Ontological Expansion

Definition

Ontological expansion is the process by which the skill ontology grows through composition or discovery of new skills.

Mechanisms

1. Compositional Closure

Given primitives ${\mathcal{S}}_0$, the generated ontology is: $⟨{\mathcal{S}}_0⟩={\bigcup }_{n=0}^{\infty }{\mathcal{S}}_0^{\circ n}$

Expansion occurs when new primitives are added: ${\mathcal{S}}_0\to {\mathcal{S}}_0\cup \{S_{new}\}$

2. Scale-Dependent Realization

Following Arora & Goyal, skill compositions only become “real” past certain thresholds:

• Algebraic ontology: All possible compositions
• Realized ontology: Skills with $\varphi (S,T)>0$ for some task
• Emergent ontology: Skills within model’s current emergence boundary

3. Automated Discovery

Per ACD, models can discover new skills through self-exploration.

In This Project

Three Levels

Algebraic ontology: ⟨𝒮₀⟩ (all possible compositions)
       ↓ (filter by task competence)
Realized ontology: ⋃ₜ 𝒮ₜᵋ (skills with nonzero fitness)
       ↓ (filter by model scale)
Emergent ontology: {S : λ(S) ≤ λ*} ∩ ⋃ₜ 𝒮ₜᵋ

Constraints on Growth

1. Finite generation: $\mathcal{S}=⟨{\mathcal{S}}_0{⟩}_{\circ }$ for finite ${\mathcal{S}}_0$
2. Bounded by domain: Fitness threshold filters meaningless compositions
3. Emergence thresholds: Scale determines which compositions manifest

Connection to SKILL-MIX

For skill set size $N$ and composition degree $k$, there are $\left(\genfrac{}{}{0ex}{}{N}{k}\right)$ potential compositions, but models only exhibit competence on a subset. The realized ontology ⊂ algebraic ontology.

Related Concepts

• ontology — What expands
• skills-algebra — The algebraic structure
• emergence — Scale-dependent expansion
• scaling-laws — What drives expansion

References

• A mereological reconciliation
• Clarifying “Algebra of Skills Ontology”