Composition Operator (∘)

Formal Definition

The composition operator (∘) combines skills to create new, more complex skills. It is a fundamental operation in the skills algebra that enables building higher-order capabilities from existing ones.

Formally, the composition operator is defined as:

${\mathcal{S}}_1{\circ }_{\alpha }{\mathcal{S}}_2={\mathcal{M}}_c({\mathcal{S}}_1,{\mathcal{S}}_2,\alpha )$

Where:

• ${\mathcal{S}}_1,{\mathcal{S}}_2\in \mathcal{S}$: Skills to be composed
• $\alpha \in \mathcal{A}$: Composition type
• ${\mathcal{M}}_c$: Composition metaskill
• Result: A new skill ${\mathcal{S}}_{new}\in \mathcal{S}$

Composition Types

The composition type $\alpha$ determines how skills are combined:

Sequential Composition (${\alpha }_{seq}$)

Skills are applied one after another: ${\mathcal{S}}_1{\circ }_{{\alpha }_{seq}}{\mathcal{S}}_2$

Example: “Data collection” ∘ “Data analysis” for research pipelines

Parallel Composition (${\alpha }_{par}$)

Skills are applied simultaneously: ${\mathcal{S}}_1{\circ }_{{\alpha }_{par}}{\mathcal{S}}_2$

Example: “Visual processing” ∘ “Audio processing” for multimodal understanding

Conditional Composition (${\alpha }_{cond}$)

Skills are applied based on conditions: ${\mathcal{S}}_1{\circ }_{{\alpha }_{cond}}{\mathcal{S}}_2$

Example: “Classification” ∘ “Specialized handling” based on classification results

Key Properties and Characteristics

1. Closure (Axiom 1)

The composition of any two skills yields a valid skill: $\forall {\mathcal{S}}_1,{\mathcal{S}}_2\in \mathcal{S},\exists {\mathcal{S}}_3\in \mathcal{S}:{\mathcal{S}}_1\circ {\mathcal{S}}_2={\mathcal{S}}_3$

2. Associativity (Axiom 6)

Composition is associative: $\forall {\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3\in \mathcal{S}:({\mathcal{S}}_1\circ {\mathcal{S}}_2)\circ {\mathcal{S}}_3={\mathcal{S}}_1\circ ({\mathcal{S}}_2\circ {\mathcal{S}}_3)$

3. Identity Element (Axiom 5)

There exists an identity skill ${\mathcal{S}}_1$: $\exists {\mathcal{S}}_1\in \mathcal{S}:\forall \mathcal{S}\in \mathcal{S},\mathcal{S}\circ {\mathcal{S}}_1=\mathcal{S}$

4. Semigroup Structure

These properties mean $(\mathcal{S},\circ )$ forms a semigroup (see Theorem 1).

Research Context and Applications

The composition operator is fundamental to skill-based learning and development:

• Skill Building: Systematically construct complex capabilities from simpler ones
• Curriculum Design: Design learning paths through progressive composition
• Capability Transfer: Combine skills from different domains
• Automated Skill Discovery: Algorithmically explore skill compositions

In LLM research:

• Prompt chaining as skill composition
• Tool composition for complex workflows
• Multi-step reasoning as sequential skill composition
• Ensemble methods as parallel skill composition

Connections to Other Concepts

• Skills (𝒮): The elements being composed
• Composition Metaskill (${\mathcal{M}}_c$): Implements the composition operator
• Superskills: Created through composition of subskills
• Decomposition Operator (↓): Inverse operation that breaks skills apart
• Skill Composition Semigroup: Algebraic structure formed by composition
• Skill Lattice: Composition creates upper bounds in the lattice

Open Research Questions

1. Primitive Operators: Is the set ${\mathcal{A}}_{primitive}=\{{\alpha }_{seq},{\alpha }_{par},{\alpha }_{cond}\}$ complete and minimal?
2. Commutativity: Under what conditions is composition commutative?
3. Optimal Composition: How to find optimal composition sequences for a target skill?
4. Composition Cost: What is the computational cost of composing skills?
5. Composition Learning: Can agents learn new composition types beyond primitives?
6. Inverse Operations: Does every composition have a unique decomposition?
7. Distributed Composition: How to compose skills across distributed agents?