Cancellation-free circuits in unbounded and bounded depth

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/log2⁡n)Ω(n/log2⁡n). 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))Ω(nlog⁡(n)) lower bound.