Computation of fixed points in MAX and MIN multi-state networks

Abstract

In this work, we study the fixed points of multi-state networks over a complement-closed set $X$, a wide generalization of Boolean networks and some types of multi-state networks. We particularly focus on MAX (and MIN) multi-state networks, whose corresponding graph is undirected and all self-loop, and where the global evolution operator is induced by a disjunction (or conjunction) of direct and complemented variables, independently of the updating scheme (synchronous, asynchronous or mixed). In this context, we characterize the exact configurations of fixed points and provide the best possible lower and upper bounds for their number analytically. As a corollary, we prove that a Fixed Point Theorem is not possible for non-binary MAX (and MIN) multi-state networks, i.e., when $\left|X\right|>2$.