The expressive limitations of message-passing Graph Neural Networks (GNNs) have motivated a wide range of more powerful graph learning architectures. We advocate Deep Homomorphism Networks (DHNs) as a model particularly well-suited for learning over relational databases, due to their close connection to important fragments of SQL such as conjunctive queries. We study the precise expressive power of DHNs by relating them to various natural fragments and extensions of first-order logic (FO). For DHNs with max, sum, and mean aggregations, we establish connections to the unary negation fragment (UNFO) and to the extensions of UNFO with counting quantifiers and with ratio quantifiers. We further relate sum-aggregation DHNs to the unary quantifier alternation fragment of FO and to an extension of FO with expressive counting. Through the classical correspondence between FO and SQL, these results also illuminate the relation between DHNs and SQL. They also enable us to study the decidability of two fundamental static analysis problems for DHNs, the emptiness problem and the subsumption problem. Finally, we confirm through experiments that the established differences in expressive power are reflected in the performance on suitable prediction tasks.

翻译：消息传递图神经网络（GNNs）的表达能力局限性促使研究者开发出多种更强大的图学习架构。我们主张深度同态网络（DHNs）是一种特别适合关系数据库学习的模型，因为它与SQL的重要片段（如合取查询）紧密相关。通过将DHNs与一阶逻辑（FO）的各种自然片段及其扩展联系起来，我们研究了DHNs的精确表达能力。对于使用最大值、求和和均值聚合的DHNs，我们建立了其与一元否定片段（UNFO）以及带有计数量词和比率量词的UNFO扩展之间的联系。我们进一步将求和聚合的DHNs与FO的一元量词交替片段以及带有表达性计数的FO扩展联系起来。通过FO与SQL之间的经典对应关系，这些结果也阐明了DHNs与SQL之间的关联。它们还使我们能够研究DHNs的两个基本静态分析问题——空值问题和包含问题的可判定性。最后，我们通过实验证实，所建立的表达能力差异在合适的预测任务性能中得到了体现。