arXiv subjects

Ryan Cotterell

Publications and source records attributed to Ryan Cotterell.

At least 37 records · Page 2Linked to original sources

arXiv2026

Understanding the Parameter Space Geometry of Transformers Encoding Boolean Functions

Transformers consistently fail to learn certain simple functions that are provably expressible with specific parameter settings. This gap between learnability and expressivity is particularly prominent for sensitive functions -- functions whose output is likely to change if a single bit of the input is flipped -- for example, PARITY. While prior work has established that transformers exhibit a bias toward functions with low average sensitivity, the precise mechanism underlying this bias remains poorly understood. To shed light on this phenomenon, we study the geometry of transformers' parameter space. We show that sensitive functions -- even when representable -- occupy a vanishingly small region that random initialization is very likely to miss. Specifically, we shift the focus from average sensitivity to the full sensitivity profile -- the distribution of sensitivity values across all inputs -- and prove that randomly initialized transformers almost surely compute functions which have low-sensitivity strings. Consequently, any function that lacks such strings is provably unlearnable.

cs.LG↗

arXiv2026

Context-Free Recognition with Transformers

Transformers excel empirically on tasks that process well-formed inputs according to some grammar, such as natural language and code. However, it remains unclear how they can process grammatical syntax. In fact, under standard complexity conjectures, standard transformers cannot recognize context-free languages (CFLs), a canonical formalism to describe syntax, or even regular languages, a subclass of CFLs. Past work has shown that $\mathcal{O}(\log(N))$ looping layers (w.r.t. input length $N$) allow transformers to recognize regular languages, but the question of context-free recognition with looped transformers remained open. In this work, we show that looped transformers with $\mathcal{O}(\log(N))$ looping layers and $\mathcal{O}(N^6)$ padding symbols can recognize all CFLs. However, training and inference with $\mathcal{O}(N^6)$ padding symbols is potentially impractical. Fortunately, we show that, for natural subclasses such as unambiguous CFLs, the recognition problem on transformers becomes more tractable, requiring $\mathcal{O}(N^3)$ padding. Empirically, looped and padded transformers perform better than fixed-depth transformers in recognizing CFLs. Overall, our results shed light on the intricacy of CFL recognition by transformers: while general recognition may require an intractable amount of padding, natural constraints such as unambiguity yield efficient recognition algorithms.

cs.LG↗

arXiv2026

Post-Training Language Models for Crosslingual Consistency

Language models often respond inconsistently to translation-equivalent prompts across languages, undermining the reliability of multilingual systems. To quantify this, we give an information-theoretic definition of crosslingual consistency as a divergence bound between a model's response distribution and its round-trip pushforward across languages. We then introduce penalized consistency optimization (PCO), a post-training procedure that couples this divergence with a Kullback-Leibler penalty to a fixed reference language model. Because direct optimization of PCO requires expensive on-policy roll-outs, we propose a tractable surrogate, direct consistency optimization (DCO), which can be optimized off-policy. Across diverse language models and 26 languages, DCO significantly improves crosslingual consistency, outperforms existing methods, and enables targeted alignment of low-resource languages.

cs.CL↗

arXiv2026

Revisiting Padded Transformer Expressivity: Which Architectural Choices Matter and Which Don't

Recent work describes what transformers can and cannot compute through connections to boolean circuits, but existing results lack exact characterizations and are sensitive to modeling choices. Padded transformers -- to whose input filler symbols such as ``...'' are appended -- emerge as a useful gadget for establishing equivalences to circuit classes by providing polynomial space for adaptive parallel computation. However, only a limited set of padded transformer idealizations has been studied, leaving open how robustly these equivalences hold under changes to attention type, model width, and uniformity. We find that, under practical assumptions, padded transformers are surprisingly robust to all of these, and identify numeric precision and model depth as the main factors affecting expressivity. Concretely, we prove that polynomially padded $\text{L-uniform}$ constant-precision transformers are equivalent to $\text{L-uniform AC}^0$, while growing-precision ones achieve $\text{L-uniform TC}^0$ regardless of width. Furthermore, looping enables sequential processing analogous to circuits: $\log^d N$-looped constant-precision transformers reach $\text{FO-uniform AC}^d$, and growing-precision ones reach $\text{FO-uniform TC}^d$. Interestingly, growing width or precision beyond logarithmic does not increase expressivity, and all our results hold for both softmax and average hard attention transformers.

cs.LG↗

arXiv2026

Probability Distributions Computed by Autoregressive Transformers

Most expressivity results for transformers treat them as language recognizers -- devices that accept or reject strings -- rather than as they are used in practice: as language models that generate strings autoregressively and probabilistically. We characterize the probability distributions that transformer language models can express. We show that making transformer language recognizers autoregressive can sometimes increase their expressivity, and that making them probabilistic can break equivalences that hold in the non-probabilistic case. Our overall contribution is to tease apart what functions transformers are capable of expressing in their most common use case as language models.

cs.CL↗

arXiv2026

Learning to Reason Efficiently with A* Post-Training

Many applications of large language models (LLMs) require deductive reasoning, yet models frequently produce incorrect or redundant inference steps. We frame natural language inference as a search problem where the final answer is the valid proof itself, requiring a reasoning procedure in which intermediate inferences are correct. Specifically, we investigate whether LLMs can learn to generate correct and efficient proofs with guidance from A* search -- an algorithm that guarantees an optimally efficient path to a goal. We explore two training techniques: supervised fine-tuning on execution traces from A* and reinforcement learning with A*-informed process reward models. Empirically, we find that Llama-3.2 models in the 1B--3B range benefit substantially from A* post training, going from near-zero accuracy to outperforming DeepSeek-V3.2 -- a much larger model. Our analysis uncovers a trade-off: while simple correctness rewards maximize accuracy, A*-informed signals strike a balance between accuracy and efficiency. Furthermore, we find that on larger search spaces, models trained with imperfect heuristics exhibit superior accuracy. Our results demonstrate a promising direction towards reasoning guided by principles derived from classical search algorithms.

cs.AI↗

arXiv2026

Transformers are Inherently Succinct

We study succinctness as a measure of the expressive power of transformers. Succinctness -- how compactly a formalism can describe a language relative to other formalisms -- is a classical notion in logic and automata theory. We prove that fixed-precision transformers are remarkably succinct: they can be exponentially more succinct than both linear temporal logic (LTL) and recurrent neural networks, and, by extension, state-space models, and doubly exponentially more succinct than finite automata. In other words, there exist families of languages describable by polynomial-size transformers whose smallest equivalent LTL formula or recurrent neural network is exponentially large, and whose smallest equivalent automaton is doubly exponentially large. We also establish matching upper bounds, showing that any fixed-precision transformer can be converted to an LTL formula with at most an exponential blow-up -- improving a prior doubly exponential translation. As a consequence of this succinctness, we show that basic verification problems for transformers, such as emptiness and equivalence, are provably intractable: specifically, EXPSPACE-complete.

cs.FL↗

arXiv2026

On the Proper Treatment of Units in Surprisal Theory

Surprisal theory links human processing effort to the predictability of an upcoming linguistic unit, but empirical work often leaves the notion of a unit underspecified. In practice, experimental stimuli are segmented into linguistically motivated units (e.g., words), while pretrained language models assign probability mass to a fixed token alphabet that typically does not align with those units. As a result, surprisal-based predictors depend implicitly on ad hoc procedures that conflate two distinct modeling choices: the definition of the unit of analysis and the choice of regions of interest over which predictions are evaluated. In this paper, we disentangle these choices and give a unified framework for reasoning about surprisal over arbitrary unit inventories. We argue that surprisal-based analyses should make these choices explicit and treat tokenization as an implementation detail rather than a scientific primitive.

cs.CL↗

arXiv2026

Prefix Parsing is Just Parsing

Prefix parsing asks whether an input prefix can be extended to a complete string generated by a given grammar. In the weighted setting, it also provides prefix probabilities, which are central to context-free language modeling, psycholinguistic analysis, and syntactically constrained generation from large language models. We introduce the prefix grammar transformation, an efficient reduction of prefix parsing to ordinary parsing. Given a grammar, our method constructs another grammar that generates exactly the prefixes of its original strings. Prefix parsing is then solved by applying any ordinary parsing algorithm on the transformed grammar without modification. The reduction is both elegant and practical: the transformed grammar is only a small factor larger than the input, and any optimized implementation can be used directly, eliminating the need for bespoke prefix-parsing algorithms. We also present a strategy-based on algorithmic differentiation-for computing the next-token weight vector, i.e., the prefix weights of all one-token extensions, enabling efficient prediction of the next token. Together, these contributions yield a simple, general, and efficient framework for prefix parsing.

cs.CL↗

arXiv2026

Probing for Reading Times

Probing has shown that language model representations encode rich linguistic information, but it remains unclear whether they also capture cognitive signals about human processing. In this work, we probe language model representations for human reading times. Using regularized linear regression on two eye-tracking corpora spanning five languages (English, Greek, Hebrew, Russian, and Turkish), we compare the representations from every model layer against scalar predictors -- surprisal, information value, and logit-lens surprisal. We find that the representations from early layers outperform surprisal in predicting early-pass measures such as first fixation and gaze duration. The concentration of predictive power in the early layers suggests that human-like processing signatures are captured by low-level structural or lexical representations, pointing to a functional alignment between model depth and the temporal stages of human reading. In contrast, for late-pass measures such as total reading time, scalar surprisal remains superior, despite its being a much more compressed representation. We also observe performance gains when using both surprisal and early-layer representations. Overall, we find that the best-performing predictor varies strongly depending on the language and eye-tracking measure.

cs.CL↗

arXiv2026

Faster LLM Inference via Sequential Monte Carlo

Speculative decoding (SD) accelerates language model inference by drafting tokens from a cheap proposal model and verifying them against an expensive target model via rejection sampling. Because rejection truncates the draft block at the first error, throughput degrades when draft and target diverge. Rather than rejecting draft tokens outright, we propose to reweight them. To this end, we introduce sequential Monte Carlo speculative decoding (SMC-SD), which replaces token-level rejection with importance-weighted resampling over a population of draft particles. SMC-SD is a principled approximate inference scheme that trades exactness for additional speed, while preserving theoretical bounds on its per-step approximation error. Because LLM inference is memory bandwidth-bound, the arithmetic needed to draft particles and to score them in parallel comes nearly for free -- SMC-SD uses idle compute to turn verification into a vectorized, fixed-size operation with no rollback. Empirically, SMC-SD achieves 2.36x speed-up over speculative decoding and a 5.2x speed-up over autoregressive decoding, while remaining within 3% of the target model's accuracy on reasoning, instruction-following, and coding benchmarks.

cs.LG↗

arXiv2026

Transducing Language Models

Modern language models define distributions over strings, but downstream tasks often require different output formats. For instance, a model that generates byte-pair strings does not directly produce word-level predictions, and a DNA model does not directly produce amino-acid sequences. In such cases, a deterministic string-to-string transformation can convert the model's output to the desired form. This is a familiar pattern in probability theory: applying a function $f$ to a random variable $X\sim p$ yields a transformed random variable $f(X)$ with an induced distribution. While such transformations are occasionally used in language modeling, prior work does not treat them as yielding new, fully functional language models. We formalize this perspective and introduce a general framework for language models derived from deterministic string-to-string transformations. We focus on transformations representable as finite-state transducers -- a commonly used state-machine abstraction for efficient string-to-string mappings. We develop algorithms that compose a language model with an FST to *marginalize* over source strings mapping to a given target, propagating probabilities through the transducer without altering model parameters and enabling *conditioning* on transformed outputs. We present an exact algorithm, an efficient approximation, and a theoretical analysis. We conduct experiments in three domains: converting language models from tokens to bytes, from tokens to words, and from DNA to amino acids. These experiments demonstrate inference-time adaptation of pretrained language models to match application-specific output requirements.

cs.CL↗

arXiv2026

Ensembling Language Models with Sequential Monte Carlo

Practitioners have access to an abundance of language models and prompting strategies for solving many language modeling tasks; yet prior work shows that modeling performance is highly sensitive to both choices. Classical machine learning ensembling techniques offer a principled approach: aggregate predictions from multiple sources to achieve better performance than any single one. However, applying ensembling to language models during decoding is challenging: naively aggregating next-token probabilities yields samples from a locally normalized, biased approximation of the generally intractable ensemble distribution over strings. In this work, we introduce a unified framework for composing $K$ language models into $f$-ensemble distributions for a wide range of functions $f\colon\mathbb{R}_{\geq 0}^{K}\to\mathbb{R}_{\geq 0}$. To sample from these distributions, we propose a byte-level sequential Monte Carlo (SMC) algorithm that operates in a shared character space, enabling ensembles of models with mismatching vocabularies and consistent sampling in the limit. We evaluate a family of $f$-ensembles across prompt and model combinations for various structured text generation tasks, highlighting the benefits of alternative aggregation strategies over traditional probability averaging, and showing that better posterior approximations can yield better ensemble performance.

cs.CL↗

arXiv2026

BabyLM Turns 4 and Goes Multilingual: Call for Papers for the 2026 BabyLM Workshop

The goal of the BabyLM is to stimulate new research connections between cognitive modeling and language model pretraining. We invite contributions in this vein to the BabyLM Workshop, which will also include the 4th iteration of the BabyLM Challenge. As in previous years, the challenge features two ``standard'' tracks (Strict and Strict-Small), in which participants must train language models on under 100M or 10M words of data, respectively. This year, we move beyond our previous English-only pretraining datasets with a new Multilingual track, focusing on English, Dutch, and Chinese. For the workshop, we call for papers related to the overall theme of BabyLM, which includes training efficiency, small-scale training datasets, cognitive modeling, model evaluation, and architecture innovation.

cs.CL↗

arXiv2026

Are Language Models Efficient Reasoners? A Perspective from Logic Programming

Modern language models (LMs) exhibit strong deductive reasoning capabilities, yet standard evaluations emphasize correctness while overlooking a key aspect of reasoning: efficiency. In real-world reasoning scenarios, much of the available information is irrelevant, and effective deductive inference requires identifying and ignoring such distractions. We propose a framework for assessing LM reasoning efficiency through the lens of logic programming, introducing a simple method to align proofs written in natural language -- as generated by an LM -- with shortest proofs found by executing the logic program. Efficiency is quantified by measuring how well a model avoids unnecessary inference. Empirically, we construct a dataset of math word problems injected with various number of irrelevant axioms that vary in semantic overlap with the goal theorem. We find that current LMs show marked accuracy declines under such conditions -- even with minimal, domain-consistent distractions -- and the proofs they generate frequently exhibit detours through irrelevant inferences.

cs.CL↗

arXiv2025

Automating the Analysis of Parsing Algorithms (and other Dynamic Programs)

Much algorithmic research in NLP aims to efficiently manipulate rich formal structures. An algorithm designer typically seeks to provide guarantees about their proposed algorithm -- for example, that its running time or space complexity is upper-bounded as a certain function of its input size. They may also wish to determine the necessary properties of the quantities derived by the algorithm to synthesize efficient data structures and verify type errors. In this paper, we develop a system for helping programmers to perform these types of analyses. We apply our system to a number of NLP algorithms and find that it successfully infers types, dead and redundant code, and parametric runtime and space complexity bounds.

cs.PL↗

arXiv2025

Characterizing the Expressivity of Fixed-Precision Transformer Language Models

Transformer-based language models (LMs) have achieved widespread empirical success, but their theoretical expressive power remains only partially understood. In this work, we analyze a restricted idealization of fixed-precision transformers with strict future masking, soft attention, and no positional encodings. We establish that this class of models is exactly as expressive as a specific fragment of linear temporal logic that contains only a single temporal operator: the past operator. We further connect this fragment to established classes in formal language theory, automata theory, and algebra, yielding a unified framework for understanding transformer expressivity under this idealization. Finally, we present empirical results that align closely with our theory: transformers trained on languages within their characterized expressive capacity generalize reliably across sequence lengths, while they consistently fail to generalize on languages beyond it.

cs.CL↗

arXiv2025

Unique Hard Attention: A Tale of Two Sides

Understanding the expressive power of transformers has recently attracted attention, as it offers insights into their abilities and limitations. Many studies analyze unique hard attention transformers, where attention selects a single position that maximizes the attention scores. When multiple positions achieve the maximum score, either the rightmost or the leftmost of those is chosen. In this paper, we highlight the importance of this seeming triviality. Recently, finite-precision transformers with both leftmost- and rightmost-hard attention were shown to be equivalent to Linear Temporal Logic (LTL). We show that this no longer holds with only leftmost-hard attention -- in that case, they correspond to a \emph{strictly weaker} fragment of LTL. Furthermore, we show that models with leftmost-hard attention are equivalent to \emph{soft} attention, suggesting they may better approximate real-world transformers than right-attention models. These findings refine the landscape of transformer expressivity and underscore the role of attention directionality.

cs.LG↗