Clarifying Algebra of Skills Ontology

1. Comparison with the Literature

Your framework in the attached document and what’s in the literature share some core intuitions but differ significantly in formalization strategy.

Arora & Goyal’s emergence theory treats skills as nodes in a bipartite “skill graph” where text-pieces connect to $k$-tuples of skills. Their key insight: competence emerges statistically via random graph theory arguments, not through explicit algebraic operations. They deliberately avoid formalizing what skills are or how they compose: “this conservative accounting has the significant benefit of obviating the need for a mathematical formulation of what skills are, and what it means to combine skills—which is unformulated.”

SKILL-MIX operationalizes this by sampling $k$-tuples from a skill list $\mathcal{S}$ and testing whether a model can generate text exhibiting all $k$ skills simultaneously. The composition is implicit—it’s the model’s problem, not a formal operation in the evaluation framework.

The Transformers meta-skills paper comes closest to your approach: they define meta-skills as learned operators over basic function classes (addition, maximum, multiplexing) and show transformers can generalize to unseen compositions like $\mathcal{A}+\mathcal{C}$ after training on $\mathcal{A}+\mathcal{B}$ and $\mathcal{B}+\mathcal{C}$.

SELF framework defines metaskills procedurally (self-feedback, self-refinement) as learned behaviors enabling self-evolution, not as algebraic entities.

Your framework is more ambitious: you’re attempting to axiomatize $(\mathcal{S},\circ )$ as a semigroup with identity, define metaskills as higher-order functions $\mathcal{M}:2^{\mathcal{S}}\times 2^{\mathcal{S}}\times \mathcal{K}\to 2^{\mathcal{S}}$, and establish compositional closure. This is algebraically richer but raises questions about empirical grounding.

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2–3. Clarifying “Algebra of Skills Ontology”

You’re right that the phrase sounds wrong. Here’s why, and how to fix it.

The problem: An ontology is a specification of a conceptualization—it declares what exists and how entities relate. An algebra is a structure with operations—it specifies how entities combine and transform. These are different layers:

$\text{Ontology}⟶\text{what exists, what relations hold}$ $\text{Algebra}⟶\text{how things compose, transform}$

What you have is better described as:

Option A: An algebraic theory over a skills ontology. The ontology fixes the sorts (Skill, Task, Metaskill, Agent) and primitive relations (part-of, prerequisite, fitness). The algebra $(\mathcal{S},\circ ,1)$ then operates within the ontological framework, defining how skills combine.

Option B: A skills algebra with an induced ontological commitment. Here the algebra is primary; the ontology is derived from algebraic structure (e.g., the partial order $⪯$ induced by divisibility under composition, or the lattice structure from meets/joins).

The literature actually supports Option B more naturally. Arora & Goyal note that “prior attempts to formalize ‘combinations’ got highly technical, e.g., involving category theory.” Category-theoretic approaches treat composition as primitive and derive structure from it.

Suggested terminology: Instead of “algebra of skills ontology,” use:

• “Compositional algebra of skills” (if focusing on the algebraic structure)
• “Algebraic framework for skill ontology” (if the ontology is primary)
• “Skills calculus” (if you want to emphasize the operational aspect)

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4–5. Fixed Ontologies vs. Open-Ended Skill Generation

You’re correct that traditional ontologies are fixed—BFO, SNOMED CT, Gene Ontology all presuppose a stable domain of entities. Your framework is fundamentally different: Axiom 1 (closure under composition) guarantees that composition generates new skills.

This is a generative ontology or ontology with ontological expansion. The key distinction:

	Fixed Ontology	Generative Ontology
Entity set	Pre-specified, finite	Potentially infinite, constructively generated
Relations	Declared a priori	Induced by operations
Closure	Extensionally given	Intensionally guaranteed by axioms

Your Axiom 1 states: $\forall {\mathcal{S}}_1,{\mathcal{S}}_2\in \mathcal{S},\exists {\mathcal{S}}_3\in \mathcal{S}:{\mathcal{S}}_1\circ {\mathcal{S}}_2={\mathcal{S}}_3$

This makes $\mathcal{S}$ a closure under $\circ$—but what constrains the growth? Three possibilities:

1. Finite generation: There exists a finite set ${\mathcal{S}}_0$ such that $\mathcal{S}=⟨{\mathcal{S}}_0{⟩}_{\circ }$ (the closure of primitives under composition). This makes $\mathcal{S}$ countable and gives you something like a free monoid quotient.

2. Bounded by domain: The fitness function ${\varphi }_S(\mathcal{T})>ϵ$ for some task constrains which “composite skills” are actually meaningful. Most random compositions have $\varphi \approx 0$.

3. Emergence thresholds: Following Arora & Goyal, skill $k^{\prime }$-tuples only become “real” (i.e., exhibit nonzero competence) past certain scaling thresholds. This makes the effective ontology scale-dependent.

The SKILL-MIX paper explicitly connects to this: for skill set size $N$ and composition degree $k$, there are $\left(\genfrac{}{}{0ex}{}{N}{k}\right)$ potential $k$-compositions, but models only exhibit competence on a subset. The ontology realized by a model is smaller than the ontology permitted by the algebra.

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6. Mereology and Skills

Mereology (the formal theory of parts and wholes) is indeed relevant, but with caveats.

Classical mereology (Leśniewski, Goodman) axiomatizes part-of ($⊑$) with:

• Reflexivity: $x⊑x$
• Antisymmetry: $x⊑y\wedge y⊑x\Rightarrow x=y$
• Transitivity: $x⊑y\wedge y⊑z\Rightarrow x⊑z$

Plus principles like:

• Weak supplementation: Proper parts require additional parts
• Strong supplementation: Difference requires a distinguishing part
• Unrestricted fusion: Any collection has a mereological sum

Where mereology fits your framework:

Your subskill/superskill relation $⪯$ defined by ${\mathcal{S}}_{sub}⪯{\mathcal{S}}_{super}⟺{\mathcal{S}}_{sub}\in \mathcal{P}({\mathcal{S}}_{super})$ is a partial order, which is mereology-like. But there’s a tension:

• Mereological composition is idempotent: $x\bigsqcup x=x$
• Your composition need not be: $\mathcal{S}\circ \mathcal{S}\ne \mathcal{S}$ in general (think: “apply metaphor twice” vs. “apply metaphor once”)

This suggests skills have both mereological structure (part-whole) and algebraic structure (composition as operation), and these may not coincide.

A cleaner approach: Distinguish:

1. Mereological structure: Which skills are constitutive parts of which (your $⪯$)
2. Operational structure: How skills combine to produce new capabilities (your $\circ$)

These give you two different partial orders on $\mathcal{S}$:

• ${⊑}_{mereological}$: “is a part of”
• ${\le }_{operational}$: induced by $x\le y⟺\exists z:x\circ z=y$

Whether these coincide, one refines the other, or they’re independent is an empirical/design question.

Recommendation: Mereology is useful for the static aspect (skill taxonomies, decomposition), but insufficient for the dynamic aspect (composition, emergence). You need both. This is analogous to how BFO separates continuants (mereological structure) from occurrents (processual structure).

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7. Should You Use BFO or an Upper Ontology?

Arguments for:

1. Disambiguation of skill vs. skill-exercise: In BFO terms, a skill is a disposition (a dependent continuant), while applying a skill to a task is a process (an occurrent). Your framework conflates these—${\varphi }_S(\mathcal{T})$ mixes the capacity (disposition) with its realization (process). BFO would force you to separate:

   • Skill-as-disposition: the capacity inhering in an agent
   • Skill-realization: the process of applying the skill
   • Task: the goal-directed process the skill enables

2. Interoperability: If you want your skill ontology to connect with other ontologies (cognitive ontologies, task ontologies, competency frameworks), upper ontology alignment helps.

3. Preventing category errors: BFO’s strict continuant/occurrent divide prevents you from treating a skill (persisting entity) as identical to a skill-application (temporal entity).

Arguments against:

1. Your framework is fundamentally algebraic, not ontological. BFO is designed for classification, not generation. Its categories are meant to be exhaustive and stable. Your closure axiom violates this.

2. BFO’s realism may conflict with your operationalism. BFO commits to entities existing in reality independent of representation. Skills in LLMs are arguably functional or dispositional in a way that doesn’t map cleanly to BFO’s substance ontology.

3. Overhead without payoff: BFO’s distinctions (site vs. spatial region, quality vs. relational quality, etc.) may not carve your domain at its joints.

My recommendation: Don’t adopt BFO wholesale, but borrow specific distinctions:

• Disposition vs. realization: Separate skill-as-capacity from skill-exercise
• Bearer-dependence: Skills inhere in agents; make this explicit
• Function vs. role: Distinguish skills from the roles they play in task structures

A lightweight ontological commitment might look like:

Agent ──bears──► Skill (disposition)
                    │
                    ▼ realizes-in
                 Process (skill application)
                    │
                    ▼ achieves
                  Task-goal

This gives you BFO’s key insight (continuant/occurrent separation) without the full apparatus.

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Summary

Aspect	Your Framework	Literature	Recommendation
Core structure	Semigroup $(\mathcal{S},\circ )$	Bipartite skill-graph	Keep algebraic, but ground primitives empirically
Skill generation	Closed under composition	Implicit in model capabilities	Add emergence thresholds à la Arora & Goyal
Metaskills	Higher-order functions	Procedural (SELF) or compositional (Transformers)	Distinguish meta-composition from meta-evaluation
Ontological status	Fixed-but-generative	Statistical/emergent	Frame as “induced ontology” from algebra
Mereology	Implicit in $⪯$	Not used	Separate part-of from composes-to
Upper ontology	None	None	Borrow disposition/realization distinction