We study the statistical behavior of reasoning probes in a stylized model of iterative computation inspired by neural algorithmic reasoning. The underlying computation is given by a looped Boolean circuit whose graph is a perfect $ν$-ary tree ($ν\ge 2$), with outputs recursively fed back as inputs across computation rounds. A probe observes a sampled subset of internal nodes and seeks to infer the latent operation at each node, represented as a probability distribution over a finite set of admissible Boolean gates. This partial observability induces a transductive generalization problem on a structured computation graph. We show that when the probe is parameterized by a graph convolutional network and queries $N$ nodes, the worst-case generalization error decays at the optimal rate $\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$ with probability at least $1-δ$. Our analysis combines metric embedding techniques with tools from optimal transport. A key insight is that this rate is achievable independently of the size of the computation graph, enabled by a low-distortion one-dimensional snowflake embedding of the induced graph metric. These results highlight a geometric mechanism underlying statistical efficiency in probing structured, iterative computations.

翻译：我们研究神经算法推理启发下迭代计算简约模型中推理探针的统计行为。底层计算由完美ν叉树（ν≥2）结构的循环布尔电路实现，其输出在计算轮次间递归反馈作为输入。探针观测内部节点的采样子集，旨在推断每个节点的潜在操作，该操作表示为有限可容许布尔门集合上的概率分布。这种部分可观测性在结构化计算图上引发转导泛化问题。我们证明：当探针由图卷积网络参数化并查询N个节点时，最坏情况泛化误差以最优速率\(\mathcal{O}(\sqrt{\log(2/\delta)}/\sqrt{N})\)衰减（概率至少为\(1-\delta\)）。我们的分析将度量嵌入技术与最优输运工具相结合。关键洞见在于：通过诱导图度量的低畸变一维雪花嵌入，该速率可独立于计算图规模实现。这些结果揭示了支撑结构化迭代计算统计效率的几何机制。