On the Limits of Self-Improving in LLMs and Why AGI, ASI and the Singularity Are Not Near Without Symbolic Model Synthesis

2026-01-05

Robotics

AI Summary

This work addresses the inherent limitations of large language models in purely self-supervised recursive self-training, where reliance on self-generated data inevitably leads to mode collapse and semantic drift, thereby hindering sustained self-improvement. By modeling the process as a discrete-time dynamical system, the study provides the first information-theoretic characterization of two fundamental failure mechanisms: entropy decay and variance amplification, rigorously demonstrating that distributional learning alone cannot support long-term evolution. To overcome these barriers, the authors propose a neuro-symbolic program synthesis framework that integrates symbolic regression with algorithmic probability guidance. This approach leverages symbolic priors to circumvent the constraints imposed by the data processing inequality, establishing a theoretically viable pathway toward sustainable self-improvement.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Natural Language Processing: (Large) Language Models
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We formalise recursive self-training in Large Language Models (LLMs) and Generative AI as a discrete-time dynamical system and prove that, as training data become increasingly self-generated (αt → 0), the system undergoes inevitably degenerative dynamics. We derive two fundamental failure modes: (1) Entropy Decay, where finite sampling effects cause a monotonic loss of distributional diversity (mode collapse), and (2) Variance Amplification, where the loss of external grounding causes the model’s representation of truth to drift as a random walk, bounded only by the support diameter. We show these behaviours are not contingent on architecture but are consequences of distributional learning on finite samples. We further argue that Reinforcement Learning with imperfect verifiers suffers similar semantic collapse. To overcome these limits, we propose a path involving symbolic regression and program synthesis guided by Algorithmic Probability. The Coding Theorem Method (CTM) allows for identifying generative mechanisms rather than mere correlations, escaping the data-processing inequality that binds standard statistical learning. We conclude that while purely distributional learning leads to model collapse, hybrid neurosymbolic approaches offer a coherent framework for sustained self-improvement.

Problem

• self-improving
• model collapse
• distributional learning
• semantic drift
• LLMs

Innovation

• symbolic regression
• algorithmic probability
• model collapse
• neurosymbolic AI
• Coding Theorem Method