The Logical Expressiveness of Temporal GNNs via Two-Dimensional Product Logics

TL;DR

This paper explores the logical expressiveness of temporal GNNs by linking them to two-dimensional product logics, revealing how different architectures capture various logical properties over space and time.

Contribution

It introduces the first logical characterization of temporal GNNs, analyzing how their architecture influences their expressive power in temporal and spatial domains.

Findings

Temporal GNNs applying static GNNs recursively over time can express all properties in the product logic of PTL and K.

Architectures like TGNNs and global TGNNs have limited expressiveness, constrained to specific logical fragments.

This work provides foundational insights into the logical capabilities of temporal GNN architectures.

Abstract

In recent years, the expressive power of various neural architectures -- including graph neural networks (GNNs), transformers, and recurrent neural networks -- has been characterised using tools from logic and formal language theory. As the capabilities of basic architectures are becoming well understood, increasing attention is turning to models that combine multiple architectural paradigms. Among them particularly important, and challenging to analyse, are temporal extensions of GNNs, which integrate both spatial (graph-structure) and temporal (evolution over time) dimensions. In this paper, we initiate the study of logical characterisation of temporal GNNs by connecting them to two-dimensional product logics. We show that the expressive power of temporal GNNs depends on how graph and temporal components are combined. In particular, temporal GNNs that apply static GNNs recursively over time can capture all properties definable in the product logic of (past) propositional temporal logic PTL and the modal logic K. In contrast, architectures such as graph-and-time TGNNs and global TGNNs can only express restricted fragments of this logic, where the interaction between temporal and spatial operators is syntactically constrained. These provide us with the first results on the logical expressiveness of temporal GNNs.