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 possibibility 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, 21].

翻译：图神经网络（GNN）的表达性可以通过一阶逻辑的适当片段完全刻画。具体而言，任何在带标记图上解释的分级模态逻辑的两变量片段（GC2）中的查询，都可以用规模仅取决于查询深度的GNN表达。正如[Barcelo等，2020；Grohe，2021]所指出的，这一描述适用于一类激活函数，从而为根据所选激活函数可被GNN表达的逻辑层级留下了可能性。在本文中，我们通过证明GC2查询无法被带多项式激活函数的GNN表达，展示了这种层级确实存在。这意味着多项式激活与流行的非多项式激活（如修正线性单元）之间存在分离，并回答了[Grohe，21]提出的一个开放性问题。