On the Role of Directionality in Structural 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.

The Cognitive Categorical Transformer (CCT) is a 306M-parameter architecture that augments a pretrained GPT-2 Small backbone with cognitively grounded components derived from category theory and several inspirations from cognitive science. Under a matched-step protocol (215,000 optimizer steps, matched data, matched optimizer and schedule) on WikiText-103, CCT reaches 21.27 validation perplexity, compared with 24.19 for an identically fine-tuned GPT-2 Small baseline. The architecture therefore contributes a 2.92 PPL (12% relative) reduction beyond what in-domain fine-tuning alone provides. A retrain-from-scratch ablation that holds GT-Full simplicial message passing bypassed across the entire seven-phase activation schedule reaches 23.72 PPL, localizing 84% of the architectural improvement (2.45 of 2.92 PPL) to GT-Full. We present the first ablation-validated evidence that simplicial message passing improves language-model perplexity at the 306M-parameter scale on WikiText-103. Published GPT-2 Large reaches 22.05 zero-shot PPL on WikiText-103 with 6.2x more parameters than GPT-2 Small; this paper treats that number as an external published reference, not as the architectural benchmark. Three negative results on consistency-style categorical priors (sheaf smoothing, adjunction round-trip, curvature regularization) and the joint structural-prior result for GT-Full and PrecisionWeightedPP together support an empirical pattern termed the structure/consistency distinction, in which categorical priors that add new topology improve language modeling and those that enforce a consistency identity do not.

Positional embeddings (PE) in Transformers encode token distance and order but are largely agnostic to \textit{syntactic structure}. We introduce \textbf{S}yntax-\textbf{i}nformed \textbf{P}ositional \textbf{E}mbeddings (\textbf{SiPE}), which learns a lightweight syntactic prior from dependency parses during pretraining and injects it across all three dominant PE families (absolute, relative, rotary), for both encoders and decoders, leaving self-attention and the rest of the architecture untouched. We isolate \emph{where} and \emph{how} the prior should enter the model, and find it depends on the architecture: for autoregressive decoders that use relative PE, the prior is strongest when coupled multiplicatively with the relative-position term of the attention score, outperforming injection into the input embeddings, into self-attention, or into the positional and attention terms jointly---while for encoders it is best added directly to the input embeddings, composing with each encoder's native positional mechanism. We find that models pre-trained with SiPE improve on the SyntaxGym benchmark by up to $10.3\%$ while simultaneously reducing perplexity by $9.0\%$ over a base model with no syntactic supervision---a metric nearly every existing syntax-injection method instead degrades. Crucially, these gains extend beyond syntactic generalization: SiPE also improves real-world language understanding, raising scores on the GLUE benchmark by up to $8.2\%$ over a model trained without it. Unlike existing syntactic language models that marginalize over many parses at inference or discard syntax at runtime, SiPE conditions on a single parse, establishing a new Pareto frontier between syntactic supervision and inference cost.