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的沙堆自动机（Phys. Rev. Lett., 1987）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定下预测其演化。尽管经过数十年的研究，二维沙堆自动机该问题的分类仍未解决。Goles等人最近提出的真菌自动机（Phys. Lett. A, 2020）是该模型的衍生版本，其扩散过程根据所谓的更新方案沿水平$(H)$或垂直$(V)$方向进行。Goles等人证明了采用$H^4V^4$更新方案的该模型预测问题是$\textbf{P}$-完全的。该结果随后被Modanese和Worsch改进（Algorithmica, 2024），他们证明即使采用更简单的$HV$更新方案，该问题同样是$\textbf{P}$-完全的。在本研究中，我们填补了现有空白，证明对于任何至少包含一次$H$和一次$V$的更新方案，其预测问题均为$\textbf{P}$-完全的。