We study the statistical behaviour of reasoning probes in a stylized model of looped reasoning, given by Boolean circuits whose computational graph is a perfect $ν$-ary tree ($ν\ge 2$) and whose output is appended to the input and fed back iteratively for subsequent computation rounds. A reasoning probe has access to a sampled subset of internal computation nodes, possibly without covering the entire graph, and seeks to infer which $ν$-ary Boolean gate is executed at each queried node, representing uncertainty via a probability distribution over a fixed collection of $\mathtt{m}$ admissible $ν$-ary gates. This partial observability induces a generalization problem, which we analyze in a realizable, transductive setting. We show that, when the reasoning probe is parameterized by a graph convolutional network (GCN)-based hypothesis class and queries $N$ nodes, the worst-case generalization error attains the optimal rate $\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$ with probability at least $1-δ$, for $δ\in (0,1)$. Our analysis combines snowflake metric embedding techniques with tools from statistical optimal transport. A key insight is that this optimal rate is achievable independently of graph size, owing to the existence of a low-distortion one-dimensional snowflake embedding of the induced graph metric. As a consequence, our results provide a sharp characterization of how structural properties of the computational graph govern the statistical efficiency of reasoning under partial access.

翻译：我们研究循环推理风格化模型中推理探针的统计行为，该模型由布尔电路给出，其计算图是完美的$ν$元树（$ν\ge 2$），其输出被附加到输入并迭代反馈给后续计算轮次。推理探针可访问内部计算节点的采样子集（可能未覆盖整个图），并试图推断每个查询节点执行的是何种$ν$元布尔门，通过固定集合$\mathtt{m}$个可容许$ν$元门上的概率分布来表示不确定性。这种部分可观测性引出了一个泛化问题，我们在可实现的转导设置下对其进行分析。我们证明，当推理探针由基于图卷积网络（GCN）的假设类参数化并查询$N$个节点时，最坏情况泛化误差以至少$1-δ$的概率达到最优速率$\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$，其中$δ\in (0,1)$。我们的分析结合了雪花度量嵌入技术与统计最优传输工具。一个关键见解是，由于诱导图度量存在低失真的一维雪花嵌入，该最优速率可独立于图大小实现。因此，我们的结果精确刻画了计算图的结构特性如何支配部分访问下推理的统计效率。