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.

本文分析了三种Transformer编码器的形式模型，比较它们的自注意力机制：unique hard attention（UHAT）、generalized unique hard attention（GUHAT）和averaging hard attention（AHAT）。我们发现，UHAT和GUHAT Transformer只能识别AC$^0$复杂性类中的正式语言，而AHAT网络可以识别UHAT和GUHAT不能识别的语言。

利用Hard Attention Transformers进行正式语言识别：来自电路复杂性的视角