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

TL;DR

This paper analyzes a probabilistic cellular automaton that interpolates between certain Wolfram rules, deriving analytical expressions for its long-term behavior and contrasting it with deterministic cases, with applications to opinion dynamics.

Contribution

It provides an analytical framework for understanding probabilistic cellular automata that interpolate between specific Wolfram rules, including explicit probability formulas and behavior analysis.

Findings

Asymptotic state probabilities are independent of initial conditions for 0<p,r<1.

Deterministic rules exhibit additional periodic asymptotic states.

Analytical solutions for small size cases using Markov processes.

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 Finally, we discuss applying this kind of PCA to describe opinion dynamics involving hesitant agents.