Deductive Kernel (DK)
This directory contains the implementation and specifications for the Deductive Kernel component of the SPaDE project.
Overview
The Deductive Kernel is the core logical inference engine that performs formal reasoning within contexts provided by the Knowledge Repository. It represents a significant evolution beyond the traditional LCF paradigm, designed for reflexive self-improvement and AI integration.
Key Features
- Abstract Logical Engine: Completely divorced from concrete syntax and persistent theory representation.
- Reflexive Capabilities: The kernel admits the direct use of derived rules which are proven consistent with the specified derivability relation.
- Derived Rules Instead of Tactics: Derived rules play the role previously assumed by tactics, yielding the theorem which would normally be supplied by the tactic.
- AI Integration: Designed to work with AI which will use neural net heuristics in proof search and will implement derived rules for common inference patterns.
- Efficient Execution: Supports direct execution of verified algorithms
Architecture
The Deductive Kernel operates on contexts from the Knowledge Repository and delivers theorems derivable in those contexts.
Unlike traditional LCF systems, it does not manipulate the repository directly - theorems are not automatically added to contexts.
Core Concepts
- Context: The logical environment for inference (provided by KR)
- Theorem: A proposition proven to be derivable in a context
- Tactic: A verified algorithm for proof construction
- Metatheory: The theory of the logical system itself
- Reflexive Reasoning: The ability to reason about the system within itself
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Documentation
Development Status
Preliminary prototyping plan sketched.
- Plan1 - Deductive Kernel Development Plan
Key Innovations
1. LCF Evolution
The kernel evolves beyond the traditional LCF paradigm by:
- Separating logical inference from theory management
- Supporting verified tactics that can be trusted without detailed proof
- Enabling reflexive reasoning about the system itself
2. Reflexive Capabilities
The kernel is designed to support:
- Self-analysis through metatheory
- Verification of its own algorithms
- Self-improvement through reflexive reasoning
- AI-assisted development of its own capabilities
3. AI Integration
Designed to intersect with AI capabilities:
- AI-assisted proof generation
- Verified tactics that can be trusted
- Human-AI collaboration models
- Automated verification of AI-generated proofs
- Knowledge Repository (kr/): Provides contexts for logical inference
- API Layer: Interfaces for accessing kernel capabilities
- Tools: Development and verification tools
Development Priorities
- Complete formal specifications for the logical system
- Implement core inference engine (type checking, proof checking)
- Design metatheory framework for reflexive reasoning
- Create verified tactic system
- Build AI integration interfaces
Technical Challenges
- Formalizing the theory of the logical system within itself
- Ensuring consistency of reflexive reasoning
- Managing the complexity of self-reference
- Efficient theorem proving for large knowledge bases
- Optimized tactic execution
- Scalable proof checking
3. AI Integration
- Interfaces for AI-assisted proof generation
- Verification of AI-generated proofs
- Human-AI collaboration protocols
Contributing
See the main project CONTRIBUTING.md for general guidelines.
For DK-specific development:
- Review the formal specifications in
specs/
- Follow the architecture defined in
docs/
- Add tests for all new functionality
- Update documentation for any API changes
References