Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

Chain-of-thought reasoning and scratchpads have emerged as critical tools for enhancing the computational capabilities of transformers. While theoretical results show that polynomial-length scratchpads can extend transformers' expressivity from $TC^0$ to $PTIME$, their required length remains poorly understood. Empirical evidence even suggests that transformers need scratchpads even for many problems in $TC^0$, such as Parity or Multiplication, challenging optimistic bounds derived from circuit complexity. In this work, we initiate the study of systematic lower bounds for the number of CoT steps across different algorithmic problems, in the hard-attention regime. We study a variety of algorithmic problems, and provide bounds that are tight up to logarithmic factors. Overall, these results contribute to emerging understanding of the power and limitations of chain-of-thought reasoning.

Emergent Stack Representations in Modeling Counter Languages Using Transformers

Transformer architectures are the backbone of most modern language models, but understanding the inner workings of these models still largely remains an open problem. One way that research in the past has tackled this problem is by isolating the learning capabilities of these architectures by training them over well-understood classes of formal languages. We extend this literature by analyzing models trained over counter languages, which can be modeled using counter variables. We train transformer models on 4 counter languages, and equivalently formulate these languages using stacks, whose depths can be understood as the counter values. We then probe their internal representations for stack depths at each input token to show that these models when trained as next token predictors learn stack-like representations. This brings us closer to understanding the algorithmic details of how transformers learn languages and helps in circuit discovery.

A Formal Framework for Understanding Length Generalization in Transformers

A major challenge for transformers is generalizing to sequences longer than those observed during training. While previous works have empirically shown that transformers can either succeed or fail at length generalization depending on the task, theoretical understanding of this phenomenon remains limited. In this work, we introduce a rigorous theoretical framework to analyze length generalization in causal transformers with learnable absolute positional encodings. In particular, we characterize those functions that are identifiable in the limit from sufficiently long inputs with absolute positional encodings under an idealized inference scheme using a norm-based regularizer. This enables us to prove the possibility of length generalization for a rich family of problems. We experimentally validate the theory as a predictor of success and failure of length generalization across a range of algorithmic and formal language tasks. Our theory not only explains a broad set of empirical observations but also opens the way to provably predicting length generalization capabilities in transformers.

Separations in the Representational Capabilities of Transformers and Recurrent Architectures

Transformer architectures have been widely adopted in foundation models. Due to their high inference costs, there is renewed interest in exploring the potential of efficient recurrent architectures (RNNs). In this paper, we analyze the differences in the representational capabilities of Transformers and RNNs across several tasks of practical relevance, including index lookup, nearest neighbor, recognizing bounded Dyck languages, and string equality. For the tasks considered, our results show separations based on the size of the model required for different architectures. For example, we show that a one-layer Transformer of logarithmic width can perform index lookup, whereas an RNN requires a hidden state of linear size. Conversely, while constant-size RNNs can recognize bounded Dyck languages, we show that one-layer Transformers require a linear size for this task. Furthermore, we show that two-layer Transformers of logarithmic size can perform decision tasks such as string equality or disjointness, whereas both one-layer Transformers and recurrent models require linear size for these tasks. We also show that a log-size two-layer Transformer can implement the nearest neighbor algorithm in its forward pass; on the other hand recurrent models require linear size. Our constructions are based on the existence of $N$ nearly orthogonal vectors in $O(\log N)$ dimensional space and our lower bounds are based on reductions from communication complexity problems. We supplement our theoretical results with experiments that highlight the differences in the performance of these architectures on practical-size sequences.

The Expressive Capacity of State Space Models: A Formal Language Perspective

Recently, recurrent models based on linear state space models (SSMs) have shown promising performance in language modeling (LM), competititve with transformers. However, there is little understanding of the in-principle abilities of such models, which could provide useful guidance to the search for better LM architectures. We present a comprehensive theoretical study of the capacity of such SSMs as it compares to that of transformers and traditional RNNs. We find that SSMs and transformers have overlapping but distinct strengths. In star-free state tracking, SSMs implement straightforward and exact solutions to problems that transformers struggle to represent exactly. They can also model bounded hierarchical structure with optimal memory even without simulating a stack. On the other hand, we identify a design choice in current SSMs that limits their expressive power. We discuss implications for SSM and LM research, and verify results empirically on a recent SSM, Mamba.

InversionView: A General-Purpose Method for Reading Information from Neural Activations

The inner workings of neural networks can be better understood if we can fully decipher the information encoded in neural activations. In this paper, we argue that this information is embodied by the subset of inputs that give rise to similar activations. Computing such subsets is nontrivial as the input space is exponentially large. We propose InversionView, which allows us to practically inspect this subset by sampling from a trained decoder model conditioned on activations. This helps uncover the information content of activation vectors, and facilitates understanding of the algorithms implemented by transformer models. We present three case studies where we investigate models ranging from small transformers to GPT-2. In these studies, we demonstrate the characteristics of our method, show the distinctive advantages it offers, and provide causally verified circuits.

Linguistic Structure from a Bottleneck on Sequential Information Processing

Human language is a unique form of communication in the natural world, distinguished by its structured nature. Most fundamentally, it is systematic, meaning that signals can be broken down into component parts that are individually meaningful -- roughly, words -- which are combined in a regular way to form sentences. Furthermore, the way in which these parts are combined maintains a kind of locality: words are usually concatenated together, and they form contiguous phrases, keeping related parts of sentences close to each other. We address the challenge of understanding how these basic properties of language arise from broader principles of efficient communication under information processing constraints. Here we show that natural-language-like systematicity arises from minimization of excess entropy, a measure of statistical complexity that represents the minimum amount of information necessary for predicting the future of a sequence based on its past. In simulations, we show that codes that minimize excess entropy factorize their source distributions into approximately independent components, and then express those components systematically and locally. Next, in a series of massively cross-linguistic corpus studies, we show that human languages are structured to have low excess entropy at the level of phonology, morphology, syntax, and semantics. Our result suggests that human language performs a sequential generalization of Independent Components Analysis on the statistical distribution over meanings that need to be expressed. It establishes a link between the statistical and algebraic structure of human language, and reinforces the idea that the structure of human language may have evolved to minimize cognitive load while maximizing communicative expressiveness.

Why are Sensitive Functions Hard for Transformers?

Empirical studies have identified a range of learnability biases and limitations of transformers, such as a persistent difficulty in learning to compute simple formal languages such as PARITY, and a bias towards low-degree functions. However, theoretical understanding remains limited, with existing expressiveness theory either overpredicting or underpredicting realistic learning abilities. We prove that, under the transformer architecture, the loss landscape is constrained by the input-space sensitivity: Transformers whose output is sensitive to many parts of the input string inhabit isolated points in parameter space, leading to a low-sensitivity bias in generalization. We show theoretically and empirically that this theory unifies a broad array of empirical observations about the learning abilities and biases of transformers, such as their generalization bias towards low sensitivity and low degree, and difficulty in length generalization for PARITY. This shows that understanding transformers' inductive biases requires studying not just their in-principle expressivity, but also their loss landscape.

A Cross-Linguistic Pressure for Uniform Information Density in Word Order

While natural languages differ widely in both canonical word order and word order flexibility, their word orders still follow shared cross-linguistic statistical patterns, often attributed to functional pressures. In the effort to identify these pressures, prior work has compared real and counterfactual word orders. Yet one functional pressure has been overlooked in such investigations: the uniform information density (UID) hypothesis, which holds that information should be spread evenly throughout an utterance. Here, we ask whether a pressure for UID may have influenced word order patterns cross-linguistically. To this end, we use computational models to test whether real orders lead to greater information uniformity than counterfactual orders. In our empirical study of 10 typologically diverse languages, we find that: (i) among SVO languages, real word orders consistently have greater uniformity than reverse word orders, and (ii) only linguistically implausible counterfactual orders consistently exceed the uniformity of real orders. These findings are compatible with a pressure for information uniformity in the development and usage of natural languages.

A Theory of Emergent In-Context Learning as Implicit Structure Induction

Scaling large language models (LLMs) leads to an emergent capacity to learn in-context from example demonstrations. Despite progress, theoretical understanding of this phenomenon remains limited. We argue that in-context learning relies on recombination of compositional operations found in natural language data. We derive an information-theoretic bound showing how in-context learning abilities arise from generic next-token prediction when the pretraining distribution has sufficient amounts of compositional structure, under linguistically motivated assumptions. A second bound provides a theoretical justification for the empirical success of prompting LLMs to output intermediate steps towards an answer. To validate theoretical predictions, we introduce a controlled setup for inducing in-context learning; unlike previous approaches, it accounts for the compositional nature of language. Trained transformers can perform in-context learning for a range of tasks, in a manner consistent with the theoretical results. Mirroring real-world LLMs in a miniature setup, in-context learning emerges when scaling parameters and data, and models perform better when prompted to output intermediate steps. Probing shows that in-context learning is supported by a representation of the input's compositional structure. Taken together, these results provide a step towards theoretical understanding of emergent behavior in large language models.