Statistical Guarantees for Reasoning Probes on Looped Boolean Circuits

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 $\nu$-ary tree ($\nu\ge 2$), 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 $\mathcal{O}(\sqrt{\log(2/\delta)}/\sqrt{N})$ with probability at least $1-\delta$. 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.