Universality in Freezing Cellular Automata

TL;DR

This paper explores the concept of universality in freezing cellular automata, analyzing conditions under which such automata can simulate all others, with results varying based on dimension, neighborhood, and rule monotonicity.

Contribution

It establishes the conditions for universality in freezing cellular automata, highlighting the impact of dimension, neighborhood type, and rule monotonicity on their simulation capabilities.

Findings

No universal FCA in 1D.

Universal FCA exists in 2D with certain conditions.

Monotonic rules hinder universality.

Abstract

Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.