On the decomposition of stochastic cellular automata

TL;DR

This paper explores properties of stochastic cellular automata, including methods to compute cell-wise probabilities and a decomposition into deterministic automata, aiding in understanding their complex dynamics.

Contribution

It introduces two properties: one for calculating cell-wise probability distributions and another for decomposing stochastic automata into deterministic components.

Findings

Cell-wise probability distributions can be computed for stochastic automata.

Stochastic automata are equivalent to mixtures of deterministic automata.

Decomposition helps analyze automaton behavior more effectively.

Abstract

In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular automaton can be decomposed into a set of deterministic cellular automata, each of which contributes to the behavior of the stochastic cellular automaton.