Separating OR, SUM, and XOR circuits

Given a boolean n×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 separating OR-circuits from the two other models in terms of circuit complexity:We show how to obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size Ω(n^3/2/log² n).We consider the task of 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.