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.

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.