Logical Characterizations of Recurrent Graph Neural Networks with Reals and Floats

2024-05-23

Neural Information Processing Systems

AI Summary

This paper precisely characterizes the expressive power of Recurrent Graph Neural Networks (Recurrent GNNs) over the real numbers and floating-point numbers. To this end, it introduces, for the first time, non-relativized logical characterizations: infinitary modal logic with counting (L∞,ω[C]) over reals, and a finite-rule modal logic (FML) over floating-point numbers. Theoretically, both logics are proven expressively equivalent on MSO-definable graph properties. Moreover, a bidirectional correspondence is established between Recurrent GNNs and distributed automata. The key contributions are: (i) the first non-relativized, exact logical characterization of Recurrent GNNs; (ii) the demonstration that floating-point GNNs are fully captured by a finite logic—thereby circumventing the conventional reliance on real-number approximations; and (iii) a unified logical framework bridging numerical precision, automata theory, and GNN expressivity. These results fundamentally advance the theoretical understanding of GNNs under practical numerical representations.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In pioneering work from 2019, Barcel'o and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this article, we give exact logical characterizations of recurrent GNNs in two scenarios: (1) in the setting with floating-point numbers and (2) with reals. For floats, the formalism matching recurrent GNNs is a rule-based modal logic with counting, while for reals we use a suitable infinitary modal logic, also with counting. These results give exact matches between logics and GNNs in the recurrent setting without relativising to a background logic in either case, but using some natural assumptions about floating-point arithmetic. Applying our characterizations, we also prove that, relative to graph properties definable in monadic second-order logic (MSO), our infinitary and rule-based logics are equally expressive. This implies that recurrent GNNs with reals and floats have the same expressive power over MSO-definable properties and shows that, for such properties, also recurrent GNNs with reals are characterized by a (finitary!) rule-based modal logic. In the general case, in contrast, the expressive power with floats is weaker than with reals. In addition to logic-oriented results, we also characterize recurrent GNNs, with both reals and floats, via distributed automata, drawing links to distributed computing models.

Problem

• Characterize recurrent GNNs with floats using rule-based modal logic
• Characterize recurrent GNNs with reals using infinitary modal logic
• Compare expressive power of GNNs with floats and reals

Innovation

• Logical characterization of recurrent GNNs with floats
• Infinitary modal logic for GNNs with reals
• Distributed automata links to GNNs