Skills Calculus

A research project developing a formal algebraic framework for understanding skill emergence, composition, and evaluation in Large Language Models (LLMs).

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Core Knowledge Base

• Concepts — Core theoretical definitions
• Papers — Annotated research paper summaries

Research Documents

• Literature Overview & Meta-Analysis — Comprehensive synthesis of 11+ research papers
• Project Roadmap — Full 5-phase research plan
• Methodology & Theoretical Framework
  • A Mereological Reconciliation — Reconciling algebraic and part-whole structures
  • Clarifying “Algebra of Skills Ontology” — Comparison with existing literature

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About

This is an Obsidian knowledge base documenting ongoing research into how skills in LLMs can be understood through mathematical and algebraic structures. The project formalizes the compositional nature of capabilities in language models—how primitive skills combine to form complex abilities, how these abilities emerge with scale, and how they can be systematically evaluated.

Research Focus

Core Questions

• Skill Composition: How do multiple skills combine to form new, more complex capabilities? We model this through a monoid structure $(\mathcal{S},\circ ,1)$ where $\mathcal{S}$ is the set of skills, $\circ$ is the composition operation, and $1$ is the identity (null skill).
• Emergence: Why do certain abilities appear suddenly at specific model scales? We connect scaling laws to skill competence thresholds and ontological expansion.
• Evaluation: How can we assess whether models genuinely understand versus merely memorize? We explore metrics that distinguish compositional generalization from pattern matching.
• Ontology: What is the “natural” taxonomy of skills? We investigate task-relative ontologies and Galois connections between skills and tasks.

Key Concepts

Concept	Description
Skills Algebra	Formal structure $(\mathcal{S},\circ ,1,\mathcal{T},\varphi )$ where $\mathcal{T}$ is the task set and $\varphi :\mathcal{S}\times \mathcal{T}\to [0,1]$ is the fitness function
Mereology	Part-whole relations complementing algebraic composition—distinguishing “what a skill is made of” from “what it produces when combined”
Meta-skills	Second-order capabilities for composing, decomposing, and evaluating skills
Ontological Expansion	How the skill ontology grows through composition and scale-dependent realization
Scaling Laws	Power-law relationships connecting model scale to skill competence

Related Literature

This project builds on and synthesizes work from:

• Arora & Goyal (2023) — Theory for emergence of complex skills
• SKILL-MIX (2023) — Evaluating skill composition ability
• Wei et al. (2022) — Documenting emergent abilities
• Michaud et al. (2024) — Quantization model of neural scaling
• Fan et al. (2024) — Meta-skills for task generalization
• Lu et al. (2024) — SELF: Self-evolution with language feedback