Skill Hierarchy

Formal Definition

The skill hierarchy defines the structural relationships between skills through a partial order relation. It captures prerequisite dependencies and compositional structure.

The partial order relation $⪯$ on skills is defined as:

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

Where:

• $⪯$: “is a prerequisite of” or “is less complex than”
• $\mathcal{P}({\mathcal{S}}_{super})$: Set of prerequisite skills for superskill ${\mathcal{S}}_{super}$

Key Properties and Characteristics

1. Reflexivity

Every skill is a prerequisite of itself (trivially): $\forall \mathcal{S}\in \mathcal{S}:\mathcal{S}⪯\mathcal{S}$

2. Antisymmetry

If two skills are prerequisites of each other, they are the same skill: ${\mathcal{S}}_1⪯{\mathcal{S}}_2\wedge {\mathcal{S}}_2⪯{\mathcal{S}}_1⟹{\mathcal{S}}_1={\mathcal{S}}_2$

3. Transitivity

Prerequisites are transitive through the hierarchy: ${\mathcal{S}}_1⪯{\mathcal{S}}_2\wedge {\mathcal{S}}_2⪯{\mathcal{S}}_3⟹{\mathcal{S}}_1⪯{\mathcal{S}}_3$

4. Lattice Structure

The skill space $(\mathcal{S},⪯)$ forms a lattice (Theorem 2):

• Every pair of skills has a least upper bound (through composition)
• Every pair has a greatest lower bound (common prerequisites)

Hierarchical Levels

Skills can be organized into hierarchical levels:

1. Primitive Skills: Base-level skills with no prerequisites
2. Intermediate Skills: Composed from primitive skills
3. Complex Skills: Higher-order compositions
4. Expert Skills: Highly specialized, complex compositions

Research Context and Applications

The skill hierarchy is fundamental for:

• Learning Path Design: Ordering skills from simple to complex
• Curriculum Development: Structuring learning sequences
• Prerequisite Analysis: Identifying what must be learned first
• Capability Mapping: Understanding agent competence levels
• Skill Transfer: Identifying common foundational skills

In LLM research:

• Understanding capability emergence during training
• Designing progressive fine-tuning strategies
• Identifying capability dependencies
• Planning multi-stage learning approaches

Visualization

The hierarchy can be visualized as a directed acyclic graph (DAG):

         [Complex Skill]
              / \
             /   \
    [Skill A]   [Skill B]
         |       /    \
         |      /      \
    [Skill C] [Skill D]

Where edges point from prerequisites to dependent skills.

Connections to Other Concepts

• Skills (𝒮): Elements of the hierarchy
• Subskills (𝒮_sub): Lower levels in the hierarchy
• Superskills (𝒮_super): Higher levels formed from composition
• Skill Lattice: Mathematical structure formed by the partial order
• Composition Operator (∘): Creates upper levels from lower levels
• Decomposition Operator (↓): Reveals hierarchical structure

Operations on Hierarchies

Finding Prerequisites

Given a skill, find all its prerequisites: $\text{prereq}(\mathcal{S})=\{{\mathcal{S}}^{\prime }\in \mathcal{S}:{\mathcal{S}}^{\prime }⪯\mathcal{S}\}$

Finding Dependents

Given a skill, find all skills that depend on it: $\text{dependent}(\mathcal{S})=\{{\mathcal{S}}^{\prime }\in \mathcal{S}:\mathcal{S}⪯{\mathcal{S}}^{\prime }\}$

Path Length

The complexity distance between skills: $\text{distance}({\mathcal{S}}_1,{\mathcal{S}}_2)=\min \{n:\exists \text{ path of length }n\text{ from }{\mathcal{S}}_1\text{ to }{\mathcal{S}}_2\}$

Open Research Questions

1. Hierarchy Depth: What is the typical depth of skill hierarchies in LLMs?

2. Branching Factor: What is the average number of prerequisites per skill?

3. Primitive Skills: How to identify and enumerate primitive skills?

4. Dynamic Hierarchy: How does the hierarchy evolve during learning?

5. Multiple Hierarchies: Can different valid hierarchies exist for the same skill set?

6. Hierarchy Discovery: How to automatically discover hierarchical structure from behavior?

7. Cross-Domain Hierarchies: How do hierarchies differ across domains?