将任意布尔电路嵌入到具有任意更新序列的菌丝自动机中
Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences
AI总结：
本研究证明了在任意包含H和V的更新序列下，菌丝自动机的预测问题为P-完全。
摘要：
Bak、Tang和Wiesenfeld（Phys. Rev. Lett., 1987）的沙堆自动机是空间中粒子扩散的简单模型。与该模型复杂性相关的根本问题在于预测其在并行设置下的演变。尽管数十年的努力，二维沙堆自动机的此问题分类仍悬而未决。Goles等人（Phys. Lett. A, 2020）最近提出菌丝自动机作为该模型的衍生模型，其中扩散发生在水平（H）或垂直（V）方向，根据所谓的更新方案。Goles等人证明了对于更新方案H⁴V⁴的此模型预测问题是P-完全的。随后，Modanese和Worsch（Algorithmica, 2024）进一步改进了这一结果，证明对于更简单的更新方案HV，此问题同样为P-完全。在本工作中，我们填补了空白，并证明对于任何包含至少一次H和V的更新方案，预测问题都是P-完全的。
英文摘要：
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.
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