Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

TL;DR

This paper establishes a complexity classification for counting fixed points in boolean dynamical systems, showing when the problem is #P-complete or polynomial-time solvable based on function and graph classes.

Contribution

It provides the first dichotomy theorems for fixed point counting in boolean dynamical systems with functions given by lookup tables or formulas.

Findings

#P-complete for certain function and graph classes

Polynomial-time computable in other cases

Dichotomy theorems for lookup tables and formulas

Abstract

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.