On Solomonoff Induction in Large Language Models and the Limits of Self-Improving: The Singularity Is Not Near Without Symbolic Model Synthesis

Abstract

On the one hand, the question of whether large language models (LLMs) are Solomonoff induction estimators has become an explicit question at the intersection of Algorithmic Information Theory (AIT) and Machine Learning (ML) of great interest. On the other hand, the now old idea of an AI Singularity that requires a reliable positive-feedback process in which a system can generate, evaluate and retain genuine improvements to itself continues to come up and is a recurrent concept in the discussion of AGI. We connect and provide some answers to these issues based on current assumptions and future developments of neurosymbolic ML. We will demonstrate that cross-entropy, negative log-likelihood and cognate next-token objectives do not or cannot, by themselves, implement Solomonoff induction: they optimise fit to a supplied conditional distribution rather than a program-weighted universal mixture. While more compute within a fixed objective can improve fit without changing the inductive principle, additional computational resources do not intrinsically without external hyper-parameter or architectural changes, behave as optimal predictors in the Solomonoff and Levin sense. While the data-processing inequality (DPI) and Levin non-growth remain valid, we will show that for finite learners and finite observers, theoretical boundaries have less relevancy and generate a drift between possible approaches. To this end, we interpret different resource-bounded estimators as finite tools for mechanism search that show divergence, not violation, of (algorithmic) information conservation laws. A neurosymbolic direction taken by current frontier-model developers points towards the adoption of model synthesis and no longer purely statistical approaches to LLMs but where Solomonoff estimators are possible.

31 pages. Update: DPI and Levin's non-growth is not violated explanation when it comes to finite learners