Transformers in Uniform TC$^0$

David Chiang

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arXiv · 2409.13629

Transformers in Uniform TC$^0$

David Chiang

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Abstract

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$^0$.

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David Chiang. 2024-09-20. Transformers in Uniform TC$^0$. https://arxiv.org/abs/2409.13629

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