On relative OR-complexity of Boolean matrices and their complements

We construct explicit Boolean square matrices whose rectifier complexity (OR-complexity) differs significantly from the complexity of the complement matrices. This note can be viewed as an addition to the material of [2, §5.6]. Recall that rectifier (m,n)-circuit is an oriented graph with n vertices labeled as inputs and m vertices labeled as outputs. Rectifier circuit (ORcircuit) implements a Boolean m× n matrix A = (A[i, j]) iff for any i and j the value A[i, j] indicates the existence of an oriented path from j-th input to i-th output. Complexity of a circuit is the number of edges in it, circuit depth is the maximal length of an oriented path. See details in [2, 5]. We denote by OR(A) the complexity of an edge-minimal circuit implementing a given matrix A; if we speak about circuits of depth ≤ d, then the corresponding complexity is denoted by ORd(A). It was proved in [2] via method [3] the existence of n × n-matrices A satisfying OR(Ā)/OR(A) = Ω(n/ log n).