Depth behavior of per-step success probability in the Diligent Learner framework

Determine whether, as multi-step reasoning proceeds over longer horizons, the Diligent Learner stepwise success probability γ—defined as the probability that the generator policy proposes a next-step extension that keeps the current reasoning prefix completable—remains bounded below by a positive constant independent of depth, or whether there exist categories of problems for which γ degrades catastrophically with increasing depth.

Background

The Diligent Learner framework formalizes reasoning as validator-guided depth-first search, where the critical parameter γ is the per-step probability mass that the policy assigns to "good" next steps that keep the current prefix completable. If γ stays bounded away from zero with depth, search achieves controlled overhead; if γ collapses with depth, guarantees become vacuous.

This work introduces a GF(2) circuit reconstruction benchmark designed to operationalize and measure γ at each step by ensuring a unique correct continuation and eliminating data-only or history-only shortcuts. The authors observe that smaller LLMs exhibit superlinear declines in γ with depth, while frontier models—especially with tool use—show partial robustness. The open question asks for a principled determination of γ’s behavior as depth grows across tasks, beyond the specific benchmark.