Graph neural networks and MSO
Veeti Ahvonen, Damian Heiman, Antti Kuusisto
TL;DR
The paper addresses the expressiveness gap between recurrent Graph Neural Networks over $\mathbb{R}$ and a logical formalism by establishing that, restricted to MSO-definable properties, $\mathrm{GNN}[\mathbb{R}]$ and $\mathrm{GMSC}$ have equivalent expressive power. It delivers an alternative proof to the known result by replacing parity-tree automata with distributed message-passing automata (CMPA) and by showing how MSO-definable node properties translate into bounded FCMPA, which in turn correspond to $\mathrm{GMSC}$ and $\mathrm{GNN}[\mathbb{R}]$. The approach relies on constructing automata with fixed-point accepting conditions, transforming non-deterministic forgetful automata into deterministic ones, and using $k$-extensions of trees to capture MSO truth conditions. The findings reinforce the deep connections between logic and graph-based neural models, and they highlight the robustness of the $\mathrm{GMSC}$-based characterization across various acceptance paradigms. The results have implications for understanding the theoretical limits of GNNs in logic-guided graph reasoning and suggest avenues for automata-theoretic analyses of neural architectures.
Abstract
We give an alternative proof for the existing result that recurrent graph neural networks working with reals have the same expressive power in restriction to monadic second-order logic MSO as the graded modal substitution calculus. The proof is based on constructing distributed automata that capture all MSO-definable node properties over trees. We also consider some variants of the acceptance conditions.