Separating OR, SUM, and XOR Circuits

TL;DR

This paper investigates the complexity differences between OR, SUM, and XOR circuits for matrix transformations, providing new lower bounds and showing the difficulty of circuit rewriting under computational complexity assumptions.

Contribution

It introduces new separation results between different circuit models and establishes the computational hardness of rewriting circuits, advancing understanding of circuit complexity.

Findings

Matrices with small OR-circuits require large SUM-circuits.

Existence of matrices with small XOR-circuits but large OR-circuits.

Rewriting OR-circuits as XOR-circuits is computationally hard under ETH.

Abstract

Given a boolean n by n matrix A we consider arithmetic circuits for computing the transformation x->Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on \emph{separating} these models in terms of their circuit complexities. We give three results towards this goal: (1) We prove a direct sum type theorem on the monotone complexity of tensor product matrices. As a corollary, we obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size \Omega(n^{3/2}/\log^2n). (2) We construct so-called \emph{k-uniform} matrices that admit XOR-circuits of size O(n), but require OR-circuits of size \Omega(n^2/\log^2n). (3) We consider the task of \emph{rewriting} a given OR-circuit as a XOR-circuit and prove that any subquadratic-time algorithm for this task violates the strong exponential time hypothesis.