Aggregate-Combine-Readout GNNs Are More Expressive Than Logic C2

TL;DR

This paper proves that aggregate-combine-readout GNNs have strictly greater logical expressiveness than the guarded fragment of C2 logic, resolving a key open problem in understanding GNN capabilities.

Contribution

We demonstrate that aggregate-combine-readout GNNs surpass the expressive power of C2 logic, providing new insights into their logical and computational capabilities.

Findings

Aggregate-combine-readout GNNs are more expressive than C2 logic.

The result applies to both directed and undirected graphs.

Implications for understanding the limits of GNN expressiveness.

Abstract

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 Barcel\'o 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.