Abstract
We study the notion of “cancellation-free” circuits. This is a restriction of XOR circuits, but can be considered as being equivalent to previously studied models of computation. The notion was coined by Boyar and Peralta in a study of heuristics for a particular circuit minimization problem. They asked how large a gap there can be between the smallest cancellation-free circuit and the smallest XOR circuit. We present a new proof showing that the difference can be a factor Ω(n/log2n). Furthermore, our proof holds for circuits of constant depth. We also study the complexity of computing the Sierpinski matrix using cancellation-free circuits and give a tight Ω(nlog(n)) lower bound.
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