Aggregate-Combine-Readout GNNs Are More Expressive Than Logic C2
This resolves a foundational question in theoretical machine learning about the expressive power of GNNs, with implications for understanding their capabilities relative to logical frameworks.
The paper tackled the open problem of whether aggregate-combine-readout graph neural networks (GNNs) are characterized by the logical language C2, proving that these GNNs are strictly more expressive than C2 over undirected and directed graphs.
In recent years, there has been growing interest in understanding the expressive power of graph neural networks (GNNs) by relating them to logical languages. This research has been been initialised by an influential result of Barceló et al. (2020), who showed that the graded modal logic (or a guarded fragment of the logic C2), characterises the logical expressiveness of aggregate-combine GNNs. As a ``challenging open problem'' they left the question whether full C2 characterises the logical expressiveness of aggregate-combine-readout GNNs. This question has remained unresolved despite several attempts. In this paper, we solve the above open problem by proving that the logical expressiveness of aggregate-combine-readout GNNs strictly exceeds that of C2. This result holds over both undirected and directed graphs. Beyond its implications for GNNs, our work also leads to purely logical insights on the expressive power of infinitary logics.