# Technical Document Extraction ## Section 1: Identify Hidden Predicates ### Diagram Description - **Input**: `X` (represented by a cluster of nodes on the left) - **Output**: `Ŷ` (represented by a single node on the right) - **Architecture**: - Multiple hidden layers (depicted as interconnected nodes) - Connections between input, hidden, and output layers - **Legend**: - **Red circles**: Predicate `p₁` - **Blue circles**: Predicate `p₂` - **White circles**: Predicate `p₃` - **Yellow circles**: Predicate `p₄` - **Text**: - "Find the predicates for each class via purity metric" ### Spatial Grounding - Legend positioned at the bottom of the diagram. - Colors in the legend match the node colors in the neural network. --- ## Section 2: Determine the Logical Rules ### Table Structure | | `p₁` | `p₂` | `p₃` | `p₄` | |-------|------|------|------|------| | `x₁` | 1 | 0 | 1 | 0 | | `x₂` | 0 | 1 | 1 | 0 | | `x₃` | 0 | 1 | 1 | 0 | | `x₄` | 1 | 1 | 0 | 1 | | `x₅` | 1 | 0 | 0 | 1 | | `x₆` | 1 | 0 | 0 | 1 | ### Logical Rules (LaTeX) 1. `∀x, (p₁ ∧ p₃) ∨ (¬p₁ ∧ p₂ ∧ p₃ ∧ ¬p₄) ⇒ C₁` 2. `∀x, (p₁ ∧ p₂ ∧ ¬p₃ ∧ p₄) ∨ (p₁ ∧ p₄) ⇒ C₂` ### Notes - Table entries use binary values (1 = presence, 0 = absence). - Logical rules combine predicates using Boolean operators (`∧`, `∨`, `¬`). --- ## Section 3: Ground Predicates to Input Space ### Diagram Description - **Components**: - **Predicate Set `P`**: Contains `p₃` (red cloud) and `p₄` (blue cloud). - **Input Domain `X`**: Represented as a 2D plane with overlapping regions. - **Process**: - "Learn mapping from `P` to `X` via interpretable surrogate model" - **Legend**: - **Red cloud**: Predicate `p₃` - **Blue cloud**: Predicate `p₄` ### Spatial Grounding - Legend positioned at the top-right of the diagram. - Colors in the legend match the cloud colors in the input domain. --- ## Summary 1. **Hidden Predicates**: Identified via neural network architecture and purity metric. 2. **Logical Rules**: Derived from binary predicate assignments in a table. 3. **Input Mapping**: Predicates `p₃` and `p₄` are grounded to input space using a surrogate model. No non-English text detected. All labels, tables, and diagrams have been transcribed with spatial and contextual grounding.