Preservation Theorems for Unravelling-Invariant Classes: A Uniform Approach for Modal Logics and Graph Neural Networks
Przemysław Andrzej Wałęga, Bernardo Cuenca Grau
TL;DR
The paper develops a uniform, finite-model-theoretic framework for preservation theorems of unravelling-invariant classes under embeddings, injective homomorphisms, and homomorphisms in modal logic, establishing precise definability as existential or existential-positive fragments of $\mathcal{GML}$ and $\mathcal{ML}$. Central to the approach is a well-quasi-ordering result for tree-shaped models of bounded height, which yields a finite minimal-tree argument and allows explicit syntactic characterisations as finite disjunctions of modal formulae. The framework recovers Rosen’s finite Embedding Preservation Theorem for basic modal logic and yields new results for graded modal logic, while providing novel insights into the expressive power of graph neural networks: monotonic GNNs correspond exactly to $\exists^+\mathcal{GML}$ and monotonic-MAX GNNs to $\exists^+\mathcal{ML}$. This builds a bridge between finite model theory and ML on graphs, with potential extensions to richer logics and more complex GNN architectures.
Abstract
We study preservation theorems for modal logics over finite structures with respect to three fundamental semantic relations: embeddings, injective homomorphisms, and homomorphisms. We focus on classes of pointed Kripke models that are invariant under bounded unravellings, a natural locality condition satisfied by modal logics and by graph neural networks (GNNs). We show that preservation under embeddings coincides with definability in existential graded modal logic; preservation under injective homomorphisms with definability in existential positive graded modal logic; and preservation under homomorphisms with definability in existential positive modal logic. A key technical contribution is a structural well-quasi-ordering result. We prove that the embedding relation on classes of tree-shaped models of uniformly bounded height forms a well-quasi-order, and that the bounded-height assumption is essential. This well-quasi-ordering yields a finite minimal-tree argument leading to explicit syntactic characterisations via finite disjunctions of (graded) modal formulae. As an application, we derive consequences for the expressive power of GNNs. Using our preservation theorem for injective homomorphisms, we obtain a new logical characterisation of monotonic GNNs, showing that they capture exactly existential-positive graded modal logic, while monotonic GNNs with MAX aggregation correspond precisely to existential-positive modal logic.