Symbol-Equivariant Recurrent Reasoning Models

Richard Freinschlag, Timo Bertram, Erich Kobler, Andreas Mayr, Günter Klambauer
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:31618-31636, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-freinschlag26a, title = {Symbol-Equivariant Recurrent Reasoning Models}, author = {Freinschlag, Richard and Bertram, Timo and Kobler, Erich and Mayr, Andreas and Klambauer, G\"{u}nter}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {31618--31636}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/freinschlag26a/freinschlag26a.pdf}, url = {https://proceedings.mlr.press/v306/freinschlag26a.html}, abstract = {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$25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning.} }
Endnote
%0 Conference Paper %T Symbol-Equivariant Recurrent Reasoning Models %A Richard Freinschlag %A Timo Bertram %A Erich Kobler %A Andreas Mayr %A Günter Klambauer %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-freinschlag26a %I PMLR %P 31618--31636 %U https://proceedings.mlr.press/v306/freinschlag26a.html %V 306 %X Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9$\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.
APA
Freinschlag, R., Bertram, T., Kobler, E., Mayr, A. & Klambauer, G.. (2026). Symbol-Equivariant Recurrent Reasoning Models. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:31618-31636 Available from https://proceedings.mlr.press/v306/freinschlag26a.html.

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