## Line Chart: pass@1-with-n-queries
### Overview
The image is a line chart titled "pass@1-with-n-queries". It plots the performance of four different automated theorem-proving methods, measured by the cumulative number of theorems successfully proved as a function of the number of queries (n) made. The chart demonstrates how each method's success rate scales with increased computational effort (queries).
### Components/Axes
* **Chart Title:** `pass@1-with-n-queries` (Top center)
* **Y-Axis:**
* **Label:** `Number of Theorems Proved` (Left side, vertical)
* **Scale:** Linear, from 0 to 70. Major tick marks are at intervals of 10 (0, 10, 20, 30, 40, 50, 60, 70).
* **X-Axis:**
* **Label:** `Number of Queries (n)` (Bottom center)
* **Scale:** Linear, from 0 to approximately 3700. Major tick marks are at intervals of 500 (0, 500, 1000, 1500, 2000, 2500, 3000, 3500).
* **Legend:** Located in the bottom-right quadrant of the chart area. It contains four entries, each with a colored line sample and a text label.
1. **Orange Line:** `COPRA (GPT-4-turbo) (with Retrieval)`
2. **Blue Line:** `ReProver (with Retrieval)`
3. **Green Line:** `COPRA (GPT-4 turbo) (without Retrieval)`
4. **Red Line:** `ReProver (without Retrieval)`
### Detailed Analysis
The chart displays four distinct data series, each showing a rapid initial increase that plateaus at a different level.
1. **COPRA (GPT-4-turbo) with Retrieval (Orange Line):**
* **Trend:** Exhibits the steepest initial ascent, reaching its maximum value extremely quickly (within the first ~100 queries). After this point, the line is perfectly horizontal, indicating no further theorems are proved with additional queries.
* **Key Data Point:** Plateaus at approximately **71 theorems proved**.
2. **COPRA (GPT-4 turbo) without Retrieval (Green Line):**
* **Trend:** Also shows a very sharp initial rise, similar to but slightly less steep than its "with Retrieval" counterpart. It reaches its plateau very early (within ~100 queries) and remains constant thereafter.
* **Key Data Point:** Plateaus at approximately **65 theorems proved**.
3. **ReProver with Retrieval (Blue Line):**
* **Trend:** Shows a more gradual, step-wise increase compared to the COPRA methods. It continues to prove new theorems up to around 1000 queries before leveling off. The ascent is less steep initially but sustains growth for longer.
* **Key Data Point:** Plateaus at approximately **61 theorems proved** after ~1000 queries.
4. **ReProver without Retrieval (Red Line):**
* **Trend:** Follows a similar step-wise growth pattern to the blue line but with a lower slope and a lower final plateau. It also levels off around 500-1000 queries.
* **Key Data Point:** Plateaus at approximately **54 theorems proved**.
### Key Observations
* **Performance Hierarchy:** The final performance ranking from highest to lowest is: COPRA with Retrieval > COPRA without Retrieval > ReProver with Retrieval > ReProver without Retrieval.
* **Impact of Retrieval:** For both COPRA and ReProver, the "with Retrieval" variant outperforms the "without Retrieval" variant. The performance gap is larger for COPRA (~6 theorem difference) than for ReProver (~7 theorem difference).
* **Efficiency vs. Effort:** COPRA methods are highly efficient, achieving nearly all their potential proofs with very few queries (<100). ReProver methods require more queries (500-1000) to reach their maximum potential, suggesting a different, possibly more exhaustive, search strategy.
* **Plateau Behavior:** All methods exhibit a clear plateau, indicating a finite set of solvable theorems within the test suite. No method shows continuous improvement across the entire query range.
### Interpretation
This chart provides a comparative analysis of two theorem-proving systems (COPRA and ReProver) under two conditions (with and without a retrieval mechanism). The data suggests several key insights:
1. **Superiority of COPRA:** The COPRA architecture, especially when augmented with retrieval, demonstrates both higher final performance and greater sample efficiency (requiring fewer queries to reach peak performance) compared to ReProver on this task.
2. **Value of Retrieval:** Incorporating a retrieval component consistently improves the number of theorems proved for both systems. This implies that accessing a knowledge base or lemma library is a beneficial strategy for automated reasoning, helping the models overcome limitations in their parametric knowledge or reasoning depth.
3. **Diminishing Returns:** The sharp plateaus indicate a point of diminishing returns. After a certain number of queries, additional computational effort does not yield more proofs. This could be due to the inherent difficulty of the remaining theorems or limitations in the models' capabilities.
4. **Strategic Differences:** The contrast between COPRA's rapid plateau and ReProver's more gradual ascent hints at fundamental differences in their algorithms. COPRA may employ a more direct or heuristic-driven approach that quickly solves easier problems, while ReProver might use a more systematic but slower search process.
In summary, the chart is evidence that for this specific benchmark, the COPRA system with retrieval is the most effective and efficient approach, and that retrieval-augmented generation is a valuable technique for enhancing the capabilities of language models in formal reasoning tasks.
