We study the numerical and Boolean expressiveness of MPLang, a declarative language that captures the computation of graph neural networks (GNNs) through linear message passing and activation functions. We begin with A-MPLang, the fragment without activation functions, and give a characterization of its expressive power in terms of walk-summed features. For bounded activation functions, we show that (under mild conditions) all eventually constant activations yield the same expressive power - numerical and Boolean - and that it subsumes previously established logics for GNNs with eventually constant activation functions but without linear layers. Finally, we prove the first expressive separation between unbounded and bounded activations in the presence of linear layers: MPLang with ReLU is strictly more powerful for numerical queries than MPLang with eventually constant activation functions, e.g., truncated ReLU. This hinges on subtle interactions between linear aggregation and eventually constant non-linearities, and it establishes that GNNs using ReLU are more expressive than those restricted to eventually constant activations and linear layers.

翻译：我们研究了MPLang的数值和布尔表达能力，这是一种通过线性消息传递和激活函数捕捉图神经网络计算的声明性语言。我们首先分析无激活函数的片段A-MPLang，并基于游走求和特征刻画了其表达能力。对于有界激活函数，我们证明（在温和条件下）所有最终常值的激活函数具有相同的数值和布尔表达能力，且该能力包含了先前为具有最终常值激活函数但无线性层的GNN所建立的逻辑。最后，我们首次证明了在线性层存在时，无界激活与有界激活之间的表达能力分离：对于数值查询，采用ReLU的MPLang严格强于采用最终常值激活函数（如截断ReLU）的MPLang。这一结果依赖于线性聚合与最终常值非线性之间的微妙交互，并确立了使用ReLU的GNN在表达能力上优于受限于最终常值激活和线性层的GNN。