Description

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 #Pcomplete 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.

Date of creation, presentation, or exhibit

3-30-2015

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Comments

This is the post-print of an article published by Elsevier. Copyright 2015 Elsevier B.V. The final, published version is located here: https://doi.org/10.1016/j.tcs.2015.01.040

A preliminary version of this paper was presented at the 10th Italian Conference on Theoretical Computer Science, October 2007.

Document Type

Conference Proceeding

Department, Program, or Center

Computer Science (GCCIS)

Campus

RIT – Main Campus

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