We characterise the computational power of recurrent graph neural networks (GNNs) in terms of arithmetic circuits over the real numbers. Our networks are not restricted to aggregate-combine GNNs or other particular types. Generalising similar notions from the literature, we introduce the model of recurrent arithmetic circuits, which can be seen as arithmetic analogues of sequential or logical circuits. These circuits utilise so-called memory gates which are used to store data between iterations of the recurrent circuit. While (recurrent) GNNs work on labelled graphs, we construct arithmetic circuits that obtain encoded labelled graphs as real valued tuples and then compute the same function. For the other direction we construct recurrent GNNs which are able to simulate the computations of recurrent circuits. These GNNs are given the circuit-input as initial feature vectors and then, after the GNN-computation, have the circuit-output among the feature vectors of its nodes. In this way we establish an exact correspondence between the expressivity of recurrent GNNs and recurrent arithmetic circuits operating over real numbers. Our results both deepen our understanding of the capabilities of trained neural networks and open new approaches to study recurrent neural networks using the lens of circuit complexity theory.

翻译：我们从实数域算术电路的角度刻画了递归图神经网络（GNN）的计算能力。所研究的网络并不局限于聚合-组合型GNN或其他特定类型。通过泛化文献中类似概念，我们引入了递归算术电路模型，该模型可视为序列或逻辑电路在算术领域的对应。这类电路利用所谓的记忆门，在递归电路的迭代运算间存储数据。尽管（递归）GNN作用于标记图，我们构建的算术电路可将编码后的标记图作为实数值元组获取，并计算相同函数。反之，我们构造了能够模拟递归电路计算的递归GNN——这些GNN将电路输入作为初始特征向量，在完成GNN计算后，其节点特征向量中即包含电路输出。通过这种双向对应，我们建立了递归GNN与基于实数运算的递归算术电路表达力之间的精确等价关系。该成果既深化了我们对训练后神经网络能力的认知，也为借助电路复杂度理论视角研究递归神经网络开辟了新途径。