## Scatter Plot: Reasoning Skill and Embeddings ### Overview The image presents four scatter plots visualizing the distribution of different reasoning skills and embeddings. Each plot represents a different aspect: reasoning skill of (Q, R), reasoning skill of Q, raw question embedding, and raw rationale embedding. The points in each plot are color-coded and shaped according to the type of reasoning skill, as defined in the legend on the right. ### Components/Axes * **Plots:** Four scatter plots arranged in a 2x2 grid. * Top-left: "Reasoning skill of (Q, R)" * Top-right: "Reasoning skill of Q" * Bottom-left: "Raw question embedding" * Bottom-right: "Raw rationale embedding" * **Axes:** The axes are not explicitly labeled with scales or units. The plots appear to be projections of high-dimensional data into two dimensions, likely using a dimensionality reduction technique like t-SNE or PCA. * **Legend (Right side of the image):** * **Title:** "Reasoning skills" * **Categories:** * Black circle: Compute statistics * Purple down-triangle: Compute rate of change * Blue 'x': Compute money cost * Teal 'x': Filter tree leaves * Blue down-triangle: Addition/subtraction * Teal 'x': Search minimum/maximum * Green circle: Multiplication * Green down-triangle: Filter table entries * Light green 'x': Compute probability * Yellow down-triangle: Shortage or surplus? * Yellow 'x': Reason time schedule * Red 'x': Compare numbers * Red circle: Others ### Detailed Analysis **1. Reasoning skill of (Q, R) [Top-Left]** * **Compute statistics (Black circles):** Clustered in the top-right. * **Compute rate of change (Purple down-triangles):** Scattered in the top-right and bottom-left. * **Compute money cost (Blue 'x'):** Predominantly in the left half, forming a large cluster. * **Filter tree leaves (Teal 'x'):** Scattered throughout the plot. * **Addition/subtraction (Blue down-triangles):** Clustered in the center-right. * **Search minimum/maximum (Teal 'x'):** Scattered throughout the plot. * **Multiplication (Green circles):** Clustered in the top-left. * **Filter table entries (Green down-triangles):** Clustered in the top-left. * **Compute probability (Light green 'x'):** Clustered in the top-left. * **Shortage or surplus? (Yellow down-triangles):** Clustered in the top-left. * **Reason time schedule (Yellow 'x'):** Scattered in the top-left. * **Compare numbers (Red 'x'):** Scattered in the left half. * **Others (Red circle):** Located near the center. **2. Reasoning skill of Q [Top-Right]** * **Compute statistics (Black circles):** Clustered in the top-center. * **Compute rate of change (Purple down-triangles):** Scattered in the top-left. * **Compute money cost (Blue 'x'):** Predominantly in the right half, forming a large cluster. * **Filter tree leaves (Teal 'x'):** Scattered throughout the plot. * **Addition/subtraction (Blue down-triangles):** Clustered in the center. * **Search minimum/maximum (Teal 'x'):** Scattered throughout the plot. * **Multiplication (Green circles):** Clustered in the bottom-center. * **Filter table entries (Green down-triangles):** Clustered in the top-left. * **Compute probability (Light green 'x'):** Clustered in the top-left. * **Shortage or surplus? (Yellow down-triangles):** Clustered in the bottom-center. * **Reason time schedule (Yellow 'x'):** Scattered in the bottom-center. * **Compare numbers (Red 'x'):** Scattered in the top-left. * **Others (Red circle):** Located near the center. **3. Raw question embedding [Bottom-Left]** * **Compute statistics (Black circles):** Clustered in the center. * **Compute rate of change (Purple down-triangles):** Scattered in the center-left. * **Compute money cost (Blue 'x'):** Predominantly in the right half, forming a large cluster. * **Filter tree leaves (Teal 'x'):** Scattered throughout the plot. * **Addition/subtraction (Blue down-triangles):** Clustered in the center. * **Search minimum/maximum (Teal 'x'):** Scattered throughout the plot. * **Multiplication (Green circles):** Clustered in the center. * **Filter table entries (Green down-triangles):** Clustered in the center. * **Compute probability (Light green 'x'):** Clustered in the left. * **Shortage or surplus? (Yellow down-triangles):** Clustered in the bottom-left. * **Reason time schedule (Yellow 'x'):** Scattered in the bottom-left. * **Compare numbers (Red 'x'):** Scattered in the left. * **Others (Red circle):** Located near the left. **4. Raw rationale embedding [Bottom-Right]** * **Compute statistics (Black circles):** Clustered in the bottom-left. * **Compute rate of change (Purple down-triangles):** Scattered in the bottom-left. * **Compute money cost (Blue 'x'):** Predominantly in the top-right, forming a large cluster. * **Filter tree leaves (Teal 'x'):** Scattered throughout the plot. * **Addition/subtraction (Blue down-triangles):** Clustered in the top-center. * **Search minimum/maximum (Teal 'x'):** Scattered throughout the plot. * **Multiplication (Green circles):** Clustered in the top-right. * **Filter table entries (Green down-triangles):** Clustered in the top-right. * **Compute probability (Light green 'x'):** Clustered in the top-right. * **Shortage or surplus? (Yellow down-triangles):** Clustered in the top-center. * **Reason time schedule (Yellow 'x'):** Scattered in the top-center. * **Compare numbers (Red 'x'):** Scattered in the bottom-left. * **Others (Red circle):** Located near the bottom-left. ### Key Observations * **Clustering:** Points representing the same reasoning skill tend to cluster together in each plot, suggesting that the embeddings capture some semantic similarity between questions requiring the same reasoning skill. * **Distribution Differences:** The distribution of points varies across the four plots, indicating that the embeddings capture different aspects of the questions and rationales. * **Compute Money Cost:** The "Compute money cost" skill (blue 'x') consistently forms a large, distinct cluster in the right half of the plots, suggesting it has a unique embedding signature. * **Compute Statistics:** The "Compute statistics" skill (black circles) is clustered in all plots. ### Interpretation The scatter plots visualize how different reasoning skills are represented in the embedding space of questions and rationales. The clustering of points with the same reasoning skill suggests that the embeddings are able to capture the semantic similarity between questions requiring the same type of reasoning. The differences in distribution across the four plots indicate that the embeddings capture different aspects of the questions and rationales, such as the reasoning skill required, the question content, and the rationale content. The plot "Reasoning skill of (Q, R)" likely represents the combined embedding of the question (Q) and the rationale (R). Comparing this plot to "Reasoning skill of Q" shows how the inclusion of the rationale affects the representation of the reasoning skill. The "Raw question embedding" and "Raw rationale embedding" plots show the distribution of the question and rationale embeddings separately, providing insights into how each contributes to the overall representation of the reasoning skill. The consistent clustering of "Compute money cost" suggests that this skill has a distinct embedding signature, possibly due to the presence of specific keywords or patterns in the questions and rationales related to money. The scattered distribution of other skills may indicate that they are more complex or have more diverse representations in the embedding space.

\n ## Scatter Plots: Reasoning Skill Analysis ### Overview The image presents four scatter plots, arranged in a 2x2 grid. Each plot visualizes the distribution of different reasoning skills, represented by various marker types and colors. The plots are labeled "Reasoning skill of (Q, R)", "Reasoning skill of Q", "Raw question embedding", and "Raw rationale embedding". A legend in the top-right corner identifies the reasoning skills corresponding to each marker type and color. ### Components/Axes Each scatter plot has two implicit axes, representing the x and y coordinates of the data points. The axes are not explicitly labeled with numerical scales or titles, suggesting the focus is on the relative distribution of skills rather than precise values. The legend, positioned in the top-right corner, lists the following reasoning skills: * Compute statistics (Black circle) * Compute rate of change (Purple triangle) * Compute money cost (Grey 'x') * Filter tree leaves (Blue plus) * Addition/subtraction (Green diamond) * Search minimum/maximum (Brown star) * Multiplication (Orange circle) * Filter table entries (Light blue inverted triangle) * Compute probability (Yellow circle) * Shortage or surplus? (Red triangle) * Reason time schedule (Purple 'x') * Compare numbers (Pink diamond) * Others (Grey circle) ### Detailed Analysis or Content Details **1. Reasoning skill of (Q, R)** * The plot shows a relatively even distribution of most skills. * 'Compute statistics' (black circles) are concentrated in the bottom-left quadrant. * 'Addition/subtraction' (green diamonds) are scattered throughout the plot, with a slight concentration in the top-right. * 'Compute probability' (yellow circles) are clustered in the top-left. * 'Shortage or surplus?' (red triangles) are sparsely distributed. * 'Others' (grey circles) are scattered throughout the plot. **2. Reasoning skill of Q** * 'Compute statistics' (black circles) are concentrated in the bottom-left quadrant. * 'Addition/subtraction' (green diamonds) are scattered throughout the plot, with a slight concentration in the top-right. * 'Compute probability' (yellow circles) are clustered in the top-left. * 'Shortage or surplus?' (red triangles) are sparsely distributed. * 'Others' (grey circles) are scattered throughout the plot. **3. Raw question embedding** * 'Compute statistics' (black circles) are concentrated in the bottom-left quadrant. * 'Addition/subtraction' (green diamonds) are scattered throughout the plot, with a slight concentration in the top-right. * 'Compute probability' (yellow circles) are clustered in the top-left. * 'Shortage or surplus?' (red triangles) are sparsely distributed. * 'Others' (grey circles) are scattered throughout the plot. **4. Raw rationale embedding** * 'Compute statistics' (black circles) are concentrated in the bottom-left quadrant. * 'Addition/subtraction' (green diamonds) are scattered throughout the plot, with a slight concentration in the top-right. * 'Compute probability' (yellow circles) are clustered in the top-left. * 'Shortage or surplus?' (red triangles) are sparsely distributed. * 'Others' (grey circles) are scattered throughout the plot. ### Key Observations * The distribution of reasoning skills appears broadly similar across all four plots. * 'Compute statistics' consistently appears in the bottom-left quadrant in all plots. * 'Addition/subtraction' tends to be more dispersed, with a slight bias towards the top-right. * 'Compute probability' consistently appears in the top-left quadrant in all plots. * 'Shortage or surplus?' is the least represented skill in all plots. ### Interpretation The plots likely represent a visualization of the distribution of reasoning skills required or demonstrated in a dataset of questions and rationales. The consistent patterns across the four plots suggest that the underlying distribution of skills is relatively stable, regardless of whether the analysis focuses on (Q, R), Q alone, the raw question embedding, or the raw rationale embedding. The concentration of 'Compute statistics' in the bottom-left quadrant could indicate that this skill is often associated with lower-level or more fundamental reasoning tasks. Conversely, the dispersion of 'Addition/subtraction' might suggest that this skill is involved in a wider range of reasoning scenarios. The scarcity of 'Shortage or surplus?' could imply that this skill is less frequently required or assessed in the dataset. The fact that the plots are not labeled with numerical scales suggests that the primary goal is to compare the *relative* prevalence of different skills, rather than to quantify their absolute frequency. The use of scatter plots allows for a visual assessment of the relationships between different skills, although without explicit axes, it is difficult to draw definitive conclusions about these relationships.

## Scatter Plot Matrix: Reasoning Skill Embeddings ### Overview The image displays a 2x2 grid of scatter plots visualizing the clustering of different reasoning skills within various embedding spaces. A shared legend on the right maps specific symbols and colors to 13 distinct reasoning skill categories. The plots compare how these skills group based on different representations: combined question and rationale skill, question skill alone, raw question embeddings, and raw rationale embeddings. ### Components/Axes * **Titles:** * Top-Left Plot: "Reasoning skill of (Q, R)" * Top-Right Plot: "Reasoning skill of Q" * Bottom-Left Plot: "Raw question embedding" * Bottom-Right Plot: "Raw rationale embedding" * **Legend (Positioned to the right of the top-right plot):** * Title: "Reasoning skills" * Entries (Symbol - Label): 1. Black Circle (●) - "Compute statistics" 2. Purple Downward Triangle (▼) - "Compute rate of change" 3. Blue 'x' (x) - "Compute money cost" 4. Blue Circle (●) - "Filter tree leaves" 5. Cyan Downward Triangle (▼) - "Addition/subtraction" 6. Cyan 'x' (x) - "Search minimum/maximum" 7. Green Circle (●) - "Multiplication" 8. Green Downward Triangle (▼) - "Filter table entries" 9. Light Green 'x' (x) - "Compute probability" 10. Yellow Circle (●) - "Shortage or surplus?" 11. Orange Downward Triangle (▼) - "Reason time schedule" 12. Red 'x' (x) - "Compare numbers" 13. Red Circle (●) - "Others" * **Axes:** None of the four scatter plots have labeled X or Y axes or numerical scales. They are abstract 2D projections of embedding spaces. ### Detailed Analysis **1. Reasoning skill of (Q, R) [Top-Left Plot]:** * **Trend/Clustering:** Skills form distinct, separated clusters. * **Spatial Distribution:** * **Top-Left Quadrant:** A dense cluster of green circles ("Multiplication") and green downward triangles ("Filter table entries"). * **Top-Center:** A cluster of orange downward triangles ("Reason time schedule"). * **Top-Right Quadrant:** A dense cluster of black circles ("Compute statistics"). * **Center-Left:** A large, dispersed cluster dominated by blue 'x's ("Compute money cost"). * **Center:** Scattered cyan 'x's ("Search minimum/maximum") and blue circles ("Filter tree leaves"). * **Center-Right:** A cluster of purple downward triangles ("Compute rate of change"). * **Bottom-Left:** A small cluster of light green 'x's ("Compute probability"). * **Bottom-Center:** A small cluster of yellow circles ("Shortage or surplus?"). * **Isolated Points:** A few red 'x's ("Compare numbers") and red circles ("Others") are scattered sparsely. **2. Reasoning skill of Q [Top-Right Plot]:** * **Trend/Clustering:** Clusters are more dispersed and less distinct than in the (Q,R) plot. * **Spatial Distribution:** * **Top-Left Quadrant:** A cluster of purple downward triangles ("Compute rate of change"). * **Top-Center:** A dense cluster of black circles ("Compute statistics"). * **Top-Right Quadrant:** A large, dispersed cluster of blue 'x's ("Compute money cost"). * **Center:** Scattered cyan 'x's ("Search minimum/maximum"), blue circles ("Filter tree leaves"), and green downward triangles ("Filter table entries"). * **Bottom-Center:** A cluster of orange downward triangles ("Reason time schedule"). * **Bottom-Right Quadrant:** A cluster of green circles ("Multiplication"). * **Isolated Points:** A few red 'x's ("Compare numbers") and a red circle ("Others") are present. **3. Raw question embedding [Bottom-Left Plot]:** * **Trend/Clustering:** Shows two primary, large clusters with some internal structure. * **Spatial Distribution:** * **Left Cluster:** A dense, vertical cluster containing a mix of blue 'x's ("Compute money cost"), cyan 'x's ("Search minimum/maximum"), blue circles ("Filter tree leaves"), green downward triangles ("Filter table entries"), and scattered points from other categories. * **Right Cluster:** A dense cluster containing black circles ("Compute statistics"), purple downward triangles ("Compute rate of change"), green circles ("Multiplication"), and yellow circles ("Shortage or surplus?"). * **Bottom:** A distinct, small cluster of orange downward triangles ("Reason time schedule"). * **Isolated Points:** Red 'x's ("Compare numbers") and red circles ("Others") are scattered. **4. Raw rationale embedding [Bottom-Right Plot]:** * **Trend/Clustering:** Shows a more continuous, elongated distribution compared to the other plots. * **Spatial Distribution:** * **Top-Right to Bottom-Left Diagonal:** A broad, dispersed band containing a mix of many skills, notably blue 'x's ("Compute money cost"), cyan 'x's ("Search minimum/maximum"), and green circles ("Multiplication"). * **Top-Left Quadrant:** A cluster of black circles ("Compute statistics"). * **Center-Left:** A cluster of purple downward triangles ("Compute rate of change"). * **Bottom-Center:** A cluster of blue circles ("Filter tree leaves"). * **Bottom-Left Quadrant:** A cluster of light green 'x's ("Compute probability"). * **Isolated Points:** Red 'x's ("Compare numbers"), red circles ("Others"), and yellow circles ("Shortage or surplus?") are scattered. ### Key Observations 1. **Cluster Coherence:** The "Reasoning skill of (Q, R)" plot shows the most distinct and separated clusters for each skill, suggesting the combined representation best disentangles these reasoning abilities. 2. **Skill Similarity:** "Compute statistics" (black circles) and "Compute rate of change" (purple triangles) often cluster near each other, implying similarity in their required reasoning. 3. **Dominant Skills:** "Compute money cost" (blue 'x') and "Search minimum/maximum" (cyan 'x') are highly prevalent and form large, dispersed clusters across multiple plots. 4. **Embedding Shift:** The transition from "Raw question embedding" to "Raw rationale embedding" shows a significant change in the spatial organization of skills, indicating the rationale provides different or additional information. 5. **Outliers:** The "Others" (red circle) and "Compare numbers" (red 'x') categories are consistently sparse and scattered, suggesting they are either rare or not well-defined within this embedding space. ### Interpretation This visualization demonstrates how different reasoning skills are represented in an AI model's internal embedding space. The clear clustering in the "Reasoning skill of (Q, R)" plot indicates the model has learned to distinguish these skills when considering both the question and its rationale. The proximity of certain clusters (e.g., statistics and rate of change) suggests the model perceives an underlying relationship between these tasks. The stark difference between the raw question and rationale embeddings highlights that the rationale (the step-by-step reasoning) carries distinct semantic information crucial for task classification. The dispersion of skills like "Compute money cost" may indicate it is a broad category encompassing varied sub-tasks. Overall, the plots provide evidence that the model's internal representations align with human-defined reasoning skill categories, with the combined (Q,R) representation being the most discriminative.

## Scatter Plot Grid: Reasoning Skills and Embeddings ### Overview The image contains a 2x2 grid of scatter plots comparing reasoning skills and embeddings. Each plot uses distinct colored markers (circles, triangles, crosses) to represent different reasoning skills, with a shared legend on the right. The plots explore relationships between reasoning skills in different contexts: (Q, R), Q alone, raw question embeddings, and raw rationale embeddings. ### Components/Axes - **Legend**: Located on the right side, mapping 13 reasoning skills to colors and symbols: - **Black circles**: Compute statistics - **Purple triangles**: Compute rate of change - **Blue crosses**: Compute money cost - **Teal triangles**: Filter tree leaves - **Dark blue circles**: Addition/subtraction - **Green triangles**: Search minimum/maximum - **Yellow circles**: Multiplication - **Light green crosses**: Filter table entries - **Red circles**: Compute probability - **Orange triangles**: Shortage or surplus? - **Pink triangles**: Reason time schedule - **Red crosses**: Compare numbers - **Black dots**: Others - **Subplots**: 1. **Top-left**: "Reasoning skill of (Q, R)" – Dense clusters of multiple skills. 2. **Top-right**: "Reasoning skill of Q" – Dominated by black circles (compute statistics) and purple triangles (compute rate of change). 3. **Bottom-left**: "Raw question embedding" – Overlapping distributions with teal triangles (filter tree leaves) and blue crosses (compute money cost) forming distinct clusters. 4. **Bottom-right**: "Raw rationale embedding" – Linear trend with black circles (compute statistics) and orange triangles (shortage or surplus?) concentrated in the lower-left. ### Detailed Analysis - **Top-left Plot**: - **Distribution**: Scattered points across the plot, with teal triangles (filter tree leaves) and green triangles (search minimum/maximum) forming tight clusters in the upper-right quadrant. - **Trends**: No clear linear relationship; skills are broadly dispersed. - **Top-right Plot**: - **Distribution**: Black circles (compute statistics) dominate the center, while purple triangles (compute rate of change) cluster in the upper-right. Blue crosses (compute money cost) are sparse. - **Trends**: Compute statistics is the most frequent skill, followed by compute rate of change. - **Bottom-left Plot**: - **Distribution**: Teal triangles (filter tree leaves) and blue crosses (compute money cost) form distinct clusters in the lower-left and upper-right, respectively. Overlap occurs in the central region. - **Trends**: Filter tree leaves and compute money cost are spatially separated, suggesting distinct embeddings. - **Bottom-right Plot**: - **Distribution**: Black circles (compute statistics) and orange triangles (shortage or surplus?) form a linear trend from lower-left to upper-right. Red crosses (compare numbers) cluster in the lower-right. - **Trends**: Compute statistics and shortage/surplus? show a positive correlation. ### Key Observations 1. **Dominant Skills**: Compute statistics (black circles) appears most frequently across all plots. 2. **Clustering**: Skills like filter tree leaves (teal triangles) and compute money cost (blue crosses) show spatial separation in raw embeddings. 3. **Linear Relationship**: Compute statistics and shortage/surplus? exhibit a positive correlation in raw rationale embeddings. 4. **Symbol Consistency**: Symbols (circles, triangles, crosses) align with legend labels but lack explicit explanation for their meaning (e.g., data type or group). ### Interpretation The plots suggest that reasoning skills are context-dependent: - **Q vs. (Q, R)**: Skills like compute rate of change (purple triangles) are more prominent when considering only Q, while (Q, R) introduces additional skills like filter tree leaves (teal triangles). - **Embedding Relationships**: Raw question and rationale embeddings reveal distinct clusters for certain skills (e.g., filter tree leaves vs. compute money cost), indicating separable features in the data. - **Linear Correlation**: The positive trend between compute statistics and shortage/surplus? in raw rationale embeddings implies these skills may share underlying features or are often co-occurring in reasoning tasks. **Uncertainties**: - Axis labels are missing, preventing quantification of trends. - Symbol meanings (circles vs. triangles vs. crosses) are not explained, limiting interpretation of groupings. - Exact numerical values cannot be extracted without axis scales.