一句话：用描述复杂性与电路复杂性精确刻画图神经网络的表达能力，证明其对应守卫片段 GFO+C 与 TC0。
本文用布尔电路复杂性与描述复杂性精确刻画图神经网络（GNN）的表达能力。作者证明，多项式规模、有界深度的 GNN 族所能计算的图查询恰好是带计数和内置关系的守卫一阶逻辑片段 GFO+C 可定义的查询，从而将 GNN 置于电路类 TC0；该结果允许任意实数权重和包括 ReLU、sigmoid、tanh 在内的广泛激活函数。若允许随机初始化和全局读出，GNN 族恰好计算与带阈值门的有界深度布尔电路相同的查询，即 TC0 中的查询。此外，单个具有分段线性激活和有理权重的 GNN 所计算查询可在无内置关系的 GFO+C 中定义，因此属于一致 TC0。
We analyse the power of graph neural networks (GNNs) in terms of Boolean circuit complexity and descriptive complexity.We prove that the graph queries that can be computed by a polynomial-size bounded-depth family of GNNs are exactly those definable in the guarded fragment GFO+C of first-order logic with counting and with built-in relations. This puts GNNs in the circuit complexity class TC0. Remarkably, the GNN families may use arbitrary real weights and a wide class of activation functions that includes the standard ReLU, logistic "sigmoid", and hyperbolic tangent functions. If the GNNs are allowed to use random initialisation and global readout (both standard features of GNNs widely used in practice), they can compute exactly the same queries as bounded depth Boolean circuits with threshold gates, that is, exactly the queries in TC0.Moreover, we show that queries computable by a single GNN with piecewise linear activations and rational weights are definable in GFO+C without built-in relations. Therefore, they are contained in uniform TC0.