Dichotomy results for fixed point counting in boolean dynamical systems

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}{0,1}. For a class FF of boolean functions and a class GG of graphs, an (F,G)(F,G)-system is a boolean dynamical system with local transitions functions lying in FF and graphs in GG. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let FF be a class of boolean functions closed under superposition and let GG be a graph class closed under taking minors. If FF contains all min-functions, all max-functions, or all self-dual and monotone functions, and GG contains all planar graphs, then it is #P#P-complete to compute the number of fixed points in an (F,G)(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.