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.

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.

We consider transformer encoders with hard attention (in which all attention is focused on exactly one position) and strict future masking (in which each position only attends to positions strictly to its left), and prove that the class of languages recognized by these networks is exactly the star-free languages. Adding position embeddings increases the class of recognized languages to other well-studied classes. A key technique in these proofs is Boolean RASP, a variant of RASP that is restricted to Boolean values. Via the star-free languages, we relate transformers to first-order logic, temporal logic, and algebraic automata theory.

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.