Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize from just training on 9x9 to smaller 4x4 and larger 16x16 and 25x25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning. Code is available at https://github.com/ml-jku/SE-RRM.

翻译：诸如数独和ARC-AGI等推理问题对神经网络而言仍具挑战性。循环推理模型（RRMs）这一结构化问题求解架构家族，包括分层推理模型（HRM）和微型递归模型（TRM），为大型语言模型提供了一种紧凑的替代方案，但目前仅通过代价高昂的数据增强隐式处理符号对称性。我们引入了符号等变循环推理模型（SE-RRMs），该模型通过符号等变层在架构层面强制实现置换等变性，从而保证在符号或颜色置换下得到相同的解。SE-RRMs在9x9数独上超越了先前的RRMs，并且仅通过9x9的训练即可泛化至更小的4x4以及更大的16x16和25x25实例，而现有的RRMs无法外推至这些规模。在ARC-AGI-1和ARC-AGI-2上，SE-RRMs以显著更少的数据增强和仅200万参数实现了有竞争力的性能，这表明显式编码对称性提升了神经推理的鲁棒性和可扩展性。代码发布于 https://github.com/ml-jku/SE-RRM。