Composition

Definition

Composition is the operation by which multiple skills combine to form new, more complex capabilities. In our framework: $S_1\circ S_2=S_3$.

In the Literature

SKILL-MIX

• Tests simultaneous application of $k$ skills
• Prompts: “Generate text demonstrating all $k$ skills on topic $t$”
• Performance follows $Y\approx X^2$ (single → composite)

Fan et al. (2024)

• Function composition operations:
  • Addition: $f(x)=f_1(x)+f_2(x)$
  • Maximum: $f(x)=\max (f_1(x),f_2(x))$
  • Multiplexing: conditional selection
• Key finding: Training on partial compositions enables generalization

Arora & Goyal

• $k^{\prime }$-tuples: skills co-occurring in text
• Competence on compositions emerges at similar scale as primitives
• “Slingshot generalization” — models handle unseen combinations

In This Project

Algebraic Composition

Our monoid $(\mathcal{S},\circ ,1)$ satisfies:

• Closure: $S_1\circ S_2\in \mathcal{S}$
• Associativity: $(S_1\circ S_2)\circ S_3=S_1\circ (S_2\circ S_3)$
• Identity: $S\circ 1=S$

Compositional Structures

• Complexity filtration: ${\mathcal{S}}_{\le n}$ = skills with composition depth ≤ $n$
• Divisibility: $S_1⊴S_2$ if $S_1$ is a compositional factor of $S_2$

vs. Mereological Structure

Composition (algebraic) ≠ Part-of (mereological):

• Composition: “what does $S$ produce when combined?”
• Part-of: “what is $S$ made of?”

Related Concepts

• skills — The units being composed
• skills-algebra — Formal framework
• mereology — Contrasting structure
• meta-skills — Skills for composing

References

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