On the Expressive Power of GNNs for Boolean Satisfiability

2026-02-09

AI Summary

This study investigates the expressive limitations of graph neural networks (GNNs) in solving Boolean satisfiability (SAT) problems. Building upon the Weisfeiler–Leman (WL) test framework, it establishes—for the first time—that WL tests of any fixed order cannot reliably distinguish between satisfiable and unsatisfiable SAT instances, thereby revealing a fundamental theoretical barrier for GNN-based approaches to SAT solving. Through rigorous theoretical analysis and empirical evaluation on benchmarks including G4SAT and instances from international SAT competitions, the work demonstrates that random SAT instances are often separable by low-order WL tests, whereas industrial instances typically demand higher expressive power to effectively predict satisfying assignments. This highlights a critical performance bottleneck in current sequential assignment-based SAT solvers when handling complex, real-world instances.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Machine learning approaches to solving Boolean Satisfiability (SAT) aim to replace handcrafted heuristics with learning-based models. Graph Neural Networks have emerged as the main architecture for SAT solving, due to the natural graph representation of Boolean formulas. We analyze the expressive power of GNNs for SAT solving through the lens of the Weisfeiler-Leman (WL) test. As our main result, we prove that the full WL hierarchy cannot, in general, distinguish between satisfiable and unsatisfiable instances. We show that indistinguishability under higher-order WL carries over to practical limitations for WL-bounded solvers that set variables sequentially. We further study the expressivity required for several important families of SAT instances, including regular, random and planar instances. To quantify expressivity needs in practice, we conduct experiments on random instances from the G4SAT benchmark and industrial instances from the International SAT Competition. Our results suggest that while random instances are largely distinguishable, industrial instances often require more expressivity to predict a satisfying assignment.

Problem

• Boolean Satisfiability
• Graph Neural Networks
• Expressive Power
• Weisfeiler-Leman Test
• SAT Solving

Innovation

• Graph Neural Networks
• Boolean Satisfiability
• Weisfeiler-Leman test
• Expressive Power
• SAT Solving