Comment on "Modeling rapid language learning by distilling Bayesian priors into artificial neural networks"

McCoy & Griffiths (2025, henceforth M&G) suggest that a Bayesian prior can be distilled into Artificial Neural Networks (ANNs) through Model-Agnostic Meta-Learning (MAML, Finn et al., 2017). They support this empirically by showing that meta-trained networks demonstrate formal language learning abilities comparable to Yang & Piantadosi (2023)'s Bayesian learner, significantly outperforming standard ANNs. We point out that under the standard interpretation of a prior, M&G's procedure does not actually instill one; it merely initializes network weights favorably, leaving the objective function unchanged. We then consider a more permissive interpretation, where the system as a whole can be seen as implementing a Bayesian learner even without an explicit prior in the objective. We show that this interpretation faces nontrivial challenges. Finally, we assess how well MAML approximates the empirical results of Bayesian learning, showing that unlike genuine Bayesian learners, M&G's model overfits and generalizes poorly to unseen data.

Disentangling the Expressivity of RoPE

Two accounts recur in explanations of the success of rotary position embeddings (RoPE). Expressivity studies associate periodic position information with modular predicates, whereas mechanistic and long-context studies emphasize positional anchors and local offsets. We formalize both accounts for fully uniform, finite-precision soft-attention transformers. We find that, if every rotary component is periodic, RoPE transformers recognize exactly the languages definable in past temporal logic with modular predicates. Conventional RoPE is different: The rotations it computes never repeat. This yields a precision-dependent bounded simulation of fixed-offset look-back operators, rather than an all-length modular characterization. Controlled experiments match this separation: Constructed periodic schedules length-generalize on modular languages, while conventional RoPE behaves more like a bounded locality bias and can impair tasks requiring position-invariant access to distant context. Altogether, our findings shed light on RoPE transformers, bringing theoretical expressivity characterizations closer to models used in practice.

Causally Evaluating the Learnability of Formal Language Tasks

Language models, as multi-task learners, acquire a wide range of abilities during training. A fundamental question is how much task-specific data is needed to learn a given task. Answering this for natural language is difficult: tasks are hard to delineate and can confound one another. To rigorously investigate the relationship between data frequency and learnability, we turn to a controlled setting using formal languages induced from probabilistic finite automata. These serve as a methodological testbed to demonstrate that standard correlational evaluation practices are inherently flawed. To enable causal analysis, we introduce the binning semiring, an algebraic object that lets us control how often a targeted property occurs in a sampled corpus. We formulate the experimental pipeline as a causal graphical model and derive decomposed Kullback-Leibler divergence metrics to measure the learnability of specific sub-tasks. Our experiments show that evaluating learnability without causal intervention leads to incorrect conclusions due to confounders in correlational analysis, and serve as a warning about correlational pitfalls in natural-language settings.

Generating in the Limit with Infinitely Many Hallucinations

The classic paradigm of language identification in the limit models learning as a game between an adversary, who reveals strings from an unknown target language, and a learner tasked with identifying that language. The recently introduced framework of language generation in the limit shifted the objective to better reflect modern language modeling, requiring the learner to produce valid, unseen strings from the target language. Related work highlighted a fundamental tension: a broad coverage of the target often comes at the cost of validity. We introduce a new notion of precision and recast this problem as the classic recall-precision trade-off. We analyze generation in the limit under varying constraints on enumeration, novelty, and validity, aimed at reflecting settings closer to those encountered by large language models. A key contribution is our analysis of learners that are not eventually valid: we allow infinitely many mistakes, provided their frequency tends to zero so that precision remains one. We show that this relaxation can strictly increase recall when the adversary permanently withholds a large portion of the target language. We also study a continuous relaxation of the novelty constraint that requires only a fixed fraction of outputs to be novel. Taken together, our results move toward a more realistic model of language generation where occasional errors and repetitions are unavoidable, but their rates are controlled.

Causal Evidence of Stack Representations in Modeling Counter Languages Using Transformers

Formal languages have proven to be effective conduits to understand the inner mechanisms of transformers. Past work has shown that transformers trained on next token prediction over counter languages learn representations consistent with an underlying stack structure. Beyond representational analysis, this paper investigates the causal role of these representations. Linear probes are trained to predict the stack depth at each token from the model's hidden states, and a principal representation direction is extracted from the probe. Ablation of this direction from the model causes sequential accuracy to collapse to near 0%, providing strong empirical evidence that the stack representation is not just learned, but is causally necessary for model performance.

Tractable Hierarchical Control of Autoregressive Language Models

Constraining the generation of autoregressive large language models (LLMs) is an important component of integrating language models into formal systems. In the generation of code and data for tasks like program synthesis, ensuring that language models produce syntactically valid output is a prerequisite for processing such output. These languages (such as SQL or JSON) are often designed as $LR(k)$ context-free grammars. By distilling the LLM to a tractable probabilistic model, its autoregressive generation can be steered and masked to incorporate the probability of satisfying logical constraints, ensuring high quality output that is guaranteed to be valid. This paper demonstrates that the satisfaction of any $LR(k)$ grammar of finite duration can be calculated in polynomial time, an improvement over the exponential time of applying previous methods to such grammars. This result enables efficient constraint and steering of LLM generation towards output that better satisfies formal syntactic constraints.