Simulating Hard Attention Using Soft Attention

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 variants of linear temporal logic, whose formulas have been previously been shown to be computable using hard attention transformers. 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 a large subclass of average-hard attention transformers, those that have what we call the uniform-tieless property.

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.

Transformers as Recognizers of Formal Languages: A Survey on Expressivity

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 questions such as this will help to compare transformers with other models, and transformer variants with one another, for various tasks. 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.

Average-Hard Attention Transformers are Constant-Depth Uniform Threshold Circuits

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.