Anej Svete

Anej Svete, Benjamin Dayan, Tim Vieira et al. · allen-ai, eth-zurich

Weighted finite-state automata (WSFAs) are commonly used in NLP. Failure transitions are a useful extension for compactly representing backoffs or interpolation in $n$-gram models and CRFs, which are special cases of WFSAs. The pathsum in ordinary acyclic WFSAs is efficiently computed by the backward algorithm in time $O(|E|)$, where $E$ is the set of transitions. However, this does not allow failure transitions, and preprocessing the WFSA to eliminate failure transitions could greatly increase $|E|$. We extend the backward algorithm to handle failure transitions directly. Our approach is efficient when the average state has outgoing arcs for only a small fraction $s \ll 1$ of the alphabet $Σ$. We propose an algorithm for general acyclic WFSAs which runs in $O{\left(|E| + s |Σ| |Q| T_\text{max} \log{|Σ|}\right)}$, where $Q$ is the set of states and $T_\text{max}$ is the size of the largest connected component of failure transitions. When the failure transition topology satisfies a condition exemplified by CRFs, the $T_\text{max}$ factor can be dropped, and when the weight semiring is a ring, the $\log{|Σ|}$ factor can be dropped. In the latter case (ring-weighted acyclic WFSAs), we also give an alternative algorithm with complexity $\displaystyle O{\left(|E| + |Σ| |Q| \min(1,sπ_\text{max}) \right)}$, where $π_\text{max}$ is the size of the longest failure path.

Clément Guerner, Tianyu Liu, Anej Svete et al. · allen-ai, eth-zurich

The linear subspace hypothesis (Bolukbasi et al., 2016) states that, in a language model's representation space, all information about a concept such as verbal number is encoded in a linear subspace. Prior work has relied on auxiliary classification tasks to identify and evaluate candidate subspaces that might give support for this hypothesis. We instead give a set of intrinsic criteria which characterize an ideal linear concept subspace and enable us to identify the subspace using only the language model distribution. Our information-theoretic framework accounts for spuriously correlated features in the representation space (Kumar et al., 2022) by reconciling the statistical notion of concept information and the geometric notion of how concepts are encoded in the representation space. As a byproduct of this analysis, we hypothesize a causal process for how a language model might leverage concepts during generation. Empirically, we find that linear concept erasure is successful in erasing most concept information under our framework for verbal number as well as some complex aspect-level sentiment concepts from a restaurant review dataset. Our causal intervention for controlled generation shows that, for at least one concept across two languages models, the concept subspace can be used to manipulate the concept value of the generated word with precision.

William Merrill, Yanhong Li, Tyler Romero et al. · allen-ai, eth-zurich

Recent work has demonstrated the potential of non-transformer language models, especially linear recurrent neural networks (RNNs) and hybrid models that mix recurrence and attention. Yet there is no consensus on whether the potential benefits of these new architectures justify the risk and effort of scaling them up. To address this, we provide evidence for the advantages of hybrid models over pure transformers on several fronts. First, theoretically, we show that hybrid models do not merely inherit the expressivity of transformers and linear RNNs, but can express tasks beyond both, such as code execution. Putting this theory to practice, we train Olmo Hybrid, a 7B-parameter model largely comparable to Olmo 3 7B but with the sliding window layers replaced by Gated DeltaNet layers. We show that Olmo Hybrid outperforms Olmo 3 across standard pretraining and mid-training evaluations, demonstrating the benefit of hybrid models in a controlled, large-scale setting. We find that the hybrid model scales significantly more efficiently than the transformer, explaining its higher performance. However, its unclear why greater expressivity on specific formal problems should result in better scaling or superior performance on downstream tasks unrelated to those problems. To explain this apparent gap, we return to theory and argue why increased expressivity should translate to better scaling efficiency, completing the loop. Overall, our results suggest that hybrid models mixing attention and recurrent layers are a powerful extension to the language modeling paradigm: not merely to reduce memory during inference, but as a fundamental way to obtain more expressive models that scale better during pretraining.

Ryan Cotterell, Anej Svete, Clara Meister et al. · allen-ai, eth-zurich

Large language models have become one of the most commonly deployed NLP inventions. In the past half-decade, their integration into core natural language processing tools has dramatically increased the performance of such tools, and they have entered the public discourse surrounding artificial intelligence. Consequently, it is important for both developers and researchers alike to understand the mathematical foundations of large language models, as well as how to implement them. These notes are the accompaniment to the theoretical portion of the ETH Zürich course on large language models, covering what constitutes a language model from a formal, theoretical perspective.

Anej Svete, William Merrill, Ryan Cotterell et al.

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.

Anej Svete, Ryan Cotterell · allen-ai, eth-zurich

Studying language models (LMs) in terms of well-understood formalisms allows us to precisely characterize their abilities and limitations. Previous work has investigated the representational capacity of recurrent neural network (RNN) LMs in terms of their capacity to recognize unweighted formal languages. However, LMs do not describe unweighted formal languages -- rather, they define \emph{probability distributions} over strings. In this work, we study what classes of such probability distributions RNN LMs can represent, which allows us to make more direct statements about their capabilities. We show that simple RNNs are equivalent to a subclass of probabilistic finite-state automata, and can thus model a strict subset of probability distributions expressible by finite-state models. Furthermore, we study the space complexity of representing finite-state LMs with RNNs. We show that, to represent an arbitrary deterministic finite-state LM with $N$ states over an alphabet $\alphabet$, an RNN requires $Ω\left(N |Σ|\right)$ neurons. These results present a first step towards characterizing the classes of distributions RNN LMs can represent and thus help us understand their capabilities and limitations.

Franz Nowak, Anej Svete, Li Du et al. · allen-ai, eth-zurich

This work investigates the computational expressivity of language models (LMs) based on recurrent neural networks (RNNs). Siegelmann and Sontag (1992) famously showed that RNNs with rational weights and hidden states and unbounded computation time are Turing complete. However, LMs define weightings over strings in addition to just (unweighted) language membership and the analysis of the computational power of RNN LMs (RLMs) should reflect this. We extend the Turing completeness result to the probabilistic case, showing how a rationally weighted RLM with unbounded computation time can simulate any deterministic probabilistic Turing machine (PTM) with rationally weighted transitions. Since, in practice, RLMs work in real-time, processing a symbol at every time step, we treat the above result as an upper bound on the expressivity of RLMs. We also provide a lower bound by showing that under the restriction to real-time computation, such models can simulate deterministic real-time rational PTMs.

Selim Jerad, Anej Svete, Sophie Hao et al. · allen-ai, eth-zurich

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 proves that $\mathcal{O}(\log(n))$ looping layers (w.r.t. input length n) allows transformers to recognize regular languages, but the question of context-free recognition remained open. In this work, we show that looped transformers with $\mathcal{O}(\log(n))$ looping layers and $\mathcal{O}(n^6)$ padding tokens can recognize all CFLs. However, training and inference with $\mathcal{O}(n^6)$ padding tokens 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. We empirically validate our results and show that looping helps on a language that provably requires logarithmic depth. 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.

Alexandra Butoi, Ghazal Khalighinejad, Anej Svete et al. · allen-ai, cambridge

Characterizing the computational power of neural network architectures in terms of formal language theory remains a crucial line of research, as it describes lower and upper bounds on the reasoning capabilities of modern AI. However, when empirically testing these bounds, existing work often leaves a discrepancy between experiments and the formal claims they are meant to support. The problem is that formal language theory pertains specifically to recognizers: machines that receive a string as input and classify whether it belongs to a language. On the other hand, it is common instead to evaluate language models on proxy tasks, e.g., language modeling or sequence-to-sequence transduction, that are similar in only an informal sense to the underlying theory. We correct this mismatch by training and evaluating neural networks directly as binary classifiers of strings, using a general method that can be applied to a wide variety of languages. As part of this, we extend an algorithm recently proposed by Snæbjarnarson et al. (2025) for efficient length-controlled sampling of strings from regular languages. We provide results on a variety of languages across the Chomsky hierarchy for three neural architectures: a simple RNN, an LSTM, and a causally-masked transformer. We find that the RNN and LSTM often outperform the transformer, and that auxiliary training objectives such as language modeling can help, although no single objective uniformly improves performance across languages and architectures. Our contributions will facilitate theoretically sound empirical testing of language recognition claims in future work. We have released our datasets as a benchmark called FLaRe (Formal Language Recognition), along with our code.

Andy Yang, Anej Svete, Jiaoda Li et al.

Most expressivity results for transformers treat them as language recognizers (which accept or reject strings), and not as they are used in practice, as language models (which generate strings autoregressively and probabilistically). Here, 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.

Anej Svete, Ryan Cotterell · allen-ai, eth-zurich

Existing work has analyzed the representational capacity of the transformer architecture by means of formal models of computation. However, the focus so far has been on analyzing the architecture in terms of language \emph{acceptance}. We contend that this is an ill-suited problem in the study of \emph{language models} (LMs), which are definitionally \emph{probability distributions} over strings. In this paper, we focus on the relationship between transformer LMs and $n$-gram LMs, a simple and historically relevant class of language models. We show that transformer LMs using the hard or sparse attention mechanisms can exactly represent any $n$-gram LM, giving us a concrete lower bound on their probabilistic representational capacity. This provides a first step towards understanding the mechanisms that transformer LMs can use to represent probability distributions over strings.

Shauli Ravfogel, Anej Svete, Vésteinn Snæbjarnarson et al. · allen-ai, eth-zurich

Understanding and manipulating the causal generation mechanisms in language models is essential for controlling their behavior. Previous work has primarily relied on techniques such as representation surgery -- e.g., model ablations or manipulation of linear subspaces tied to specific concepts -- to \emph{intervene} on these models. To understand the impact of interventions precisely, it is useful to examine \emph{counterfactuals} -- e.g., how a given sentence would have appeared had it been generated by the model following a specific intervention. We highlight that counterfactual reasoning is conceptually distinct from interventions, as articulated in Pearl's causal hierarchy. Based on this observation, we propose a framework for generating true string counterfactuals by reformulating language models as a structural equation model using the Gumbel-max trick, which we called Gumbel counterfactual generation. This reformulation allows us to model the joint distribution over original strings and their counterfactuals resulting from the same instantiation of the sampling noise. We develop an algorithm based on hindsight Gumbel sampling that allows us to infer the latent noise variables and generate counterfactuals of observed strings. Our experiments demonstrate that the approach produces meaningful counterfactuals while at the same time showing that commonly used intervention techniques have considerable undesired side effects.

Luca Malagutti, Andrius Buinovskij, Anej Svete et al. · allen-ai, eth-zurich

For nearly three decades, language models derived from the $n$-gram assumption held the state of the art on the task. The key to their success lay in the application of various smoothing techniques that served to combat overfitting. However, when neural language models toppled $n$-gram models as the best performers, $n$-gram smoothing techniques became less relevant. Indeed, it would hardly be an understatement to suggest that the line of inquiry into $n$-gram smoothing techniques became dormant. This paper re-opens the role classical $n$-gram smoothing techniques may play in the age of neural language models. First, we draw a formal equivalence between label smoothing, a popular regularization technique for neural language models, and add-$λ$ smoothing. Second, we derive a generalized framework for converting any $n$-gram smoothing technique into a regularizer compatible with neural language models. Our empirical results find that our novel regularizers are comparable to and, indeed, sometimes outperform label smoothing on language modeling and machine translation.

Selim Jerad, Anej Svete, Jiaoda Li et al. · allen-ai, eth-zurich

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.

Anej Svete, Robin Shing Moon Chan, Ryan Cotterell · allen-ai, eth-zurich

Recent work by Hewitt et al. (2020) provides an interpretation of the empirical success of recurrent neural networks (RNNs) as language models (LMs). It shows that RNNs can efficiently represent bounded hierarchical structures that are prevalent in human language. This suggests that RNNs' success might be linked to their ability to model hierarchy. However, a closer inspection of Hewitt et al.'s (2020) construction shows that it is not inherently limited to hierarchical structures. This poses a natural question: What other classes of LMs can RNNs efficiently represent? To this end, we generalize Hewitt et al.'s (2020) construction and show that RNNs can efficiently represent a larger class of LMs than previously claimed -- specifically, those that can be represented by a pushdown automaton with a bounded stack and a specific stack update function. Altogether, the efficiency of representing this diverse class of LMs with RNN LMs suggests novel interpretations of their inductive bias.

Anej Svete, Ashish Sabharwal · allen-ai, eth-zurich

Masked diffusion models (MDMs) for text offer a compelling alternative to traditional autoregressive language models. Parallel generation makes them efficient, but their computational capabilities and the limitations inherent to their parallelism remain largely unexplored. To this end, we characterize what types of reasoning problems MDMs can provably solve and how efficiently. We do this by connecting MDMs to the well-understood reasoning frameworks of chain of thought (CoT) and padded looped transformers (PLTs) in the finite-precision log-width setting: We show that MDMs and polynomially-padded PLTs are, in fact, equivalent in this setting, and that MDMs can solve all problems that CoT-augmented transformers can. Moreover, we showcase classes of problems (including regular languages) for which MDMs are inherently more efficient than CoT transformers, where parallel generation allows for substantially faster reasoning.

Andy Yang, Christopher Watson, Anton Xue et al. · allen-ai, eth-zurich

We present the transformer cookbook: a collection of techniques for directly encoding algorithms into a transformer's parameters. This work addresses the steep learning curve of such endeavors, a problem exacerbated by a fragmented literature where key results are scattered across numerous papers. In particular, we synthesize this disparate body of findings into a curated set of recipes that demonstrate how to implement everything from basic arithmetic in feed-forward layers to complex data routing via self-attention. Our mise en place of formulations is for both newcomers seeking an accessible entry point and experts in need of a systematic reference. This unified presentation of transformer constructions provides a foundation for future work spanning theoretical research in computational complexity to empirical investigations in architecture design and interpretability.

Taiga Someya, Anej Svete, Brian DuSell et al. · allen-ai, eth-zurich

Inductive biases are inherent in every machine learning system, shaping how models generalize from finite data. In the case of neural language models (LMs), debates persist as to whether these biases align with or diverge from human processing constraints. To address this issue, we propose a quantitative framework that allows for controlled investigations into the nature of these biases. Within our framework, we introduce $m$-local entropy$\unicode{x2013}$an information-theoretic measure derived from average lossy-context surprisal$\unicode{x2013}$that captures the local uncertainty of a language by quantifying how effectively the $m-1$ preceding symbols disambiguate the next symbol. In experiments on both perturbed natural language corpora and languages defined by probabilistic finite-state automata (PFSAs), we show that languages with higher $m$-local entropy are more difficult for Transformer and LSTM LMs to learn. These results suggest that neural LMs, much like humans, are highly sensitive to the local statistical structure of a language.

Clemente Pasti, Talu Karagöz, Anej Svete et al. · allen-ai, eth-zurich

Extracting finite state automata (FSAs) from black-box models offers a powerful approach to gaining interpretable insights into complex model behaviors. To support this pursuit, we present a weighted variant of Angluin's (1987) $\mathbf{L^*}$ algorithm for learning FSAs. We stay faithful to the original algorithm, devising a way to exactly learn deterministic weighted FSAs whose weights support division. Furthermore, we formulate the learning process in a manner that highlights the connection with FSA minimization, showing how $\mathbf{L^*}$ directly learns a minimal automaton for the target language.

Franz Nowak, Anej Svete, Alexandra Butoi et al.

The performance of modern language models (LMs) has been improved by chain-of-thought (CoT) reasoning, i.e., the process of generating intermediate results that guide the model towards a final answer. A possible explanation for this improvement is that CoT reasoning extends an LM's computational power, as RNNs and transformers with additional scratch space are known to be Turing complete. Comparing LMs to Turing machines, however, introduces a category error - Turing machines decide language membership, whereas LMs define distributions over strings. To bridge this gap, we formalize CoT reasoning in a probabilistic setting. We present several results on the representational capacity of recurrent and transformer LMs with CoT reasoning, showing that they can represent the same family of distributions over strings as probabilistic Turing machines.

Naaman Tan, Josef Valvoda, Tianyu Liu et al.

The relationship between the quality of a string, as judged by a human reader, and its probability, $p(\boldsymbol{y})$ under a language model undergirds the development of better language models. For example, many popular algorithms for sampling from a language model have been conceived with the goal of manipulating $p(\boldsymbol{y})$ to place higher probability on strings that humans deem of high quality. In this article, we examine the probability--quality relationship in language models explicitly aligned to human preferences, e.g., through reinforcement learning through human feedback. We show that, when sampling corpora from an aligned language model, there exists a trade-off between the strings' average reward and average log-likelihood under the prior language model, i.e., the same model before alignment with human preferences. We provide a formal treatment of this phenomenon and demonstrate how a choice of sampling adaptor allows for a selection of how much likelihood we exchange for the reward.

Nadav Borenstein, Anej Svete, Robin Chan et al.

What can large language models learn? By definition, language models (LM) are distributions over strings. Therefore, an intuitive way of addressing the above question is to formalize it as a matter of learnability of classes of distributions over strings. While prior work in this direction focused on assessing the theoretical limits, in contrast, we seek to understand the empirical learnability. Unlike prior empirical work, we evaluate neural LMs on their home turf-learning probabilistic languages-rather than as classifiers of formal languages. In particular, we investigate the learnability of regular LMs (RLMs) by RNN and Transformer LMs. We empirically test the learnability of RLMs as a function of various complexity parameters of the RLM and the hidden state size of the neural LM. We find that the RLM rank, which corresponds to the size of linear space spanned by the logits of its conditional distributions, and the expected length of sampled strings are strong and significant predictors of learnability for both RNNs and Transformers. Several other predictors also reach significance, but with differing patterns between RNNs and Transformers.

Robin SM Chan, Reda Boumasmoud, Anej Svete et al.

Pre-trained language encoders -- functions that represent text as vectors -- are an integral component of many NLP tasks. We tackle a natural question in language encoder analysis: What does it mean for two encoders to be similar? We contend that a faithful measure of similarity needs to be \emph{intrinsic}, that is, task-independent, yet still be informative of \emph{extrinsic} similarity -- the performance on downstream tasks. It is common to consider two encoders similar if they are \emph{homotopic}, i.e., if they can be aligned through some transformation. In this spirit, we study the properties of \emph{affine} alignment of language encoders and its implications on extrinsic similarity. We find that while affine alignment is fundamentally an asymmetric notion of similarity, it is still informative of extrinsic similarity. We confirm this on datasets of natural language representations. Beyond providing useful bounds on extrinsic similarity, affine intrinsic similarity also allows us to begin uncovering the structure of the space of pre-trained encoders by defining an order over them.