MCP-Logic

# TechStack ## Purpose and Overview The mcp-logic server provides a critical bridge between modern AI systems and formal logical reasoning capabilities. It integrates Prover9 (an automated theorem prover) and Mace4 (a finite model builder) into the Model Context Protocol framework, enabling: 1. Automated theorem proving for various logical systems: - First-order logic - Modal logic - Multi-agent reasoning - Temporal logic - Meta-logical statements 2. Counterexample generation for invalid logical statements 3. Integration with broader AI reasoning systems 4. Formal verification of knowledge and inference chains ## Step-by-Step Explanations ### Setting Up the Environment 1. Install Prover9-Mace4 binaries in a known location 2. Configure the mcp-logic server with the path to Prover9-Mace4 binaries 3. Start the server with proper configuration ### Using the Prover 1. Format logical statements using Prover9 syntax: ``` all x (P(x) -> Q(x)) # Universal quantification exists x P(x) # Existential quantification P & Q # Conjunction P | Q # Disjunction -P # Negation P -> Q # Implication ``` 2. Modal Logic Operators: ``` box(P) # Necessity operator dia(P) # Possibility operator ``` 3. Submit proofs through the MCP interface: ```json { "premises": ["all x (P(x) -> Q(x))", "P(a)"], "conclusion": "Q(a)" } ``` ### Using Mace4 for Counterexamples 1. When a proof fails, automatically attempt to find a counterexample 2. Interpret the model output to understand why the proof failed 3. Use the counterexample to refine the logical statements ## Annotated Examples ### Basic First-Order Logic Proof ``` # Proving modus ponens premises = [ "P -> Q", # If P then Q "P" # P is true ] conclusion = "Q" # Therefore Q is true ``` ### Modal Logic Example ``` # Proving necessity distributes over implication premises = [ "box(P -> Q)" # It is necessary that (if P then Q) ] conclusion = "box(P) -> box(Q)" # Therefore (necessarily P implies necessarily Q) ``` ### Multi-Agent Knowledge ``` # Agent A knows what Agent B knows premises = [ "knows(a, knows(b, P))", # A knows that B knows P "knows(X,P) -> P" # Knowledge implies truth ] conclusion = "P" # Therefore P is true ``` ### Using Mace4 for Counterexamples ``` # Finding a model where transitive closure fails clauses = [ "R(a,b)", # a is related to b "R(b,c)", # b is related to c "-R(a,c)" # a is not related to c ] # Mace4 will find a model satisfying these conditions ``` ## Contextual Notes ### Design Decisions 1. Integration with Prover9/Mace4: - Chosen for its robust support of first-order logic - Well-documented input syntax - Efficient proof search algorithms - Built-in model finding capabilities 2. MCP Interface Design: - Simple JSON-based interface for proof requests - Structured output format for both proofs and counterexamples - Support for multiple logical systems through syntax conventions ### Trade-offs 1. Syntax Restrictions: - Must follow Prover9's specific syntax requirements - Some advanced logical operators require encoding - Modal logic requires careful operator definitions 2. Performance Considerations: - Proof search can be computationally intensive - Timeout mechanisms needed for complex proofs - Balance between completeness and efficiency ### Future Developments 1. Planned Extensions: - Support for additional logical systems - Integration with other theorem provers - Enhanced proof visualization tools - Improved counterexample generation ## Actionable Advice ### Common Pitfalls 1. Syntax Issues: - Always check operator precedence in complex formulas - Avoid using reserved words as variable names - Be careful with parentheses in nested expressions - Remember that variables are implicitly universally quantified 2. Proof Strategy: - Start with simpler versions of complex proofs - Break down complex proofs into lemmas - Use Mace4 to check for counterexamples early - Keep premises minimal to improve proof search 3. Modal Logic Specifics: - Define modal operators consistently - Avoid operator overloading - Use box() and dia() for necessity and possibility - Check accessibility relation requirements 4. Error Handling: - Always check for syntax errors before proof search - Set appropriate timeouts for complex proofs - Validate input format before sending to Prover9 - Look for common typos in logical operators ### Best Practices 1. Documentation: - Comment complex logical formulas - Document encoding decisions for special operators - Keep a library of proven lemmas - Maintain examples of common patterns 2. Testing: - Create test cases with known proofs - Include counterexample cases - Test boundary conditions - Verify performance on large proofs 3. Maintenance: - Regular syntax validation - Monitor proof search performance - Update operator libraries as needed - Keep documentation in sync with capabilities