Logical Languages Accepted by Transformer Encoders with Hard Attention

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class ${\sf AC}^0$, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside ${\sf AC}^0$), but their expressive power still lies within the bigger circuit complexity class ${\sf TC}^0$, i.e., ${\sf AC}^0$-circuits extended by majority gates.We first show a negative result that there is an ${\sf AC}^0$-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of ${\sf AC}^0$-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from ${\sf AC}^0$. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. Using these results, we obtain a characterization of which counting properties are expressible by UHAT and AHAT, in relation to regular languages.