Project 8: Automated Theorem Prover (The Resolution Principle)

Build a resolution-based theorem prover for propositional logic.

Quick Reference

Attribute Value
Difficulty Level 4: Expert
Time Estimate 12-20 hours
Language Python
Prerequisites Logic basics, CNF conversion
Key Topics resolution, CNF, proof traces

1. Learning Objectives

By completing this project, you will:

  1. Convert formulas to CNF.
  2. Implement resolution rule.
  3. Detect contradictions to prove theorems.
  4. Produce proof traces.
  5. Evaluate solver performance on test cases.

2. Theoretical Foundation

2.1 Resolution Principle

Resolution proves unsatisfiability by deriving contradictions from clauses.

3. Project Specification

3.1 What You Will Build

A prover that takes logical statements, converts to CNF, and uses resolution to prove or refute.

3.2 Functional Requirements

  1. CNF converter for formulas.
  2. Clause representation and simplification.
  3. Resolution engine.
  4. Proof trace generation.
  5. Evaluation on benchmark problems.

3.3 Non-Functional Requirements

  • Deterministic proofs.
  • Readable trace output.
  • Configurable limits to avoid blowups.

4. Solution Architecture

4.1 Components

Component Responsibility
Parser Parse logical formulas
CNF Converter Normalize formulas
Resolver Apply resolution
Tracer Record proofs

5. Implementation Guide

5.1 Project Structure

SYMBOLIC_AI_AND_EXPERT_SYSTEMS_MASTERY/P08-theorem-prover/
├── src/
│   ├── parse.py
│   ├── cnf.py
│   ├── resolve.py
│   └── trace.py

5.2 Implementation Phases

Phase 1: Parsing + CNF (4-6h)

  • Parse formulas and convert to CNF.
  • Checkpoint: CNF conversion correct.

Phase 2: Resolution (4-8h)

  • Implement resolution and contradiction detection.
  • Checkpoint: simple theorems proved.

Phase 3: Tracing + eval (4-6h)

  • Add proof traces and benchmarks.
  • Checkpoint: proof report generated.

6. Testing Strategy

6.1 Test Categories

Category Purpose Examples
Unit CNF correct transformations
Integration resolver proof of contradiction
Regression trace deterministic proofs

6.2 Critical Test Cases

  1. CNF conversion preserves equivalence.
  2. Resolution finds contradiction when unsat.
  3. Trace lists clause derivations.

7. Common Pitfalls & Debugging

Pitfall Symptom Fix
Clause explosion memory blowup add limits and heuristics
CNF errors wrong proofs validate transformations
Infinite resolution no termination add depth or clause limits

8. Extensions & Challenges

Beginner

  • Add simple tautology checks.
  • Add clause simplification.

Intermediate

  • Add heuristics for clause selection.
  • Add proof visualization.

Advanced

  • Extend to first-order logic.
  • Compare with SAT solvers.

9. Real-World Connections

  • SAT solvers and verification rely on resolution.
  • Formal methods use proof traces for audit.

10. Resources

  • Logic and resolution references
  • SAT solver tutorials

11. Self-Assessment Checklist

  • I can convert formulas to CNF.
  • I can implement resolution.
  • I can generate proof traces.

12. Submission / Completion Criteria

Minimum Completion:

  • CNF conversion + resolution

Full Completion:

  • Proof traces + benchmarks

Excellence:

  • First-order extensions
  • Comparison with SAT solvers

This guide was generated from project_based_ideas/AI_AGENTS_LLM_RAG/SYMBOLIC_AI_AND_EXPERT_SYSTEMS_MASTERY.md.