Transformers in Uniform TC^0

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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