Cellular Automata and Bootstrap Percolation

TL;DR

This paper investigates the behavior of two-dimensional freezing cellular automata, especially focusing on the constraints of monotonicity and how it affects the automaton's ability to fill space, revealing undecidability and non-monotone phenomena.

Contribution

It proves that space-filling properties are undecidable for monotone automata and demonstrates the existence of automata with non-monotone space-filling behavior depending on initial density.

Findings

Undecidability of space-filling in monotone automata

Existence of automata with non-monotone density-dependent space-filling

Monotonicity constraints significantly limit automata dynamics

Abstract

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics by proving two results that do not hold in the subclass of monotone automata. First, it is undecidable whether the automaton almost surely fills the space when initialized on a Bernoulli random configuration with density $p$, for some/all $0 < p < 1$. Second, there exists an automaton whose space-filling property depends on $p$ in a non-monotone way.