\n ## Charts: Performance Comparison of Proof Strategies ### Overview The image presents two charts comparing the performance of different proof strategies. Chart (a) is a survival plot showing the number of proofs found over training time. Chart (b) is a scatter plot comparing proof lengths for two strategies. ### Components/Axes **Chart (a): Survival Plot** * **Title:** (a) Survival plot * **X-axis:** training time in seconds (logarithmic scale, from 10⁰ to 10⁴) * **Y-axis:** proofs (from 0 to 1000) * **Data Series:** * E (blue dashed line) * IL w/HER (orange solid line) * IL w/o HER (green dotted line) **Chart (b): Proof Lengths** * **Title:** (b) Proof lengths * **X-axis:** E (from 0 to 200) * **Y-axis:** IL w/HER (from 0 to 200) * **Diagonal Line:** A dashed black line representing y = x. * **Data Points:** Blue dots representing individual proof length pairs. ### Detailed Analysis or Content Details **Chart (a): Survival Plot** * **E (blue dashed line):** Starts at approximately 850 proofs at 10⁰ seconds, increases to approximately 950 proofs at 10² seconds, and plateaus around 970 proofs at 10⁴ seconds. The line shows a rapid initial increase, followed by diminishing returns. * **IL w/HER (orange solid line):** Starts at approximately 150 proofs at 10⁰ seconds, increases rapidly to approximately 850 proofs at 10² seconds, and continues to increase to approximately 950 proofs at 10⁴ seconds. This line demonstrates a slower initial growth but eventually catches up to and nearly matches the performance of 'E'. * **IL w/o HER (green dotted line):** Starts at approximately 100 proofs at 10⁰ seconds, increases to approximately 650 proofs at 10² seconds, and reaches approximately 750 proofs at 10⁴ seconds. This line shows the slowest growth among the three strategies. **Chart (b): Proof Lengths** * The scatter plot shows a concentration of blue dots clustered around the lower-left corner, indicating that for many proofs, the 'E' and 'IL w/HER' strategies produce similar proof lengths. * There is a noticeable spread of points above the diagonal line (y = x), suggesting that for a significant number of proofs, 'IL w/HER' generates longer proofs than 'E'. * The points are relatively sparse in the upper-right corner, indicating that neither strategy frequently produces very long proofs. * Approximate data points: * (10, 10) - a cluster of points * (50, 20) - a cluster of points * (100, 40) - a cluster of points * (150, 60) - a cluster of points * (180, 180) - a few points near the diagonal ### Key Observations * In Chart (a), 'E' initially finds more proofs faster, but 'IL w/HER' eventually approaches its performance. 'IL w/o HER' consistently underperforms both other strategies. * In Chart (b), 'IL w/HER' often produces longer proofs than 'E', although there is significant variability. * The logarithmic scale on the x-axis of Chart (a) emphasizes the initial rapid gains in proof finding. ### Interpretation The data suggests that the 'IL w/HER' strategy is a viable alternative to 'E', eventually achieving comparable performance in terms of the number of proofs found. However, 'IL w/HER' comes at the cost of potentially longer proof lengths, as indicated by Chart (b). The 'IL w/o HER' strategy appears to be less effective than both 'E' and 'IL w/HER'. The relationship between the two charts is that Chart (a) shows the *quantity* of proofs found over time, while Chart (b) shows the *quality* (length) of the proofs generated by two of the strategies. The diagonal line in Chart (b) serves as a benchmark: points above the line indicate longer proofs for 'IL w/HER' compared to 'E', while points below the line indicate shorter proofs. The outlier points in Chart (b) – those significantly above the diagonal – represent cases where 'IL w/HER' generates substantially longer proofs than 'E'. These cases might warrant further investigation to understand the underlying reasons for the increased proof length. The overall trend suggests a trade-off between proof-finding speed and proof length, with 'IL w/HER' offering a potential balance between the two.