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  "path": "/abs/2602.19477v1",
  "publishedAt": "2026-02-24T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Eric Goles",
    "Augusto Modanese",
    "Martín Ríos-Wilson",
    "Domingo Ruiz-Tala",
    "Thomas Worsch"
  ],
  "textContent": "**Authors:** Eric Goles, Augusto Modanese, Martín Ríos-Wilson, Domingo Ruiz-Tala, Thomas Worsch\n\nThe sandpile automata of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett., 1987) are a simple model for the diffusion of particles in space. A fundamental problem related to the complexity of the model is predicting its evolution in the parallel setting. Despite decades of effort, a classification of this problem for two-dimensional sandpile automata remains outstanding. Fungal automata were recently proposed by Goles et al. (Phys. Lett. A, 2020) as a spin-off of the model in which diffusion occurs either in horizontal $(H)$ or vertical $(V)$ directions according to a so-called update scheme. Goles et al. proved that the prediction problem for this model with the update scheme $H^4V^4$ is $\\textbf{P}$-complete. This result was subsequently improved by Modanese and Worsch (Algorithmica, 2024), who showed the problem is $\\textbf{P}$-complete also for the simpler updatenscheme $HV$. In this work, we fill in the gaps and prove that the prediction problem is $\\textbf{P}$-complete for any update scheme that contains both $H$ and $V$ at least once.",
  "title": "Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences"
}