A Little Depth Goes a Long Way: The Expressive Power of Log-Depth Transformers

Recent theoretical results show transformers cannot express sequential reasoning problems over long input lengths, intuitively because their computational depth is bounded. However, prior work treats the depth as a constant, leaving it unclear to what degree bounded depth may suffice for solving problems over short inputs, or how increasing the transformer's depth affects its expressive power. We address these questions by analyzing the expressive power of transformers whose depth can grow minimally with context length $n$. We show even highly uniform transformers with depth $\Theta(\log n)$ can express two important problems: recognizing regular languages, which captures state tracking abilities, and graph connectivity, which underlies multi-step reasoning. Notably, both of these problems cannot be expressed by fixed-depth transformers under standard complexity conjectures, demonstrating the expressivity benefit of growing depth. Moreover, our theory quantitatively predicts how depth must grow with input length to express these problems, showing that depth scaling is more efficient than scaling width or chain-of-thought steps. Empirically, we find our theoretical depth requirements for regular language recognition match the practical depth requirements of transformers remarkably well. Thus, our results clarify precisely how depth affects transformers' reasoning capabilities, providing potential practical insights for designing models that are better at sequential reasoning.

ZebraLogic: On the Scaling Limits of LLMs for Logical Reasoning

We investigate the logical reasoning capabilities of large language models (LLMs) and their scalability in complex non-monotonic reasoning. To this end, we introduce ZebraLogic, a comprehensive evaluation framework for assessing LLM reasoning performance on logic grid puzzles derived from constraint satisfaction problems (CSPs). ZebraLogic enables the generation of puzzles with controllable and quantifiable complexity, facilitating a systematic study of the scaling limits of models such as Llama, o1 models, and DeepSeek-R1. By encompassing a broad range of search space complexities and diverse logical constraints, ZebraLogic provides a structured environment to evaluate reasoning under increasing difficulty. Our results reveal a significant decline in accuracy as problem complexity grows -- a phenomenon we term the curse of complexity. This limitation persists even with larger models and increased inference-time computation, suggesting inherent constraints in current LLM reasoning capabilities. Additionally, we explore strategies to enhance logical reasoning, including Best-of-N sampling, backtracking mechanisms, and self-verification prompts. Our findings offer critical insights into the scalability of LLM reasoning, highlight fundamental limitations, and outline potential directions for improvement.

Understanding the Logic of Direct Preference Alignment through Logic

Recent direct preference alignment algorithms (DPA), such as DPO, have shown great promise in aligning large language models to human preferences. While this has motivated the development of many new variants of the original DPO loss, understanding the differences between these recent proposals, as well as developing new DPA loss functions, remains difficult given the lack of a technical and conceptual framework for reasoning about the underlying semantics of these algorithms. In this paper, we attempt to remedy this by formalizing DPA losses in terms of discrete reasoning problems. Specifically, we ask: Given an existing DPA loss, can we systematically derive a symbolic expression that characterizes its semantics? How do the semantics of two losses relate to each other? We propose a novel formalism for characterizing preference losses for single model and reference model based approaches, and identify symbolic forms for a number of commonly used DPA variants. Further, we show how this formal view of preference learning sheds new light on both the size and structure of the DPA loss landscape, making it possible to not only rigorously characterize the relationships between recent loss proposals but also to systematically explore the landscape and derive new loss functions from first principles. We hope our framework and findings will help provide useful guidance to those working on human AI alignment.

SUPER: Evaluating Agents on Setting Up and Executing Tasks from Research Repositories

Given that Large Language Models (LLMs) have made significant progress in writing code, can they now be used to autonomously reproduce results from research repositories? Such a capability would be a boon to the research community, helping researchers validate, understand, and extend prior work. To advance towards this goal, we introduce SUPER, the first benchmark designed to evaluate the capability of LLMs in setting up and executing tasks from research repositories. SUPERaims to capture the realistic challenges faced by researchers working with Machine Learning (ML) and Natural Language Processing (NLP) research repositories. Our benchmark comprises three distinct problem sets: 45 end-to-end problems with annotated expert solutions, 152 sub problems derived from the expert set that focus on specific challenges (e.g., configuring a trainer), and 602 automatically generated problems for larger-scale development. We introduce various evaluation measures to assess both task success and progress, utilizing gold solutions when available or approximations otherwise. We show that state-of-the-art approaches struggle to solve these problems with the best model (GPT-4o) solving only 16.3% of the end-to-end set, and 46.1% of the scenarios. This illustrates the challenge of this task, and suggests that SUPER can serve as a valuable resource for the community to make and measure progress.

AppWorld: A Controllable World of Apps and People for Benchmarking Interactive Coding Agents

Autonomous agents that address day-to-day digital tasks (e.g., ordering groceries for a household), must not only operate multiple apps (e.g., notes, messaging, shopping app) via APIs, but also generate rich code with complex control flow in an iterative manner based on their interaction with the environment. However, existing benchmarks for tool use are inadequate, as they only cover tasks that require a simple sequence of API calls. To remedy this gap, we built $\textbf{AppWorld Engine}$, a high-quality execution environment (60K lines of code) of 9 day-to-day apps operable via 457 APIs and populated with realistic digital activities simulating the lives of ~100 fictitious users. We then created $\textbf{AppWorld Benchmark}$ (40K lines of code), a suite of 750 natural, diverse, and challenging autonomous agent tasks requiring rich and interactive code generation. It supports robust programmatic evaluation with state-based unit tests, allowing for different ways of completing a task while also checking for unexpected changes, i.e., collateral damage. The state-of-the-art LLM, GPT-4o, solves only ~49% of our 'normal' tasks and ~30% of 'challenge' tasks, while other models solve at least 16% fewer. This highlights the benchmark's difficulty and AppWorld's potential to push the frontiers of interactive coding agents. The project website is available at https://appworld.dev/.

Answer, Assemble, Ace: Understanding How Transformers Answer Multiple Choice Questions

Multiple-choice question answering (MCQA) is a key competence of performant transformer language models that is tested by mainstream benchmarks. However, recent evidence shows that models can have quite a range of performance, particularly when the task format is diversified slightly (such as by shuffling answer choice order). In this work we ask: how do successful models perform formatted MCQA? We employ vocabulary projection and activation patching methods to localize key hidden states that encode relevant information for predicting the correct answer. We find that prediction of a specific answer symbol is causally attributed to a single middle layer, and specifically its multi-head self-attention mechanism. We show that subsequent layers increase the probability of the predicted answer symbol in vocabulary space, and that this probability increase is associated with a sparse set of attention heads with unique roles. We additionally uncover differences in how different models adjust to alternative symbols. Finally, we demonstrate that a synthetic task can disentangle sources of model error to pinpoint when a model has learned formatted MCQA, and show that an inability to separate answer symbol tokens in vocabulary space is a property of models unable to perform formatted MCQA tasks.

DiscoveryBench: Towards Data-Driven Discovery with Large Language Models

Can the rapid advances in code generation, function calling, and data analysis using large language models (LLMs) help automate the search and verification of hypotheses purely from a set of provided datasets? To evaluate this question, we present DiscoveryBench, the first comprehensive benchmark that formalizes the multi-step process of data-driven discovery. The benchmark is designed to systematically assess current model capabilities in discovery tasks and provide a useful resource for improving them. Our benchmark contains 264 tasks collected across 6 diverse domains, such as sociology and engineering, by manually deriving discovery workflows from published papers to approximate the real-world challenges faced by researchers, where each task is defined by a dataset, its metadata, and a discovery goal in natural language. We additionally provide 903 synthetic tasks to conduct controlled evaluations across task complexity. Furthermore, our structured formalism of data-driven discovery enables a facet-based evaluation that provides useful insights into different failure modes. We evaluate several popular LLM-based reasoning frameworks using both open and closed LLMs as baselines on DiscoveryBench and find that even the best system scores only 25%. Our benchmark, thus, illustrates the challenges in autonomous data-driven discovery and serves as a valuable resource for the community to make progress.

The Illusion of State in State-Space Models

State-space models (SSMs) have emerged as a potential alternative architecture for building large language models (LLMs) compared to the previously ubiquitous transformer architecture. One theoretical weakness of transformers is that they cannot express certain kinds of sequential computation and state tracking (Merrill and Sabharwal, 2023), which SSMs are explicitly designed to address via their close architectural similarity to recurrent neural networks (RNNs). But do SSMs truly have an advantage (over transformers) in expressive power for state tracking? Surprisingly, the answer is no. Our analysis reveals that the expressive power of SSMs is limited very similarly to transformers: SSMs cannot express computation outside the complexity class $\mathsf{TC}^0$. In particular, this means they cannot solve simple state-tracking problems like permutation composition. It follows that SSMs are provably unable to accurately track chess moves with certain notation, evaluate code, or track entities in a long narrative. To supplement our formal analysis, we report experiments showing that Mamba-style SSMs indeed struggle with state tracking. Thus, despite its recurrent formulation, the "state" in an SSM is an illusion: SSMs have similar expressiveness limitations to non-recurrent models like transformers, which may fundamentally limit their ability to solve real-world state-tracking problems.

Transformers as Transducers

We study the sequence-to-sequence mapping capacity of transformers by relating them to finite transducers, and find that they can express surprisingly large classes of transductions. We do so using variants of RASP, a programming language designed to help people "think like transformers," as an intermediate representation. We extend the existing Boolean variant B-RASP to sequence-to-sequence functions and show that it computes exactly the first-order rational functions (such as string rotation). Then, we introduce two new extensions. B-RASP[pos] enables calculations on positions (such as copying the first half of a string) and contains all first-order regular functions. S-RASP adds prefix sum, which enables additional arithmetic operations (such as squaring a string) and contains all first-order polyregular functions. Finally, we show that masked average-hard attention transformers can simulate S-RASP. A corollary of our results is a new proof that transformer decoders are Turing-complete.

Data-driven Discovery with Large Generative Models

With the accumulation of data at an unprecedented rate, its potential to fuel scientific discovery is growing exponentially. This position paper urges the Machine Learning (ML) community to exploit the capabilities of large generative models (LGMs) to develop automated systems for end-to-end data-driven discovery -- a paradigm encompassing the search and verification of hypotheses purely from a set of provided datasets, without the need for additional data collection or physical experiments. We first outline several desiderata for an ideal data-driven discovery system. Then, through DATAVOYAGER, a proof-of-concept utilizing GPT-4, we demonstrate how LGMs fulfill several of these desiderata -- a feat previously unattainable -- while also highlighting important limitations in the current system that open up opportunities for novel ML research. We contend that achieving accurate, reliable, and robust end-to-end discovery systems solely through the current capabilities of LGMs is challenging. We instead advocate for fail-proof tool integration, along with active user moderation through feedback mechanisms, to foster data-driven scientific discoveries with efficiency and reproducibility.