## Bar Chart: Comparison of Lean Calls by Method ### Overview The image is a vertical bar chart comparing four different computational methods based on the number of "Lean Calls" they generate. The chart uses a simple, clean design with a white background and a legend in the top-left corner. ### Components/Axes * **Y-Axis (Vertical):** Labeled "# of Lean Calls (thousand)". The scale runs from 0 to 30, with major tick marks at 0, 10, 20, and 30. The unit is thousands. * **X-Axis (Horizontal):** Labeled "Method". It displays four categorical bars representing different methods. * **Legend:** Positioned in the top-left quadrant of the chart area. It contains four entries, each with a colored box and a corresponding label: 1. **Brown Box:** `TrialMaster` 2. **Beige Box:** `DFS (t=1.5, 20 tactic)` 3. **Light Teal Box:** `DFS (t=1.5, 10 tactic)` 4. **Dark Teal Box:** `DFS (t=1.8, 10 tactic)` ### Detailed Analysis The chart presents the following approximate data points, derived from visual estimation against the y-axis scale: 1. **TrialMaster (Brown Bar):** The bar height is slightly below the 20 mark. Estimated value: **~18,000 Lean Calls**. 2. **DFS (t=1.5, 20 tactic) (Beige Bar):** The bar height is just below the 20 mark, appearing marginally taller than the TrialMaster bar. Estimated value: **~19,500 Lean Calls**. 3. **DFS (t=1.5, 10 tactic) (Light Teal Bar):** The bar height is visually identical to the beige bar (DFS t=1.5, 20 tactic). Estimated value: **~19,500 Lean Calls**. 4. **DFS (t=1.8, 10 tactic) (Dark Teal Bar):** This bar is significantly taller than the others. Its top aligns just above the 30 mark. Estimated value: **~31,000 Lean Calls**. **Visual Trend:** The first three methods (TrialMaster and both DFS variants with t=1.5) produce a similar, moderate number of Lean Calls, clustering around 18,000-19,500. The fourth method (DFS with t=1.8) shows a sharp, substantial increase, producing over 50% more calls than the others. ### Key Observations * **Significant Outlier:** The `DFS (t=1.8, 10 tactic)` method is a clear outlier, generating approximately 11,500 more Lean Calls than the next highest method. * **Parameter Sensitivity:** Changing the parameter `t` from 1.5 to 1.8 (while keeping "10 tactic" constant) appears to have a dramatic effect on the output, nearly doubling the number of Lean Calls. * **Tactic Count Insensitivity (at t=1.5):** For the DFS method with `t=1.5`, changing the tactic count from 10 to 20 shows no discernible difference in the number of Lean Calls based on this visual representation. * **Baseline Comparison:** The `TrialMaster` method serves as a baseline, performing slightly better (fewer calls) than the DFS methods with `t=1.5`. ### Interpretation This chart likely illustrates a performance or cost metric (Lean Calls) for different algorithmic strategies or configurations in a formal verification or automated reasoning context (suggested by terms like "Lean" and "tactic"). The data suggests that the `t` parameter in the DFS (Depth-First Search) method is a critical factor influencing computational cost. A higher `t` value (1.8 vs. 1.5) leads to a disproportionately large increase in Lean Calls, which could imply a less efficient search, a more exhaustive exploration, or a change in the algorithm's behavior that triggers more proof steps or interactions with the Lean theorem prover. The similarity between the two `t=1.5` DFS results indicates that, for this specific configuration, the number of tactics (10 vs. 20) is not a primary driver of the Lean Call count. The `TrialMaster` method appears to be a competitive alternative to the `t=1.5` DFS configurations, achieving a slightly lower call count. **In summary, the chart demonstrates that the choice of method and its parameters (specifically the `t` value for DFS) has a substantial impact on the associated computational workload, as measured by Lean Calls.**