## Truth Table and Weight Matrix with Probability Distributions ### Overview The image contains four components: 1. A **truth table** for logical operations (XNOR, AND, OR) with decimal equivalents 2. A **weight matrix** showing numerical relationships between variables 3. Two **probability bar charts** visualizing state distributions 4. A **legend** explaining color mappings for the charts --- ### Components/Axes #### Truth Table - **Columns**: A, B, XNOR, AND, OR, Decimal - **Rows**: All combinations of A (0/1) and B (0/1) - **Decimal Column**: Binary-to-decimal conversion (e.g., 00 = 0, 01 = 1, 10 = 2, 11 = 3) #### Weight Matrix - **Rows**: h, A, XNOR, AND, OR - **Columns**: h, A, B, XNOR, AND, OR - **Values**: Numerical weights with signs (e.g., +1, -1, +0.5) #### Probability Charts - **X-Axis**: "states" (labeled 4, 9, 17, 31 for the first chart; 9, 17 for the second) - **Y-Axis**: "Probability" (0.0 to 0.12 scale) - **Legends**: - First chart: "XNOR → 0" (blue bars) - Second chart: "A → 1" and "B → 0" (blue bars) --- ### Detailed Analysis #### Truth Table | A | B | XNOR | AND | OR | Decimal | |---|---|------|-----|----|---------| | 0 | 0 | 1 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 1 | 1 | | 1 | 0 | 0 | 0 | 1 | 2 | | 1 | 1 | 1 | 1 | 1 | 3 | #### Weight Matrix | | h | A | B | XNOR | AND | OR | |-------|---|----|----|------|-----|-----| | **h** | 0 | 0 | 0 | +1 | -1 | +1 | | **A** | 0 | 0 | -1 | +1 | +1 | +1 | | **XNOR** | 0 | -1 | 0 | 0 | +1 | +1 | | **AND** | 0 | +1 | +1 | +1 | 0 | 0 | | **OR** | 0 | +1 | +1 | +1 | +1 | 0 | #### Probability Charts 1. **First Chart (XNOR → 0)**: - States: 4 (highest bar), 9, 17, 31 - Probabilities: ~0.12 (state 4), ~0.08 (state 9), ~0.06 (state 17), ~0.04 (state 31) - Bars: Blue, aligned with legend "XNOR → 0" 2. **Second Chart (A → 1, B → 0)**: - States: 9 (highest bar), 17 - Probabilities: ~0.15 (state 9), ~0.10 (state 17) - Bars: Blue, aligned with legend "A → 1" and "B → 0" --- ### Key Observations 1. **Truth Table**: - XNOR outputs 1 only when A and B match (rows 0 and 3). - Decimal values confirm binary-to-decimal mapping (e.g., 11 = 3). 2. **Weight Matrix**: - Negative weights for XNOR and AND in row "h" suggest inhibitory relationships. - Positive weights for OR and A/B in row "h" indicate excitatory relationships. 3. **Probability Charts**: - State 9 and 17 dominate both charts, suggesting these states are most probable. - First chart shows a decay in probability from state 4 to 31. - Second chart emphasizes state 9 (A=1, B=0) as the most likely outcome. --- ### Interpretation - **Logical Relationships**: The truth table defines deterministic outputs for XNOR, AND, and OR gates. - **Weight Matrix Dynamics**: - Negative weights for XNOR and AND in row "h" may suppress their influence on the output. - Positive weights for OR and A/B suggest these variables directly contribute to the output. - **Probability Trends**: - State 9 (A=1, B=0) has the highest probability in both charts, aligning with the weight matrix's emphasis on A and B. - State 17 (A=1, B=1) appears in both charts but with lower probability, possibly due to inhibitory weights for AND/XNOR. - **Anomalies**: - State 4 (A=0, B=0) has the highest probability in the first chart despite A and B being 0, which conflicts with the weight matrix's positive weights for A/B. This may indicate a non-linear interaction or external factors not captured in the matrix. The data suggests a system where logical operations (XNOR, AND, OR) are weighted to prioritize certain states (e.g., A=1, B=0), with inhibitory/excitatory relationships shaping the probability distribution. The discrepancy in state 4's high probability warrants further investigation into the model's assumptions or data quality.