Title: Statistical Assurances for Reasoning Probes Operating on Recurrent Boolean Circuits

Abstract:

This work investigates the statistical properties of reasoning probes within a simplified framework of iterative computation, drawing inspiration from neural algorithmic reasoning. The computational process is modeled by a looped Boolean circuit structured as a perfect $\nu$-ary tree (where $\nu\ge 2$), featuring outputs that are recursively routed back as inputs across successive computation steps. The probe functions by observing a randomly sampled subset of internal nodes, aiming to deduce the latent operation at each node, which is characterized by a probability distribution over a finite collection of permissible Boolean gates. This constraint of partial observability creates a transductive generalization challenge on a structured computation graph. We demonstrate that when the probe is defined by a graph convolutional network and samples $N$ nodes, the worst-case generalization error diminishes at the optimal rate of $\mathcal{O}(\sqrt{\log(2/\delta)}/\sqrt{N})$ with a confidence level of at least $1-\delta$. Our theoretical framework integrates metric embedding methods with optimal transport techniques. A critical finding is that this convergence rate is attainable regardless of the computation graph’s scale, facilitated by a low-distortion one-dimensional snowflake embedding of the resulting graph metric. These findings underscore a geometric mechanism that drives statistical efficiency in the context of probing structured, iterative computations.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC