Transformers in Uniform TC$^0$
<|reference_start|>Transformers in Uniform TC$^0$: Previous work has shown that the languages recognized by average-hard attention transformers (AHATs) and softmax-attention transformers (SMATs) are within the circuit complexity class TC$^0$. However, these results assume limited-precision arithmetic: using floating-point numbers with O(log n) bits (where n is the length of the input string), Strobl showed that AHATs can be approximated in L-uniform TC$^0$, and Merrill and Sabharwal showed that SMATs can be approximated in DLOGTIME-uniform TC$^0$. Here, we improve these results, showing that AHATs with no approximation, SMATs with O(poly(n)) bits of floating-point precision, and SMATs with at most $2^{-O(poly(n))}$ absolute error are all in DLOGTIME-uniform TC$^0$.<|reference_end|>
@article{chiang2024transformers,
    title={Transformers in Uniform TC$^0$},
    author={David Chiang},
    journal={arXiv preprint arXiv:2409.13629},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.13629},
    primaryClass={cs.CC cs.FL cs.LG}
}