Henryk Fuks - Probabilistic initial value problem for cellular automaton rule 172

dmtcs:2761 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AL, Automata 2010 - 16th Intl. Workshop on CA and DCS - https://doi.org/10.46298/dmtcs.2761
Probabilistic initial value problem for cellular automaton rule 172Article

Authors: Henryk Fuks 1

  • 1 Department of Mathematics and Statistics [St. Catharines]

We present a method of solving of the probabilistic initial value problem for cellular automata (CA) using CA rule 172 as an example. For a disordered initial condition on an infinite lattice, we derive exact expressions for the density of ones at arbitrary time step. In order to do this, we analyze topological structure of preimage trees of finite strings of length 3. Level sets of these trees can be enumerated directly using classical combinatorial methods, yielding expressions for the number of $n$-step preimages of all strings of length 3, and, subsequently, probabilities of occurrence of these strings in a configuration obtained from the initial one after $n$ iterations of rule 172. The density of ones can be expressed in terms of Fibonacci numbers, while expressions for probabilities of other strings involve Lucas numbers. Applicability of this method to other CA rules is briefly discussed.


Volume: DMTCS Proceedings vol. AL, Automata 2010 - 16th Intl. Workshop on CA and DCS
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: cellular automata,initial value problem,preimage trees,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS],[NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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