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.

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 (total functional) 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 transductions and show that it computes exactly the first-order rational transductions (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 transductions. S-RASP adds prefix sum, which enables additional arithmetic operations (such as squaring a string) and contains all first-order polyregular transductions. Finally, we show that masked average-hard attention transformers can simulate S-RASP.

This paper analyzes three formal models of Transformer encoders that differ in the form of their self-attention mechanism: unique hard attention (UHAT); generalized unique hard attention (GUHAT), which generalizes UHAT; and averaging hard attention (AHAT). We show that UHAT and GUHAT Transformers, viewed as string acceptors, can only recognize formal languages in the complexity class AC 0 , the class of languages recognizable by families of Boolean circuits of constant depth and polynomial size. This upper bound subsumes Hahn’s ( 2020 ) results that GUHAT cannot recognize the DYCK languages or the PARITY language, since those languages are outside AC 0 (Furst et al., 1984 ). In contrast, the non-AC 0 languages MAJORITY and DYCK -1 are recognizable by AHAT networks, implying that AHAT can recognize languages that UHAT and GUHAT cannot.