The 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.

翻译：Bak、Tang和Wiesenfeld（《物理评论快报》，1987年）提出的沙堆自动机是描述粒子在空间中扩散的简单模型。与该模型复杂性相关的一个基本问题是在并行设置下预测其演化过程。尽管经过数十年的努力，针对二维沙堆自动机的该问题分类仍未得到解决。Goles等人（《物理快报A》，2020年）最近提出了真菌自动机作为该模型的衍生版本，其中扩散根据所谓的更新方案沿水平方向$(H)$或垂直方向$(V)$进行。Goles等人证明了采用更新方案$H^4V^4$的该模型预测问题是$\textbf{P}$-完全的。这一结果随后被Modanese和Worsch（《算法学》，2024年）改进，他们证明对于更简单的更新方案$HV$，该问题同样是$\textbf{P}$-完全的。在本工作中，我们填补了空白，证明对于任何至少包含一次$H$和一次$V$的更新方案，该预测问题都是$\textbf{P}$-完全的。