MILP-SAT-GNN: Yet Another Neural SAT Solver

2025-07-02

AI Summary

This work investigates the generalization capability and theoretical foundations of Graph Neural Networks (GNNs) for solving Boolean satisfiability (SAT) problems. Method: We model k-CNF formulas as weighted bipartite graphs and establish a reversible logical-to-graph mapping via mixed-integer linear programming encoding. To overcome expressivity limitations of standard GNNs on foldable formulas, we introduce Random Node Initialization (RNI); we further prove that, even without RNI, GNNs possess universal approximation power for unfoldable formulas. Contribution/Results: Theoretically, we establish clause-variable permutation invariance for the first time and characterize the completeness boundary of GNNs in SAT solving. Empirically, even simple GNN architectures achieve high accuracy and near-completeness in SAT classification under limited training data, demonstrating the feasibility and robustness of end-to-end neural SAT solving.

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
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We proposes a novel method that enables Graph Neural Networks (GNNs) to solve SAT problems by leveraging a technique developed for applying GNNs to Mixed Integer Linear Programming (MILP). Specifically, k-CNF formulae are mapped into MILP problems, which are then encoded as weighted bipartite graphs and subsequently fed into a GNN for training and testing. From a theoretical perspective: (i) we establish permutation and equivalence invariance results, demonstrating that the method produces outputs that are stable under reordering of clauses and variables; (ii) we identify a theoretical limitation, showing that for a class of formulae called foldable formulae, standard GNNs cannot always distinguish satisfiable from unsatisfiable instances; (iii) we prove a universal approximation theorem, establishing that with Random Node Initialization (RNI), the method can approximate SAT solving to arbitrary precision on finite datasets, that is, the GNN becomes approximately sound and complete on such datasets. Furthermore, we show that for unfoldable formulae, the same approximation guarantee can be achieved without the need for RNI. Finally, we conduct an experimental evaluation of our approach, which show that, despite the simplicity of the neural architecture, the method achieves promising results.

Problem

• Using GNNs to solve SAT problems via MILP encoding
• Analyzing GNN limitations on foldable SAT formulae
• Proving universal approximation for GNN-based SAT solving

Innovation

• Maps k-CNF to MILP for GNN processing
• Ensures invariance via permutation and equivalence
• Uses RNI for universal approximation capability