Cancellation-Free Circuits in Unbounded and Bounded Depth

TL;DR

This paper investigates the power and limitations of cancellation-free linear Boolean circuits, establishing significant lower bounds and analyzing their complexity for specific matrices like the Sierpinski matrix, with implications for circuit minimization.

Contribution

It proves a near-quadratic gap between cancellation-free and general linear circuits and provides tight bounds for computing the Sierpinski matrix with cancellation-free circuits.

Findings

The gap between cancellation-free and linear circuits can be as large as Ω(n/ log^2 n).

A tight Ω(n log n) lower bound is established for computing the Sierpinski matrix.

The results hold for circuits of constant depth, extending previous work.

Abstract

We study the notion of "cancellation-free" circuits. This is a restriction of linear Boolean circuits (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 linear circuit. We show that the difference can be a factor $\Omega(n/\log^{2}n)$. This improves on a recent result by Sergeev and Gashkov who have studied a similar problem. 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 $\Omega(n\log n)$ lower bound.