{
  "id": "2110.00656",
  "version": "v2",
  "published": "2021-10-01T21:24:22.000Z",
  "updated": "2022-04-19T11:44:28.000Z",
  "title": "Cellular Automata and Bootstrap Percolation",
  "authors": [
    "Ville Salo",
    "Guillaume Theyssier",
    "Ilkka Törmä"
  ],
  "comment": "18 pages, 3 figures",
  "categories": [
    "math.PR",
    "cs.DM",
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-19T11:44:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B15",
      "68Q80",
      "60K35"
    ],
    "keywords": [
      "bootstrap percolation",
      "two-dimensional freezing cellular automata",
      "binary state set",
      "bernoulli random configuration",
      "non-monotone way"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}