Local negative circuits and fixed points in non-expansive Boolean networks

Abstract

AbstractGiven a Boolean function F:{0,1}n→{0,1}n, and a point x in {0,1}n, we represent the discrete Jacobian matrix of F at point x by a signed directed graph GF(x). We then focus on the following open problem: Is the absence of a negative circuit in GF(x) for every x in {0,1}n a sufficient condition for F to have at least one fixed point? As result, we give a positive answer to this question under the additional condition that F is non-expansive with respect to the Hamming distance