On the Computational Complexity of Structural Generalization
This paper formally defines structural generalization, and proves that, under standard complexity assumptions, pure Transformers cannot learn it, while neuro-symbolic systems achieve high scores by hardcoding semantic projections.
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[Submitted on 21 Jul 2026]
Title:On the Computational Complexity of Structural Generalization
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Abstract:Structural generalization has been measured repeatedly by several benchmarks, yet it has never been formally defined. We give a definition that translates the two premises (compositional structure and unbounded generalization) into mathematical language. The definition itself is neutral: a compiler that hard-codes the rules satisfies it just as well. But structural generalization becomes a scientific question only insofar as the capacity can autonomously emerge from finite data. This question pits the computational lower bound $\mathrm{NC}^1$ against the learnable ceiling $\mathrm{TC}^0$ of pure Transformers. Under a Montagovian instantiation, each compositional rule splits into two projections: a syntactic face ($F_\gamma$) and a semantic face ($G_\gamma$). Tree evaluation on the $G_\gamma$ side is an instantiation of BFVP, which is $\mathrm{NC}^1$-complete (Buss, 1987). A pure Transformer must learn both faces at once, but Kraus et al. (2026) prove that its learnable class $\subseteq \mathrm{TC}^0$. Under the standard assumption $\mathrm{TC}^0 \neq \mathrm{NC}^1$, a pure Transformer cannot learn structural generalization. Neuro-symbolic systems achieve the best benchmark scores precisely because they inject $G_\gamma$, sidestepping the genuinely hard half. Benchmark scores cannot distinguish "learned" from "given." This is what this paper sets out to make clear.
Subjects:
Computation and Language (cs.CL); Machine Learning (cs.LG)
Cite as: arXiv:2607.19573 [cs.CL]
(or arXiv:2607.19573v1 [cs.CL] for this version)
https://doi.org/10.48550/arXiv.2607.19573
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Zichao Wei [view email] [v1] Tue, 21 Jul 2026 21:00:32 UTC (225 KB)
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