Preservation Theorems for Unravelling-Invariant Classes: A Uniform Approach for Modal Logics and Graph Neural Networks

2026-02-02

AI Summary

This work investigates the definability of modal logic fragments over finite structures under three semantic preservation relations—embeddings, injective homomorphisms, and homomorphisms—and establishes a precise correspondence with the expressive power of graph neural networks (GNNs). By introducing structural well-quasi-ordering theory, it proves for the first time that bounded-height tree-like Kripke models form a well-quasi-order under embeddings, enabling a unified framework that aligns these semantic relations with existential graded modal logic, existential positive graded modal logic, and existential positive modal logic, respectively. Furthermore, it demonstrates that monotone GNNs and their MAX-aggregation variants exactly capture the latter two logics, thereby establishing the first fine-grained correspondence between logical definability and GNN expressiveness.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Machine Learning: Statistical Relational/Logic Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• preservation theorems
• modal logic
• graph neural networks
• unravelling-invariance
• well-quasi-ordering

Innovation

• well-quasi-ordering
• unravelling-invariance
• preservation theorems
• graded modal logic
• graph neural networks