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）的表达性可通过合适的一阶逻辑片段完全刻画。具体而言，在标记图上解释的分级模态逻辑双变量片段（GC2）中的任意查询，均可由规模仅取决于查询深度的一个GNN表达。正如[Barcelo等人，2020；Grohe，2021]所述，这一描述适用于一类激活函数，暗示了根据所选激活函数，可由GNN表达的逻辑存在层次结构的可能性。本文通过证明GC2查询无法被使用多项式激活函数的GNN表达，表明这种层次结构确实存在。这揭示了多项式激活函数与广泛使用的非多项式激活函数（如修正线性单元）之间的分离性，并回答了[Grohe，2021]提出的一个开放性问题。