Abstract
A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearly-exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic (linear-size lower bounds for general circuits [30], nearly cubic lower bounds for formula size [23], nearly nloglogn size lower bounds for branching programs [12], \documentclass[12pt]{minimal}
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\begin{document}$n^{1+c_d}$\end{document} for depth d threshold circuits [26]). Here, we present two instances where “pathetic” lower bounds of the form n1 + ε would suffice to derandomize interesting classes of probabilistic algorithms. We show:
If the word problem over S5 requires constant-depth threshold circuits of size n1 + ε for some ε> 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.)If no constant-depth arithmetic circuits of size n1 + ε can multiply a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC0 circuits of subexponential size).