Symbol-Equivariant Recurrent Reasoning Models

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 9$\times$9 Sudoku and generalize from just training on 9$\times$9 to smaller 4$\times$4 and larger 16$\times$16 and 25$\times$25 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.

AI models often struggle with permutation equivariant logic puzzles because they treat different symbols differently, even when the underlying rules stay the same. For example, if the numbers in Sudoku are shuffled, the solution of the puzzle will be shuffled in the same way. We introduce a small AI model that mathematically guarantees symbol equivariance. For a permutation equivariant problem, such as Sudoku, this means that if a solution to one permutation is learned, all possible permutations can be solved correctly. This greatly reduces the need for data augmentation and improves generalization to new settings.