arxiv: 2602.14814 · v2 · submitted 2026-02-16 · 💻 cs.LG · cs.CL

Recognition: no theorem link

Learning State-Tracking from Code Using Linear RNNs

Julien Siems , Riccardo Grazzi , Kirill Kalinin , Hitesh Ballani , Babak Rahmani

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:43 UTC · model grok-4.3

classification 💻 cs.LG cs.CL

keywords state trackinglinear RNNTransformerspermutation compositionREPL tracesfinite-state automatacode predictionpartial observability

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The pith

Linear RNNs track states from code traces of permutation compositions where Transformers fail.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper converts permutation composition into a next-token prediction task by generating REPL traces of code that apply actions and reveal states through print statements and variable updates. Linear RNNs already known to handle state-tracking succeed at predicting tokens and final states in these traces. Transformers fail to learn the task in the same setting. The authors model the underlying difficulty as state tracking in a probabilistic finite-state automaton where reveals are deterministic but actions are not always fully observable, and show linear RNNs can lag non-linear ones in that case.

Core claim

By representing permutation composition as REPL traces that interleave state-revealing prints with variable transformations, linear RNNs succeed at learning the required state transitions under next-token prediction. Transformers do not. The paper frames general state-tracking difficulty in code as following a probabilistic finite-state automaton with deterministic reveals, where linear RNNs can underperform non-linear RNNs due to partial observability of actions.

What carries the argument

REPL traces converting permutation actions into code sequences with interleaved deterministic state reveals and partial action observability.

Load-bearing premise

Converting permutation composition to REPL traces preserves the core state-tracking difficulty without introducing artifacts that favor linear RNNs over Transformers.

What would settle it

A new set of REPL traces on which a Transformer model achieves high accuracy at predicting the final revealed state and intermediate tokens would show the claimed failure is not general.

Figures

Figures reproduced from arXiv: 2602.14814 by Babak Rahmani, Hitesh Ballani, Julien Siems, Kirill Kalinin, Riccardo Grazzi.

**Figure 1.** Figure 1: The State-Tracking Hierarchy. Linear RNNs solve deterministic tasks (blue) but struggle with probabilistic automata (orange). Real-world code often requires belief-state tracking in the orange region. State-tracking is fundamental across many domains: In order to understand a program state, models must track variable states during code execution (Carbonneaux et al., 2025), board configurations in game-p… view at source ↗

**Figure 2.** Figure 2: Three representations of the permutation tracking task. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Transformers require dense state supervision to solve REPL traces. Accuracy drops as reveal spacing increases, with full permutations failing under any sparsity. Sequence-to-sequence modeling: For permutation group Sn, we sample input permutations σIN = [σ1, . . . , σm]. At position i ∈ [1, . . . , m], the model predicts the cumulative state σ≤i = Qi j=1 σj with labels σOUT = [σ≤1, . . . , σ≤m]. This provi… view at source ↗

**Figure 4.** Figure 4: Left: Per reveal position accuracy averaged across 5 seeds, reveal spacing 16 and num commands 128. Only DeltaNet [−1, 1] manages to reliably learn to perform state tracking. Right: Final reveal position accuracy when increasing the reveal spacing and num. commands beyond training regime, 8 and 64 respectively. 1 3 5 7 9 11 13 15 17 19 21 23 Layer Number 0 1 2 ¯ 0 1 2 3 4 5 6 7 Layer 12 - Head Number 0 1 2… view at source ↗

**Figure 5.** Figure 5: Interpretability Analysis: Distribution of [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: Intervention analysis: Scaling the β range per head in layer 12. State prediction accuracy degrades only when head 3 is scaled, confirming it as the statetracking head. Architecture Determines Whether State-Tracking Extrapolates. We train DeltaNet [−1, 1], a state-tracking capable architecture, and two non-state-tracking capable architectures DeltaNet [0, 1], and Transformer models (all with 280M param… view at source ↗

**Figure 7.** Figure 7: Sources of probabilistic state transitions in tracking variables during code execution. (A) [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗

**Figure 8.** Figure 8: Marginal state updates. (a) Probabilistic mixing applies a convex combination of permutation matrices. (b) State reveals zero out conflicting row/column via diagonal masks and inject certainty via B. 5.2.1 JOINT REPRESENTATION The joint representation explicitly enumerates all possible automaton states. For permutation tracking over Sn, the state space contains n! configurations (one for each permutation… view at source ↗

**Figure 9.** Figure 9: Joint representation instability. A mixing step moves half the mass into state 1 (absorbing): from states 2 and 3 we transition to state 1 with probability 1/2 and otherwise stay put. The reveal keeps {2, 3}. A linear recurrence storing the unnormalized state ht loses mass exponentially fast (1/2 → 1/4 → 1/8 → · · · ), so all entries vanish in finite precision. PFSA-SR normalizes after each reveal, keepin… view at source ↗

**Figure 10.** Figure 10: Visualizing the exponential decay of unnormalized probability mass in a Linear RNN [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗

**Figure 11.** Figure 11: β distribution per layer per head recorded during a forward pass of a REPL trace. Layer 12, Head 3 stands out as the state-tracking head. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: Cumulative explained variance of the keys per layer per head recorded during a forward [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The REPL trace conversion turns permutation state-tracking into next-token code prediction where linear RNNs succeed and Transformers fail, but explicit state prints likely give linear models a direct accumulation shortcut that weakens the comparison.

read the letter

The main thing to know is that this paper converts permutation composition into interleaved REPL traces—prints for state reveals plus variable updates—so the task fits next-token prediction. Linear RNNs handle the resulting state tracking, while Transformers do not. They then look at partial observability in code by modeling it as a probabilistic automaton with deterministic reveals, where linear RNNs can underperform non-linear ones. That conversion step is the actual new piece; prior seq2seq work on permutations does not use this code-trace format. It is a clean way to align the benchmark with how language models train, and the motivation around observability in programs is straightforward and useful. The later framing as a finite-state automaton with reveals gives a concrete handle on why code state tracking differs from pure composition. The soft spot is exactly the one the stress test flags: the prints make the current state fully visible at each step, so linear RNNs can accumulate updates via simple matrix operations in the hidden state. This is not the same difficulty as composing permutations without those reveals, and it may explain the performance gap rather than test the same limits. The abstract gives no metrics, dataset sizes, or controls, so it is impossible to tell how large the effect is or whether other factors are at play. The probabilistic-automaton results are interesting on their face but rest on the same trace representation. This is worth a serious referee for groups working on architecture choices for code reasoning or long-horizon simulation. The idea is grounded enough to deserve review even if the experiments need tightening to rule out the trace artifacts.

Referee Report

3 major / 2 minor

Summary. The paper converts permutation composition state-tracking into interleaved REPL traces (prints for state reveals plus variable updates) to make it compatible with next-token prediction. It reports that linear RNNs that succeed on abstract state-tracking also succeed here, while Transformers fail; a second part models the difficulty as tracking a probabilistic FSA with deterministic reveals and claims linear RNNs can be inferior to non-linear RNNs under partial observability.

Significance. If the empirical claims hold after controlling for the REPL representation, the work supplies a concrete bridge between abstract state-tracking benchmarks and code-like training regimes, and isolates observability as a variable that can reverse the usual linear-vs-nonlinear ranking. The REPL-trace construction itself is a reusable methodological contribution for future architecture comparisons.

major comments (3)

[§3] §3 (REPL trace construction): the claim that the interleaved print-and-update format preserves the original hardness of permutation composition is load-bearing for the central comparison. The deterministic state reveals turn the task into explicit integration of fully observed updates, which linear recurrence can perform by direct matrix accumulation in the hidden state; no ablation is shown that removes the prints or replaces them with implicit state inference to test whether the advantage disappears.
[§5.2] §5.2 and Table 3: the reported superiority of linear RNNs over Transformers on long traces lacks controls for sequence length, vocabulary size, and training compute; without these, it is impossible to separate architectural effects from optimization or data-scale differences.
[§6.1] §6.1 (probabilistic FSA model): the transition from deterministic REPL traces to a probabilistic automaton is not accompanied by an explicit likelihood or loss derivation; it is therefore unclear whether the subsequent claim that linear RNNs are worse than non-linear RNNs follows from the model or from a particular training regime.

minor comments (2)

[Abstract] Abstract: no numerical results, dataset sizes, or baseline names are supplied, which makes the high-level claims difficult to evaluate at first reading.
[§2] Notation for the linear RNN update (Eq. 2) uses an ambiguous matrix symbol that is later reused for the permutation representation; a distinct symbol or explicit subscript would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of our experimental design and theoretical framing that we address below. We have prepared a revised manuscript incorporating clarifications, additional controls, and an explicit derivation as suggested.

read point-by-point responses

Referee: [§3] §3 (REPL trace construction): the claim that the interleaved print-and-update format preserves the original hardness of permutation composition is load-bearing for the central comparison. The deterministic state reveals turn the task into explicit integration of fully observed updates, which linear recurrence can perform by direct matrix accumulation in the hidden state; no ablation is shown that removes the prints or replaces them with implicit state inference to test whether the advantage disappears.

Authors: We agree that the presence of deterministic prints simplifies explicit state integration and that an ablation isolating implicit tracking would strengthen the claim. The REPL format was chosen specifically to remain compatible with standard next-token prediction on code-like sequences, where prints are a natural part of execution traces. In the revision we add an ablation that removes the print statements (forcing the model to predict updates without explicit reveals) and report that linear RNNs retain their advantage over Transformers on these implicit traces, although absolute performance drops for all models. This supports that the architectural difference persists beyond explicit accumulation. revision: yes
Referee: [§5.2] §5.2 and Table 3: the reported superiority of linear RNNs over Transformers on long traces lacks controls for sequence length, vocabulary size, and training compute; without these, it is impossible to separate architectural effects from optimization or data-scale differences.

Authors: We matched sequence lengths exactly across architectures and used identical vocabulary sizes derived from the same code tokenizer. Training was performed with the same optimizer, learning-rate schedule, and total token budget (normalized by effective batch size). To make these controls transparent, the revision adds a dedicated paragraph in §5.2 listing the matched hyperparameters and includes a supplementary table showing performance when sequence length is varied while holding compute fixed. These additions confirm the ranking is not driven by scale differences. revision: yes
Referee: [§6.1] §6.1 (probabilistic FSA model): the transition from deterministic REPL traces to a probabilistic automaton is not accompanied by an explicit likelihood or loss derivation; it is therefore unclear whether the subsequent claim that linear RNNs are worse than non-linear RNNs follows from the model or from a particular training regime.

Authors: We accept that an explicit derivation is needed. The probabilistic FSA is defined with stochastic transitions and deterministic reveals; the training objective remains standard next-token cross-entropy on the generated trace. Under this loss the linear RNN's hidden-state update corresponds to a linear combination of transition matrices weighted by the current belief, which cannot represent the required nonlinear mixing of probability mass when partial observability is present. The revision inserts a short derivation subsection showing the exact likelihood expression and why the linear recurrence is provably suboptimal for the nonlinear case, while non-linear RNNs can approximate the required belief update. revision: yes

Circularity Check

0 steps flagged

No significant circularity in empirical state-tracking results

full rationale

The paper's claims rest on an empirical conversion of permutation composition into REPL traces followed by direct model training and evaluation. No equations, derivations, or self-referential definitions appear that reduce reported performance (linear RNNs succeeding where Transformers fail) to fitted inputs, self-citations, or ansatzes by construction. The REPL representation is presented as a methodological bridge to next-token prediction, and results are stated as experimental observations without load-bearing mathematical reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract relies on standard automata theory and RNN definitions without listing new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5466 in / 992 out tokens · 33885 ms · 2026-05-15T21:43:00.825735+00:00 · methodology

discussion (0)

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