Skill Lattice (Theorem 2)

Theorem Statement

Theorem 2: The skill space $(\mathcal{S},⪯)$ forms a lattice structure under the subskill partial order.

Formal Proof

Proof: Every pair of skills has a least upper bound (composition) and greatest lower bound (common subskills). $\square$

Detailed Proof

To show that $(\mathcal{S},⪯)$ is a lattice, we must verify that for any two skills ${\mathcal{S}}_1,{\mathcal{S}}_2\in \mathcal{S}$:

1. Existence of Join (Least Upper Bound):

The join ${\mathcal{S}}_1\vee {\mathcal{S}}_2$ is the least skill that has both ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ as prerequisites:

${\mathcal{S}}_1\vee {\mathcal{S}}_2={\mathcal{S}}_1\circ {\mathcal{S}}_2$

This is the composition of the two skills, which by closure (Axiom 1) exists and is a valid skill.

2. Existence of Meet (Greatest Lower Bound):

The meet ${\mathcal{S}}_1\wedge {\mathcal{S}}_2$ is the greatest skill that is a prerequisite of both ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$:

${\mathcal{S}}_1\wedge {\mathcal{S}}_2={\max }_{⪯}\{\mathcal{S}:\mathcal{S}⪯{\mathcal{S}}_1\wedge \mathcal{S}⪯{\mathcal{S}}_2\}$

This represents the most specific common prerequisite skill.

Since both join and meet exist for any pair of skills, $(\mathcal{S},⪯)$ forms a lattice.

Lattice Operations

Join (∨): Least Upper Bound

${\mathcal{S}}_1\vee {\mathcal{S}}_2=\text{smallest skill containing both }{\mathcal{S}}_1\text{ and }{\mathcal{S}}_2$

Interpretation: The minimal skill that encompasses both skills.

Meet (∧): Greatest Lower Bound

${\mathcal{S}}_1\wedge {\mathcal{S}}_2=\text{largest common prerequisite of }{\mathcal{S}}_1\text{ and }{\mathcal{S}}_2$

Interpretation: The most specific skill shared as a foundation.

Key Properties and Characteristics

Lattice Properties

1. Idempotence:

   • $\mathcal{S}\vee \mathcal{S}=\mathcal{S}$
   • $\mathcal{S}\wedge \mathcal{S}=\mathcal{S}$

2. Commutativity:

   • ${\mathcal{S}}_1\vee {\mathcal{S}}_2={\mathcal{S}}_2\vee {\mathcal{S}}_1$
   • ${\mathcal{S}}_1\wedge {\mathcal{S}}_2={\mathcal{S}}_2\wedge {\mathcal{S}}_1$

3. Associativity:

   • $({\mathcal{S}}_1\vee {\mathcal{S}}_2)\vee {\mathcal{S}}_3={\mathcal{S}}_1\vee ({\mathcal{S}}_2\vee {\mathcal{S}}_3)$
   • $({\mathcal{S}}_1\wedge {\mathcal{S}}_2)\wedge {\mathcal{S}}_3={\mathcal{S}}_1\wedge ({\mathcal{S}}_2\wedge {\mathcal{S}}_3)$

4. Absorption:

   • ${\mathcal{S}}_1\vee ({\mathcal{S}}_1\wedge {\mathcal{S}}_2)={\mathcal{S}}_1$
   • ${\mathcal{S}}_1\wedge ({\mathcal{S}}_1\vee {\mathcal{S}}_2)={\mathcal{S}}_1$

Special Lattice Elements

Top Element (⊤): Most complex skill (if it exists) $\forall \mathcal{S}\in \mathcal{S}:\mathcal{S}⪯\top$

Bottom Element (⊥): Most primitive skill (if it exists) $\forall \mathcal{S}\in \mathcal{S}:\perp ⪯\mathcal{S}$

Research Context and Applications

The lattice structure enables:

• Structured Learning: Natural progression from simple to complex skills
• Common Ground Identification: Finding shared prerequisites
• Skill Generalization: Identifying abstract common patterns
• Capability Planning: Understanding minimal required skills
• Optimization: Finding optimal skill combinations

In LLM research:

• Understanding capability hierarchies
• Designing incremental learning approaches
• Identifying capability gaps
• Planning skill acquisition strategies
• Analyzing emergence patterns

Visual Representation

A Hasse diagram can visualize the lattice:

        ⊤ (Most Complex)
       / \
      /   \
    S₃   S₄
     \   /
      \ /
      S₂
      |
      S₁
      |
      ⊥ (Most Primitive)

Connections to Other Concepts

• Skill Hierarchy: The partial order $⪯$ defining the lattice
• Composition Operator (∘): Implements the join operation
• Decomposition Operator (↓): Reveals meet structure
• Subskills: Lower elements in the lattice
• Superskills: Upper elements formed by joins
• Skill Composition Semigroup: Related algebraic structure

Lattice Types

Possible Specializations

1. Distributive Lattice: Does distributivity hold? ${\mathcal{S}}_1\wedge ({\mathcal{S}}_2\vee {\mathcal{S}}_3)\stackrel{?}{=}({\mathcal{S}}_1\wedge {\mathcal{S}}_2)\vee ({\mathcal{S}}_1\wedge {\mathcal{S}}_3)$

2. Complemented Lattice: Does each skill have a complement? $\exists {\mathcal{S}}^{\prime }:\mathcal{S}\vee {\mathcal{S}}^{\prime }=\top \wedge \mathcal{S}\wedge {\mathcal{S}}^{\prime }=\perp$

3. Modular Lattice: Does modularity hold?

Open Research Questions

1. Completeness: Is the lattice complete (do infinite joins/meets exist)?

2. Boundedness: Do universal top and bottom elements exist?

3. Distributivity: Is the skill lattice distributive?

4. Complementation: Do skills have complements in the lattice?

5. Density: Is the lattice dense (between any two skills, is there another)?

6. Finite vs. Infinite: Is the skill lattice finite or infinite?

7. Sublattices: What interesting sublattices exist (e.g., domain-specific skills)?