# Strategy 2: Witness Predicate Implementation Archive

[← Back to Archive](../README.md) | [Documentation →](docs/) | [Exclusion Theory →](../../README.md)

## Directory Structure

```
strategy2_witness/
├── README.md               # This file - witness predicate archive overview
├── __init__.py            # Module initialization
├── semantic.py            # Witness-aware semantic framework (1183 lines)
├── operators.py           # Exclusion operators with witnesses (5 operators, 656 lines)
├── examples.py            # Main test examples (39 examples)
├── examples_fine.py       # CB vs Fine comparison tests
└── docs/                  # Strategy documentation
    ├── PLAN_2.md          # Witness predicate implementation plan
    ├── SEED_2.md          # Initial conceptual seed
    └── WITNESS.md         # Comprehensive witness predicate explanation
```

## Overview

The **Witness Predicate Implementation** represents a breakthrough approach to implementing hyperintensional exclusion semantics by solving the fundamental challenge of existentially quantified functions in Z3. This archived version demonstrates how making witness functions explicit as first-class model predicates enables correct handling of complex nested formulas that were previously intractable.

Within the exclusion theory development history, this implementation achieved the first fully correct execution of all test examples by introducing witness predicates for the Champollion-Bernard (CB) preclusion semantics. The architecture provides two distinct approaches: function-based CB preclusion using witness predicates (`\unineg`) and set-based Fine preclusion without function quantification (`\set_unineg`).

This archive preserves the successful solution to the "false premise problem" that plagued earlier attempts, with all 39 test examples executing correctly and demonstrating the theoretical relationship between CB and Fine preclusion approaches.

## Quick Start

```python
# Note: This is archived code - use for reference and study
# Demonstrates the witness predicate solution

from strategy2_witness import (
    WitnessSemantics,
    WitnessProposition,
    WitnessStructure,
    witness_operators
)

# Define theory with witness predicates
theory = {
    "semantics": WitnessSemantics,
    "proposition": WitnessProposition,
    "model": WitnessStructure,
    "operators": witness_operators,
    "dictionary": {}
}

# Run examples showing witness functions in action
# ./dev_cli.py -p -z src/model_checker/theory_lib/exclusion/archive/strategy2_witness/examples.py
```

## Subdirectories

### [docs/](docs/)

Comprehensive documentation explaining the witness predicate breakthrough. Contains WITNESS.md providing an accessible introduction to witness predicates and the CB vs Fine distinction, PLAN_2.md outlining the strategic design decisions for making functions explicit, and SEED_2.md with the initial conceptual insights. Essential for understanding how this approach solved the existential quantification challenge.

## Documentation

### For Researchers

- **[Witness Predicates Explained](docs/WITNESS.md)** - Accessible introduction to the solution
- **[Implementation Strategy](docs/PLAN_2.md)** - Design decisions and architecture
- **[Theoretical Comparison](#theoretical-foundations)** - CB vs Fine preclusion analysis

### For Developers

- **[Semantic Framework](semantic.py)** - 1183-line witness-aware implementation
- **[Operator Implementations](operators.py)** - 5 operators with witness support
- **[Test Suite](examples.py)** - 39 working examples demonstrating correctness

### For Historians

- **[Exclusion Theory History](../../history/)** - Complete development narrative
- **[From Strategy 1](../strategy1_multi/)** - Evolution from multi-strategy approach
- **[Current Implementation](../../)** - How this influenced the final design

## Implementation Architecture

### Core Innovation: Witness Predicates

The key breakthrough was making existentially quantified functions explicit as model predicates:

```python
# For each \unineg(φ) formula, create witness functions:
h_pred = z3.Function(f"{formula_str}_h", z3.BitVecSort(N), z3.BitVecSort(N))
y_pred = z3.Function(f"{formula_str}_y", z3.BitVecSort(N), z3.BitVecSort(N))
```

### Semantic Components

**WitnessAwareModel**: Extended Z3 model providing witness function access
- Query methods: `get_h_witness()`, `get_y_witness()`
- Seamless integration with standard model evaluation

**WitnessRegistry**: Centralized management of witness predicates
- Automatic registration during formula traversal
- Unique witnesses per formula instance

**WitnessConstraintGenerator**: Translates CB semantics to Z3
- Encodes three CB conditions as Z3 constraints
- Handles bidirectional implications correctly

### Operator Implementation

The archive includes **5 operators**:

1. **UniNegationOperator** (`\unineg`) - CB preclusion with witnesses
2. **FineUniNegation** (`\set_unineg`) - Fine's set-based preclusion
3. **UniConjunctionOperator** (`\uniwedge`) - Standard conjunction
4. **UniDisjunctionOperator** (`\univee`) - Standard disjunction
5. **UniIdentityOperator** (`\uniequiv`) - Verifier-based identity

## Examples

### Test Categories

The archive includes **39 comprehensive test examples**:

- **18 Theorems** (TH_*): Valid principles including DeMorgan's laws, distribution, absorption
- **21 Countermodels** (CM_*): Invalid inferences correctly rejected
- **Comparison Tests**: CB vs Fine preclusion relationship validation

### Notable Successes

**DeMorgan's Laws**: All four forms now work correctly
```python
# Previously failed with false premise
de_morgan_3 = [
    ["\\unineg(A \\uniwedge B)"],  # ¬(A ∧ B)
    ["\\unineg A \\univee \\unineg B"],  # ¬A ∨ ¬B
    {"N": 3}
]
# Now correctly finds countermodel!
```

**Witness Function Output**:
```
Functions
  \unineg(A)_h: □ → □
  \unineg(A)_h: a → b
  \unineg(A)_h: b → a
  \unineg(A)_h: a.b → a.b
```

## Theoretical Foundations

### CB Preclusion (Champollion-Bernard)

**Definition**: State e precludes φ iff there exist functions h,y such that:
1. For all v verifying φ: y(v) ⊑ v and h(v) excludes y(v)
2. For all v verifying φ: h(v) ⊑ e
3. e is minimal satisfying conditions 1-2

**Implementation**: Witness predicates make h,y explicit in the model

### Fine Preclusion

**Definition**: State e precludes φ iff:
1. e covers all verifiers of φ
2. Some part of e is relevant to each verifier

**Implementation**: Direct set operations without function quantification

### Theoretical Results

Tests confirm:
- CB preclusion → Fine preclusion (always)
- Fine preclusion ↛ CB preclusion (counterexamples exist)
- CB is strictly stronger than Fine

## Testing and Validation

### Running Historical Tests

```bash
# Main test suite with witness predicates
./dev_cli.py -p -z src/model_checker/theory_lib/exclusion/archive/strategy2_witness/examples.py

# CB vs Fine comparison tests
./dev_cli.py -p -z src/model_checker/theory_lib/exclusion/archive/strategy2_witness/examples_fine.py
```

### Validation Results

- **All 39 examples**: Execute without false premises
- **Witness functions**: Correctly computed and displayed
- **Theoretical properties**: CB/Fine relationship confirmed

## Historical Significance

### Problems Solved

1. **False Premise Issue**: Completely eliminated through explicit witnesses
2. **Function Quantification**: Z3 can now handle complex nested formulas
3. **Bidirectional Constraints**: Proper if-and-only-if semantics maintained

### Architectural Insights

1. **Explicit Over Implicit**: Making functions model citizens avoids Skolemization
2. **Modularity**: Each formula gets independent witness functions
3. **Debuggability**: Witness values visible in model output

### Influence on Final Design

This implementation directly influenced:
- Recognition that simpler semantics might suffice
- Understanding of Z3's quantification limitations
- Design patterns for handling complex semantics

## References

### Related Documentation

- **[Strategy 1 Archive](../strategy1_multi/)** - Previous multi-strategy attempt
- **[Current Implementation](../../)** - Final simplified approach
- **[Implementation Story](../../history/IMPLEMENTATION_STORY.md)** - Complete narrative

### Technical Context

- Breakthrough implementation solving false premise problem
- Demonstrates power of architectural innovation
- Foundation for understanding Z3 quantification challenges

---

[← Back to Archive](../README.md) | [Documentation →](docs/) | [Exclusion Theory →](../../README.md)
