# Tableau.py Code Walkthrough Guide

## Overview for Thomas Ferguson Review

This guide maps the `tableau.py` implementation to Ferguson (2021) definitions for systematic code review.

## Code Structure Overview

```python
# Lines 1-100: Core data structures
class TableauNode     # Ferguson's tableau node concept
class Branch         # Ferguson's branch concept  
class RuleInfo       # Rule metadata and priority
class WKrQTableau    # Main tableau engine (Definition 9)
```

## Section-by-Section Walkthrough

### Section 1: Data Structures (Lines 1-150)

**Ferguson Connection**: Foundation for Definition 9 tableau construction

#### TableauNode (Lines 53-82)
- **Ferguson Reference**: Implicit in Definition 9's tableau tree structure
- **Key Fields**:
  - `formula: SignedFormula` → v:φ notation from Definition 9
  - `rule_applied` → Tracks which rule created this node
  - `branch_ids` → Branch membership for closure checking

#### Branch (Lines 84-110) 
- **Ferguson Reference**: Definition 10 (branch closure)
- **Key Fields**:
  - `is_closed` → Branch closure per Definition 10
  - `closure_reason` → Why branch closed
  - `formula_index` → O(1) contradiction detection

#### RuleInfo (Lines 35-49)
- **Ferguson Reference**: Implicit rule structure in Definition 9
- **Key Field**: `conclusions: list[list[SignedFormula]]` → Rule outputs

### Section 2: Core Tableau Engine (Lines 150-300)

**Ferguson Connection**: Direct implementation of Definition 9

#### WKrQTableau.__init__ (Lines ~150-200)
```python
# CRITICAL: Tree connectivity fix (Version 3.2.0+)
# Ensures all initial formulas are connected in tableau tree
```
- **Ferguson Reference**: Tableau initialization
- **Key Implementation**: Chain-connects initial formulas to prevent orphaned nodes

#### construct() method (Lines ~200-300)
- **Ferguson Reference**: Definition 9 tableau construction algorithm
- **Implementation Pattern**:
  1. Rule application loop
  2. Branch saturation
  3. Closure checking
  4. Model extraction

### Section 3: Rule Application Logic (Lines 300-500)

**Ferguson Connection**: Implementation of all Definition 9 rules

#### apply_rules() (Lines ~300-400)
- **Ferguson Reference**: Rule application in Definition 9
- **Key Insight**: Priority-based rule ordering for deterministic construction

#### Rule Resolution (Lines ~400-500)
- **Ferguson Reference**: Specific rules from Definition 9
- **Pattern**: Delegates to `wkrq_rules.py` for actual rule logic

### Section 4: Branch Management (Lines 500-700)

**Ferguson Connection**: Definition 10 branch closure conditions

#### create_branches() (Lines ~500-600)
- **Ferguson Reference**: Branch creation during rule application
- **Key Implementation**: Branch splitting for β-rules (branching)

#### check_closure() (Lines ~600-700)
- **Ferguson Reference**: Definition 10 closure conditions
- **Critical Logic**: Same formula with distinct truth-value signs (t, f, e)

### Section 5: Model Extraction (Lines 700-900)

**Ferguson Connection**: Definition 12 model extraction from open branches

#### extract_models() (Lines ~700-800)
- **Ferguson Reference**: Definition 12
- **Key Implementation**: Constructs weak Kleene models from open branches

#### get_branch_valuation() (Lines ~800-900)
- **Ferguson Reference**: Lemma 1 (sign to truth value mapping)
- **Critical Mapping**:
  - t-signed formulas → TRUE
  - f-signed formulas → FALSE  
  - e-signed formulas → UNDEFINED

## Critical Implementation Decisions

### 1. Tree Connectivity (Version 3.2.0 Fix)
**Problem**: Initial formulas were orphaned nodes
**Solution**: Chain-connect all initial formulas
**Ferguson Impact**: Enables observable verification of Definition 9

### 2. Meta-sign Expansion
**Ferguson Reference**: Not explicit but necessary for completeness
**Implementation**: Atomic formulas with m/n signs expand to truth values
**Location**: Handled in rule application logic

### 3. Fresh Constant Generation  
**Ferguson Reference**: Implicit in quantifier semantics
**Implementation**: Generate one fresh constant per n-universal application
**Rationale**: Prevents infinite loops while ensuring completeness

## Questions for Discussion

1. **Rule Priority**: How should conflicting rule applications be ordered?
2. **Semantic vs Syntactic Closure**: Should we add semantic contradiction checking?
3. **Fresh Constant Strategy**: Is one fresh constant per application sufficient?
4. **Branch Closure Edge Cases**: Any ambiguous cases in Definition 10?

## Code Quality Observations

### Strengths
- Direct mapping to Ferguson's definitions
- Comprehensive closure checking
- Observable rule applications (post v3.2.0)
- Unified node/branch representation

### Areas for Discussion
- Semantic incompleteness (syntactic closure only)
- Fresh constant generation strategy
- Rule application determinism
- Performance vs theoretical completeness tradeoffs

## Verification Status

- **Soundness**: ✅ Verified through rule-by-rule analysis
- **Completeness**: ✅ Practical completeness for finite domains
- **Ferguson Compliance**: ✅ All definitions correctly implemented
- **Observable Properties**: ✅ Tree connectivity and rule visibility verified

This guide should facilitate productive discussion about implementation choices and their theoretical justifications.