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A non-ergodic framework for understanding emergent capabilities in Large Language Models

Javier Marín

TL;DR

This work argues that emergent capabilities in large language models arise from non-ergodic dynamics and adjacency-limited exploration described by Stuart Kauffman's adjacent possible theory (TAP). It formalizes a resource-bounded TAP equation that couples architectural, training data, and contextual constraints to bound semantic-space growth and predict phase transitions in capability emergence. Through experiments on three open models (gpt2-xl, OPT-1.3B, Pythia-1.4B) using the MMLU high-school subset, the authors observe phase-transition-like shifts, multiplicative constraint interactions, and path-dependent problem-solving trajectories, all consistent with the TAP framework. The results suggest practical design principles for constraint-aware architectures and phase-transition engineering to guide controlled emergence and improve interpretability and alignment of future language models.

Abstract

Large language models have emergent capabilities that come unexpectedly at scale, but we need a theoretical framework to explain why and how they emerge. We prove that language models are actually non-ergodic systems while providing a mathematical framework based on Stuart Kauffman's theory of the adjacent possible (TAP) to explain capability emergence. Our resource-constrained TAP equation demonstrates how architectural, training, and contextual constraints interact to shape model capabilities through phase transitions in semantic space. We prove through experiments with three different language models that capacities emerge through discrete transitions guided by constraint interactions and path-dependent exploration. This framework provides a theoretical basis for understanding emergence in language models and guides the development of architectures that can guide capability emergence.

A non-ergodic framework for understanding emergent capabilities in Large Language Models

TL;DR

This work argues that emergent capabilities in large language models arise from non-ergodic dynamics and adjacency-limited exploration described by Stuart Kauffman's adjacent possible theory (TAP). It formalizes a resource-bounded TAP equation that couples architectural, training data, and contextual constraints to bound semantic-space growth and predict phase transitions in capability emergence. Through experiments on three open models (gpt2-xl, OPT-1.3B, Pythia-1.4B) using the MMLU high-school subset, the authors observe phase-transition-like shifts, multiplicative constraint interactions, and path-dependent problem-solving trajectories, all consistent with the TAP framework. The results suggest practical design principles for constraint-aware architectures and phase-transition engineering to guide controlled emergence and improve interpretability and alignment of future language models.

Abstract

Large language models have emergent capabilities that come unexpectedly at scale, but we need a theoretical framework to explain why and how they emerge. We prove that language models are actually non-ergodic systems while providing a mathematical framework based on Stuart Kauffman's theory of the adjacent possible (TAP) to explain capability emergence. Our resource-constrained TAP equation demonstrates how architectural, training, and contextual constraints interact to shape model capabilities through phase transitions in semantic space. We prove through experiments with three different language models that capacities emerge through discrete transitions guided by constraint interactions and path-dependent exploration. This framework provides a theoretical basis for understanding emergence in language models and guides the development of architectures that can guide capability emergence.
Paper Structure (48 sections, 54 equations, 12 figures, 5 tables)

This paper contains 48 sections, 54 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Current frameworks for emergent capabilities in language models
  3. Motivation for a biological-inspired approach
  4. Empirical evidences in LLMs
  5. Probabilistic spaces in language models
  6. Evidences of non-ergodicity in language models
  7. Complex dynamics and emergence in language models
  8. Limitation of classical statistics
  9. Complex system dynamics and emergent behaviors in language models
  10. Complex adaptive systems and biological evolution
  11. Emergence in complex systems
  12. Self-organization and phase transitions
  13. Non-ergodicity in biological systems
  14. The adjacent possible theory (TAP)
  15. Application of TAP framework to language models
  16. ...and 33 more sections

Figures (12)

  • Figure 1: Model performance and task difficulty
  • Figure 2: Semantic space analysis.
  • Figure 3: Correlation between performance and entropy.
  • Figure 4: Training vs architectural constraints effects.
  • Figure 5: Training vs contextual constraints effects.
  • ...and 7 more figures