On the Computational Complexity of Structural Generalization

Compositional generalization$\unicode{x2013}$the ability to systematically process novel combinations of known components$\unicode{x2013}$is a hallmark of human intelligence; however, its theoretical foundation in neural networks is not yet well understood. This paper establishes a necessary and sufficient condition for provable compositional generalization, precisely characterizing its boundary. Conceptually, the condition consists of two principles: (i) structural alignment, where a model's computational graph aligns with a task's true compositional hierarchy, and (ii) unambiguous minimized representations, where each component encodes adequate but not redundant information on the training data. The result is fully proved and machine-verified in Lean 4 and holds even in few-shot and one-shot regimes. The necessity direction establishes that provable compositional generalization cannot circumvent these requirements, while the sufficiency direction yields a unified inductive bias that jointly governs architectural design, training data properties, and regularization strategies. Building on this condition, we develop an example algorithmic approach, illustrate it through a controlled minimal example, and further demonstrate the condition on the SCAN jump task. All conclusions are derived mathematically without reliance on empirical validation. Our work provides a theoretical characterization of provable compositional generalization.

Structural generalization in semantic parsing requires systems to apply learned compositional rules to novel structural combinations. Existing approaches either rely on hand-written algebraic rules (AM-Parser) or fail to generalize structurally (Transformer-based models). We present an alternative requiring no hand-written compositional rules, based on a neural cellular automaton (NCA) with a discrete bottleneck: all compositional rules are learned from data through local iteration. On the SLOG benchmark, the system achieves an overall accuracy of $67.3 \pm 0.2\%$ across 10 seeds (AM-Parser: $70.8 \pm 4.3\%$), with 11 of 17 structural generalization categories at $100\%$ type-exact match, including three where AM-Parser scores $0$--$74\%$. Analysis reveals that all 5,539 failure instances reduce to exactly two mechanisms: novel combinations of wh-extraction context with reduced verb types, and modifiers appearing on the subject side of verbs. When we decompose results by CCG structural features, each sub-pattern either succeeds on all instances or fails on all. Intermediate scores (e.g., $41.4\%$) are mixtures of structurally distinct CCG patterns, not partial generalization. These results suggest that CCG directed types provide higher resolution than SLOG's phenomenon-level categories for characterizing structural generalization, and that the success/failure boundary is determined by the coverage of directed operations in the training data.

Large language models are able to compose skills in order to perform complex tasks, many of which might not have been seen during training. The details of how exactly this composition occurs remain elusive. In this paper, we study a mechanism for compositional generalization in transformers by considering a simple controlled setting involving variable assignment and modular addition. By partitioning our training data into disjoint sets, we observe that small transformers are able to generalize to previously unseen combinations of variables and numbers. Our mechanistic analysis shows that the same ``modular addition'' MLP module is used whether the inputs are given directly or indirectly through a separate variable assignment mechanism. We also analyze the training dynamics from an empirical lens, which reveals three phases of learning: first, modular addition is learned, then the structure required for variable assignment, and finally a refinement phase where the model generalizes to some hard sequences not seen in training. Finally, we provide a theoretical framework to explain how compositionality emerges from training dynamics. These results suggest that compositional generalization can be a natural consequence of the compositionality of internal mechanisms in~transformers.