The expressive power of voting polynomials
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Reduced order LQG controllers for linear time varying plants
Systems & Control Letters
Threshold circuits of bounded depth
Journal of Computer and System Sciences
On Optimal Depth Threshold Circuits for Multiplication andRelated Problems
SIAM Journal on Discrete Mathematics
Simulating Threshold Circuits by Majority Circuits
SIAM Journal on Computing
On Small Depth Threshold Circuits
SWAT '92 Proceedings of the Third Scandinavian Workshop on Algorithm Theory
On Small Depth Threshold Circuits
SWAT '92 Proceedings of the Third Scandinavian Workshop on Algorithm Theory
Depth efficient neural networks for division and related problems
IEEE Transactions on Information Theory
SIAM Journal on Computing
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We study the circuit complexity of the powering function, defined as POW"m(Z)=Z^m for an n-bit integer input Z and an integer exponent m==2. Specifically, we prove a 2^@W^(^n^/^l^o^g^n^) lower bound on the size of any depth-2 majority circuit that computes POW"m. This work generalizes Wegener's earlier result that the squaring function (i.e., POW"m for the special case m=2) is not in LT@?"2. Our depth lower bound is optimal due to Siu and Roychowdhury's matching upper bound: POW"m@?LT@?"3. The second part of this research note presents several counterintuitive findings about the membership of arithmetic functions in the circuit classes LT@?"1 and LT@?"2. For example, we construct a function f(Z) such that f@?LT@?"1 but 5f@?LT@?"1. We obtain similar findings for LT@?"2. This apparent brittleness of LT@?"1 and LT@?"2 highlights a difficulty in proving lower bounds for arithmetic functions.