Decidability of Graph Neural Networks via Logical Characterizations

TL;DR

This paper explores the expressiveness and decidability of graph neural networks (GNNs) using logical frameworks, establishing connections with decidable logics and analyzing verification problems.

Contribution

It introduces a logical approach to measure GNN expressiveness and provides decision procedures for certain verification tasks, highlighting both decidability and undecidability results.

Findings

Exact correspondences between GNN expressiveness and logical formalisms.

Decision procedures for verification problems over GNNs.

Undecidability results for static analysis and verification tasks.

Abstract

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving "Presburger quantifiers". We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems.