Subskill (𝒮_sub)

Formal Definition

A subskill is a component skill that contributes to more complex skills. Subskills represent the building blocks that, when composed together, form higher-order capabilities.

Formally, a subskill is a skill ${\mathcal{S}}_{sub}\in \mathcal{S}$ that satisfies the prerequisite relation for at least one superskill:

${\mathcal{S}}_{sub}⪯{\mathcal{S}}_{super}⟺{\mathcal{S}}_{sub}\in \mathcal{P}({\mathcal{S}}_{super})$

Where:

• ${\mathcal{S}}_{sub}$: The subskill (component skill)
• ${\mathcal{S}}_{super}$: A superskill that requires this subskill
• $\mathcal{P}({\mathcal{S}}_{super})$: Set of prerequisite skills for the superskill
• $⪯$: Partial order relation (“is a prerequisite of”)

Additional Structural Properties:

• $part\_of:{\mathcal{S}}_{sub}\to 2^{{\mathcal{S}}_{super}}$ - Mapping to all superskills requiring this subskill
• $composition\_role\in \{sequential,parallel,conditional\}$ - How subskill contributes to composition

Key Properties and Characteristics

1. Prerequisite Relationship: Subskills are prerequisites for superskills
2. Composition Role: Each subskill has a specific role in the composition (sequential, parallel, or conditional)
3. Reusability: A single subskill can contribute to multiple superskills
4. Partial Order: Subskills form a partial order relation $⪯$ with superskills

The partial order relation is defined as: ${\mathcal{S}}_{sub}⪯{\mathcal{S}}_{super}⟺{\mathcal{S}}_{sub}\in \mathcal{P}({\mathcal{S}}_{super})$

Where $\mathcal{P}({\mathcal{S}}_{super})$ is the set of prerequisite skills for the superskill.

Research Context and Applications

Subskills are critical for understanding skill decomposition and learning:

• Skill Breakdown: Complex tasks can be decomposed into manageable subskills
• Learning Paths: Understanding prerequisite relationships guides learning sequences
• Modular Design: Subskills enable modular approaches to capability development
• Transfer Learning: Subskills learned in one context transfer to other superskills

In LLM research:

• Identifying subskills helps understand how complex behaviors emerge
• Enables targeted improvement of specific capability components
• Facilitates curriculum learning approaches

Connections to Other Concepts

• Skills (𝒮): Subskills are a subset of skills with additional structural constraints
• Superskills (𝒮_super): Higher-order skills composed from subskills
• Skill Hierarchy: Defines the partial order relation between sub and superskills
• Decomposition Operator (↓): Breaks complex skills into constituent subskills
• Composition Operator (∘): Combines subskills to form superskills

Open Research Questions

1. Minimal Decomposition: What is the minimal set of subskills for any given superskill?
2. Composition Uniqueness: Is the decomposition of a superskill into subskills unique?
3. Role Assignment: How to determine the optimal composition role (sequential, parallel, conditional) for each subskill?
4. Granularity: What is the appropriate level of granularity for subskill definition?
5. Dynamic Decomposition: Can decomposition strategies adapt based on context or agent capabilities?