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 ReLUs, sigmoid and hyperbolic tan and others) and answers an open question formulated by [Grohe, 2021].

翻译：图神经网络（GNNs）的表达能力可以完全由一阶逻辑的适当片段来刻画。具体而言，任何在标记图上解释的分阶模态逻辑的两变量片段（GC2）中的查询，都可以通过一个大小仅取决于查询深度的GNN来表达。正如[Barcelo等，2020，Grohe，2021]所指出的，这一描述适用于一系列激活函数，这意味着根据所选激活函数的不同，GNN所能表达的逻辑层次可能存在差异。在本文中，我们通过证明带有多项式激活函数的GNN无法表达GC2查询，证实了这种层级结构确实存在。这一定理表明了多项式激活与流行的非多项式激活（如ReLU、Sigmoid、双曲正切等）之间的分离，并回答了[Grohe，2021]提出的一个开放性问题。