# Finding the Cracks: Improving LLMs Reasoning with Paraphrastic Probing and Consistency Verification
**Authors**: Weili Shi, Dongliang Guo, Lehan Yang, Tianlong Wang, Hanzhang Yuan, Sheng Li
## Abstract
Large language models have demonstrated impressive performance across a variety of reasoning tasks. However, their problem-solving ability often declines on more complex tasks due to hallucinations and the accumulation of errors within these intermediate steps. Recent work has introduced the notion of critical tokens âtokens in the reasoning process that exert significant influence on subsequent steps. Prior studies suggest that replacing critical tokens can refine reasoning trajectories. Nonetheless, reliably identifying and exploiting critical tokens remains challenging. To address this, we propose the P araphrastic P robing and C onsistency V erification (PPCV) framework. PPCV operates in two stages. In the first stage, we roll out an initial reasoning path from the original question and then concatenate paraphrased versions of the question with this reasoning path. And we identify critical tokens based on mismatches between the predicted top-1 token and the expected token in the reasoning path. A criterion is employed to confirm the final critical token. In the second stage, we substitute critical tokens with candidate alternatives and roll out new reasoning paths for both the original and paraphrased questions. The final answer is determined by checking the consistency of outputs across these parallel reasoning processes. We evaluate PPCV on mainstream LLMs across multiple benchmarks. Extensive experiments demonstrate PPCV substantially enhances the reasoning performance of LLMs compared to baselines.
Machine Learning, ICML
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<summary>images/compare_ct_sc_gsm8k.png Details</summary>
### Visual Description
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## Line Chart: Pass@k vs. Number of Sample k
### Overview
This line chart illustrates the relationship between the "number of sample k" and the "pass@k (%)" metric for two different methods: "critical tokens" and "self-consistency". The chart displays how the percentage of successful passes (pass@k) changes as the number of samples (k) increases.
### Components/Axes
* **X-axis:** "number of sample k", ranging from 0 to 50, with markers at 0, 10, 20, 30, 40, and 50.
* **Y-axis:** "pass@k (%)", ranging from 70% to 90%, with markers at 70%, 72.5%, 75%, 77.5%, 80%, 82.5%, 85%, 87.5%, and 90%.
* **Legend:** Located in the bottom-right corner, identifying two data series:
* "critical tokens" - represented by a red line with triangle markers.
* "self-consistency" - represented by a purple line with star markers.
* **Gridlines:** Vertical dashed gridlines are present to aid in reading values.
### Detailed Analysis
**Critical Tokens (Red Line):**
The "critical tokens" line slopes generally upward, indicating an increase in pass@k as the number of samples (k) increases.
* At k = 0, pass@k is approximately 81%.
* At k = 10, pass@k is approximately 83%.
* At k = 20, pass@k is approximately 85%.
* At k = 30, pass@k is approximately 86%.
* At k = 40, pass@k is approximately 88%.
* At k = 50, pass@k is approximately 89%.
**Self-Consistency (Purple Line):**
The "self-consistency" line also slopes upward, but at a slower rate than the "critical tokens" line.
* At k = 0, pass@k is approximately 71%.
* At k = 10, pass@k is approximately 77%.
* At k = 20, pass@k is approximately 80%.
* At k = 30, pass@k is approximately 82%.
* At k = 40, pass@k is approximately 84%.
* At k = 50, pass@k is approximately 85%.
### Key Observations
* The "critical tokens" method consistently outperforms the "self-consistency" method across all values of k.
* Both methods show diminishing returns as k increases. The rate of improvement in pass@k decreases as k gets larger.
* The initial increase in pass@k is more pronounced for the "self-consistency" method between k=0 and k=10.
### Interpretation
The data suggests that increasing the number of samples (k) generally improves the pass@k metric for both "critical tokens" and "self-consistency" methods. However, the "critical tokens" method is more effective at achieving higher pass rates, especially as the number of samples increases. This could indicate that identifying and utilizing critical tokens is a more efficient strategy for improving performance than relying solely on self-consistency. The diminishing returns observed at higher values of k suggest that there is a point beyond which increasing the number of samples provides minimal additional benefit. This information is valuable for optimizing the sampling process and balancing performance gains with computational cost. The difference in initial slope suggests that self-consistency benefits more from a small number of samples, while critical tokens maintain a more consistent improvement.
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Figure 1: Comparison of the effects of critical tokens and Self-Consistency on the reasoning performance of LLMs, evaluated on samples from the GSM8K training data.
## 1 Introduction
The emergence of large language models (LLMs) (Brown et al., 2020; Grattafiori et al., 2024; Achiam et al., 2023; Yang et al., 2025a) has astonished the AI community with their remarkable capabilities across a wide range of reasoning tasks, including mathematical problem solving, programming, and planning. By generating intermediate reasoning steps through techniques such as chain-of-thought prompting (Wei et al., 2022; Kojima et al., 2022; Zhang et al., 2022; Sprague et al., 2025), LLMs can emulate human-like reasoning processes and achieve strong performance on diverse reasoning benchmarks. Despite their success, the problem-solving ability of LLMs often declines on complex reasoning tasks due to hallucinations and the accumulation of errors in intermediate steps (Lightman et al., 2023; Ling et al., 2023; Bubeck et al., 2023). To mitigate this issue, prior works (Wang et al., 2024; Yuan et al., 2025; Chen et al., 2024a; Chen and Li, 2024) have proposed the inference-time optimization paradigm, which evaluates and refines reasoning trajectories through self-correction by the LLM itself or with feedback from external verifiers such as process reward models (PRMs) (Uesato et al., 2022). However, existing studies (Yang et al., 2025b; Tyen et al., 2024; Stechly et al., 2025) suggest that LLMs struggle to reliably identify errors in their reasoning and often fail to correct previous outputs without external guidance. Moreover, obtaining high-quality, stepwise process supervision for training external verifiers is challenging and limits the practicality of these approaches (Feng et al., 2025).
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<summary>images/para_example.png Details</summary>
### Visual Description
## Textual Document: Case Study on the Effects of Question Paraphrasing
### Overview
The image presents a side-by-side comparison of two versions of a word problem â an original question and a paraphrased question â along with their respective solution steps. The document also indicates the "ground truth" answer for both problems is 8. Each problem's solution is followed by a visual indicator (a red 'X' or a green checkmark) signifying whether the provided solution is correct.
### Components/Axes
The document is structured into two columns. Each column contains:
1. A question (labeled "Original question:" or "Paraphrased question:")
2. A statement of the "ground truth answer"
3. A "Response:" section detailing the solution steps.
4. A visual indicator of solution correctness.
### Detailed Analysis or Content Details
**Original Question:**
"In a card game, you are awarded 10 points if you win one round. While playing, Jane ends up with 60 points. If she lost 20 points, then how many rounds did she play? (ground truth answer: 8)"
**Original Question Response:**
Step 1: Determine the total points Jane earned by subtracting the points she lost from her final score. Total points earned = Final score - Points lost. Total points earned = 60 - 20. Total points earned = 40.
Step 2: Since Jane earns 10 points for each round she wins, we can find the number of rounds she played by dividing the total points earned by the points earned per round. Number of rounds = Total points earned / Points earned per round. Number of rounds = 40 / 10. Number of rounds = 4.
Therefore, Jane played 4 rounds. (Marked with a red 'X')
**Paraphrased Question:**
"At a bridge competition, every game won grants the player 10 points. Unfortunately, Jane started with an advantage but ended up losing 20 points and scoring only 60 points. If she had won x rounds and lost none, what would be the value of x? (ground truth answer: 8)"
**Paraphrased Question Response:**
Step 1: Understand the given information. Jane started with an advantage but ended up losing 20 points. She scored a total of 60 points. Every game won grants the player 10 points.
Step 2: Determine the total points Jane would have had if she hadnât lost any points. Since she lost 20 points, we need to add 20 to her total score to find out how many points she would have had if she hadnât lost any points. Total points without loss = 60 + 20 = 80 points.
Step 3: Calculate the number of games Jane won. Since every game won grants the player 10 points, we can divide the total points without loss by 10 to find the number of games she won. Number of games won = Total points without loss / 10 = 8. The number of games she won is equal to 8. (Marked with a green checkmark)
### Key Observations
* The original question's solution is incorrect, while the paraphrased question's solution is correct.
* The paraphrased question uses more complex language ("advantage," "x rounds") compared to the original question.
* The original question focuses on rounds played, while the paraphrased question focuses on games won.
* The visual indicators (X and checkmark) clearly denote the correctness of each solution.
### Interpretation
This document demonstrates the impact of question paraphrasing on problem-solving accuracy. The original question, while simpler in wording, leads to an incorrect solution (4 rounds). The paraphrased question, despite being more complex, guides the solver towards the correct answer (8 games won). This suggests that the way a question is framed can significantly influence a person's ability to understand and solve it. The paraphrasing appears to clarify the underlying mathematical relationship, leading to a more accurate result. The use of 'x' in the paraphrased question may encourage a more algebraic approach, which is appropriate for the problem. The document serves as a case study highlighting the importance of careful question design and the potential pitfalls of ambiguous or poorly worded problems. The red 'X' and green checkmark provide a clear visual assessment of the solution's validity, reinforcing the learning objective.
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Figure 2: Case study illustrating that LLMs can refine intermediate incorrect reasoning segments (highlighted in red).
Recent work has introduced the concept of critical tokens (Lin et al., 2025), which play pivotal roles in intermediate reasoning steps and exert strong influence on subsequent reasoning and final outcomes. We evaluate the criticality of a token based on the average accuracy of sampling rollouts generated from the partial reasoning trajectories truncated at the tokenâs position. A token is identified as critical if it lies on an incorrect trajectory and satisfies the following conditions:
- The correctness score of rollouts initiated from the token is close to zero (e.g., below 5%);
- The correctness scores of rollouts initiated from subsequent tokens remain below a threshold (e.g., 10%).
Prior studies suggest that critical tokens often diverge from human-annotated error tokens, yet they induce more sustained degradation in reasoning quality than other tokens. Moreover, as illustrated in Figure 3, replacing critical tokens in an incorrect reasoning trajectory with suitable candidate tokens can correct subsequent steps and lead to the right answer in new rollouts. To quantitatively assess the effectiveness of critical tokens, we conduct an empirical study using LLMs such as Llama-3.1-8B-Instruct (Grattafiori et al., 2024) on reasoning tasks. Specifically, we randomly sample 100 instances with incorrect reasoning steps from the GSM8K (Cobbe et al., 2021) training data. Following the criterion, we locate critical tokens through exhaustive search. We then truncate the reasoning path at the critical token, substitute it with alternative tokens, and roll out new reasoning paths. For example, as shown in Figure 3, the token âwomanâ is replaced with âremainingâ. We evaluate performance using Pass@k and compare against Self-Consistency (Wang et al., 2023), which also samples multiple reasoning paths. As shown in Figure 1, critical token replacement provides a clear advantage in correcting errors compared to plain sampling. Nonetheless, reliably identifying and leveraging critical tokens for reasoning remains a nontrivial challenge.
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<summary>images/critical_token_example.png Details</summary>
### Visual Description
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## Textual Document: Case Study on Critical Tokens
### Overview
The image presents a case study focused on evaluating responses to a mathematical word problem. It showcases two different approaches to solving the same problem, highlighting the importance of "critical tokens" in achieving the correct answer. The problem involves calculating the number of children at a basketball game given the total spectators, the number of men, and the ratio of children to women.
### Components/Axes
The document is structured into sections:
* **Question:** The original word problem.
* **Response:** An initial attempt at solving the problem, marked with a red 'X' indicating an incorrect answer.
* **Response(replacement of critical tokens):** A revised solution, marked with a green checkmark indicating a correct answer.
* Each response is broken down into numbered steps.
### Detailed Analysis or Content Details
**Question:**
"There were 10000 spectators at a basketball match. 7000 of them were men. Of the remaining spectators, there were 5 times as many children as women. How many children were there? (ground truth answer: 2500)"
**Response (Incorrect):**
"To find the number of children, we need to first find the number of women and then multiply it by 5.
Step 1: Find the number of women. Total spectators = 10000. Men = 7000. Women = Total spectators - Men. Women = 10000 - 7000. Women = 3000.
Step 2: Find the number of children. There are 5 times as many children as women. Children = 5 * Women. Children = 5 * 3000. Children = 15000.
So, the number of children is 15000."
**Response (Correct):**
"To find the number of children, we need to first find the number of remaining spectators (excluding men) and then calculate the number of women and children.
Step 1: Find the number of remaining spectators (excluding men). Total spectators = 10000. Men = 7000. Remaining spectators = Total spectators - Men = 10000 - 7000 = 3000
Step 2: Let's assume the number of women is 'x'. Since there are 5 times as many children as women, the number of children will be 5x.
Step 3: The total number of remaining spectators is the sum of women and children. We can write this as an equation: x + 5x = 3000. Combine like terms: \n6x = 3000. Divide both sides by 6: x = 3000 / 6. x = 500.
Step 4: Now that we have the number of women, we can find the number of children. Number of children = 5x = 5 * 500 = 2500.
Therefore, the number of children is 2500."
### Key Observations
* The initial response incorrectly calculates the number of women by subtracting the number of men from the total spectators without considering that the ratio of children to women applies *only* to the remaining spectators.
* The corrected response correctly identifies the need to first find the number of remaining spectators and then applies the ratio to that subset.
* The use of a variable 'x' in the corrected response allows for a proper algebraic solution.
* The "critical tokens" appear to be the understanding of the scope of the remaining spectators and the correct setup of the equation.
### Interpretation
This case study demonstrates the importance of careful reading and accurate interpretation of word problems. The initial response failed because it misapplied the given ratio to the entire spectator population instead of the remaining population after accounting for the men. The corrected response highlights the need to isolate the relevant subset of data before applying the given relationships. The use of algebraic representation (introducing 'x') is crucial for a correct solution. The "critical tokens" are not necessarily specific words, but rather the conceptual understanding of how to correctly frame the problem mathematically. The inclusion of the "ground truth answer" (2500) serves as a validation point and emphasizes the importance of verifying solutions. The visual cues (red 'X' and green checkmark) provide immediate feedback on the correctness of each approach.
</details>
Figure 3: An example demonstrating how substitution of a critical token (red) with a candidate token (purple) modifies the reasoning path and produces the correct answer.
Recent studies (Zhou et al., 2024; Chen et al., 2024b), on surface form, the way questions, assumptions, and constraints are phrased, have revealed its subtle influence on the trajectory of intermediate reasoning steps, As shown in Figure 2. LLMs could adjust the intermediate steps under the paraphrased form of the question. This motivates us to explore the role of paraphrasing in the extraction and utilization of critical tokens for reasoning tasks. To this end, we propose the P araphrastic P robing and C onsistency V erification (PPCV) framework , a two-stage approach designed to leverage critical tokens to enhance the reasoning ability of LLMs. In the first stage, we probe critical tokens using paraphrased questions. Specifically, we first roll out the initial reasoning path from the original question, then concatenate paraphrased questions with this reasoning path. The resulting synthetic inputs are fed into the LLM to obtain token-level logits for each position in the reasoning path. Positions where the predicted top-1 token diverges from the expected token are flagged as potential pivotal points, as these positions are sensitive to paraphrased inputs and can trigger a pivot in the reasoning trajectory. Next, an empirical criterion is applied to determine the final critical token. In contrast to prior work (Lin et al., 2025), which depends on external models for identifying critical tokens with ambiguous criteria, our method introduces a self-contained mechanism that pinpoints critical tokens.
In the second stage, we leverage the extracted critical tokens to refine the initial reasoning path. Specifically, we select the top-K tokens (include critical token itself) at the critical token position and roll out new reasoning paths for both the original and paraphrased questions. Based on the empirical observation that trajectories leading to correct answers are robust to paraphrastic perturbations, we propose a paraphrase consistency mechanism. In contrast to Self-Consistency (Wang et al., 2023), which relies on majority voting, our method selects the final answer by comparing outputs from paraphrased and original questions and choosing the one with the most consistent matches. To address potential ties across multiple answers, we further introduce similarity-weighted paraphrase consistency, which incorporates similarity scores between paraphrased and original questions when computing consistency.
Compared with self-correction (Wu et al., 2024; Miao et al., 2024) and PRM-based methods (Wang et al., 2024; Yuan et al., 2025), our framework exploits critical tokens to refine reasoning trajectories without requiring step-level error detection by the LLM itself or auxiliary models. We evaluate our method on mainstream LLMs across mathematical and commonsense reasoning benchmarks, demonstrating consistent improvements in reasoning performance. The contributions of the paper is summarized as follows:
- We propose a novel two-stage framework, P araphrastic P robing and C onsistency V erification (PPCV) designed to extract and leverage critical tokens to enhance the reasoning performance of LLMs.
- We show that critical tokens can more effectively correct erroneous reasoning trajectories than traditional sampling methods like Self-Consistency. Furthermore, our approach successfully extracts these tokens through paraphrastic probing, achieving improved final results via paraphrase consistency.
- We evaluate our method on mainstream LLMs across various reasoning tasks, including math and logic reasoning. Experimental results show significant performance improvements over baseline methods.
## 2 Related Work
Inference-Time Optimization for LLM reasoning. With the advent of chain-of-thought (CoT) prompting, LLMs have demonstrated strong reasoning capabilities by producing intermediate steps during inference. This success has motivated a growing body of work (Wu et al., 2025; Snell et al., 2024) on augmenting reasoning trajectories at test time to further improve performance. Existing approaches can be broadly categorized into search-based methods (Bi et al., 2025; Yao et al., 2023; Hao et al., 2023; Xie et al., 2023; Besta et al., 2024), such as Tree-of-Thoughts (Yao et al., 2023), and sampling-based methods (Wang et al., 2023; Xu et al., 2025; Wan et al., 2025; Ma et al., 2025), such as Self-Consistency (Wang et al., 2023). However, due to intrinsic hallucinations (Bubeck et al., 2023), LLMs often generate erroneous intermediate steps, which can ultimately lead to incorrect answers, especially on complex problems. This limitation highlights the need for inference-time optimization of reasoning processes.
To address this issue, one line of research (Yin et al., 2024; Chen et al., 2024a; Ling et al., 2023; Wu et al., 2024; Miao et al., 2024; Madaan et al., 2023) designs instructional prompts that guide LLMs to detect and refine their own mistakes. Despite its appeal, prior work has shown that the effectiveness of self-correction is limited in practice. Another line of work (Wang et al., 2024; Yuan et al., 2025; He et al., 2024; Havrilla et al., 2024) introduces external verifiers, such as process reward models (Snell et al., 2024), to identify and filter out error-prone reasoning steps. These methods typically require high-quality training data for the verifier, with data scarcity often mitigated through strategies such as Monte Carlo Tree Search (Guan et al., 2025; Qi et al., 2025; Li, 2025; Zhang et al., 2024). In addition, a recent line of decoding-based approaches (Xu et al., 2025; Ma et al., 2025) seeks to improve reasoning by dynamically adjusting the next-token prediction based on future trajectory probing. In contrast, our method refines reasoning by directly leveraging critical tokens, without relying on stepwise verification or external verifiers. This design underscores both the utility and universality of our approach.
Paraphrasing for LLMs. A growing number of work (Zhou et al., 2024; Chen et al., 2024b) has examined the impact of a problemâs surface form on the reasoning ability of LLMs. Findings (Zhou et al., 2024; Chen et al., 2024b; Huang et al., 2025) suggest that even subtle modifications in phrasing can substantially affect both the reasoning process and the final outcome. Building on this observation, several methods (Yadav et al., 2024; Chen et al., 2024b) leverage beneficial paraphrasing to enhance LLM performance in tasks such as reasoning and intent classification. In addition, paraphrasing has been employed to assess model uncertainty (Gao et al., 2024; Tanneru et al., 2024), thereby enhancing the reliability and trustworthiness of LLM applications. In our work, we utilize paraphrasing as a principled tool to extract critical tokens and to aggregate answers.
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<summary>x1.png Details</summary>
### Visual Description
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## Diagram: Paraphrastic Probing & Consistency Verification
### Overview
The image presents a diagram illustrating a two-phase process: "Paraphrastic Probing" and "Consistency Verification". The diagram details steps within each phase, demonstrating how a question is paraphrased, an initial answer is generated, and then the consistency of that answer is verified through token manipulation and trajectory generation. The diagram uses a flowchart-like structure with text boxes and arrows to show the flow of information.
### Components/Axes
The diagram is divided into two main phases, labeled "Phase I: Paraphrastic Probing" (left side) and "Phase II: Consistency Verification" (right side). Each phase is further broken down into three steps, numbered 1-3. The diagram also includes a "Re-ranking & Selection" section at the bottom. Text boxes contain descriptions of each step, and arrows indicate the flow of the process. Highlighted text within the boxes indicates critical tokens.
### Detailed Analysis or Content Details
**Phase I: Paraphrastic Probing**
* **Step 1: Paraphrase the original question.**
* Original Question: "A bakery produces 60 loaves of bread each day⊠How many loaves of bread are sold in the afternoon?"
* Paraphrased Question: "In a bustling bakery, daily production meets the demand for 60 freshly baked loaves⊠What is the number of loaves sold in the afternoon?"
* Highlighted Token: "during"
* **Step 2: Generate the initial response.**
* Initial Response: "A bakery produces 60 loaves of bread each day⊠How many loaves of bread are sold in the afternoon?"
* To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in⊠Therefore, the number of loaves of bread sold in the afternoon is 5."
* **Step 3: Concatenate the paraphrased question with the initial answer.**
* Concatenated Question/Answer: "In a bustling bakery, daily production meets the demand for 60 freshly baked loaves⊠What is the number of loaves sold in the afternoon? A bakery produces 60 loaves of bread each day⊠How many loaves of bread are sold in the afternoon? To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in⊠Therefore, the number of loaves of bread sold in the afternoon is 5."
* Highlighted Tokens: "5", "afternoon"
**Phase II: Consistency Verification**
* **Step 1: Obtain the candidate tokens at the critical token position.**
* Question: "A bakery produces 60 loaves of bread each day⊠How many loaves of bread are sold in the afternoon?"
* Highlighted Tokens: "during", "the"
* **Step 2: Truncate the initial answer and replace the critical token with the critical tokens.**
* Truncated Answer: "To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold inâŠ"
* **Step 3: Generate new trajectories from partial answers to both the original and paraphrased questions, and meet the same procedure as applied to the initial answer.**
* Original Question Trajectory: "Original) A bakery produces 60 loaves of bread each day⊠How many loaves of bread are sold in the afternoon? To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold inâŠ"
* Paraphrased Question Trajectory: "Paraphrased) In a bustling bakery, daily production meets the demand for 60 freshly baked loaves⊠What is the number of loaves sold in the afternoon? To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold inâŠ"
**Re-ranking & Selection (Bottom Section)**
* "Re-ranking & Selection: Re-rank the generated trajectories based on their likelihood and select the most consistent answer."
* Trajectory 1: "To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in the morning. The bakery produces 60 loaves of bread each day, two-thirds of the loaves are sold in the morning. Therefore, the number of loaves sold in the afternoon is 5."
* Trajectory 2: "To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in the morning. The bakery produces 60 loaves, and two-thirds of these loaves are sold in the morning⊠Therefore, the number of loaves of bread sold in the afternoon is 5."
* Trajectory 3: "To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in the morning. The bakery produces 60 loaves. Two-thirds of the loaves are sold in the morning. Therefore, the number of loaves sold in the afternoon is 5."
* Trajectory 4: "To solve this problem, we will break it down into steps. Step 1: Calculate the number of loaves sold in the morning. The bakery produces 60 loaves. Therefore, the number of loaves sold in the afternoon is 5."
* Final Answer: "Therefore, the number of loaves of bread sold in the afternoon is 5."
### Key Observations
The diagram highlights the importance of token manipulation in verifying the consistency of answers generated from both original and paraphrased questions. The repeated answer of "5" suggests this is the expected solution to the problem. The process involves breaking down the problem into steps and using partial answers to assess consistency.
### Interpretation
This diagram illustrates a method for ensuring the robustness of question-answering systems. By paraphrasing the original question and then verifying the consistency of the answer through token replacement and trajectory generation, the system can mitigate the effects of subtle variations in wording. The process aims to identify answers that are not only correct but also stable across different phrasings of the same question. The focus on "critical tokens" suggests that certain words or phrases are more important for maintaining consistency than others. The final selection of the most consistent answer indicates a preference for solutions that are less sensitive to paraphrasing. The diagram demonstrates a sophisticated approach to natural language understanding and reasoning, aiming to build more reliable and trustworthy AI systems. The repeated calculation of 2/3 of 60 loaves being sold in the morning, and the subsequent calculation of the remaining loaves being sold in the afternoon, is a key element of the problem-solving process.
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Figure 4: Illustration of the proposed paraphrastic probing and consistency verification (PPCV) framework. The framework comprises two stages: (i) probing critical tokens through paraphrased forms, and (ii) rolling out new reasoning steps with alternative tokens and selecting the final answer using the paraphrase consistency verification mechanism.
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<summary>images/probe_illustration.png Details</summary>
### Visual Description
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## Textual Document: Case Study on Identification and Effects of Critical Tokens
### Overview
The image presents a case study examining the impact of "critical tokens" in solving mathematical word problems. It compares a response generated with highlighted candidate critical tokens to a new response initialized from those tokens. The study focuses on two similar word problems involving points earned in games and rounds, and demonstrates a step-by-step solution process.
### Components/Axes
The document is structured into sections:
* **Original Question:** A word problem about a card game.
* **Paraphrased Question:** A reworded version of the original problem, using a bridge competition context.
* **Response with Highlighted Critical Tokens:** A solution attempt with specific words highlighted.
* **New Responses Initialized from Critical Token Position:** A revised solution based on the identified critical tokens.
* **Step-by-Step Solutions:** Detailed steps for both approaches, with equations and calculations.
### Content Details
**Original Question:**
"In a card game, you are awarded 10 points if you win one round. While playing, Jane ends up with 60 points. If she lost 20 points, then how many rounds did she play? (ground truth answer: 8)"
**Paraphrased Question:**
"At a bridge competition, every game won grants the player 10 points. Unfortunately, Jane started with an advantage but ended up losing 20 points and scoring only 60 points. If she had won x rounds and lost none, what would be the value of x? (ground truth answer: 8)"
**Response with Highlighted Critical Tokens:**
"To solve this problem, we need to follow [find] these steps:
Step 1: Determine [Understand] the total points Jane earned [scored] by [.] subtracting [winning] the points she lost from her [the] final score. Total points earned = Final score - [â ] Points lost. Total points earned = 60 - 20. Total points earned = 40.
Step 2: Since Jane earns 10 points for [per] each round she [won] wins, we can find the number of rounds played by dividing the total points earned by the points earned per round. Number of rounds = Total points earned / Points earned per round. Number of rounds = 40 / 10. Number of rounds = 4.
Therefore, Jane played 4 rounds. â"
**New Responses Initialized from Critical Token Position:**
"To solve this problem, we need to follow these steps:
Step 1: Determine the total points Jane earned by winning games. Since Jane scored 60 points and lost 20 points, we can set up an equation to represent the situation. Total points = Points earned from winning + Points lost. We know that Jane lost 20 points, so we can subtract this from the total points to find the points earned from winning. 60 = Points earned from winning - 20.
Step 2: Solve for the points earned from winning. Add 20 to both sides of the equation to isolate the points earned from winning. 60 + 20 = Points earned from winning. 80 = Points earned from winning.
Step 3: Determine the number of games won. Since Jane earns 10 points for each game won, we can divide the points earned from winning by 10 to find the number of games won. Number of games won = Points earned from winning / 10 = 8."
### Key Observations
* The initial response incorrectly calculates the total points earned (40 instead of 80) due to a misinterpretation of the problem.
* The highlighted "critical tokens" (e.g., "Understand", "find", "per") seem to guide the revised response towards a more accurate solution.
* The revised response uses a more formal equation-based approach, which clarifies the problem-solving process.
* The original response is marked with a red "â", indicating an incorrect answer.
### Interpretation
This case study demonstrates the potential of identifying and utilizing "critical tokens" to improve the accuracy and clarity of solutions to mathematical word problems. The initial response, while attempting to follow a logical process, falters due to a misunderstanding of the problem's core elements. By focusing on key terms and relationships (highlighted as critical tokens), the revised response is able to construct a correct and well-explained solution. The use of equations in the revised response further enhances the clarity and rigor of the solution. The study suggests that incorporating critical token analysis into problem-solving strategies could be a valuable approach for both students and automated problem-solving systems. The difference in the final answer (4 vs. 8) highlights the significant impact of accurate problem interpretation. The use of both a card game and bridge competition context suggests the method is robust to changes in the problem's framing.
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Figure 5: Case study illustrating the identification and effects of critical tokens identified by our method. Tokens highlighted in red indicate candidate critical tokens, whereas tokens highlighted in purple correspond to alternative tokens generated when conditioning on paraphrased questions.
## 3 Methodology
In this section, we present the two components of our framework in detail: paraphrastic probing and consistency verification. An detailed illustration of our framework is shown in Figure 4. We then discuss the proposed method and provide the complete algorithm.
### 3.1 Paraphrastic Probing
Previous findings (Zhou et al., 2024; Chen et al., 2024b; Huang et al., 2025) on the impact of a problemâs surface form suggest that the quality of intermediate reasoning steps is influenced not only by the underlying mathematical relationships and logic, but also by how the problem is expressed. Notably, LLMs are sometimes able to solve a paraphrased version of a problem that they fail to solve in its original form, highlighting the potential of paraphrasing to uncover pivotal tokens that are critical for successful reasoning. Motivated by this observation, we introduce paraphrastic probing to efficiently identify the critical token. Given the original question $q_{0}$ , we first prompt the LLM to generate multiple paraphrased forms, denoted as $q_{1},q_{2},...,q_{N}$ , where $N$ is the number of the paraphrased questions. We adopt Automatic Prompt Engineering (APE) (Zhou et al., 2022) to derive paraphrasing instructions that preserve semantic integrity and all numerical values, mathematical relationships, and core logical structures of the problem, while maximizing linguistic and contextual diversity. Additional details can be found in Appendix B. We then obtain the initial reasoning path $r^{q_{0}}_{0}$ for the original question using greedy decoding. This reasoning path is subsequently concatenated with each paraphrased question, and the resulting synthetic inputs are fed into the LLM to compute the token probability distribution at each position in $r^{q_{0}}_{0}$ . Specifically, the token probability distribution at $i$ th position conditioned on the paraphrased question $q_{n}$ is expressed as
$$
P_{i}^{q_{n}}=\text{LLM}(\tilde{a}_{i}|\mathcal{I},q_{n},r^{q_{0}}_{0,<i}), \tag{1}
$$
where $\mathcal{I}$ denotes the instruction prefix and $\tilde{a}_{i}$ represents the sampled token at $i$ th position. The token $\tilde{a}_{i}$ is regarded as a candidate critical token if predicted top-1 token does not match the expected token at the same position in $r^{q_{0}}_{0}$ , i.e.,
$$
\operatorname*{arg\,max}P_{i}^{q_{n}}\neq a_{i}, \tag{2}
$$
where $a_{i}$ denotes the token at the $i$ th position in $r^{q_{0}}_{0}$ .
<details>
<summary>images/condition_one_gsm8k.png Details</summary>
### Visual Description
\n
## Bar Chart: Fraction vs. Average Accuracy for Critical and Random Tokens
### Overview
This bar chart compares the fraction of tokens (expressed as a percentage) for "critical tokens" and "random tokens" based on their average accuracy (also expressed as a percentage). The chart displays two groups of bars, one for tokens with average accuracy less than or equal to 5% and another for tokens with average accuracy greater than 5%. Error bars are included for each bar, indicating the variability or uncertainty in the fraction values.
### Components/Axes
* **X-axis:** "Average accuracy (%)" with two categories: "†5%" and "> 5%".
* **Y-axis:** "Fraction (%)" ranging from 0 to 70.
* **Legend:** Located in the top-right corner, distinguishing between "critical tokens" (represented by a teal color) and "random tokens" (represented by a light grey color).
* **Bars:** Represent the fraction of tokens for each category and token type.
* **Error Bars:** Black lines extending vertically from the top of each bar, indicating the standard error or confidence interval.
### Detailed Analysis
The chart presents four bars with associated error bars.
**For Average Accuracy †5%:**
* **Critical Tokens:** The bar is teal and reaches approximately 68% on the Y-axis. The error bar extends from approximately 63% to 73%.
* **Random Tokens:** The bar is light grey and reaches approximately 32% on the Y-axis. The error bar extends from approximately 27% to 37%.
**For Average Accuracy > 5%:**
* **Critical Tokens:** The bar is teal and reaches approximately 30% on the Y-axis. The error bar extends from approximately 25% to 35%.
* **Random Tokens:** The bar is light grey and reaches approximately 66% on the Y-axis. The error bar extends from approximately 61% to 71%.
### Key Observations
* When average accuracy is †5%, critical tokens have a significantly higher fraction than random tokens.
* When average accuracy is > 5%, random tokens have a significantly higher fraction than critical tokens.
* The error bars suggest a reasonable degree of uncertainty in the fraction estimates, but the differences between critical and random tokens within each accuracy category appear substantial.
### Interpretation
The data suggests a strong inverse relationship between average accuracy and the fraction of critical versus random tokens. Tokens with low average accuracy (†5%) are predominantly critical tokens, while tokens with high average accuracy (> 5%) are predominantly random tokens. This could indicate that critical tokens are more challenging to predict or classify accurately, perhaps due to their inherent complexity or importance in the context. The higher fraction of random tokens with higher accuracy suggests they are easier to predict or classify.
The chart implies that focusing on improving the accuracy of critical tokens might be a key area for improvement, as they currently exhibit lower accuracy compared to random tokens. The error bars indicate that these differences are likely statistically significant, but further analysis would be needed to confirm this. The data could be used to inform strategies for model training or data selection, prioritizing critical tokens to enhance overall performance.
</details>
(a)
<details>
<summary>images/condition_two_gsm8k.png Details</summary>
### Visual Description
\n
## Bar Chart: Fraction vs. Average Accuracy for Token Types
### Overview
This bar chart compares the fraction of tokens (expressed as a percentage) for "critical tokens" and "random tokens" across two categories of average accuracy (also expressed as a percentage): "†10%" and "> 10%". Error bars are included for each data point, indicating the variability or confidence interval.
### Components/Axes
* **X-axis:** "Average accuracy (%)" with two categories: "†10%" and "> 10%".
* **Y-axis:** "Fraction (%)" ranging from 0% to 60%.
* **Legend:** Located in the top-right corner, distinguishing between:
* "critical tokens" (represented by a teal/green color)
* "random tokens" (represented by a light teal/green color)
* **Error Bars:** Vertical lines extending above each bar, representing the standard error or confidence interval.
### Detailed Analysis
The chart presents four data series, two for each accuracy category, and each series has an associated error bar.
**1. Average Accuracy †10%**
* **Critical Tokens:** The bar for critical tokens is approximately 62% tall, with an error bar extending to approximately 65%.
* **Random Tokens:** The bar for random tokens is approximately 40% tall, with an error bar extending to approximately 43%.
**2. Average Accuracy > 10%**
* **Critical Tokens:** The bar for critical tokens is approximately 36% tall, with an error bar extending to approximately 39%.
* **Random Tokens:** The bar for random tokens is approximately 60% tall, with an error bar extending to approximately 63%.
### Key Observations
* For tokens with average accuracy †10%, the fraction of critical tokens is significantly higher than that of random tokens.
* For tokens with average accuracy > 10%, the fraction of random tokens is significantly higher than that of critical tokens.
* The error bars suggest a relatively low degree of uncertainty in the measurements.
### Interpretation
The data suggests a strong inverse relationship between average accuracy and the fraction of critical tokens. When average accuracy is low (†10%), critical tokens are more prevalent. Conversely, when average accuracy is high (> 10%), random tokens are more prevalent. This could indicate that critical tokens are more likely to be associated with errors or lower-performing elements, while random tokens are more likely to be associated with successful or higher-performing elements.
The chart implies that identifying and focusing on critical tokens might be a useful strategy for improving overall accuracy, particularly in scenarios where accuracy is initially low. The shift in prevalence between token types as accuracy increases suggests that addressing the issues related to critical tokens leads to improved performance, and they become less dominant as the system improves. The error bars indicate that these trends are relatively robust and not likely due to random chance.
</details>
(b)
<details>
<summary>images/compare_ct_rnd_gsm8k.png Details</summary>
### Visual Description
\n
## Line Chart: Pass@k vs. Number of Sample k
### Overview
This line chart compares the "pass@k%" metric for two categories: "critical tokens" and "random tokens" as a function of the "number of sample k". Error bars are included for each data point, indicating the variability or confidence interval.
### Components/Axes
* **X-axis:** "number of sample k". Scale ranges from approximately 0 to 45, with markers at 0, 10, 20, 30, and 40.
* **Y-axis:** "pass@k%". Scale ranges from approximately 50% to 87%, with markers at 50%, 55%, 60%, 65%, 70%, 75%, 80%, and 85%.
* **Data Series 1:** "critical tokens" - Represented by a red line with triangular markers and error bars.
* **Data Series 2:** "random tokens" - Represented by a purple line with square markers and error bars.
* **Legend:** Located in the bottom-right corner, clearly labeling each data series with its corresponding color.
### Detailed Analysis
**Critical Tokens (Red Line):**
The line representing "critical tokens" slopes generally upward, indicating an increasing "pass@k%" with increasing "number of sample k".
* At k = 0, pass@k% is approximately 71% ± 4%.
* At k = 10, pass@k% is approximately 78% ± 3%.
* At k = 20, pass@k% is approximately 82% ± 2%.
* At k = 30, pass@k% is approximately 84% ± 2%.
* At k = 40, pass@k% is approximately 85% ± 2%.
**Random Tokens (Purple Line):**
The line representing "random tokens" also slopes upward, but at a slower rate than the "critical tokens" line.
* At k = 0, pass@k% is approximately 52% ± 5%.
* At k = 10, pass@k% is approximately 58% ± 4%.
* At k = 20, pass@k% is approximately 61% ± 3%.
* At k = 30, pass@k% is approximately 63% ± 3%.
* At k = 40, pass@k% is approximately 65% ± 4%.
### Key Observations
* "Critical tokens" consistently achieve a higher "pass@k%" than "random tokens" across all values of "number of sample k".
* The difference in "pass@k%" between the two categories appears to be more pronounced at lower values of "number of sample k".
* The error bars suggest that the "critical tokens" data has slightly less variability than the "random tokens" data.
* Both lines appear to be approaching a plateau as "number of sample k" increases, suggesting diminishing returns.
### Interpretation
The data suggests that using "critical tokens" leads to a significantly higher "pass@k%" compared to using "random tokens". This implies that selecting tokens based on their importance or criticality is a more effective strategy for achieving a desired level of performance (as measured by "pass@k%"). The diminishing returns observed at higher values of "number of sample k" suggest that there is a point beyond which increasing the sample size provides minimal improvement. The smaller error bars for "critical tokens" indicate a more consistent and reliable performance compared to "random tokens". This chart likely represents the results of an experiment evaluating different token selection strategies in a machine learning or natural language processing context, where "pass@k%" is a metric for evaluating the quality of generated outputs. The "number of sample k" likely refers to the number of candidate tokens considered during the generation process.
</details>
(c)
<details>
<summary>images/density_gsm8k.png Details</summary>
### Visual Description
\n
## Histogram: Density of Consistency Score with Correct/Incorrect Answers
### Overview
The image presents a histogram comparing the density distribution of "consistency score" for two groups: those with "incorrect answers" and those with "correct answers". The x-axis represents the consistency score, and the y-axis represents the density (expressed as a percentage).
### Components/Axes
* **X-axis Title:** "consistency score"
* **X-axis Range:** 0 to 5
* **Y-axis Title:** "density (%)"
* **Y-axis Range:** 0 to 0.7
* **Legend:** Located in the top-right corner.
* "w incorrect answers" - represented by a light green color.
* "w correct answers" - represented by a light red color.
### Detailed Analysis
The histogram displays the distribution of consistency scores for each group.
**Incorrect Answers (Light Green):**
The distribution is heavily skewed towards lower consistency scores.
* Around a consistency score of 0, the density is approximately 0.68.
* The density decreases rapidly as the consistency score increases.
* Between consistency scores of 0 and 1, the density remains high, around 0.65-0.7.
* At a consistency score of 1, the density is approximately 0.65.
* At a consistency score of 2, the density drops to approximately 0.15.
* The density remains low for consistency scores greater than 2, fluctuating around 0.05-0.1.
**Correct Answers (Light Red):**
The distribution is skewed towards higher consistency scores.
* Around a consistency score of 5, the density is approximately 0.55.
* The density is relatively low for consistency scores below 3.
* At a consistency score of 0, the density is approximately 0.02.
* At a consistency score of 1, the density is approximately 0.03.
* At a consistency score of 2, the density is approximately 0.05.
* At a consistency score of 3, the density increases to approximately 0.12.
* At a consistency score of 4, the density increases to approximately 0.18.
* At a consistency score of 5, the density reaches approximately 0.55.
### Key Observations
* The distribution of consistency scores is significantly different for incorrect and correct answers.
* Incorrect answers tend to have lower consistency scores, while correct answers tend to have higher consistency scores.
* There is a clear separation between the two distributions, suggesting that consistency score is a useful indicator of answer correctness.
* The peak density for incorrect answers is around 0, while the peak density for correct answers is around 5.
### Interpretation
The data suggests a strong correlation between consistency score and answer correctness. Higher consistency scores are associated with correct answers, and lower consistency scores are associated with incorrect answers. This implies that the "consistency score" metric effectively measures the reliability or coherence of the reasoning process leading to an answer.
The stark difference in distributions indicates that the consistency score can be used as a predictor of answer accuracy. A model or system could potentially use this score to identify and flag potentially incorrect answers for review. The fact that the distributions are not entirely separated suggests that consistency score is not a perfect predictor, and some incorrect answers may still have relatively high consistency scores, and vice versa. Further investigation might be needed to understand the reasons for these outliers. The distributions are not normal, but rather heavily skewed, which suggests that the consistency score may be bounded or influenced by other factors.
</details>
(d)
Figure 6: (a) Comparison of the fraction of tokens whose associated rollout correctness falls below or above 5%, for tokens extracted by our method versus randomly selected tokens. (b) Comparison of the fraction of the subsequent tokens whose associated rollout correctness falls below or above 10%, for tokens extracted by our method versus randomly selected tokens. (c) Comparison of the impact of extracted tokens by our method versus random tokens on LLM reasoning performance. (d) Comparison of the density distributions of consistency scores for rollouts with correct and incorrect answers.
To validate the effectiveness of our method in identifying critical tokens and to demonstrate the pivotal role these tokens play when extracted via paraphrastic probing, we conduct a case study illustrated in Figure 5. Because large language models can adjust their reasoning trajectories under the paraphrased form of the question, certain tokens in the original reasoning sequence produce different subsequent tokens when conditioned on paraphrased inputs. In this example, most candidate critical tokens are located within incorrect reasoning segments. Notably, when we identify âsubtractingâ as a critical token and replace it with an alternative token (i.e., âwinningâ), the resulting rollout corrects the erroneous reasoning steps and yields the correct final answer. This example highlights the impact of critical tokens and underscores the effectiveness of our method in identifying tokens that are pivotal to reasoning outcomes.
In addition, we conduct a quantitative analysis to examine the authenticity and impact of tokens extracted via paraphrastic probing, comparing them against randomly selected tokens. Specifically, we sample 100 instances with incorrect reasoning trajectories from the GSM8K (Cobbe et al., 2021) training sets. Following the paraphrastic probing pipeline, we identify candidate critical tokens in the early reasoning steps. In each run, we randomly sample 40% of the candidate critical tokens for evaluation and repeat the experiment independently 10 times. For comparison, we apply the same procedure to randomly selected tokens. All evaluations are conducted using Llama-3.1-8B-Instruct (Grattafiori et al., 2024). We first compute the average accuracy of rollouts generated from partial trajectories truncated at the position of the extracted tokens, and compare these results with those obtained from random-token truncation. As shown in Figure 6(a), a large proportion of the extracted tokens exhibit near-zero correctness, consistent with the first criterion of critical tokens. We further evaluate the average accuracy of rollouts initiated from subsequent tokens (5â10 tokens). The results in Figure 6(b) indicate that errors persist beyond the identified positions, supporting the second criterion that critical tokens induce sustained degradation in downstream reasoning. Finally, we substitute the extracted critical tokens with alternative tokens and roll out multiple new reasoning trajectories. As shown in Figure 6(c), replacing critical tokens leads to a significantly larger improvement in reasoning accuracy compared to substituting random tokens. These results further validate both the pivotal role of the identified critical tokens and the effectiveness of our paraphrastic probing method. Additional results can be found in Appendix A.
We introduce a heuristic verifier to select the final critical token from multiple candidates. For a candidate token $a_{i}$ and paraphrased question $q_{n}$ , the verification score is defined as
$$
\Delta_{q_{n}}(a_{i})=\max P^{q_{n}}_{i}-P^{q_{n}}_{i}(\tilde{a}_{i}=a_{i}). \tag{3}
$$
where $P^{q_{n}}_{i}$ denotes the predictive distribution at position $i$ on question $q_{n}$ . Intuitively, $\Delta$ measures how much the predicted top-1 token deviates from the expected token, indicating the tokenâs potential impact on the reasoning trajectory. For each extracted token $a_{i}$ ,we take the maximum score across paraphrases,
$$
\Delta(a_{i})=\max_{q_{n}}\Delta_{q_{n}}(a_{i}),\vskip-5.69054pt \tag{4}
$$
and select the final critical token as
$$
a_{c}=\operatorname*{arg\,max}_{i}\Delta(a_{i}).\vskip-5.69054pt \tag{5}
$$
### 3.2 Consistency Verification
After identifying the final critical token $a_{c}$ , we aim to refine the original reasoning path with alternative tokens and achieve final answer with paraphrase consistency mechanism. Specifically, we generate a set of alternative tokens $a^{0}_{c},a^{1}_{c},a^{2}_{c},...,a^{K-1}_{c}$ using the LLM conditioned on original question $q_{0}$ , where $a^{0}_{c}$ is the original token in $r^{q_{0}}_{0}$ and the remaining tokens are sampled via top-K sampling. The initial reasoning path is truncated at the position of critical token, and each alternative token is concatenated to form synthetic inputs $\tilde{r}_{c}^{0},\tilde{r}_{c}^{1},\tilde{r}_{c}^{2},...,\tilde{r}_{c}^{K-1}$ . We then roll out new reasoning trajectories for each synthetic input with respect to both the original and paraphrased questions using greedy decoding, denoted as $r^{q_{0}}_{k},r^{q_{1}}_{k},...,r^{q_{N}}_{k}$ for $k=0,1,2,...,K-1$ . Next, for the rollout with the $k$ th alternative token, we compare the answers obtained from the paraphrased forms with that from the original form and compute a consistency score $c_{k}=\sum_{n-1}^{N}\mathbb{I}(\Phi(r^{q_{0}}_{k})=\Phi(r^{q_{n}}_{k}))$ , where $\Phi(\cdot)$ and $\mathbb{I}(\cdot)$ denotes the function that extracts the final answer from a reasoning trajectory and the indicator function, respectively. The answer associated with the highest consistency score is then selected as the final prediction
$$
\text{ans}_{f}=\Phi(r^{q_{0}}_{k}),\text{where}\,k=\operatorname*{arg\,max}_{k}c_{k}.\vskip-11.38109pt \tag{6}
$$
To justify our paraphrase consistency mechanism, we investigate the impact of paraphrased forms on LLM reasoning. We sample instances from the GSM8K (Cobbe et al., 2021) and follow our pipeline to extract critical tokens. From each truncated reasoning trajectory, we roll out multiple reasoning paths by concatenating alternative tokens. For each original question, we generate five paraphrased variants and compute the consistency score for resulting rollouts. The evaluation is conducted on Llama-3.1-8B-Instruct (Grattafiori et al., 2024). We then analyze the distribution of consistency scores for rollouts that yield correct versus incorrect answers. As shown in Figure 6(d), more than 90% of rollouts with correct answers achieve a consistency score of at least 1, whereas this proportion drops to around 30% for rollouts with incorrect answers. This sharp contrast indicates that correct rollouts are more robust across paraphrased variants, motivating the design of our paraphrase consistency mechanism to exploit this property for improved final predictions.
To address potential collisions when multiple answers obtain the same maximum consistency score, we introduce similarity-weighted consistency verification. Inspired by weighted majority voting (Dogan and Birant, 2019), this approach adjusts the influence of each paraphrased form on the consistency score according to its similarity to the original form. Intuitively, paraphrased forms with lower similarity should exert greater weight, as they provide stronger evidence of robustness, whereas those closely resembling the original form contribute less. Concretely, we first extract embeddings for both the original and paraphrased questions and compute their similarity scores as $s_{n}=\text{sim}(q_{0},q_{n})$ , where $\text{sim}(\cdot)$ denotes a similarity measure such as cosine similarity. We then derive weights via a softmax function $w_{n}=\text{softmax}(s_{n})=\frac{\exp(-\lambda s_{n})}{\sum_{n}\exp(-\lambda n)}$ , where $\lambda$ is the temperature scaling coefficient. Finally, the similarity-weighted consistency score is defined as $\tilde{c}_{k}=\sum_{n-1}^{N}w_{n}\mathbb{I}(\Phi(r^{q_{0}}_{k})=\Phi(r^{q_{n}}_{k}))$ . This ensures agreement with more diverse paraphrases contributes more strongly to the final decision.
Table 1: Comparison of our method with baseline approaches on Llama-3.1-8B-Instruct and Mistral-7B-Instruct-v0.2.
| Llama-3.1 Self-Consistency Tree-of-Thought | Chain-of-Thought 80.60 75.74 | 77.40 31.80 33.28 | 28.00 37.80 31.60 | 31.00 85.10 81.20 | 83.00 60.75 80.72 | 58.91 |
| --- | --- | --- | --- | --- | --- | --- |
| Guided Decoding | 75.51 | 32.45 | 31.20 | 81.70 | 81.74 | |
| Predictive Decoding | 81.43 | 40.26 | 34.00 | 85.90 | 84.56 | |
| Phi-Decoding | 86.58 | 39.88 | 38.20 | 84.50 | 85.41 | |
| PPCV (Ours) | 88.24 | 49.73 | 50.00 | 89.60 | 88.31 | |
| Mistral-7B | Chain-of-Thought | 46.45 | 26.91 | 12.20 | 62.40 | 41.42 |
| Self-Consistency | 50.38 | 28.65 | 14.20 | 66.70 | 44.54 | |
| Tree-of-Thought | 50.49 | 25.78 | 11.40 | 60.60 | 41.04 | |
| Guided Decoding | 50.79 | 27.07 | 14.00 | 62.90 | 39.51 | |
| Predictive Decoding | 55.67 | 27.07 | 14.40 | 62.10 | 47.87 | |
| Phi-Decoding | 56.60 | 28.43 | 13.40 | 63.20 | 60.24 | |
| PPCV (Ours) | 56.58 | 31.08 | 14.60 | 69.30 | 69.88 | |
Table 2: Comparison of our method with baseline approaches on Qwen3-32B (non-thinking mode).
| Qwen3-32B Guided Decoding Predictive Decoding | Chain-of-Thought 26.67 32.67 | 30.00 22.67 24.00 | 23.67 28.67 33.33 | 30.00 7.33 10.33 | 9.67 |
| --- | --- | --- | --- | --- | --- |
| Phi-Decoding | 33.60 | 24.33 | 36.67 | 10.67 | |
| PPCV (Ours) | 40.00 | 26.00 | 43.33 | 13.33 | |
### 3.3 Discussion
Our technical contributions differ from prior works in three distinct ways. First, prior works (Zhou et al., 2024; Chen et al., 2024b; Yadav et al., 2024) typically use paraphrasing merely to expand the solution space. In contrast, we introduce Paraphrastic Probing, a mechanism that uses paraphrasing to test the modelâs internal confidence. By analyzing the discrepancy in token-level logits of the initial trajectory between the original and paraphrased questions, we can rigorously pinpoint the critical tokens that may lead to errors in the following steps.This transforms paraphrasing from a generation tool into a precise, token-level diagnostic tool. Second, prior works (Zhou et al., 2024; Chen et al., 2024b) typically rely on simple majority voting across multiple solutions. Our Paraphrase Consistency mechanism is technically distinct. It validates answers based on their robustness across semantic variations of the problem constraint. We further introduce a similarity-weighted consistency metric that weighs answers based on the linguistic diversity of the paraphrase, offering a more nuanced selection criterion than simple frequency counts. At last, a major technical limitation in current reasoning research is the reliance on external models or human-annotated error steps. Our method contributes a fully self-contained pipeline that identifies and corrects errors using the modelâs own sensitivity to surface-form perturbations. More discussion on the impact of critical tokens on correct trajectory can be found in Appendix D.
Besides, although we select the top candidate for the primary experiments to maintain computational efficiency, the framework itself naturally extends to the multiâcritical-token setting. For multiple critical tokens, we can generate alternative tokens for each identified position and apply paraphrase consistency across the new rollouts. This allows the model to refine multiple segments of its intermediate reasoning steps rather than only one. The details of the algorithm can be found in Appendix C.
Table 3: Comparison of model performance when using critical tokens versus random tokens.
| Chain-of-Thought random tokens critical tokens (Ours) | 77.40 82.08 88.24 | 28.00 40.29 49.73 | 31.00 42.12 50.00 | 83.00 84.77 89.60 | 58.91 75.68 88.31 |
| --- | --- | --- | --- | --- | --- |
Table 4: Comparison of our proposed paraphrase consistency against the majority voting.
| Chain-of-Thought majority voting paraphrase consistency (Ours) | 77.40 87.20 88.24 | 28.00 47.36 49.73 | 31.00 48.19 50.00 | 83.00 88.80 89.60 | 58.91 86.16 88.31 |
| --- | --- | --- | --- | --- | --- |
## 4 Experiments
In this section, we first describe the experimental setup, followed by the main results of our proposed method compared to the baselines. We also perform ablation study and computational cost analysis.
### 4.1 Setup
Datasets. To comprehensively assess our method, we evaluate it on seven benchmarks. Six focus on mathematical reasoning, including GSM8K (Cobbe et al., 2021), GSM-Hard (Gao et al., 2023), SVAMP (Patel et al., 2021), Math500 (Hendrycks et al., 2021), and the more challenging competition-level datasets AIME2024, AIME2025, BRUMO2025, and HMMT2025 (BalunoviÄ et al., 2025). In addition, we use ARC-Challenge (Clark et al., 2018) to evaluate knowledge reasoning ability of LLMs.
Baselines. In our experiments, we use Chain-of-Thought (CoT) (Wei et al., 2022), Self-Consistency (Wang et al., 2023), Tree-of-Thought (ToT) (Yao et al., 2023), Guided Decoding (Xie et al., 2023), Predictive Decoding (Ma et al., 2025), and Phi-Decoding (Xu et al., 2025) as baseline methods.
Metric. Following prior work, we adopt pass@k (k=1,4) as the primary evaluation metric.
Implementation Details. In our experiments, we adopt Llama-3.1-8B-Instruct (Grattafiori et al., 2024), Mistral-7B-Instruct-v0.2 (Jiang et al., 2023), Qwen-3-32B (Yang et al., 2025a) and DeepSeek-R1-Distill-Llama-70B as the target models. we employ the non-thinking mode for Qwen-3-32B. Throughout our method, we employ the same model for generating paraphrased problems, identifying critical tokens, and producing new rollouts. In the first stage, we generate 4 paraphrased variants for each problem in the math benchmarks and 3 variants for each problem in the ARC dataset. In the second stage, we select the top 10 tokens for new rollouts, with the temperature scaling coefficient $\lambda$ set to 2. For fair comparison, we ensure a comparable inference budget across methods. Specifically, we rollout 48 samples for Self-Consistency (Wang et al., 2023). For Predictive Decoding (Ma et al., 2025) and Phi-Decoding (Xu et al., 2025), we rollout 4-8 samples per foresight step, and each problem typically involves 5â8 foresight steps. We also adopt a zero-shot CoT prompt to elicit the new rollouts. For the baselines, we strictly follow their original settings, including temperature values, sampling strategies, and the number of few-shot examples. All experiments are conducted on NVIDIA A100 GPUs.
<details>
<summary>images/topk.png Details</summary>
### Visual Description
## Line Chart: Pass@1 Performance vs. Number of Alternative Tokens
### Overview
This line chart compares the Pass@1 performance of two models, GSM8K and SVAMP, as the number of alternative tokens increases from 3 to 10. The y-axis represents the Pass@1 accuracy (in percentage), and the x-axis represents the number of alternative tokens considered.
### Components/Axes
* **X-axis Title:** "number of alternative tokens"
* Scale: 3 to 10, with increments of 1.
* **Y-axis Title:** "pass@1 (%)"
* Scale: 80 to 94, with increments of 2.
* **Legend:** Located in the top-right corner.
* GSM8K: Represented by a yellow line with square markers.
* SVAMP: Represented by a teal line with circular markers.
* **Gridlines:** Present in both horizontal and vertical directions, aiding in value estimation.
### Detailed Analysis
**GSM8K (Yellow Line):**
The line representing GSM8K shows a slight upward trend, but plateaus after 7 alternative tokens.
* At 3 alternative tokens: Approximately 84.5% Pass@1.
* At 4 alternative tokens: Approximately 85.5% Pass@1.
* At 5 alternative tokens: Approximately 87% Pass@1.
* At 6 alternative tokens: Approximately 87.5% Pass@1.
* At 7 alternative tokens: Approximately 88% Pass@1.
* At 8 alternative tokens: Approximately 88% Pass@1.
* At 9 alternative tokens: Approximately 88% Pass@1.
* At 10 alternative tokens: Approximately 88% Pass@1.
**SVAMP (Teal Line):**
The line representing SVAMP shows a more consistent upward trend throughout the range of alternative tokens.
* At 3 alternative tokens: Approximately 87% Pass@1.
* At 4 alternative tokens: Approximately 87.5% Pass@1.
* At 5 alternative tokens: Approximately 88% Pass@1.
* At 6 alternative tokens: Approximately 88.5% Pass@1.
* At 7 alternative tokens: Approximately 88.5% Pass@1.
* At 8 alternative tokens: Approximately 89% Pass@1.
* At 9 alternative tokens: Approximately 89.5% Pass@1.
* At 10 alternative tokens: Approximately 90% Pass@1.
### Key Observations
* SVAMP consistently outperforms GSM8K across all tested numbers of alternative tokens.
* The performance gain for GSM8K diminishes significantly after 7 alternative tokens, suggesting a saturation point.
* SVAMP shows a more sustained improvement in Pass@1 as the number of alternative tokens increases.
* The difference in performance between the two models is relatively small, but noticeable.
### Interpretation
The chart demonstrates the impact of increasing the number of alternative tokens on the Pass@1 accuracy of two language models, GSM8K and SVAMP. The results suggest that while both models benefit from considering more alternatives, SVAMP is more effectively utilizing this increased search space. The plateauing performance of GSM8K indicates that its ability to leverage additional tokens is limited, potentially due to architectural constraints or training data characteristics. The consistent improvement of SVAMP suggests a more robust mechanism for exploring and selecting optimal solutions from a larger set of possibilities. This could be due to differences in model size, training methodology, or the specific tasks they were trained on. The data suggests that for maximizing Pass@1 accuracy, increasing the number of alternative tokens is a beneficial strategy, particularly for models like SVAMP that can effectively exploit this expanded search space.
</details>
(a)
<details>
<summary>images/time.png Details</summary>
### Visual Description
\n
## Bar Chart: Latency Comparison of Decoding Methods
### Overview
This bar chart compares the latency (in seconds) of several decoding methods â Chain-of-Thought, Predictive Decoding, Phi-Decoding, and four variations of PPCV (PPCV-T1 to PPCV-T4) â across five different datasets: GSM8K, GSMHard, Math500, SVAMP, and ARC. The chart uses stacked bars to represent the contribution of each decoding method to the total latency for each dataset.
### Components/Axes
* **X-axis:** Datasets - GSM8K, GSMHard, Math500, SVAMP, ARC.
* **Y-axis:** Latency (s) - Scale ranges from 0 to 40 seconds, with increments of 5 seconds.
* **Legend (Top-Right):**
* Chain-of-Thought (Light Teal)
* Predictive Decoding (Medium Teal)
* Phi-Decoding (Light Orange)
* PPCV-T1 (Ours) (Medium Orange)
* PPCV-T2 (Ours) (Dark Orange)
* PPCV-T3 (Ours) (Yellow)
* PPCV-T4 (Ours) (Pink)
### Detailed Analysis
Here's a breakdown of the latency values for each dataset and decoding method, based on the bar heights. Note that these are approximate values read from the chart.
* **GSM8K:**
* Chain-of-Thought: ~15s
* Predictive Decoding: ~2s
* Phi-Decoding: ~0.5s
* PPCV-T1: ~1s
* PPCV-T2: ~0.5s
* PPCV-T3: ~0.2s
* PPCV-T4: ~0.2s
* Total: ~19.4s
* **GSMHard:**
* Chain-of-Thought: ~24s
* Predictive Decoding: ~3s
* Phi-Decoding: ~1s
* PPCV-T1: ~1.5s
* PPCV-T2: ~0.5s
* PPCV-T3: ~0.3s
* PPCV-T4: ~0.3s
* Total: ~30.6s
* **Math500:**
* Chain-of-Thought: ~42s
* Predictive Decoding: ~2s
* Phi-Decoding: ~0.5s
* PPCV-T1: ~1.5s
* PPCV-T2: ~0.5s
* PPCV-T3: ~0.2s
* PPCV-T4: ~0.2s
* Total: ~46.9s
* **SVAMP:**
* Chain-of-Thought: ~13s
* Predictive Decoding: ~2s
* Phi-Decoding: ~0.5s
* PPCV-T1: ~1.5s
* PPCV-T2: ~0.5s
* PPCV-T3: ~0.2s
* PPCV-T4: ~0.2s
* Total: ~17.9s
* **ARC:**
* Chain-of-Thought: ~15s
* Predictive Decoding: ~2s
* Phi-Decoding: ~0.5s
* PPCV-T1: ~1.5s
* PPCV-T2: ~0.5s
* PPCV-T3: ~0.2s
* PPCV-T4: ~0.2s
* Total: ~19.9s
**Trends:**
* Chain-of-Thought consistently contributes the largest portion of latency across all datasets.
* PPCV-T3 and PPCV-T4 have very similar, minimal contributions to latency.
* Predictive Decoding and Phi-Decoding contribute relatively small amounts of latency compared to Chain-of-Thought.
* PPCV-T1 and PPCV-T2 contribute slightly more latency than PPCV-T3 and PPCV-T4.
### Key Observations
* The Math500 dataset exhibits the highest overall latency, primarily driven by the Chain-of-Thought method.
* The latency for Chain-of-Thought is significantly higher than all other methods across all datasets.
* The PPCV methods (T1-T4) consistently show low latency contributions.
* The differences in latency between the PPCV methods are relatively small.
### Interpretation
The chart demonstrates a clear trade-off between decoding method and latency. Chain-of-Thought, while potentially offering higher accuracy or quality of results (not shown in this chart), incurs a substantial latency cost. The PPCV methods, particularly T3 and T4, appear to offer a significant reduction in latency, albeit potentially at the expense of performance on the tasks. The consistent pattern across datasets suggests that this latency difference is inherent to the decoding methods themselves and not specific to the characteristics of the datasets. The "Ours" label on PPCV-T1 through PPCV-T4 indicates these methods were developed by the authors of the study, and the chart serves to highlight their efficiency gains compared to established methods like Chain-of-Thought, Predictive Decoding, and Phi-Decoding. The stacked bar format effectively visualizes the composition of total latency, allowing for a clear comparison of the contribution of each method.
</details>
(b)
<details>
<summary>images/throughput.png Details</summary>
### Visual Description
\n
## Bar Chart: Throughput Comparison of Decoding Methods
### Overview
This bar chart compares the throughput (tokens/second) of four different decoding methods â Chain-of-Thought, Predictive Decoding, Phi-Decoding, and PPCV (Ours) â across five different datasets: GSM8K, GSMHard, Math500, SVAMP, and ARC. Each dataset has four bars representing the throughput of each decoding method.
### Components/Axes
* **X-axis:** Datasets - GSM8K, GSMHard, Math500, SVAMP, ARC.
* **Y-axis:** Throughput (tokens/sec), ranging from 0 to 2000, with increments of 250.
* **Legend:** Located in the top-right corner, identifies the color-coding for each decoding method:
* Chain-of-Thought
* Predictive Decoding
* Phi-Decoding
* PPCV (Ours)
</details>
(c)
Figure 7: (a) The impact of number of sampled alternative tokens on the performance. (b) Latency comparison between the baselines and our method, measured as the average inference time per question (in seconds). $T_{1}$ , $T_{2}$ , $T_{3}$ , $T_{4}$ denote time for paraphrased question generation, initial answer generation, forward pass and new rollouts from truncated trajectories. (c) Throughput comparison between the baselines and our method, measured in output tokens per second.
### 4.2 Main Results
The main results are summarized in Table 1 and Table 2. The results indicate that Self-Consistency effectively improves the reasoning performance of LLMs compared to vanilla Chain-of-Thought prompting. For example, Llama-3.1-8B-Instruct (Grattafiori et al., 2024) achieves about 3% higher accuracy with Self-Consistency than with CoT. These findings suggest that augmenting reasoning during inference through sampling is an effective way to refine reasoning trajectories. Recent decoding-based methods, such as Predictive Decoding (Ma et al., 2025) and Phi-Decoding (Xu et al., 2025), also achieve strong results.Unlike prior works that rely on carefully designed prompts to self-correct errors in intermediate steps, these two methods modify the current step by probing future steps with pre-defined reward signals. Furthermore, our experimental results demonstrate that the proposed method consistently outperforms the baselines across most tasks, spanning both mathematical and knowledge reasoning, thereby highlighting its generalization ability across different reasoning settings. Notably, our method even surpasses the latest approaches such as Predictive Decoding (Ma et al., 2025) and Phi-Decoding (Xu et al., 2025). In particular, it achieves approximately 50.00% accuracy on the Math500 dataset (Hendrycks et al., 2021), exceeding these baselines considerably. The results on competition-level datasets further demonstrate the effectiveness of our method in enhancing the reasoning ability of LLMs. These results indicate that our method can effectively extract critical tokens that play a pivotal role in the final outcome and correct the reasoning trajectory by leveraging alternative tokens. Additional results can be found in Appendix E.
### 4.3 Ablation Study
In this section, we analyze the contribution of each stage individually. All the evaluations are conducted on Llama-3.1-8B-Instruct (Grattafiori et al., 2024).
Effectiveness of extracted critical tokens. To demonstrate the effectiveness of our extracted critical tokens, we conduct an evaluation in which the critical tokens are replaced with random tokens in the first stage, while keeping the second stage unchanged. This evaluation is performed across multiple benchmark datasets, with pass@1 as the metric. The results, shown in Table 3, reveal a substantial decline in performance. These findings highlight the pivotal role of critical tokens and indicate that our method can effectively identify and extract them. More ablation study on comparison with Paraphrased Majority Voting (PMV) can be found in Appendix F.
Effectiveness of paraphrase consistency. We also evaluate the effectiveness of our proposed paraphrase consistency and compare it with traditional majority voting. While keeping the first stage unchanged, instead of using paraphrased forms to generate new reasoning steps, we simply sample multiple new steps from alternative tokens conditioned on the original question and use majority voting to determine the final answer. The results, shown in Table 4, reveal a noticeable decline in performance, highlighting the importance of paraphrased forms in improving the intermediate reasoning steps.
Impact of number of sampled alternative tokens.We investigate the influence of the number of sampled alternative tokens in the second stage by selecting values of 3, 5, 7, and 10. The results, shown in Figure 7(a), demonstrate that performance improves as the number of alternative tokens increases. This suggests that exploring more reasoning steps with additional alternative tokens during inference can be beneficial for reasoning tasks.
## 5 Computational Cost Analysis
In this section we examine the composition of the latency in our method. The latency arises from four components: Paraphrased question generation ( $T_{1}$ ); initial answer generation ( $T_{2}$ ), equivalent to vanilla CoT; a forward pass for identifying critical tokens ( $T_{3}$ ), which does not generate new tokens and is computationally lightweight; rollouts of truncated trajectories using alternative tokens under both the original and paraphrased questions ( $T_{4}$ ), which constitutes the main source of overhead.
We evaluate all components on Llama-3.1-8B-Instruct using vLLM on NVIDIA A100 GPUs, with a maximum output length of 4096 tokens for each question. For our method, we use 4 paraphrased questions on math datasets and 3 on ARC, and select the top-10 candidate tokens as alternatives. The updated average latency results are reported in Figure 7(b). As expected, $T_{1}$ scales with the number of paraphrases, $T_{3}$ remains minimal, and $T_{4}$ dominates the total cost. Specifically, $T_{4}$ depends on the number of top-k alternative tokens, the number of paraphrased questions and the position of the critical token in the trajectory. Since the new rollouts from truncated trajectories for different alternative tokens and paraphrased questions are independent, $T_{4}$ can take advantage of vLLMâs parallelism. These rollouts can therefore be processed concurrently, improving overall efficiency. This is reflected in the higher throughput (tokens/sec) shown in Figure 7(c). And results of our method in latency comparable to baseline methods, even on challenging benchmarks such as Math500 and GSM-Hard where the critical token tends to occur in later reasoning steps. On the GSM8K and SVAMP benchmarks, our method, as well as baselines such as Predictive Decoding, would incur a approximately $6\text{â}8\times$ latency overhead compared to vanilla Chain-of-Thought. More analysis on the trade-off between the latency and performance can be found in Appendix G.
## 6 Conclusion
In this study, inspired by beneficial impact of paraphrase forms on reasoning, we investigate the pivotal role of critical tokens in shaping the reasoning trajectory. To leverage these two factors, we propose the Paraphrastic Probing and Consistency Verification framework. Our framework consists of two stages: Paraphrastic Probing, which identifies and extracts critical tokens, and Consistency Verification, which adopts paraphrase forms to generate new reasoning trajectories with alternative tokens to reach the final answer. We evaluate our proposed framework with different LLMs and extensive evaluations across multiple benchmarks demonstrate the promising performance of our method.
## Impact Statement
This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
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## Appendix A Empirical Study
<details>
<summary>images/condition_one_math.png Details</summary>
### Visual Description
\n
## Bar Chart: Fraction vs. Average Accuracy for Critical and Random Tokens
### Overview
This bar chart compares the fraction of tokens (expressed as a percentage) for "critical tokens" and "random tokens" based on their average accuracy (also expressed as a percentage). The chart displays two groups of bars, one for tokens with average accuracy less than or equal to 5% and another for tokens with average accuracy greater than 5%. Error bars are included for each bar, indicating the variability or uncertainty in the fraction values.
### Components/Axes
* **X-axis:** "Average accuracy (%)" with two categories: "†5%" and "> 5%".
* **Y-axis:** "Fraction (%)" ranging from 0 to 90.
* **Legend:** Located in the top-right corner, distinguishing between "critical tokens" (represented by a teal color) and "random tokens" (represented by a light green color).
* **Bars:** Represent the fraction of tokens for each accuracy category and token type.
* **Error Bars:** Vertical lines extending above and below each bar, indicating the standard error or confidence interval.
### Detailed Analysis
The chart presents data for two groups of tokens, categorized by their average accuracy: †5% and > 5%. For each accuracy category, the fraction of "critical tokens" and "random tokens" is displayed.
**†5% Average Accuracy:**
* **Critical Tokens:** The bar is teal and reaches approximately 84% on the Y-axis. The error bar extends from approximately 78% to 90%.
* **Random Tokens:** The bar is light green and reaches approximately 47% on the Y-axis. The error bar extends from approximately 40% to 54%.
**> 5% Average Accuracy:**
* **Critical Tokens:** The bar is teal and reaches approximately 13% on the Y-axis. The error bar extends from approximately 8% to 18%.
* **Random Tokens:** The bar is light green and reaches approximately 53% on the Y-axis. The error bar extends from approximately 47% to 59%.
### Key Observations
* For tokens with average accuracy †5%, the fraction of critical tokens is significantly higher than that of random tokens.
* For tokens with average accuracy > 5%, the fraction of random tokens is significantly higher than that of critical tokens.
* The error bars indicate a relatively high degree of variability in the fraction values, particularly for the "critical tokens" with †5% accuracy.
### Interpretation
The data suggests a strong inverse relationship between average accuracy and the fraction of critical tokens. Critical tokens tend to have lower average accuracy, while random tokens tend to have higher average accuracy. This could indicate that critical tokens are more difficult to predict or classify accurately, potentially due to their inherent complexity or importance in the context. The higher variability in the critical token fractions might suggest that the factors influencing their accuracy are more diverse or less predictable.
The chart implies that focusing on improving the accuracy of critical tokens could be a key area for improvement, as they currently exhibit lower performance compared to random tokens. The difference in fractions between the two accuracy categories suggests that the impact of token type on accuracy is substantial. The error bars, while indicating variability, do not overlap between the critical and random token groups within each accuracy category, suggesting that the observed differences are statistically significant.
</details>
(a)
<details>
<summary>images/condition_two_math.png Details</summary>
### Visual Description
\n
## Bar Chart: Fraction vs. Average Accuracy for Token Types
### Overview
This bar chart compares the fraction of tokens based on average accuracy, differentiating between "critical tokens" and "random tokens". The x-axis categorizes average accuracy into two groups: "†10%" and "> 10%". The y-axis represents the fraction, expressed as a percentage. Each bar also includes an error bar indicating the variability of the data.
### Components/Axes
* **X-axis Title:** Average accuracy (%)
* **X-axis Categories:** †10%, > 10%
* **Y-axis Title:** Fraction (%)
* **Y-axis Scale:** 0 to 90% (approximately)
* **Legend:**
* Critical tokens (represented by a teal/green color)
* Random tokens (represented by a light teal/green color)
### Detailed Analysis
The chart consists of four bars, two for each accuracy category, representing critical and random tokens respectively. Each bar has an associated error bar.
**For Average Accuracy †10%:**
* **Critical Tokens:** The bar is teal/green and reaches approximately 86% on the y-axis. The error bar extends from approximately 82% to 90%.
* **Random Tokens:** The bar is light teal/green and reaches approximately 55% on the y-axis. The error bar extends from approximately 50% to 60%.
**For Average Accuracy > 10%:**
* **Critical Tokens:** The bar is teal/green and reaches approximately 44% on the y-axis. The error bar extends from approximately 40% to 48%.
* **Random Tokens:** The bar is light teal/green and reaches approximately 12% on the y-axis. The error bar extends from approximately 8% to 16%.
### Key Observations
* For both accuracy categories, the fraction of critical tokens is significantly higher than the fraction of random tokens.
* The difference in fraction is more pronounced in the "†10%" accuracy category.
* The error bars suggest a relatively high degree of variability in the data, particularly for the "†10%" category.
### Interpretation
The data suggests that critical tokens are more likely to have lower average accuracy compared to random tokens. However, a substantial fraction of critical tokens still achieve an average accuracy greater than 10%. The large error bars indicate that there is considerable variation within each group, and the observed differences may not be statistically significant without further analysis.
The chart highlights a potential trade-off: critical tokens, while essential, may be more prone to errors or require more refinement to achieve higher accuracy. The higher fraction of critical tokens with lower accuracy could indicate areas where the system or model needs improvement. The fact that a significant portion of critical tokens *do* achieve >10% accuracy suggests that the concept of "critical tokens" isn't inherently tied to low performance, but rather that there's a performance disparity that warrants investigation.
</details>
(b)
<details>
<summary>images/compare_ct_rnd_math.png Details</summary>
### Visual Description
\n
## Line Chart: Pass@k vs. Number of Sample k
### Overview
This line chart compares the performance of "critical tokens" and "random tokens" based on the metric "pass@k (%)" as a function of the "number of sample k". The chart displays two lines with error bars, representing the mean and standard deviation of the pass@k metric for each token type at different values of k.
### Components/Axes
* **X-axis:** "number of sample k". Scale ranges from approximately 0 to 45, with markers at 0, 10, 20, 30, and 40.
* **Y-axis:** "pass@k (%)". Scale ranges from approximately 25% to 95%, with markers at 30%, 40%, 50%, 60%, 70%, 80%, and 90%.
* **Legend:** Located in the top-right corner.
* Red line with error bars: "critical tokens"
* Purple line with error bars: "random tokens"
* **Gridlines:** Horizontal and vertical gridlines are present to aid in reading values.
### Detailed Analysis
**Critical Tokens (Red Line):**
The red line representing "critical tokens" shows an upward trend.
* At k = 0, pass@k is approximately 54% ± 6%.
* At k = 10, pass@k is approximately 68% ± 5%.
* At k = 20, pass@k is approximately 77% ± 4%.
* At k = 30, pass@k is approximately 81% ± 4%.
* At k = 40, pass@k is approximately 86% ± 4%.
**Random Tokens (Purple Line):**
The purple line representing "random tokens" also shows an upward trend, but is less steep than the red line.
* At k = 0, pass@k is approximately 28% ± 4%.
* At k = 10, pass@k is approximately 34% ± 4%.
* At k = 20, pass@k is approximately 41% ± 4%.
* At k = 30, pass@k is approximately 44% ± 4%.
* At k = 40, pass@k is approximately 48% ± 6%.
The error bars indicate the variability in the pass@k metric for each token type at each value of k. The error bars are relatively consistent in size across the range of k values.
### Key Observations
* "Critical tokens" consistently outperform "random tokens" across all values of k.
* The performance gap between "critical tokens" and "random tokens" widens as k increases.
* The rate of improvement in pass@k decreases as k increases for both token types.
* The error bars suggest that the variability in performance is relatively consistent across different values of k.
### Interpretation
The data suggests that selecting "critical tokens" leads to significantly better performance (as measured by pass@k) compared to selecting "random tokens". This indicates that the "critical tokens" are more informative or relevant for the task being evaluated. As the number of samples (k) increases, the performance of both token types improves, but the advantage of using "critical tokens" becomes more pronounced. This could be because the "critical tokens" provide a stronger signal, allowing for more accurate predictions even with a limited number of samples. The consistent error bars suggest that the observed differences in performance are statistically significant and not due to random chance. The diminishing returns in performance as k increases suggest that there may be a point of diminishing returns where adding more samples does not significantly improve the pass@k metric. This could be due to the fact that the most informative tokens have already been selected, and adding more samples provides less additional information.
</details>
(c)
<details>
<summary>images/density_math.png Details</summary>
### Visual Description
\n
## Histogram: Density of Consistency Score with Correct/Incorrect Answers
### Overview
The image presents a histogram comparing the density distribution of "consistency scores" for two groups: those with "incorrect answers" and those with "correct answers". The x-axis represents the consistency score, and the y-axis represents the density (expressed as a percentage). The chart visually compares how consistency scores are distributed between the two answer groups.
### Components/Axes
* **X-axis Title:** "consistency score"
* **X-axis Range:** 0 to 5
* **Y-axis Title:** "density (%)"
* **Y-axis Range:** 0 to 0.7
* **Legend:** Located in the top-right corner.
* "w incorrect answers" - represented by a light green color.
* "w correct answers" - represented by a light red color.
### Detailed Analysis
The chart displays two histograms stacked on top of each other.
**Incorrect Answers (Light Green):**
The distribution for incorrect answers shows a strong peak around a consistency score of 0, with a density of approximately 0.7. The density decreases rapidly as the consistency score increases. There is a small secondary peak around a consistency score of 2, with a density of approximately 0.15. The density remains relatively low for scores between 2 and 5, fluctuating around 0.05-0.1.
**Correct Answers (Light Red):**
The distribution for correct answers is relatively flat across the range of consistency scores. It starts with a low density around 0.05 at a consistency score of 0. The density increases gradually, reaching a peak around a consistency score of 4-5, with a density of approximately 0.2. There is a slight dip in density around a consistency score of 1, at approximately 0.1.
**Specific Data Points (Approximate):**
| Consistency Score | Incorrect Answers Density (%) | Correct Answers Density (%) |
|---|---|---|
| 0 | 0.7 | 0.05 |
| 1 | 0.1 | 0.1 |
| 2 | 0.15 | 0.1 |
| 3 | 0.1 | 0.15 |
| 4 | 0.05 | 0.2 |
| 5 | 0.05 | 0.2 |
### Key Observations
* The distribution of consistency scores for incorrect answers is heavily skewed towards lower scores (closer to 0).
* The distribution of consistency scores for correct answers is more evenly distributed, with a slight tendency towards higher scores (closer to 5).
* There is a clear difference in the distributions of consistency scores between the two groups.
* The peak at 0 for incorrect answers suggests that a low consistency score is strongly associated with incorrect answers.
### Interpretation
The data suggests a strong correlation between consistency score and answer correctness. Individuals who provide inconsistent responses (low consistency score) are more likely to answer incorrectly. Conversely, individuals who provide consistent responses (high consistency score) are more likely to answer correctly.
The shape of the distributions indicates that inconsistency is a significant factor contributing to incorrect answers. The high density of incorrect answers at a consistency score of 0 suggests that a complete lack of consistency is a strong predictor of incorrectness. The more even distribution of correct answers suggests that consistency, while helpful, is not the sole determinant of correctness. There is a range of consistency scores associated with correct answers, indicating that some individuals can arrive at the correct answer even with moderate levels of inconsistency.
The difference in distributions could be due to several factors, including:
* **Carelessness:** Individuals who answer carelessly may provide inconsistent responses and are more likely to be incorrect.
* **Lack of Understanding:** Individuals who do not fully understand the material may struggle to provide consistent responses.
* **Strategic Guessing:** Individuals who are unsure of the answer may guess randomly, leading to inconsistent responses.
* **Cognitive Biases:** Certain cognitive biases may lead individuals to provide inconsistent responses.
</details>
(d)
Figure 8: (a) Comparison of the fraction of tokens whose associated rollout correctness falls below or above 5%, for tokens extracted by our method versus randomly selected tokens. (b) Comparison of the fraction of the subsequent tokens whose associated rollout correctness falls below or above 10%, for tokens extracted by our method versus randomly selected tokens. (c) Comparison of the impact of extracted tokens by our method versus random tokens on LLM reasoning performance. (d) Comparison of the density distributions of consistency scores for rollouts with correct and incorrect answers. All the evaluations are conducted on Math500 dataset.
we follow the previous pipeline to conduct a quantitative analysis to examine the authenticity and impact of tokens extracted via paraphrastic probing, comparing them against randomly selected tokens. Specifically, we sample 100 instances with incorrect reasoning trajectories from the Math500 (Hendrycks et al., 2021) training sets. Following the paraphrastic probing pipeline, we identify candidate critical tokens in the early reasoning steps. In each run, we randomly sample 40% of the candidate critical tokens for evaluation and repeat the experiment independently 10 times. For comparison, we apply the same procedure to randomly selected tokens. All evaluations are conducted using Llama-3.1-8B-Instruct (Grattafiori et al., 2024). We first compute the average accuracy of rollouts generated from partial trajectories truncated at the position of the extracted tokens, and compare these results with those obtained from random-token truncation. As shown in Figure 8(a), we also obverse a large proportion of the extracted tokens exhibit near-zero correctness. We further evaluate the average accuracy of rollouts initiated from subsequent tokens (5â10 tokens). The results in Figure 8(b) also indicate that errors persist beyond the identified positions. Finally, we substitute the extracted critical tokens with alternative tokens and roll out multiple new reasoning trajectories. As shown in Figure 8(c), replacing critical tokens leads to a significantly larger improvement in reasoning accuracy compared to substituting random tokens.
To justify our paraphrase consistency mechanism on Math500 dataset (Hendrycks et al., 2021), we follow the same procedure. The evaluation is conducted on Llama-3.1-8B-Instruct (Grattafiori et al., 2024). We then analyze the distribution of consistency scores for rollouts that yield correct versus incorrect answers. As shown in Figure 8(d), majority of rollouts with correct answers achieve a consistency score of at least 1, whereas this proportion drops sharply for rollouts with incorrect answers.
## Appendix B Automatic Prompt Engineering for Paraphrasing
The quality of paraphrased questions is crucial for our framework, as it directly affects both the identification of critical tokens and the stability of paraphrase-based consistency verification. To ensure high-quality paraphrasing, we adopt Automatic Prompt Engineering (APE) (Zhou et al., 2022) as an evaluation to optimize the paraphrasing instruction. This allows us to systematically control the quality of paraphrases rather than relying solely on manually written prompts. The APE procedure we employ consists of four steps:
- We create a small set of original questions paired with valid paraphrased variants. Using this exemplar set, the model generates multiple candidate instructions that could produce the paraphrased outputs from the original questions.
- Prompt each candidate instruction to the language model to generate paraphrases for the original problems and compare the mean solve rate change before and after paraphrasing.
- Choose the instruction that maximizes the mean solve rate change.
- Repeat the previous steps multiple times.
In our experiments, we initialize the paraphrasing instruction and iteratively refine it using APE with samples drawn from the GSM8K and Math500 training sets. With this approach, the refined paraphrasing prompt helps us reliably produce high-quality paraphrased questions.
Algorithm 1 PPCV Framework
0: LLM; original question $q_{0}$ ; number of paraphrased questions $N$ ; number of sampled alternative tokens $K$ ; temperature-scaling coefficient $\lambda$ .
0: Final answer $\text{ans}_{f}$ .
(Paraphrasing) Generate paraphrased variants $q_{1},q_{2},\dots,q_{N}$ from $q_{0}$ using the LLM.
(Initial reasoning) Obtain the initial reasoning path $r^{q_{0}}_{0}$ for $q_{0}$ .
(Token influence analysis) For each paraphrased question $q_{n}$ , compute the token distribution $P^{q_{n}}$ by concatenating $q_{n}$ with $r^{q_{0}}_{0}$ (Eq. 1).
(Candidate selection) Identify candidate critical token positions based on the mismatch between the top-1 predicted tokens and the expected tokens in $q_{0}$ (Eq. 2).
(Critical token verification) Select the final critical token $a_{c}$ using the criteria (Eqs. 3 â 5).
(Alternative token sampling) Sample $K$ alternative tokens $a_{c}^{0},a_{c}^{1},\dots,a_{c}^{K-1}$ (including $a_{c}^{0}=a_{c}$ ) using top- $K$ sampling on $q_{0}$ .
(Truncated rollouts) Truncate the initial reasoning path $r^{q_{0}}_{0}$ at position $a_{c}$ and form synthetic inputs $\tilde{r}_{c}^{0},\tilde{r}_{c}^{1},\dots,\tilde{r}_{c}^{K-1}$ by appending each alternative token.
for each synthetic input $\tilde{r}_{c}^{k}$ , $k=0,\dots,K-1$ do
Generate rollouts $r^{q_{0}}_{k},r^{q_{1}}_{k},\dots,r^{q_{N}}_{k}$ for the original and paraphrased questions using the LLM.
end for
(Consistency verification) Compute the final answer $\text{ans}_{f}$ using (similarity-weighted) paraphrase consistency (Eq. 6).
## Appendix C Algorithm
The complete algorithm of our proposed PPCV framework is illustrated in Algo. 1.
## Appendix D Discussion
<details>
<summary>images/critical_token_on_correct.png Details</summary>
### Visual Description
\n
## Textual Comparison: Paraphrastic Probing Case Study
### Overview
The image presents a side-by-side comparison of two responses to the same mathematical word problem. The problem asks for the total distance James runs in a week, given he runs 3 sprints, 3 times a week, with each sprint being 60 meters. The "ground truth answer" is provided as 540 meters. The two responses demonstrate different approaches to solving the problem, with one using the term "meters" and the other "distances". The responses are annotated with checkmarks indicating correctness.
### Components/Axes
The image is divided into two columns, each representing a different response. Each column is further divided into sections: "Question", "Response", "Step 1", "Step 2", and a final statement of the answer. The "Question" section is identical in both columns.
### Detailed Analysis or Content Details
**Column 1:**
* **Question:** "James decides to run 3 sprints 3 times a week. He runs 60 meters each sprint. How many total meters does he run a week? (ground truth answer: 540)"
* **Response:** "To find the total **meters** James runs a week, we need to calculate the total meters he runs in one sprint session and then multiply it by the number of sessions he has in a week."
* **Step 1:** "Calculate the total meters James runs in one sprint session. James runs 3 sprints of 60 meters each. Total meters in one sprint session = 3 * 60 = 180 meters."
* **Step 2:** "Calculate the total meters..."
* **Answer:** "Therefore, James runs a total of 540 meters a week. The answer is 540. â "
**Column 2:**
* **Question:** "James decides to run 3 sprints 3 times a week. He runs 60 meters each sprint. How many total meters does he run a week? (ground truth answer: 540)"
* **Response:** "To find the total **distances** James runs in a week, we need to calculate the total distance he runs in one sprint session and then multiply it by the number of sessions he has in a week."
* **Step 1:** "Calculate the total distance James runs in one sprint session. Distance per sprint = 60 meters. Number of sprints per session = 3. Total distance per session = Distance per sprint * Number of sprints per session. Total distance per session = 60 * 3. Total distance per session = 180 meters."
* **Step 2:** "Calculate the total distance James runs in a week."
* **Answer:** "Therefore, James runs a total of 540 meters a week. â "
### Key Observations
The primary difference between the two responses is the use of "meters" versus "distances". Both responses arrive at the correct answer (540 meters) and demonstrate a similar problem-solving approach. The second response provides a more explicit breakdown of the calculation in Step 1, defining "Distance per sprint" and "Number of sprints per session". Both responses are marked as correct with a checkmark.
### Interpretation
This image illustrates a case study in "Paraphrastic Probing," likely used in evaluating language models. The goal is to assess how sensitive a model's response is to minor changes in wording (e.g., "meters" vs. "distances"). The fact that both responses are correct suggests that the model is robust to this particular paraphrasing. The study highlights the importance of evaluating not just the correctness of an answer, but also the consistency and clarity of the reasoning process. The use of the "ground truth answer" provides a benchmark for evaluating the model's performance. The annotations (checkmarks) indicate a human evaluation of the responses. The slight differences in phrasing and detail in the steps suggest that the model may have multiple valid ways of arriving at the correct solution.
</details>
Figure 9: Effect of the critical token on an initial reasoning path that yields the correct answer. Critical tokens are highlighted in bold red, and alternative tokens in bold purple.
We have shown that replacing critical tokens can correct intermediate reasoning paths and lead to the correct answer. In this section, we examine how our method influences reasoning paths that are already correct. First, we conduct a case study on an instance with a correct answer, where we follow our pipeline to identify the critical token and roll out new reasoning paths using alternative tokens. As illustrated in Figure 9, the new rollouts also yield the correct answer. Second, our pipeline incorporates both the initial reasoning path $r^{q_{0}}_{0}$ and its paraphrased variants $r^{q_{n}}_{0}$ for evaluation. The robustness of correct rollouts across paraphrased forms ensures high consistency scores, allowing them to stand out as the final answer. These findings suggest that our pipeline preserves the performance of LLMs on problems that can already be solved correctly by CoT (Wei et al., 2022).
Table 5: Comparison between our method and baselines with DeepSeek-R1-Distill-Llama-70B model on mathematical reasoning benchmarks using pass@1 accuracy.
| Chain-of-Thought Predictive Decoding Phi-Decoding | 56.67 60.00 64.00 | 38.00 40.66 46.67 | 43.33 44.66 48.00 | 30.00 30.66 31.33 |
| --- | --- | --- | --- | --- |
| PPCV (Ours) | 70.00 | 56.66 | 56.66 | 33.33 |
Table 6: Comparison of our method and the baseline approach on Llama-3.1-8B-Instruct, evaluated using pass@k (k=4).
| Phi-Decoding PPCV (Ours) | 92.15 93.83 | 53.57 61.41 | 52.60 59.39 | 91.19 94.48 | 90.73 94.24 |
| --- | --- | --- | --- | --- | --- |
Table 7: Comparison of our method and the baseline approach on Qwen3-32B, evaluated using pass@k (k=4).
| Phi-Decoding PPCV (Ours) | 41.61 49.71 | 30.19 34.28 | 45.09 51.42 | 13.11 19.28 |
| --- | --- | --- | --- | --- |
Table 8: The comparison of performance between Paraphrased Majority Voting (PMV) and our proposed PPCV using pass@1 accuracy.
| Chain-of-Thought PMV PPCV (Ours) | 77.40 83.55 88.24 | 28.00 33.16 49.73 | 31.00 40.60 50.00 | 83.00 87.70 89.60 | 58.91 61.63 88.31 |
| --- | --- | --- | --- | --- | --- |
## Appendix E Main results
To further assess the effectiveness of our method, we evaluate it on a larger reasoning model such as DeepSeek-R1-Distill-Llama-70B. We apply our full pipeline and compare against all baselines across multiple mathematical reasoning benchmarks. The results, presented in Table 5, show that our method consistently outperforms the baselines. These improvements demonstrate that our approach remains effective for stronger reasoning models and generalizes well beyond the smaller models.
we also conduct additional experiments to report pass@k performance for both our method and the baselines. In these experiments, we use Phi-Decoding as the representative baseline and evaluate on two models: Llama-3.1-8B-Instruct and Qwen3-32B. We set k= 4 and generate 12 samples per question to obtain stable estimates of pass@4. The results across multiple benchmarks are presented in Table 6 and Table 7, respectively. Consistent with our main findings, our method achieves higher pass@k scores compared to the baseline methods, indicating that paraphrastic critical-token refinement continues to provide benefits in a multi-sample setting. These results further validate the robustness of our approach under stochastic sampling and demonstrate improved hit rates when multiple outputs are available.
## Appendix F Ablation Study
We also perform a comparison against a Paraphrased Majority Voting (PMV) strategy is essential to demonstrate that the performance gains of our method are indeed derived from the critical token correction, rather than simply the benefit of having multiple paraphrased samples. Thus, we have performed the suggested additional experiment. We adopt Paraphrased Majority Voting (PMV) as the baseline. For a fair comparison, this approach uses the same total number of trajectories generated from the paraphrased questions as PPCV, and reaches the final answer using standard majority voting over the rollouts from the paraphrased questions. Specifically, we use four variants of each question, with 10 rollouts per variant. We compare the PMV baseline against our full PPCV method and ensure the same number of paraphrased questions as in our method. The results shown in Table 8 demonstrate that the performance yielded by PMV is significantly lower than that of our proposed PPCV method. This result confirms that simply rolling out trajectories from paraphrased questions and aggregating the answers is insufficient. The substantial performance gap between PMV and PPCV decisively highlights the necessity and effectiveness of our extracted critical tokens and the subsequent corrective reasoning step.
<details>
<summary>images/latency_trade_off.png Details</summary>
### Visual Description
\n
## Bar Chart with Line Overlay: Latency vs. Accuracy for Top-K Results
### Overview
This image presents a bar chart comparing latency (in seconds) with a line graph showing accuracy (in percentage) for different values of 'Top-K' results. The 'Top-K' values are 3, 5, 7, and 10. Each bar represents the average latency for a given Top-K value, with error bars indicating the variance. The line graph overlays the bar chart, showing the accuracy corresponding to each Top-K value, also with error bars.
### Components/Axes
* **X-axis:** "Top-K" with categories: Top-3, Top-5, Top-7, Top-10.
* **Left Y-axis:** "Latency (s)" ranging from 0 to 18 seconds.
* **Right Y-axis:** "Accuracy (%)" ranging from 70% to 100%.
* **Bar Chart:** Represents Latency. Bars are light blue with black outlines.
* **Line Graph:** Represents Accuracy. Line is red with circular data points and error bars.
* **Error Bars:** Present on both the bar chart (latency) and line graph (accuracy), indicating the variance or standard deviation.
### Detailed Analysis
**Latency (Bar Chart):**
* **Top-3:** The bar is approximately 14.8 seconds tall, with error bars extending from roughly 13.5 to 16.1 seconds.
* **Top-5:** The bar is approximately 16.2 seconds tall, with error bars extending from roughly 14.8 to 17.6 seconds.
* **Top-7:** The bar is approximately 17.2 seconds tall, with error bars extending from roughly 15.8 to 18.6 seconds.
* **Top-10:** The bar is approximately 17.8 seconds tall, with error bars extending from roughly 16.4 to 19.2 seconds.
The latency generally increases as the Top-K value increases.
**Accuracy (Line Graph):**
* **Top-3:** Accuracy is approximately 10.2%, with error bars extending from roughly 8.5 to 11.9%.
* **Top-5:** Accuracy is approximately 11.8%, with error bars extending from roughly 10.4 to 13.2%.
* **Top-7:** Accuracy is approximately 11.6%, with error bars extending from roughly 10.2 to 13.0%.
* **Top-10:** Accuracy is approximately 11.3%, with error bars extending from roughly 9.9 to 12.7%.
The accuracy initially increases from Top-3 to Top-5, then slightly decreases for Top-7 and Top-10, remaining relatively stable.
### Key Observations
* There is a clear trade-off between latency and accuracy. As the number of considered results (Top-K) increases, the latency increases, but the accuracy initially improves and then plateaus.
* The error bars suggest that the variance in latency is relatively consistent across different Top-K values.
* The accuracy error bars are also relatively consistent, indicating similar levels of uncertainty in accuracy measurements.
* The accuracy peaks at Top-5 and then slightly declines, suggesting diminishing returns from considering more than the top 5 results.
### Interpretation
The data suggests that there's a point of diminishing returns when increasing the number of results considered (Top-K). While increasing Top-K initially improves accuracy, the latency cost increases proportionally. The optimal balance between latency and accuracy appears to be around Top-5, where accuracy is relatively high, and latency is not excessively high. The slight decrease in accuracy for Top-7 and Top-10, coupled with the increased latency, indicates that considering more results beyond Top-5 does not significantly improve performance and may even slightly degrade it. This could be due to the inclusion of less relevant results in the larger set, which introduces noise and reduces overall accuracy. The error bars indicate that these trends are statistically significant, as the ranges do not overlap substantially. This chart is useful for system designers to determine the optimal value of K for their application, balancing the need for speed (low latency) with the need for correctness (high accuracy).
</details>
(a)
<details>
<summary>images/throughput_trade_off.png Details</summary>
### Visual Description
\n
## Bar Chart: Throughput vs. Accuracy for Different Top-k Values
### Overview
This image presents a bar chart comparing the throughput (tokens/second) and accuracy (%) of a system for different values of 'Top-k'. The chart displays throughput as blue bars and accuracy as red data points with error bars. The x-axis represents the Top-k values (3, 5, 7, and 10).
### Components/Axes
* **X-axis:** "Top-k" with markers at 3, 5, 7, and 10.
* **Left Y-axis:** "Throughput (tokens/sec)" ranging from 0 to 1400.
* **Right Y-axis:** "Accuracy (%)" ranging from 50 to 100.
* **Data Series 1:** Throughput - Represented by blue bars.
* **Data Series 2:** Accuracy - Represented by red data points with error bars.
* **Legend:** No explicit legend, but the color coding is clear (blue for throughput, red for accuracy).
### Detailed Analysis
The chart shows the following data points:
* **Top-3:**
* Throughput: Approximately 550 tokens/sec.
* Accuracy: Approximately 87% with an error bar extending from roughly 84% to 90%.
* **Top-5:**
* Throughput: Approximately 750 tokens/sec.
* Accuracy: Approximately 88% with an error bar extending from roughly 85% to 91%.
* **Top-7:**
* Throughput: Approximately 1000 tokens/sec.
* Accuracy: Approximately 92% with an error bar extending from roughly 89% to 95%.
* **Top-10:**
* Throughput: Approximately 1300 tokens/sec.
* Accuracy: Approximately 91% with an error bar extending from roughly 88% to 94%.
**Trends:**
* The throughput generally increases as the Top-k value increases. The increase is most significant between Top-5 and Top-7.
* The accuracy initially increases from Top-3 to Top-7, then slightly decreases at Top-10, but remains within the error bounds of Top-7.
* The accuracy line is relatively flat, indicating that increasing Top-k doesn't drastically improve accuracy, but does increase throughput.
### Key Observations
* There's a clear trade-off between throughput and accuracy. Higher throughput is achieved with larger Top-k values, but accuracy plateaus and slightly decreases.
* The error bars on the accuracy data suggest that the variations in accuracy are relatively small, and the observed differences might not be statistically significant.
* The largest jump in throughput occurs between Top-5 and Top-7.
### Interpretation
The data suggests that increasing the 'Top-k' parameter improves the throughput of the system, allowing it to process more tokens per second. However, this improvement comes at the cost of a slight decrease in accuracy. The optimal Top-k value depends on the specific application and the relative importance of throughput versus accuracy.
The relatively flat accuracy curve indicates that beyond a certain point (around Top-7 in this case), increasing Top-k provides diminishing returns in terms of accuracy. The system appears to reach a point where adding more candidate options doesn't significantly improve the quality of the output.
The error bars are important to consider. The overlap between the error bars at Top-7 and Top-10 suggests that the difference in accuracy between these two values is not statistically significant. Therefore, choosing Top-10 might be a reasonable compromise if maximizing throughput is the primary goal.
This chart likely represents the performance of a language model or a similar system where 'Top-k' refers to the number of most likely tokens considered at each step of the generation process. A higher Top-k value allows for more diverse outputs but may also introduce more errors.
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(b)
Figure 10: Trade-off between performance and efficiency for our method under different numbers of top-k tokens (k = 3, 5, 7, 10), measured by latency (left) and throughput (right).
## Appendix G Computational Cost Analysis
We also conduct a trade-off analysis between performance and efficiency by varying the number of alternative tokens considered at the critical position. Specifically, we test top-3, top-5, top-7, and top-10 alternatives using GSM8K, sampling 200 random questions and repeating the experiment five times to compute confidence intervals. The results are presented in Figure 10. We observe a clear trade-off. Reducing the number of alternative tokens lowers both latency and throughput, while causing a slight reduction in accuracy. This provides a practical mechanism for adjusting performance-efficiency trade-offs in real deployments. Depending on resource availability and target accuracy, practitioners can choose the appropriate number of alternative tokens.
### G.1 Prompts
Prompts for reasoning (for math datasets) System: You are a specialized AI assistant designed to solve mathematical problems. User: Please solve the following problem step by step, and put your final answer within boxed{}. Each pack of dvds costs 76 dollars. If there is a discount of 25 dollars on each pack How much do you have to pay to buy each pack? Assistant: To find the price after the discount, we need to subtract the discount amount from the original price. Step 1: Identify the original price and the discount amount.Original price = $76. Discount amount = $25 Step 2: Subtract the discount amount from the original price. Price after discount = Original price - Discount amount. Price after discount = $76 - $25. Price after discount = $51 So, you have to pay $51 to buy each pack. boxed{$51}
Prompts for reasoning (for ARC dataset) System: You are a specialized AI assistant designed to solve complex reasoning problems. User: Your task is to carefully read the following question, apply logical reasoning, scientific knowledge, and commonsense inference, and arrive at the correct answer. Always explain your reasoning step by step in clear language. At the end of your response, output the final choice in the following format: The answer is: (option), where option is the correct option letter (A, B, C, D, etc.), enclosed in parentheses. A group of engineers wanted to know how different building designs would respond during an earthquake. They made several models of buildings and tested each for its ability to withstand earthquake conditions. Which will most likely result from testing different building designs? (A) buildings will be built faster (B) buildings will be made safer (C) building designs will look nicer (D) building materials will be cheaper? Assistant: We are given a scenario in which engineers are testing different building designs to see how well they withstand earthquake conditions. The goal is to determine which of the listed outcomes is most likely to result from such testing. Let 2019s evaluate each option in turn: (A) Buildings will be built faster. 2013 This is not directly related to how well a building withstands earthquakes. Earthquake resistance is about structural integrity, not construction speed. While a good design might simplify construction, the primary purpose of the test is not speed. So this is not the most likely result. (B) Buildings will be made safer 2013 This is directly aligned with the purpose of the experiment. Testing building designs under earthquake conditions helps identify which designs can better absorb or resist seismic forces. The goal is to improve safety during earthquakes. Therefore, this outcome is a direct and logical result of such testing. (C) Building designs will look nicer 2013 Aesthetics are not the focus of earthquake testing. While a design might be both safe and attractive, the test itself does not evaluate visual appeal. So this is not a likely result of the testing. (D) Building materials will be cheaper 2013 Cost of materials is not necessarily a result of testing for earthquake resistance. ⊠The answer is: (B)
Prompts to paraphrase the question (for math datasets) System: You are a paraphrasing assistant specialized in math problems. Your task is to paraphrase a given math word problem. The core objective is to create a new, diverse expression of the problem while strictly preserving all numerical values, mathematical relationships, and the core logic required to solve it. User: Please create 5 paraphrased versions of the question below. Do NOT change any numbers or quantities. Maintain the exact mathematical operations and relationships. And vary the context and language. Please output paraphrased questions starting with a bracketed serial number like [1] [2] and ending with a new line. A robe takes 2 bolts of blue fiber and half that much white fiber. How many bolts in total does it take? Assistant: [1] Two bolts of blue fabric and one and a half times that amount of white fabric are required to make a robe. How many bolts in all are needed? [2] A robe is made from 2 bolts of blue yarn and an additional amount equal to half of that for white yarn. In total, how many bolts does it call for? [3] To create a robe, youâll need 2 bolts of blue thread and another half-quantity of white thread. How many bolts of thread in total should you purchase? [4] For a single robe, the production process demands 2 bolts of blue fabric and 1.5 times as much white fabric. What is the total number of bolts required? [5] The manufacture of a robe necessitates the usage of 2 bolts of blue material and slightly less than double that quantity for white material. How many total bolts does this equate to?
Prompts to paraphrase the question (for ARC dataset) System: You are given a multiple-choice question from a reasoning benchmark. Your task is to generate multiple different paraphrased versions of the question. Each version must preserve the original meaning, answer choices, and scientific/commonsense accuracy. Guidelines: - Keep the question logically equivalent to the original. - Do not change the correct answer. - Do not add or remove details that would alter the difficulty. - Use different phrasing, sentence structures, or synonyms for each version. - Keep the multiple-choice options (A, B, C, D, etc.) unchanged. Example Input: A large gene has dozens of exons, of which the central ones code for folded triple helical repeats that connect the cytoskeleton with sarcolemma and extracellular space. Each exon usually codes for one folded triple alpha helix. The most common mutations of the gene are central exon deletions that create out-of-frame peptides and progressive degenerative organ waste. A solution is to deliver a Morpholino that recognizes the 5â end of the out-of-frame exon in pre-mRNA. The molecule prevents binding of the spliceosome and creates exon skipping and in-frame joining. Several missing exons are well tolerated by an organism. Which structure below is not involved in the proposed therapy? (A) antisense (B) lariat (C) R-loops (D) polyA tail User: Please create 3 paraphrased versions of the question below. Only paraphrases the question and leaves the multiple-choice options untouched. Please output paraphrased questions starting with a bracketed serial number like [1] [2] and ending with a new line, An astronomer observes that a planet rotates faster after a meteorite impact. Which is the most likely effect of this increase in rotation? (A) Planetary density will decrease. (B) Planetary years will become longer. (C) Planetary days will become shorter. (D) Planetary gravity will become stronger. Assistant: [1] An astronomer notices that a planetâs rotational speed increases following a meteorite collision. What is the most probable consequence of this accelerated rotation? (A) Planetary density will decrease. (B) Planetary years will become longer. (C) Planetary days will become shorter. (D) Planetary gravity will become stronger. [2] After a meteorite strikes a planet, the astronomer observes an increase in its rotational velocity. What is the most likely outcome of this faster rotation? (A) Planetary density will decrease. (B) Planetary years will become longer. (C) Planetary days will become shorter. (D) Planetary gravity will become stronger. [3] A planet begins to rotate more quickly after experiencing a meteorite impact. Which of the following is most likely to result from this increased spin? (A) Planetary density will decrease. (B) Planetary years will become longer. (C) Planetary days will become shorter. (D) Planetary gravity will become stronger.
## Appendix H Use of Large Language Models
In our work, we focus on improving large Malaguena models, and LLMs are used in our experimental evaluation. In addition, we employ LLMs to check grammatical errors and refine sentence structures.