The UNDO Flip-Flop: A Controlled Probe for Reversible Semantic State Management in State Space Model

Abstract

State space models (SSMs) have been shown to possess the theoretical capacity to model both star-free sequential tasks and bounded hierarchical structures Sarrof et al. (2024). However, formal expressivity results do not guarantee that gradient-based optimisation will reliably discover the corresponding solutions. Existing benchmarks probe either monotonic state tracking, as in the standard Flip-Flop task, or structural nesting, as in the Dyck languages, but neither isolates reversible semantic state retrieval. We introduce the UNDO Flip-Flop task to fill this gap. By extending the standard Flip-Flop with an UNDO, the task requires a model to maintain an implicit bounded stack and recover historical states under non-monotonic update sequences. We evaluate one-layer and two-layer Mamba-2 under this framework. Both variants fail to acquire the provably expressible stack-based rollback mechanism, converging instead on a local toggle heuristic that inverts the current state rather than retrieving stored history. Under an adversarial retraction pressure test held within the training length distribution, the two-layer model collapses to 41.10% accuracy, which is below random chance. The results confirm systematic rather than incidental failure. Causal ablation shows that the bottleneck lies in retrieval, not storage. These results draw a clear line between what an architecture can in principle represent and what gradient descent reliably learns, a distinction that theoretical expressivity analyses alone cannot capture.