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Probabilistic Cellular Automata: between deterministic Wolfram's rules 23, 77, 178 and 232

Francisco J. Muñoz, Juan Carlos Nuño

Abstract

We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighborhood states, the central site takes either the majority state or the opposite and (ii) if the neighborhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters $p,r \in [0,1]$, and we analytically solve small size cases by using a Markov process formulation. We derive analytical expressions for the probability of asymptotically reaching each possible global configuration as a function of $p$ and $r$, for all initial states. We show that for $0 < p,r < 1$, the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions. This contrasts with the behavior of the deterministic Wolfram's rules 23 ($p=0,r=0$), 77 ($p=1,r=0$), 178 ($p=0,r=1$) and 232 ($p=1,r=1$), for which additional asymptotic states can occur, in particular periodic configurations.

Probabilistic Cellular Automata: between deterministic Wolfram's rules 23, 77, 178 and 232

Abstract

We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighborhood states, the central site takes either the majority state or the opposite and (ii) if the neighborhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters , and we analytically solve small size cases by using a Markov process formulation. We derive analytical expressions for the probability of asymptotically reaching each possible global configuration as a function of and , for all initial states. We show that for , the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions. This contrasts with the behavior of the deterministic Wolfram's rules 23 (), 77 (), 178 () and 232 (), for which additional asymptotic states can occur, in particular periodic configurations.

Paper Structure

This paper contains 17 sections, 41 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Deterministic rules
  3. Rule 23.
  4. Rule 77.
  5. Rule 178.
  6. Rule 232.
  7. The Probabilistic Cellular Automata
  8. Low-size Cellular Automata: an analytical approach using Markov chains
  9. $L=2$
  10. Edge $r=0$.
  11. Edge $r=1$.
  12. $L=3$
  13. Edge $r=0$.
  14. Edge $r=1$.
  15. Edge $p=0$.
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Probabilistic parameter space defined by $p,r \in [0,1]$. The four vertices correspond to the deterministic Wolfram's rules: 23 $(p=0,r=0)$, 77 $(p=1,r=0)$, 178 $(p=0,r=1)$ and 232 $(p=1,r=1)$. The horizontal edges describe rules with non-standard ($r=0$) and standard ($r=1$) contagion in case of majority. Vertical edges represent rules that, in case of tie, keep the state of the central cell ($p=0$) or change this state ($p=1$). In the interior of the square, $0<p,r<1$, the PCA exhibits stochastic dynamics that depends on the probabilistic parameters.
  • Figure 2: Stationary coordinates of the vector $v_\infty$ as functions of the parameters $p$ and $r$. Its entries can be described by the two expressions given in Eq. \ref{['eq23']}, where $v_1(\infty) = v_8(\infty)$ [gray] and $v_2(\infty) = v_3(\infty) = \dots = v_7(\infty)$ [black]. It is worth noting that both surfaces share a common line of tangency when $r = p$.
  • Figure 3: Examples of dynamics of the PCA in the case $r=0$, starting from $s_5=(1,0,0)$ initial configuration, with $L = 3$, white cells correspond to 0 and black cells to 1: a) $p=0$, b) $p=1$, c) $0<p<1$ (time evolves from top to bottom).