The $\mathsf{NC}$ versus $\mathsf{P}$-hard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second, we can highlight some necessary and sufficient elements to the dynamical complexity of sandpiles, with a surprisingly crucial role of cells with two grains.

中文翻译：

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冻结沙堆和布尔阈值网络：等价性和复杂性

的 $\mathsf{数控}$ 与 $\mathsf{P}$冯·诺伊曼邻域的二维网格上沙堆的预测问题的硬分类是一个著名的开放问题。在本文中，我们通过研究其冻结变体取得了两种进展。首先，它可以与阈值布尔函数（例如多数）网络上的其他众所周知的预测问题建立牢固的联系。其次，我们可以突出显示沙粒动力学复杂性的一些必要和充分的元素，其中具有两个颗粒的细胞起着至关重要的作用。