Lena Strobl

Lena Strobl, William Merrill, Gail Weiss et al.

As transformers have gained prominence in natural language processing, some researchers have investigated theoretically what problems they can and cannot solve, by treating problems as formal languages. Exploring such questions can help clarify the power of transformers relative to other models of computation, their fundamental capabilities and limits, and the impact of architectural choices. Work in this subarea has made considerable progress in recent years. Here, we undertake a comprehensive survey of this work, documenting the diverse assumptions that underlie different results and providing a unified framework for harmonizing seemingly contradictory findings.

Lena Strobl

Transformers have emerged as a widely used neural network model for various natural language processing tasks. Previous research explored their relationship with constant-depth threshold circuits, making two assumptions: average-hard attention and logarithmic precision for internal computations relative to input length. Merrill et al. (2022) prove that average-hard attention transformers recognize languages that fall within the complexity class TC0, denoting the set of languages that can be recognized by constant-depth polynomial-size threshold circuits. Likewise, Merrill and Sabharwal (2023) show that log-precision transformers recognize languages within the class of uniform TC0. This shows that both transformer models can be simulated by constant-depth threshold circuits, with the latter being more robust due to generating a uniform circuit family. Our paper shows that the first result can be extended to yield uniform circuits as well.

Andy Yang, Lena Strobl, David Chiang et al.

We study conditions under which transformers using soft attention can simulate hard attention, that is, effectively focus all attention on a subset of positions. First, we examine several subclasses of languages recognized by hard-attention transformers, which can be defined in variants of linear temporal logic. We demonstrate how soft-attention transformers can compute formulas of these logics using unbounded positional embeddings or temperature scaling. Second, we demonstrate how temperature scaling allows softmax transformers to simulate general hard-attention transformers, using a temperature that depends on the minimum gap between the maximum attention scores and other attention scores.

Lena Strobl, Dana Angluin, David Chiang et al.

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.

Lena Strobl, Dana Angluin, Robert Frank

While transformers have proven enormously successful in a range of tasks, their fundamental properties as models of computation are not well understood. This paper contributes to the study of the expressive capacity of transformers, focusing on their ability to perform the fundamental computational task of evaluating an arbitrary function from $[n]$ to $[n]$ at a given argument. We prove that concise 1-layer transformers (i.e., with a polylog bound on the product of the number of heads, the embedding dimension, and precision) are capable of doing this task under some representations of the input, but not when the function's inputs and values are only encoded in different input positions. Concise 2-layer transformers can perform the task even with the more difficult input representation. Experimentally, we find a rough alignment between what we have proven can be computed by concise transformers and what can be practically learned.