Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

Christopher M. Homan; Sven Kosub

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arXiv · 0812.0283

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

Christopher M. Homan·Sven Kosub

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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.

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cs.CC cond-mat.dis-nn cs.DM nlin.AO nlin.CG

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BibTeX RIS

Christopher M. Homan, Sven Kosub. 2008-12-01. Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems. https://arxiv.org/abs/0812.0283

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