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We present dichotomy theorems regarding the computational complexity ofcounting fixed points in boolean (discrete) dynamical systems, i.e., finitediscrete dynamical systems over the domain {0,1}. For a class F of booleanfunctions and a class G of graphs, an (F,G)-system is a boolean dynamicalsystem with local transitions functions lying in F and graphs in G. We showthat, if local transition functions are given by lookup tables, then thefollowing complexity classification holds: Let F be a class of booleanfunctions closed under superposition and let G be a graph class closed undertaking minors. If F contains all min-functions, all max-functions, or allself-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; otherwiseit is computable in polynomial time. We also prove a dichotomy theorem for thecase that local transition functions are given by formulas (over logicalbases). This theorem has a significantly more complicated structure than thetheorem for lookup tables. A corresponding theorem for boolean circuitscoincides with the theorem for formulas.

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems