Knowledge Database - Skills Ontology

Welcome to the LLM Skills Research knowledge database. This is a comprehensive collection of formal concepts, mathematical structures, and research applications related to the skills ontology for Large Language Models.

Overview

This knowledge database provides structured documentation of the formal framework for understanding, composing, and manipulating skills in AI systems, particularly Large Language Models.

Core Concepts

Fundamental definitions and building blocks of the skills ontology:

1. Skill (𝒮) - The basic unit of capability
2. Subskill (𝒮_sub) - Component skills that form prerequisites
3. Superskill (𝒮_super) - Higher-order composed skills
4. Metaskill (𝓜) - Skills about skills (5 types)
5. Composition Operator (∘) - Combining skills
6. Decomposition Operator (↓) - Breaking down skills
7. Fitness Functions (Φ, φ) - Performance evaluation
8. Skill Hierarchy - Partial order structure
9. Task-Skill Mapping - Linking tasks to required skills

Mathematical Structures

Formal mathematical frameworks and theorems:

1. Skill Composition Semigroup - Theorem 1: $(\mathcal{S},\circ )$ forms a semigroup
2. Skill Lattice - Theorem 2: $(\mathcal{S},⪯)$ forms a lattice
3. Metaskill Fixed Points - Theorem 3: Optimal metaskills exist

Research Applications

Practical applications and research directions:

1. LLM Skill Emergence - How capabilities emerge in LLMs
2. Skills Algebra and Composition - Algebraic framework for self-improvement

Key Mathematical Notation

• 𝒮: Skill space
• 𝓜: Metaskill space
• ∘: Composition operator
• ↓: Decomposition operator
• ⊙: Metaskill application operator
• ≼: Partial order (prerequisite relation)
• Φ: Agent-level fitness function
• φ: Skill-level fitness function

Navigation by Topic

Skill Operations

• Composition Operator
• Decomposition Operator
• Skills Algebra

Skill Types

• Skill
• Subskill
• Superskill
• Metaskill

Mathematical Theory

• Skill Composition Semigroup
• Skill Lattice
• Metaskill Fixed Points

Evaluation and Measurement

• Fitness Functions
• Task-Skill Mapping

Applications

• LLM Skill Emergence
• Skills Algebra

Five Types of Metaskills

The ontology defines five fundamental metaskill types:

1. Composition Metaskills (𝓜_c): Create new skills from existing ones
2. Decomposition Metaskills (𝓜_d): Break down complex skills into components
3. Analysis Metaskills (𝓜_a): Evaluate skill fitness and relationships
4. Optimization Metaskills (𝓜_o): Improve existing skill combinations
5. Abstraction Metaskills (𝓜_ab): Create general skill patterns from specific instances

Research Goals

The overarching goal is to create an agent that can generate ever-increasing complex, higher-order new skills for tasks where it lacks the required capabilities, using a formal “skills algebra” rather than formal verifications.

This enables:

• Self-Improvement: Autonomous capability development
• Skill Composition: Building complex capabilities from simple ones
• Meta-Learning: Learning to learn through metaskill development
• Gödel Machine Alternative: Using algebra instead of formal proofs

Contributing

To add new concepts:

1. Follow the YAML frontmatter format
2. Include formal definitions with LaTeX
3. Add research context and applications
4. Link to related concepts
5. Include open research questions

Source

All concepts extracted from the formal skills ontology: ontology/skills-ontology.md

Total Concepts: 14 (9 core + 3 mathematical + 2 applications)

Last Updated: 2025-11-21