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Characterizing neural networks in terms of better-understood formal systemshas the potential to yield new insights into the power and limitations of thesenetworks. Doing so for transformers remains an active area of research.Bhattamishra and others have shown that transformer encoders are at least asexpressive as a certain kind of counter machine, while Merrill and Sabharwalhave shown that fixed-precision transformer encoders recognize only languagesin uniform $TC^0$. We connect and strengthen these results by identifying avariant of first-order logic with counting quantifiers that is simultaneouslyan upper bound for fixed-precision transformer encoders and a lower bound fortransformer encoders. This brings us much closer than before to an exactcharacterization of the languages that transformer encoders recognize.

Tighter Bounds on the Expressivity of Transformer Encoders