Expressive Power of Deep Homomorphism Networks over Relational Databases

2026-05-18

AI Summary

This work systematically investigates the expressive power of Deep Homomorphism Networks (DHNs) over relational databases, establishing precise correspondences between DHNs and fragments of first-order logic—specifically the Unary Negation Fragment (UNFO) and its extensions with counting and proportional quantifiers—as well as conjunctive queries in SQL. By integrating logical notions of quantifier alternation hierarchies and UNFO theory with graph neural network aggregation mechanisms (max, sum, and mean), the study provides the first characterization of the expressiveness boundaries of DHNs under different aggregators. It further proves the decidability of emptiness and containment problems for these networks. Theoretical findings are complemented by empirical validation, revealing how differences in expressive power impact performance on practical prediction tasks.

Abstract

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.

Problem

• expressive power
• Deep Homomorphism Networks
• relational databases
• first-order logic
• conjunctive queries

Innovation

• Deep Homomorphism Networks
• expressive power
• relational databases
• first-order logic
• conjunctive queries