Expressive Power of Graph Transformers via Logic

TL;DR

This paper investigates the expressive power of graph transformers and GPS-networks, revealing their equivalence to certain logical formalisms under different numerical settings, bridging deep learning and logic.

Contribution

It provides a formal characterization of the expressive power of graph transformers and GPS-networks in terms of logical languages, under both real and float numerical settings.

Findings

GPS-networks with reals match graded modal logic with global modality

GPS-networks with floats match graded modal logic with counting global modality

Graph transformers are characterized by propositional logic with global modalities

Abstract

Transformers are the basis of modern large language models, but relatively little is known about their precise expressive power on graphs. We study the expressive power of graph transformers (GTs) by Dwivedi and Bresson (2020) and GPS-networks by Ramp\'asek et al. (2022), both under soft-attention and average hard-attention. Our study covers two scenarios: the theoretical setting with real numbers and the more practical case with floats. With reals, we show that in restriction to vertex properties definable in first-order logic (FO), GPS-networks have the same expressive power as graded modal logic (GML) with the global modality. With floats, GPS-networks turn out to be equally expressive as GML with the counting global modality. The latter result is absolute, not restricting to properties definable in a background logic. We also obtain similar characterizations for GTs in terms of propositional logic with the global modality (for reals) and the counting global modality (for floats).