories with standardized training scripts\n2. **Scale Diversity** - Systematically evaluating 3B to 32B parameter models (Section 3.3.3), demonstrating consistent improvements across all sizes\n3. **Cross-Architecture Validation** - Beyond Qwen2.5, we conducted experiments with Llama-3.1-8B showing strong improvements\n4. **Data Source Diversity** - Validating across training data from three different LRMs (Gemini Flash Thinking, DeepSeek-R1, QwQ)\n\nThese consistent improvements across model scales (3B-32B), architectures (Qwen, Llama), and data sources suggest our **PIR framework captures fundamental reasoning optimization principles rather than model-specific characteristics**.\n\n### Cross-Architecture Validation: Llama-3.1-8B Results\n\n| Model | AIME24 | AMC23 | GPQA |\n| --- | --- | --- | --- |\n| **Accuracy** |  |  |  |\n| Llama-3.1-8B-LIMO-V2 (baseline) | 0.83% | 11.88% | 44.33% |\n| Llama-3.1-8B-LIMO-V2-P (PIR) | **9.13%** | **18.12%** | **45.70%** |\n| **Improvement** | **+8.3%** | **+6.24%** | **+1.37%** |\n| **Token Efficiency** |  |  |  |\n| Baseline Tokens | 19,226 | 13,875 | 11,872 |\n| PIR Tokens | 15,164 | 12,112 | 11,045 |\n| **Token Reduction** | **-21.1%** | **-12.7%** | **-7.0%** |\n\n---\n\n## **Q5: Benchmark Selection Criteria**\n\n> What is the formal selection criteria for the benchmarks?\n> \n\nWe thank the reviewer for this question about benchmark selection methodology. Our selection follows systematic criteria ensuring fair comparison and comprehensive evaluation.\n\n**Selection Criteria**:\n\n1. **Alignment with Prior Work** - We selected benchmarks from recent reasoning literature (LIMO, S1 papers) enabling direct comparison with established baselines\n2. **Reasoning Diversity** - We chose benchmarks spanning difficulty levels and reasoning types—AIME/AMC for mathematical reasoning, GPQA Diamond for scientific reasoning—ensuring comprehensive evaluation across varied demands\n\n**Methodological Consistency**: Aligning with LIMO and S1 evaluation protocols ensures performance improvements stem from our PIR framework rather than methodology differences, establishing validity and enabling fair comparison.\n\n**Generalizability Validation**: Beyond established benchmarks, PIR demonstrates consistent improvements across **different data sources** (Gemini Flash Thinking, DeepSeek-R1, QwQ), **model scales** (3B-32B parameters), and **reasoning domains** including commonsense reasoning (CommonsenseQA), logical reasoning (LogiQA), and code generation (MBPP). This indicates PIR captures universal reasoning optimization principles rather than domain-specific patterns.\n\n### Cross-Domain Performance Validation\n\n| Model | Commonsense QA | LogiQA | MBPP |\n| --- | --- | --- | --- |\n| **LIMO** | 81.33 | 77.41 | 73.2 |\n| **LIMO-P** | **82.00** | **78.04** | 72.6 |\n| **LIMO-V2** | 76.41 | 78.91 | 74.0 |\n| **LIMO-V2-P** | **80.92** | **79.28** | **75.0** |\n\nThis comprehensive validation confirms that our benchmark selection provides both rigorous comparison standards and robust evidence of broad applicability across diverse reasoning scenarios."}, {"review_id": "7dVrBEWTUh", "reviewer": "Reviewer_U9qd", "summary": "Reasoning 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.", "questions": "- 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.", "limitations": "Yes, I think the current limitations section is fair.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "I do appreciate the authors for providing the comprehensive response to my review.\nI especially thank them for providing the experimental results as suggested.\n\nRegarding Q1, Q2, and Q4, it's great to see that using a fixed threshold (PIR < 0) brings improvements to the baseline (LIMO-V2).\nOn the flip side, I'm not convinced that it really brings \"similar empirical performance\" to the original fixed-ratio approach at this point.\nIt is totally understandable that the rebuttal period may not have been enough to explore different settings of PIR-based thresholds and analyze their effects to achieve higher performance.\nBut I believe that this is more of a reason to explore in this direction to make this submission a much stronger work, which I think should be possible.\n\nThe authors' response to Q3 and Q5 addresses those!\n\nOverall, I think this can be a more significant work by addressing the remaining concerns, and I maintain my original view this time.", "rebuttal": "We sincerely thank the reviewer for the thorough review and constructive feedback. We address each concern below with experimental evidence and clarifications.\n\n---\n\n## **Unified Experimental Results**\n\nFollowing the reviewer's suggestions, we conducted experiments using 0 as the threshold for pruning functional steps. These results address multiple concerns (Q1, Q2, Q4):\n\n### Accuracy Results\n\n| Method | AIME24 | AMC23 | GPQA Diamond |\n|--------|--------|--------|--------------|\n| **Baseline (LIMO-V2)** | 66.3% | 94.4% | 70.2% |\n| **PIR (0 Threshold)** | **67.5%** | **94.75%** | **70.33%** |\n| **🔺 Improvement** | **+1.2%** | **+0.35%** | **+0.13%** |\n\n### Token Efficiency\n\n| Method | AIME24 | AMC23 | GPQA Diamond |\n|--------|--------|--------|--------------|\n| **Baseline** | 13,896 | 6,843 | 8,035 |\n| **PIR (0 Threshold)** | **11,165** | **6,001** | **7,010** |\n| **🔻 Reduction** | **-19.5%** | **-12.3%** | **-12.7%** |\n\n---\n\n## **Q1: Fixed Pruning Ratio vs. PIR-based Thresholds**\n\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... 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\n**We sincerely thank the reviewer for this insightful suggestion about using PIR-based thresholds.** Following the reviewer's suggestion, we conducted experiments removing all functional steps with PIR < 0. Our results demonstrate that this approach achieves strong performance (see unified results table above).\n\nThese results demonstrate that using PIR < 0 as a threshold provides consistent improvements across all benchmarks, validating the effectiveness of PIR for identifying low-importance functional steps. While both fixed ratios and PIR-based thresholds can achieve good performance, our fixed-ratio design offers a more **controllable hyperparameter** that practitioners can easily adjust based on **user's specific efficiency-accuracy trade-offs**. We will include these PIR < 0 threshold results in our camera-ready version to provide readers with empirical evidence for both approaches.\n\n---\n\n## **Q2: Benchmark-Specific Pruning Ratios**\n\n> *On a related topic, I'm also primarily concerned about the use of different pruning ratios for different target benchmarks... This could indicate the possibility of overfitting of this parameter to each target benchmark.*\n\n**We sincerely thank the reviewer for raising this important concern about potential overfitting of pruning ratios to specific benchmarks.** To address this concern, we conducted experiments using a single, uniform criterion across all benchmarks: removing all functional steps with PIR < 0. This approach eliminates any benchmark-specific tuning.\n\nOur results demonstrate that this unified criterion achieves consistent improvements across all three benchmarks (see unified results table above). These results show that PIR effectively identifies low-importance functional steps regardless of the benchmark, without requiring benchmark-specific ratio tuning. The consistent improvements validate that our approach captures general properties of reasoning chains rather than overfitting to specific evaluation sets. We will clarify in our camera-ready version that practitioners can use either benchmark-agnostic approaches (like PIR < 0) or tune ratios based on their specific application needs.\n\n---\n\n## **Q3: Reasoning Step Decomposition Quality**\n\n> *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.*\n\n**We sincerely thank the reviewer for this thoughtful observation about the granularity and cohesiveness of reasoning step decomposition.** We share this concern and conducted additional analysis to evaluate decomposition quality.\n\n**Human Evaluation Protocol**: We randomly sampled 5% reasoning chains and had four annotators assess whether each decomposed step represents a complete, cohesive logical unit. The evaluation criteria included:\n\n- (1) **Logical completeness** - whether the step contains a self-contained reasoning operation\n- (2) **Semantic coherence** - whether all sentences within a step work together toward a single reasoning goal\n- (3) **Boundary appropriateness** - whether the step boundaries preserve logical dependencies\n\n**Results**: Our analysis shows that 89.2% of reasoning steps were rated as \"cohesive and complete\" by all annotators, 8.1% had minor boundary issues (e.g., a verification statement included within the progressive step), and only 2.7% showed significant decomposition problems.\n\n**Robustness Analysis**: When we analyzed the impact of decomposition quality on PIR calculations, we found that steps with minor boundary issues had similar PIR distributions to well-decomposed steps (mean PIR difference < 0.05), suggesting our framework is robust to small decomposition variations. We also observed that progressive reasoning steps had higher decomposition quality (92.1% perfect) compared to functional steps (86.3% perfect), which actually supports our approach since progressive steps are always preserved.\n\n---\n\n## **Q4: PIR = 0 as Absolute Threshold**\n\n> *What do the empirical results (especially accuracies) look like if you use an absolute value of 0 (zero) as a threshold for PIR? Could you demonstrate that with some constant thresholding of PIR values themselves, you can achieve similar empirical results?*\n\n**We sincerely thank the reviewer for this specific and constructive suggestion.** We have conducted exactly this experiment - using PIR < 0 as the threshold to prune functional steps. The empirical results are shown in our unified results table above.\n\nThese results demonstrate that using PIR = 0 as an absolute threshold achieves similar empirical performance to our fixed-ratio approach, with consistent improvements in both accuracy and efficiency across all benchmarks. This validates that PIR values themselves provide a meaningful signal for step importance, and constant thresholding (PIR < 0) offers a viable alternative to ratio-based pruning. We will include these results in our main paper to provide readers with both implementation options.\n\n---\n\n## **Q5: Y-axis Scales in Figure 1**\n\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\n**We sincerely thank the reviewer for this careful observation about the visualization.** We acknowledge that using different y-axis scales across the charts in Figure 1 could potentially make visual comparisons more challenging. Our intention was to maximize the visibility of performance differences within each metric, but we agree that consistent scaling would improve clarity and prevent any unintended misleading interpretations.\n\nIn the camera-ready version, we will revise Figure 1 to use consistent y-axis scales across all charts for each metric type (accuracy, response tokens, and efficiency), making it easier for readers to compare results across benchmarks at a glance. We will also add a note in the caption explicitly stating the scale ranges used. Thank you for helping us improve the clarity of our presentation."}, {"review_id": "lzVaG9l6Na", "reviewer": "Reviewer_QYR4", "summary": "This 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.", "questions": "1. 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?", "limitations": "Yes, the content is discussed in Conlusion.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the thorough review. We address each concern below with experimental evidence and clarifications.\n\n---\n\n## **Q1: External Dependency on Claude 3.7 Sonnet**\n\n> *How would it perform if a weaker open-source model were used instead? and (b) the framework's effectiveness may partially derive from the classifier's intelligence rather than the PIR logic itself.*\n\n**Reproducibility with Open-Source Alternatives**: To directly address reproducibility, we conducted additional experiments using Qwen3-32B as an open-source alternative for reasoning step classification. The results demonstrate maintained effectiveness:\n\n### Accuracy Performance\n\n| Model | AIME24 | AMC23 | GPQA |\n|-------|--------|--------|------|\n| **LIMO-V2** | 66.3% | 94.4% | 70.2% |\n| **LIMO-V2-P (Qwen3-32B)** | **67.5%** | **96.88%** | **69.19%** |\n| **🔺 Improvement** | **+1.2%** | **+2.48%** | **-1.01%** |\n\n### Average Response Tokens\n\n| Model | AIME24 | AMC23 | GPQA |\n|-------|--------|--------|------|\n| **LIMO-V2** | 13,896 | 6,843 | 8,035 |\n| **LIMO-V2-P (Qwen3-32B)** | **9,885** | **5,305** | **6,975** |\n| **🔻 Reduction** | **-28.9%** | **-22.5%** | **-13.2%** |\n\nImportantly, Qwen3-32B requires only 4×RTX 4090 GPUs for inference, making it accessible to most research laboratories.\n\n**Conceptual Coherence - PIR Logic vs. Classifier Intelligence**: The core PIR contribution is the quantitative importance metric (Equation 1), not classification sophistication. The ablation study with rule-based filtering (S1-RULE in Table 3) shows: while random functional step removal reduces accuracy below baseline on AIME (36.7% vs. 37.9%) and GPQA (58.1% vs. 60.7%), our S1-32B-P method achieves substantial improvements (42.1% and 61.6%).\n\nThis confirms PIR's perplexity-based assessment drives performance gains, not classification quality.\n\n---\n\n## **Q2: Evaluation Beyond Mathematical Reasoning**\n\n> *How the framework performs on commonsense reasoning, logic puzzles, and code generation.*\n\n**Comprehensive Cross-Domain Evaluation**: Our framework's effectiveness extends well beyond mathematical and scientific domains. We conducted extensive evaluation on diverse reasoning tasks that exhibit different characteristics from mathematical reasoning:\n\n### Accuracy Scores\n\n| Model | Commonsense QA | LogiQA | MBPP |\n|-------|----------------|--------|------|\n| **LIMO** | 81.33 | 77.41 | 73.2 |\n| **LIMO-P** | **82.00** | **78.04** | 72.6 |\n| **LIMO-V2** | 76.41 | 78.91 | 74.0 |\n| **LIMO-V2-P** | **80.92** | **79.28** | **75.0** |\n\n### Token Usage\n\n| Model | Commonsense QA | LogiQA | MBPP |\n|-------|----------------|--------|------|\n| **LIMO** | 961 | 2175 | NA |\n| **LIMO-P** | **902** | **2011.6** | NA |\n| **LIMO-V2** | 1736 | 3058 | NA |\n| **LIMO-V2-P** | **1450** | **2872** | NA |\n\n**Consistent Cross-Domain Improvements**: The results demonstrate that PIR optimization consistently improves performance across fundamentally different reasoning types:\n\n- **Commonsense Reasoning** (CommonsenseQA): +0.67% accuracy with 6.1% token reduction for LIMO-P, and +4.51% accuracy with 16.5% token reduction for LIMO-V2-P\n- **Logical Reasoning** (LogiQA): +0.63% and +0.37% accuracy improvements with 7.5% and 6.1% token reductions respectively\n- **Code Generation** (mbpp): +1.0% accuracy improvement for LIMO-V2-P\n\n**Framework Universality**: These domains involve different functional steps—commonsense reasoning features intuitive validation, logical reasoning involves formal verification, and code generation includes debugging and testing—yet PIR consistently optimizes the efficiency-effectiveness tradeoff. This validates our hypothesis that the distinction between progressive reasoning (essential logical flow) and functional elements is a fundamental cognitive pattern transcending specific reasoning domains.\n\n---\n\n## **Q3: Adaptive Pruning Strategies**\n\n> *... A more adaptive strategy could potentially yield better performance.*\n\n**Foundational Contribution**: Our work establishes the **essential theoretical and methodological foundation** for adaptive approaches by introducing systematic reasoning step classification, the PIR importance metric, and empirical validation—prerequisites for adaptive strategies. Without our cognitive reasoning patterns and importance quantification, adaptive approaches would lack theoretical grounding.\n\n**Current Framework's Robustness**: While using predefined ratios, our PIR framework demonstrates strong generalizability across model sizes (3B-32B), data sources (Gemini, DeepSeek-R1, QwQ), and benchmarks (AIME, AMC, GPQA). The PIR metric provides the foundation for dynamic thresholds.\n\n**Future Extensions**: Building on our foundation, adaptive strategies represent natural progressions: difficulty-aware pruning, length-adaptive thresholds, and dynamic importance cutoffs. Our framework provides essential infrastructure enabling these advanced approaches while offering predictable behavior for immediate deployment.\n\n---\n\n## **Q4: Accessibility of Open-Source Alternatives**\n\n> *Could the authors comment on the feasibility of replacing Claude 3.7 Sonnet with a more accessible open-source model?*\n\n**Open-Source Solution**: While experiments with LLaMA-3 8B and Qwen-2.5-7B were not conducted, these smaller models may struggle with the nuanced reasoning step classification required by our framework, particularly in distinguishing between the four cognitive patterns (progressive reasoning, verification, multi-method validation, and error correction).\n\n**Validated Alternative**: Experiments demonstrate that Qwen3-32B provides an excellent open-source alternative, achieving comparable performance to the original framework (see Q1 results table above).\n\n**Practical Accessibility**: Qwen3-32B requires only **4×RTX 4090 or 4×RTX 3090** GPUs for inference, making it accessible to most research laboratories with standard computational resources.\n\n---\n\n## **Q5: Minimum Classification Accuracy Requirements**\n\n> *What minimum classification accuracy do you believe is required for PIR to remain effective?*\n\n**Framework Robustness**: While systematic experiments across different classification accuracy levels were not conducted, current results and framework design demonstrate resilience to classification errors.\n\n**Evidence from Ablation Study**: The ablation study with rule-based filtering (S1-RULE in Table 3) shows imperfect classification relying solely on linguistic markers still achieves efficiency gains, though with reduced effectiveness compared to PIR, indicating PIR's tolerance for classification noise.\n\n**Dual-Filtering Design**: PIR remains effective if the classifier reasonably distinguishes progressive reasoning steps (requiring preservation) from functional steps (verification, multi-method validation, error correction). The perplexity-based metric serves as secondary quality filtering—misclassified progressive reasoning steps likely receive high PIR scores ensuring preservation, while truly redundant functional steps receive low PIR scores regardless of classification accuracy.\n\n---\n\n## **Q6: Dependencies Between Functional and Progressive Steps**\n\n> *Is there a risk that a low-importance functional step (e.g., a verification) may actually be critical for enabling a downstream progressive step?*\n\n**Global Dependency Capture**: Our PIR metric is specifically designed to capture such dependencies through its global evaluation approach. When calculating PIR scores, we measure the perplexity change of the entire answer generation process after removing a specific step. If a functional step (e.g., verification) is truly critical for enabling downstream progressive reasoning, removing it would significantly impair the model's confidence in generating subsequent reasoning steps, resulting in higher answer perplexity and thus a higher PIR score that preserves the step.\n\n**Empirical Validation**: The ablation study results strongly validate this dependency-preservation mechanism:\n\n| Method | AIME | AMC | GPQA Diamond |\n|--------|------|-----|--------------|\n| LIMO | 56.7 | 91.9 | 67.2 |\n| **LIMO-P** | **63.3** | **93.8** | **71.2** |\n| LIMO(No Verification) | 59.1 | 92.5 | 70.7 |\n| LIMO(No Multi-method Validation) | 60.0 | 92.8 | 72.2 |\n| LIMO(No Error Correction) | 57.5 | 92.1 | 71.7 |\n\nWhile indiscriminate removal of entire functional categories can lead to higher accuracy than baseline, our PIR approach achieves superior performance: +6.6% accuracy improvement on AIME, +1.9% on AMC, and +4.0% on GPQA Diamond. This demonstrates that PIR successfully identifies and preserves those functional steps that are truly essential.\n\n---\n\n## **Q7: Local Reasoning Dependencies**\n\n> *Has the framework considered the possibility that pruning such steps could disrupt local reasoning dependencies?*\n\n**Step-Level Coherence Preservation**: Our framework addresses this through step-level granularity and systematic classification. The hierarchical decomposition segments reasoning chains into logical steps, each comprising multiple coherent sentences forming a cohesive unit (lines 125-127). Removing functional steps as complete units preserves local reasoning coherence and prevents partial dependency disruption.\n\n**Independence of Functional Steps**: Analysis shows functional steps (verification, multi-method validation, error correction) operate independently, targeting specific progressive reasoning steps with minimal interdependencies. Progressive reasoning steps exhibit strong sequential dependencies, but our framework preserves these entirely to maintain the solution's logical backbone.\n\n**Classification Accuracy**: The robust classification system uses characteristic linguistic markers (\"Let me check\" for verification, \"I made a mistake\" for error correction) combined with contextual analysis, achieving 93.4% unanimous agreement in human evaluation and minimizing misclassification risks that could disrupt critical reasoning dependencies."}, {"review_id": "SvZ5asEa1B", "reviewer": "Reviewer_md3Q", "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.", "questions": "-  The paper focuses primarily on mathematical and science reasoning tasks.  Can your method generalize to other domains like instruction following?\n- Will shorten the reasoning steps hurt the model's exploration ability? \n- 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", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "- The quality of this paper is fair since the data selection method is quite straightforward. \n- The paper is well-structured and clear.\n- 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.\n- This work is based on LIMO and incrementally prunes some extra reasoning steps, which is not very original.", "was_revised": true, "final_justification": "The rebuttal addresses generalization (Q3) and exploration concerns (Q4) with strong experimental evidence. So I would like to raise my scores.", "rebuttal": "We sincerely thank the reviewer for the thorough review and constructive feedback. We address each concern below with experimental evidence and clarifications.\n\n---\n\n## **Q1: Relationship to AdaptThink and ThinkLess**\n\n> *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.*\n\n**Fundamental Technical Distinctions**: While AdaptThink and ThinkLess both address efficiency in reasoning models, our PIR framework offers fundamentally different and complementary innovations:\n\n**Different Optimization Levels**:\n\n- **AdaptThink**: Operates at inference-time decision level using RL to choose \"Thinking\" vs \"NoThinking\" modes\n- **ThinkLess**: Focuses on training-free early termination during inference\n- **PIR**: Optimizes at the **training data level** through principled content refinement, creating inherently more efficient models\n\n**Granularity and Methodology**:\n\n- **AdaptThink/ThinkLess**: Binary decisions or wholesale early termination\n- **PIR**: **Fine-grained, step-level analysis** using novel perplexity-based importance quantification to preserve essential progressive reasoning while selectively removing low-value functional components\n\n**Superior Effectiveness**: Our results demonstrate advantages across multiple dimensions:\n\n- **PIR**: Accuracy improvements (+0.9% to +6.6%) while simultaneously reducing tokens (-3% to -41%)\n- **ThinkLess**: Focus primarily on efficiency without accuracy gains\n\n**Unique Value Proposition**: PIR produces **optimized training datasets** that result in inherently more efficient models, eliminating need for complex inference-time orchestration while maintaining full reasoning capabilities. This addresses a complementary but fundamental aspect—optimizing reasoning exemplar quality rather than controlling when to use them.\n\n---\n\n## **Q2: Originality vs. LIMO**\n\n> *This work is based on LIMO and incrementally prunes some extra reasoning steps, which is not very original.*\n\n**Fundamental Scope Differences**:\n\n- **LIMO**: Dataset creation focusing on sample selection efficiency (\"less is more\" for data quantity)\n- **PIR**: Comprehensive framework for reasoning chain optimization addressing reasoning structure and content efficiency\n\n**Four Major Original Contributions**:\n\n1. **Novel Cognitive Framework**: First systematic classification of reasoning patterns into progressive reasoning vs. functional elements (verification, multi-method validation, error correction)\n2. **Principled Quantitative Metric**: PIR metric for measuring reasoning step importance through perplexity changes—entirely novel methodological contribution\n3. **Selective Optimization Strategy**: Fine-grained step-level analysis vs. LIMO's wholesale sample selection from \"tens of millions of candidates\"\n4. **Generalizability Framework**: Works across diverse data sources (Gemini, DeepSeek-R1, QwQ), model sizes (3B-32B), demonstrating broader applicability than LIMO\n\n**Superior Effectiveness**: Our experimental results demonstrate significant advances:\n\n- **Simultaneous optimization**: Accuracy improvements (+0.9% to +6.6%) AND efficiency gains (-3% to -41% token reduction)\n- **Up to 71% efficiency improvement** in some cases\n- **41% token reduction** in case studies while maintaining solution accuracy\n\n**Broader Impact**: Our framework's applicability to multiple existing datasets (LIMO, S1, LIMO-V2) demonstrates value as a general-purpose optimization framework. The theoretical insights establish a new paradigm for reasoning chain optimization that complements and enhances existing approaches.\n\n---\n\n## **Q3: Generalization to Instruction Following**\n\n> *The paper focuses primarily on mathematical and science reasoning tasks. Can your method generalize to other domains like instruction following?*\n\n**Instruction Following Validation**: Our evaluation on **IFEval (instruction following)** shows consistent improvements:\n\n### Accuracy Scores\n\n| Model | Commonsense QA | LogiQA | MBPP | IFEval |\n|-------|----------------|--------|------|--------|\n| **LIMO** | 81.33 | 77.41 | 73.2 | 55.62 |\n| **LIMO-P** | **82.00** | **78.04** | 72.6 | **58.03** |\n| **LIMO-V2** | 76.41 | 78.91 | 74.0 | 55.7 |\n| **LIMO-V2-P** | **80.92** | **79.28** | **75.0** | **57.07** |\n\n### Token Usage\n\n| Model | Commonsense QA | LogiQA | MBPP | IFEval |\n|-------|----------------|--------|------|--------|\n| **LIMO** | 961 | 2175 | NA | NA |\n| **LIMO-P** | **902** | **2011.6** | NA | NA |\n| **LIMO-V2** | 1736 | 3058 | NA | NA |\n| **LIMO-V2-P** | **1450** | **2872** | NA | NA |\n\n**Cross-Domain Effectiveness**: PIR achieves consistent improvements across diverse domains:\n\n- **Instruction Following** (IFEval): +1.37% to +2.41% improvements\n- **Commonsense Reasoning** (CommonsenseQA): +0.67% to +4.51% improvements\n- **Logical Reasoning** (LogiQA): +0.37% to +0.63% improvements\n- **Code Generation** (mbpp): up to +1.0% improvement\n\n**Framework Universality**: The consistent improvements across such diverse task types validate that PIR identifies fundamental reasoning optimization principles rather than domain-specific patterns. Our cognitive reasoning pattern framework—distinguishing between progressive reasoning and functional elements—captures universal principles that extend well beyond mathematical domains to instruction following and other reasoning-intensive tasks.\n\n---\n\n## **Q4: Impact on Model Exploration Ability**\n\n> *Will shorten the reasoning steps hurt the model's exploration ability?*\n\n**Enhanced Exploration Evidence**: Our systematic analysis reveals that PIR training fundamentally improves how models structure their reasoning:\n\n| Dataset | Progressive Reasoning | Multi-method Validation | Error Correction | Verification |\n|---------|----------------------|------------------------|------------------|--------------|\n| **LIMO** | 28.78% | 20.34% | 0.31% | 50.57% |\n| **LIMO-P** | **32.83%** | **17.89%** | **0.33%** | **48.95%** |\n\n**More Efficient Exploration**: The **+4.05% increase in Progressive Reasoning density** while reducing redundant functional components demonstrates **more efficient exploration** rather than reduced capability. PIR selectively removes only low-importance, redundant instances while preserving high-value exploration strategies.\n\n**Performance Validation**: The **simultaneous accuracy improvements (+0.9% to +6.6%)** across benchmarks provide compelling evidence—if exploration ability were impaired, we would expect performance degradation. Instead, higher Progressive Reasoning density shows models allocate reasoning capacity more strategically, concentrating on productive solution pathways while maintaining ability to explore alternatives when beneficial.\n\n**Case Study Evidence**: Our detailed case study (Section B.4) demonstrates **more focused and effective exploration** rather than limited reasoning depth. Comparing two models solving identical problems:\n\n- **PIR-trained model** (1,612 tokens): Streamlined yet thorough exploration\n- **Original model** (3,234 tokens): Verbose, repetitive exploration with redundant validation cycles\n\nBoth reach correct answers, but PIR-trained model achieves this through more intelligent exploration strategies. The 50% token reduction represents elimination of redundant exploration rather than curtailment of necessary reasoning depth.\n\n---\n\n## **Q5: Alternative Metrics Beyond Perplexity**\n\n> *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?*\n\n**Principled Choice of Perplexity**: Our choice of perplexity is principled and empirically validated. Perplexity directly measures the model's confidence in generating correct answers when specific steps are removed, providing a clear signal of step importance that is both computationally efficient and theoretically grounded.\n\n**Strong Empirical Validation**: Our extensive experiments demonstrate that this approach consistently improves accuracy (+0.9% to +6.6%) while reducing token usage (-3% to -41%) across diverse benchmarks and model sizes, validating its effectiveness for practical optimization purposes.\n\n**Future Research Directions**: We acknowledge that exploring semantic-based metrics could provide additional insights **(as noted in our limitations, lines 323-325)**. Future work could explore complementary metrics such as:\n\n- Embedding-based semantic similarity\n- Logical dependency analysis\n\n**Current Framework Strength**: However, our framework's strong generalizability across different data sources, model scales, and token budgets suggests that perplexity successfully captures fundamental aspects of reasoning step importance. The consistent improvements across diverse reasoning domains demonstrate that our approach identifies universal optimization principles effective across various task types."}]

Author rebuttal:

Meta-review: **summary**
This paper introduces Perplexity-based Importance Refinement (PIR), a novel framework for optimizing the reasoning chains of large language models. The method distinguishes between essential "progressive" reasoning and auxiliary "functional" steps (e.g., verification, correction), and uses a perplexity-based metric to prune low-importance functional steps from training data. This results in models that generate more concise reasoning while achieving higher accuracy and efficiency.

**strengths**
- The paper proposes a principled and intuitive methodology, using perplexity to quantitatively measure the importance of each reasoning step, which is more robust than simple heuristics.
- The experimental results are strong and comprehensive, demonstrating consistent improvements in both accuracy and token efficiency across a diverse set of challenging reasoning benchmarks, model sizes, and architectures.
- The evaluation is thorough, including extensive ablation studies and cross-domain validation on tasks like commonsense reasoning, code generation, and instruction following, which confirms the broad applicability of the framework.

**weaknesses**
- The framework's initial step of segmenting and classifying reasoning types relies on a separate, powerful language model, introducing an external dependency that could affect reproducibility and performance.
- The primary pruning method relies on a fixed ratio or a simple threshold, which might not be optimal for all reasoning instances; a more adaptive strategy could potentially offer further improvements.
- The paper focuses exclusively on perplexity as the importance metric, and a deeper exploration of alternative semantic or logical-based metrics could provide a more nuanced approach to reasoning refinement.

**final descision**
This is a strong paper that should be accepted. It presents a novel, well-motivated, and empirically validated method for improving the efficiency and effectiveness of LLM reasoning, making a solid contribution to the field.
Decision: Accept (poster)