[Paper Review] Graph Neural Reasoning for 2-Quantified Boolean Formula Solvers

Zhanfu Yang, Fei Wang+3 more|arXiv (Cornell University)|Apr 27, 2019

Graph Theory and Algorithms14 references9 citations

TL;DR

This paper investigates Graph Neural Network (GNN)-based solvers and heuristics for 2-Quantified Boolean Formula (2QBF) problems, proposing GNN-enhanced CEGAR-based solvers that rank candidates and counterexamples to improve reasoning. Despite GNNs' limitations in capturing unsatisfiability, the authors achieve improved performance on 2QBF instances through hybrid GNN heuristics, with GNN3 outperforming baselines on both satisfiable and unsatisfiable formulas.

ABSTRACT

In this paper, we investigate the feasibility of learning GNN (Graph Neural Network) based solvers and GNN-based heuristics for specified QBF (Quantified Boolean Formula) problems. We design and evaluate several GNN architectures for 2QBF formulae, and conjecture that GNN has limitations in learning 2QBF solvers. Then we show how to learn a heuristic CEGAR 2QBF solver. We further explore generalizing GNN-based heuristics to larger unseen instances, and uncover some interesting challenges. In summary, this paper provides a comprehensive surveying view of applying GNN-embeddings to specified QBF solvers, and aims to offer guidance in applying ML to more complicated symbolic reasoning problems.

Motivation & Objective

• To investigate the feasibility of using GNNs to learn 2QBF solvers and heuristics for symbolic reasoning problems.
• To address the challenge that standard GNNs fail to reason about unsatisfiability in 2QBF, a key limitation for QBF solving.
• To design and evaluate GNN-based heuristics that reduce inference overhead by avoiding multiple GNN embeddings per formula.
• To explore generalization of GNN-based heuristics to larger, unseen 2QBF instances.
• To provide guidance on applying machine learning to complex symbolic reasoning tasks like 2QBF solving.

Proposed method

• Adapt GNN architectures to 2QBF by separating embeddings for ∀- and ∃-literals using distinct neural modules.
• Design a CEGAR-based solver enhanced with GNN-based heuristics to rank potential witnesses (candidates) and counterexamples.
• Use GNNs to compute ranking scores for candidates and counterexamples based on clause satisfaction and unsatisfiability core analysis.
• Integrate multiple ranking signals: hardness-based scores, unsatisfiability core detection, and clause satisfaction counts.
• Train GNNs via supervised learning on labeled 2QBF instances, using scores derived from SAT solving and core extraction.
• Combine candidate and counterexample ranking in a single GNN architecture (GNN3) to improve overall solver performance.

Experimental results

Research questions

• RQ1 Can standard GNN architectures effectively learn to solve 2QBF problems, given their inherent difficulty in reasoning about unsatisfiability?
• RQ2 How can GNN-based heuristics be designed to improve CEGAR-based 2QBF solvers while minimizing inference cost?
• RQ3 To what extent can GNN-based heuristics generalize to larger, unseen 2QBF instances?
• RQ4 Does combining candidate and counterexample ranking improve solver performance over individual ranking strategies?
• RQ5 What are the limitations of GNNs in capturing global reasoning patterns like unsatisfiability cores in 2QBF?

Key findings

• Standard GNN-based SAT solvers fail to generalize to 2QBF due to GNNs' inability to reason about unsatisfiability.
• Ranking counterexamples based on clause satisfaction improves performance on satisfiable 2QBF instances, but not on unsatisfiable ones.
• GNN3, which combines hardness-based ranking for candidates and clause-satisfaction-based ranking for counterexamples, achieves the best performance across all test sets.
• On the test set, GNN3 reduced solving time to 12.008 seconds on unsatisfiable instances and 24.295 seconds on satisfiable ones, outperforming MaxSAT and other GNN variants.
• The results suggest that reasoning about global properties like unsatisfiability cores is inherently difficult for GNNs, even when enhanced with scoring mechanisms.
• Supervised training with multiple ranking signals leads to better generalization and lower inference overhead compared to per-instance GNN inference.