The expressivity of Graph Neural Networks (GNNs) can be entirely characterized by appropriate fragments of the first-order logic. Namely, any query of the two variable fragment of graded modal logic (GC2) interpreted over labeled graphs can be expressed using a GNN whose size depends only on the depth of the query. As pointed out by [Barcelo & Al., 2020, Grohe, 2021], this description holds for a family of activation functions, leaving the possibility for a hierarchy of logics expressible by GNNs depending on the chosen activation function. In this article, we show that such hierarchy indeed exists by proving that GC2 queries cannot be expressed by GNNs with polynomial activation functions. This implies a separation between polynomial and popular non-polynomial activations (such as Rectified Linear Units) and answers an open question formulated by [Grohe, 2021].

翻译：图神经网络（GNN）的表达能力可通过一阶逻辑的适当片段完全刻画。具体而言，任何在带标签图上解释的分级模态逻辑双变量片段（GC2）的查询，均可通过规模仅取决于查询深度的GNN来表达。正如[Barcelo & Al., 2020, Grohe, 2021]所指出的，这一描述适用于一系列激活函数，从而提出了一种可能性：基于所选激活函数的不同，GNN可表达的逻辑存在层级结构。本文通过证明GC2查询无法由具有多项式激活函数的GNN表达，证实了这种层级结构的确存在。这一结果揭示了多项式激活函数与常见非多项式激活函数（如修正线性单元）之间的区分，并解答了[Grohe, 2021]提出的开放问题。