Learning to Reason with Curriculum II: Compositional Generalization

Compositional generalization, the ability to solve complex problems by combining solutions to simpler sub-problems, is a fundamental capability of both natural and artificial intelligence, and a key mechanism underlying chain-of-thought reasoning. However, the theoretical underpinnings of compositional generalization remain poorly understood: when and why does decomposing a problem into parts yield more efficient learning than solving it directly? We study this question through the canonical problem of learning to simulate semiautomata (predicting the outcome of $T$ steps of sequential computation), a model that captures state tracking, regular language recognition, and modular arithmetic. We show that an autocurriculum-based approach building on Part I of this series, recursively decomposing longer sequences into shorter sub-problems, learning to solve them, and composing the solutions, achieves dramatically better statistical complexity than direct methods. (i) For a setting inspired by supervised fine-tuning (SFT) where the learner receives interactive feedback on intermediate states of the computation, curriculum facilitates learning from only $2^{\mathcal{O}(\sqrt{\log T})}$ tokens of supervision; i.e., subpolynomial in the sequence length $T$, overcoming the $Ω(T)$ token barrier required by direct simulation. (ii) For a setting inspired by reinforcement learning with verifiable rewards (RLVR), where the learner improves a pre-trained reference model using an outcome verifier, we show that curriculum reduces the requirement on the reference model from coverage at the full sequence length $T$ to coverage at a shorter block length $B \ll T$, an exponentially weaker condition.

A Compositional Framework for Open-ended Intelligence

Open-ended intelligence is the capacity to adapt to novel problems and environments that are substantially different from those in training. A mathematics of open-ended intelligence requires two pillars: first, a minimal set of representational primitives (e.g., states, actions) and algorithmic primitives (e.g., nearest neighbor); and second, an acquired compositional grammar for selection, recursion, and branching that produces sequences of operations and recurring motifs. We formalize open-ended intelligence in terms of the compositional closure induced by a finite primitive set $P$ and a set of composition operators $C$. We characterize properties of the induced closure $\mathcal{L}(P,C)$ that support unbounded compositional generation across families of tasks and worlds. The closure of the two pillars yields infinite adaptive responses across a wide range of settings. The mathematics supports complementary research agendas, including evaluation metrics for explanation and interpretability, and novel architectures where compositional generalization is native. We propose next primitive prediction (NPP) as a novel architectural objective, where training encourages the acquisition of reusable algorithmic primitives and their compositional grammar, such that new solutions are generated through recombination. Given such an objective, curriculum learning and self-play can enable lifelong learning, expanding the closure by discovering reusable primitives and transition motifs across settings. We ground the framework through case studies in physics, evolution, and neuroscience.