In plain words: They turn permutation-composition puzzles into code sessions that print the current state, so models can learn them by predicting the next word like a chatbot. Simple RNNs that already track state solve these traces, while Transformers still fail.
Abstract
Over the last years, state-tracking tasks, particularly permutation composition, have become a testbed to understand the limits of sequence models architectures like Transformers and RNNs (linear and non-linear). However, these are often sequence-to-sequence tasks: learning to map actions (permutations) to states, which is incompatible with the next-token prediction setting commonly used to train language models. We address this gap by converting permutation composition into code via REPL traces that interleave state-reveals through prints and variable transformations. We show that linear RNNs capable of state-tracking excel also in this setting, while Transformers still fail. Motivated by this representation, we investigate why tracking states in code is generally difficult: actions are not always fully observable. We frame this as tracking the state of a probabilistic finite-state automaton with deterministic state reveals and show that linear RNNs can be worse than non-linear RNNs at tracking states in this setup.
Julien Siems, Riccardo Grazzi, Korbinian Pöppel, Kirill Kalinin, Hitesh Ballani, Babak Rahmani
arXiv:2602.14814 · cs.LG, cs.CL · submitted Feb 16, 2026 · updated Jun 25, 2026
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