How Deep Can LLMs Learn to Reason? Expressiveness Is Key

Abstract

Reinforcement learning (RL) has been applied to improve large language model (LLM) reasoning, yet the systematic study of how training scales with task difficulty has been hampered by the lack of controlled, scalable environments. Observed LLM shortcomings in long-horizon reasoning have raised the prospect that they are fundamental to the autoregressive transformer architecture. To address this, we introduce ScaleLogic, a synthetic logical reasoning framework that offers independent control over two axes of difficulty: the depth of the required proof planning (i.e., the horizon) and the expressiveness of the underlying formal system. Our proposed framework supports a wide range of logics: from simple implication-only logic ("if-then") towards more expressive first-order reasoning with conjunction (''and''), disjunction (''or''), negation (''not''), and universal quantification ("for all"). Using this framework, we show that the RL training compute $T$ follows a power law with respect to reasoning depth $D$ ($T \propto D^{\gamma}$, $R^{2}>0.99$), and the scaling exponent $\gamma$ increases monotonically with expressiveness, from $1.05$ to $2.60$. On downstream mathematics and general reasoning benchmarks, more expressive training settings yield both larger performance gains (up to $+10.66$ points) and more compute-efficient transfer compared to less expressive settings, demonstrating that what a model is trained on, not just how much it is trained, shapes downstream transfer. We further show that the power-law relationship holds across multiple RL methods, and curriculum-based training substantially improves scaling efficiency. More broadly, our results suggest that LLM shortcomings in long-horizon reasoning are not fundamental to the underlying architecture, and could be addressed by improved training methodology and data.