Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

TL;DR

This paper investigates cancellation-free linear circuits, showing they can compute any matrix with near-optimal size, and establishes superlinear lower bounds for specific matrices, advancing understanding of circuit complexity.

Contribution

It introduces cancellation-free circuits as a structured subclass, proves superlinear lower bounds for certain matrices, and connects lower bound techniques to monotone circuit proofs.

Findings

Cancellation-free circuits can compute any matrix with size close to optimal.

Superlinear lower bounds are established for the Sierpinski gasket matrix.

Lower bounds for cancellation-free circuits relate to monotone circuit techniques.

Abstract

We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellation- free circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the $n\times n$ Sierpinski gasket matrix must use at least 1/2 n logn gates, and that this is tight. This supports a conjecture by Aaronson. Furthermore we show that a proof strategy for proving lower bounds on monotone circuits can be almost directly converted to prove lower bounds on cancellation-free linear circuits. We use this together with a result from extremal graph theory due to Andreev to prove a lower bound of {\Omega}(n^(2- \epsilon)) for infinitely many $n \times n$ matrices for every $\epsilon > 0$ for. These lower bounds for concrete matrices are almost optimal since all matrices can be computed with $O(n^2/\log n)$ gates.