AI 中文总结
研究结构泛化的计算复杂性，给出其正式定义，将计算下限NC¹与纯Transformer可学习上限TC⁰对立，指出在标准假设下纯Transformer无法学习，神经符号系统因注入Gγ获最佳分数，揭示基准分数不能区分“学来的”与“给定的”。
AI 中文摘要
结构泛化已被多个基准反复衡量，但从未被正式定义。我们给出了一个将两个前提（组合结构和无界泛化）转化为数学语言的定义。该定义本身是中立的，硬编码规则的编译器也能满足它。当能力能从有限数据中自主出现时，结构泛化成为一个科学问题，此问题将计算下限NC¹与纯Transformer的可学习上限TC⁰对立起来。在蒙塔古实例化下，每个组合规则分为两个投影，对Gγ侧的树评估是BFVP的实例，它是NC¹完全的，而纯Transformer必须同时学习两个面，但Kraus等人证明其可学习类⊆TC⁰。在标准假设TC⁰≠NC¹下，纯Transformer无法学习结构泛化。神经符号系统能获得最佳基准分数，正是因为它们注入了Gγ，避开了真正困难的部分。基准分数无法区分“学习到的”和“给定的”，本文旨在阐明这一点。
英文摘要
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 ($F_γ$) and a semantic face ($G_γ$). Tree evaluation on the $G_γ$ 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 $G_γ$, sidestepping the genuinely hard half. Benchmark scores cannot distinguish "learned" from "given." This is what this paper sets out to make clear.
