Skill (𝒮)

Formal Definition

A skill is a fundamental capability that enables an agent to perform specific types of tasks. It represents the basic unit of competence in the skills ontology.

Formally, a skill is defined as a tuple:

$\mathcal{S}=⟨id,name,desc,\mathcal{D},{\lambda }_S,\mathcal{P},{\varphi }_S,\mathcal{C}⟩$

Where:

• $id\in \mathbb{N}$: Unique identifier
• $name\in {\Sigma }^{\ast }$: Descriptive name
• $desc\in {\Sigma }^{\ast }$: Natural language description
• $\mathcal{D}$: Application domain
• ${\lambda }_S\in {\mathbb{R}}^{+}$: Complexity measure
• $\mathcal{P}\subseteq 2^{\mathcal{S}}$: Set of prerequisite subskills
• ${\varphi }_S:\mathcal{T}\to [0,1]$: Fitness function for tasks $\mathcal{T}$
• $\mathcal{C}:\mathcal{S}\times \mathcal{S}\to \mathcal{S}$: Composition rules

Key Properties and Characteristics

1. Uniqueness: Each skill has a unique identifier in the skill space
2. Compositionality: Skills can be combined using composition operators
3. Hierarchy: Skills form a partial order relation through prerequisite relationships
4. Measurability: Skill effectiveness is quantifiable via fitness functions
5. Domain-Specific: Each skill operates within a specific application domain

Research Context and Applications

Skills are the foundational building blocks in the LLM skills research framework. They enable:

• Task Performance: Determining which skills are required for specific tasks
• Agent Assessment: Evaluating an agent’s capability set
• Skill Composition: Creating new skills from existing ones
• Performance Optimization: Improving task execution through skill selection

In the context of Large Language Models:

• Skills represent learned capabilities that emerge during training
• They can be identified, composed, and optimized for specific tasks
• Understanding skill structure helps design better prompting strategies

Connections to Other Concepts

• Subskills: Component skills that are prerequisites for more complex skills
• Superskills: Higher-order skills composed from multiple subskills
• Metaskills: Skills that operate on other skills (composition, analysis, optimization)
• Composition Operator (∘): Combines skills to create new skills
• Fitness Functions (φ): Evaluates skill effectiveness for tasks
• Skill Lattice: Mathematical structure formed by skill relationships

Open Research Questions

1. Skill Identification: How to automatically identify and extract skills from LLM behavior?
2. Primitive Skills: What is the minimal set of primitive skills required?
3. Skill Transfer: How do skills transfer across different domains and tasks?
4. Emergence Dynamics: How do new skills emerge from existing ones during learning?
5. Complexity Measure: How to quantify the complexity measure ${\lambda }_S$ precisely?
6. Composition Completeness: What composition operations are necessary and sufficient?