## Chart: KL Divergence vs. τ for 5-state Adder ### Overview The image is a scatter plot showing the relationship between KL Divergence and τ for a 5-state Adder. The plot contains three data points, all represented by green diamonds. The KL Divergence appears to increase sharply from τ = 0.01 to τ = 0.03, then plateaus or slightly decreases at τ = 0.05. ### Components/Axes * **X-axis (Horizontal):** τ, with tick marks at 0.01, 0.02, 0.03, 0.04, and 0.05. * **Y-axis (Vertical):** KL Divergence, with tick marks at 0.00, 0.05, 0.10, and 0.15. * **Data Points:** Three green diamond markers. * **Text Label:** "5-state Adder" is located near the bottom-right of the plot. ### Detailed Analysis * **Data Point 1:** Located at approximately (τ = 0.01, KL Divergence = 0.00). The marker is a green diamond. * **Data Point 2:** Located at approximately (τ = 0.03, KL Divergence = 0.165). The marker is a green diamond. * **Data Point 3:** Located at approximately (τ = 0.05, KL Divergence = 0.16). The marker is a green diamond. ### Key Observations * The KL Divergence increases significantly between τ = 0.01 and τ = 0.03. * The KL Divergence plateaus or slightly decreases between τ = 0.03 and τ = 0.05. * There are only three data points in the plot. ### Interpretation The plot suggests that for the 5-state Adder, the KL Divergence is highly sensitive to changes in τ at lower values (around 0.01 to 0.03). As τ increases beyond 0.03, the KL Divergence stabilizes, indicating a potential saturation point or diminishing returns in terms of divergence. The limited number of data points makes it difficult to draw definitive conclusions about the overall trend, but it highlights a region of interest for further investigation around τ = 0.01 to 0.05.

## Scatter Plot: KL Divergence vs. $\tau$ for 5-state Adder ### Overview This image is a scatter plot illustrating the relationship between a parameter $\tau$ (tau) and "KL Divergence" for a system identified as a "5-state Adder." The plot contains three distinct data points represented by green diamonds. ### Components/Axes * **Y-Axis:** Labeled "KL Divergence." The scale ranges from 0.00 to 0.15, with major tick marks at 0.00, 0.05, 0.10, and 0.15. * **X-Axis:** Labeled "$\tau$." The scale ranges from 0.01 to 0.05, with major tick marks at 0.01, 0.02, 0.03, 0.04, and 0.05. * **Data Markers:** Three large, solid green diamond-shaped markers. * **Text Label:** "5-state Adder" is positioned in the bottom-right quadrant of the plot area. ### Detailed Analysis The data series consists of three points. The visual trend shows a sharp increase in KL Divergence between $\tau = 0.01$ and $\tau = 0.03$, followed by a plateau between $\tau = 0.03$ and $\tau = 0.05$. * **Point 1 (Bottom-Left):** * **Position:** X-axis value of 0.01, Y-axis value of 0.00. * **Status:** The marker is centered exactly on the intersection of the 0.01 X-axis tick and the 0.00 Y-axis tick. * **Point 2 (Top-Middle):** * **Position:** X-axis value of 0.03, Y-axis value of approximately 0.17. * **Status:** The marker is centered above the 0.03 X-axis tick. Vertically, it sits above the 0.15 Y-axis tick mark. * **Point 3 (Top-Right):** * **Position:** X-axis value of 0.05, Y-axis value of approximately 0.17. * **Status:** The marker is centered above the 0.05 X-axis tick. It is horizontally aligned with Point 2, indicating the same Y-axis value. ### Key Observations * **Threshold Effect:** The system exhibits a distinct threshold behavior. At $\tau = 0.01$, the KL Divergence is zero, indicating perfect alignment or convergence. * **Saturation:** The KL Divergence increases significantly as $\tau$ moves from 0.01 to 0.03, but then remains constant (plateaus) as $\tau$ increases further to 0.05. * **Data Sparsity:** The plot only provides data for three specific values of $\tau$ (0.01, 0.03, 0.05), leaving the behavior between these points undefined. ### Interpretation The data demonstrates that the "5-state Adder" system is sensitive to the parameter $\tau$. * **KL Divergence Significance:** KL Divergence (Kullback-Leibler divergence) is a measure of how one probability distribution differs from a reference distribution. A value of 0.00 at $\tau = 0.01$ suggests that at this specific parameter value, the system's output distribution perfectly matches the target distribution. * **System Instability:** The sharp rise in divergence at $\tau = 0.03$ suggests that increasing $\tau$ beyond 0.01 introduces significant error or deviation in the system's state distribution. * **Steady State Error:** The plateau at $\tau = 0.03$ and $\tau = 0.05$ suggests that once the system deviates from the target distribution, it reaches a maximum level of divergence (approximately 0.17) within this parameter range. It does not appear to diverge further as $\tau$ increases from 0.03 to 0.05, implying the system reaches a stable, albeit inaccurate, state.

## Scatter Plot: KL Divergence vs. τ for 5-state Adder ### Overview The image is a scatter plot showing the relationship between **KL Divergence** (y-axis) and **τ** (x-axis) for a system labeled "5-state Adder." Three green diamond markers are plotted at specific (τ, KL Divergence) coordinates. ### Components/Axes - **X-axis (τ)**: Labeled "τ" with values ranging from 0.01 to 0.05 in increments of 0.01. - **Y-axis (KL Divergence)**: Labeled "KL Divergence" with values from 0.00 to 0.15 in increments of 0.05. - **Data Points**: Three green diamond markers at: - (τ = 0.01, KL Divergence = 0.00) - (τ = 0.03, KL Divergence = 0.15) - (τ = 0.05, KL Divergence = 0.15) - **Text Label**: "5-state Adder" is positioned in the bottom-right corner of the plot. ### Detailed Analysis - **Data Points**: - At τ = 0.01, KL Divergence is 0.00 (minimum value). - At τ = 0.03 and τ = 0.05, KL Divergence reaches its maximum value of 0.15. - **Trend**: KL Divergence increases sharply from τ = 0.01 to τ = 0.03, then remains constant at τ = 0.05. ### Key Observations - The KL Divergence is zero at τ = 0.01, indicating no divergence at this parameter value. - A significant increase in KL Divergence occurs between τ = 0.01 and τ = 0.03, suggesting a critical threshold or transition in the system. - The divergence plateaus at τ = 0.05, implying stability or saturation beyond this parameter value. ### Interpretation The plot demonstrates that the KL Divergence of the 5-state Adder is highly sensitive to τ in the range 0.01–0.03, with a sharp increase. Beyond τ = 0.03, the divergence stabilizes, indicating that the system’s behavior becomes less variable or more predictable. The "5-state Adder" label suggests this plot is part of a study on a computational or algorithmic system with five states, where τ likely represents a parameter (e.g., time constant, threshold, or input value) influencing the system’s entropy or information divergence. The absence of additional data points or error bars limits conclusions about variability or confidence intervals.