A small transformer learns the language of an infinite Coxeter group, and its attention hints at two word-theory propositions proved by hand

A very small transformer can perform a combinatorial word-theory task in the infinite Coxeter group Ã₂, predicting an element's left and right descent sets, the sets of possible first and last letters of its minimal-length representatives. Earlier mechanistic-interpretability work mostly studied fixed operations on finite groups, such as modular addition and multiplication in finite non-abelian groups, where the solution was already known; here the learning target is normal-form expressions in an infinite group. On the ShortLex data, a substantial share of test subwords for the left-descent task were unseen during training and had length at most 36, and the model reached high accuracy on them; a second run with a different split and a manually split dataset containing all 63 words of one fixed length plus random longer words also reported high accuracy on unseen subwords.

The attention patterns the model learned were used to propose two word-theory propositions, which the researchers then proved without the model's aid. Earlier mechanistic-interpretability studies typically checked whether a model learned a known algorithm; here the two conclusions suggested by the model do not follow immediately from the known Brink–Howlett finite-state algorithm. The two propositions were formulated and proved after observing the attention patterns (Figures 2 and 3) of the model trained on the ShortLex dataset, with proofs in the appendix; the authors say the statements are likely known to experts but found no explicit references.

The structure of the training data goes together with whether the model generalizes across length. Random-word data allowed adjacent generators to repeat, reduced-word data required adjacent generators to differ, and ShortLex data kept only the lexicographically first minimal-length representative. Test accuracy began deteriorating at some length on the random-word and reduced-word datasets, while it never deteriorated on the set of all 1,998 ShortLex words of length at most 36; misclassification rate by index rose rapidly with position on random and reduced words.

The demonstrated scope is a small single-attention-layer transformer on the group Ã₂, three synthetic word datasets, and left and right descent-set prediction; the authors propose next applying the methods to other families of infinite Coxeter groups and using more advanced mechanistic-interpretability tools to extract and validate circuits implementing the Brink–Howlett finite-state automata.

The authors note that the two propositions are likely known to experts but that they found no explicit references; misclassification rate by index rises rapidly on random and reduced words, indicating the model did not learn a length-generalizable algorithm; and the model-suggested conclusions still required human proof, since attention patterns alone are not a proof.