Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

Abstract

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes $${\mathcal{F}}$$ and graph classes $${\mathcal{G}}$$ , an ( $${\mathcal{F}},{\mathcal{G}}$$ )-system is a Boolean dynamical system such that all local transition functions lie in $${\mathcal{F}}$$ and the underlying graph lies in $${\mathcal{G}}$$ . Let $${\mathcal{F}}$$ be a class of Boolean functions which is closed under composition and let $${\mathcal{G}}$$ be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the planar graphs, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the graphs having vertex covers of size one, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.