Language models exhibit strong reasoning capabilities, yet adapting them to structured domains remains challenging and can yield inconsistent outcomes.
We identify representation compatibility, the extent to which a model effectively processes a representation for a structured task, as a key factor in adaptation.
We study this in chess
which provides a controlled testbed with precise semantics, computable optimal actions, and multiple state representations, including a symbolic encoding (FEN) and a spatial format (ASCII).
We find that models often process semantically equivalent inputs substantially differently, affecting both learning and generalization.
Building on this observation, we propose representation-aligned auxiliary supervision, which uses environment-derived tasks expressed in compatible representations to improve adaptation to structured domains.
Across models and representations, auxiliary supervision consistently improves optimal-move prediction relative to target-only training under identical target data.
Tasks that expose environment dynamics provide larger and most consistent gains than surface-level or static supervision, while remaining competitive with substantially increasing the amount of target-task data.
Moreover, ASCII-trained models transfer more effectively to FEN than FEN-trained models do to ASCII, even surpassing the FEN target-only baseline on FEN evaluation.
The gains also extend beyond optimal-move prediction to open-ended, factually grounded commentary generation.
Overall, our results show that auxiliary supervision in model-compatible representations can enable effective adaptation in structured domains.