Overcoming a Learning Bottleneck in Impartial Games: A Circuit Complexity Approach

Abstract

Why can neural networks struggle to learn optimal play in a game with a simple, efficient solution algorithm? Motivated by empirical difficulties with parity in impartial games, we use NIM to study a representational bottleneck and three ways around it. We show that a restricted model of shallow, fixed-scale quantised networks lies in $\mathrm{AC}^0$, a circuit class unable to compute parity on arbitrarily large inputs. This prevents optimal single-position play across all board sizes within the model. The obstruction also guides architectural design: two-frame inputs allow restoration from a known zero-nim-sum position; complementary rollout policies reveal the missing invariant; and persistent recurrence accumulates it row by row. Each route retains shallow network components while supplying additional information or processing time. Supervised experiments illustrate the benefits of two-frame restoration and multiple rollout heads; recurrence is a theoretical construction. These results establish representational possibilities, not guarantees of successful training: $\mathrm{AC}^0$ does not characterise practical learnability. Core mathematical results and constructions have been checked in Lean, with classical complexity results as explicit external inputs. The interactive edition and complete Lean sources are available online at https://sr123.github.io/impartial-games-learning-lean/.