Thinking Out of the Box: Hybrid SAT Solving by Unconstrained Continuous Optimization

2025-05-31

AI Summary

SAT solving is critical in combinatorial optimization and formal verification, yet existing solvers exhibit limited support for non-CNF hybrid constraints—such as XOR, cardinality, and Not-All-Equal (NAE) constraints. This paper introduces the first unconstrained continuous optimization framework for SAT with rigorous theoretical analysis, eliminating conventional box constraints. It encodes hybrid constraints into a smooth objective via differentiable penalty functions. Theoretically, we derive necessary conditions for penalty term design and establish, for the first time, a principled synergy between deep learning optimizers (e.g., Adam) and symbolic reasoning. Empirically, our method achieves significant performance gains across diverse hybrid SAT benchmarks—including XOR-SAT, cardinality-constrained SAT, and NAE-SAT—outperforming state-of-the-art solvers. These results demonstrate that unconstrained continuous optimization substantially enhances symbolic reasoning capabilities, paving the way for a new AI-for-SAT paradigm grounded in differentiable optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

The Boolean satisfiability (SAT) problem lies at the core of many applications in combinatorial optimization, software verification, cryptography, and machine learning. While state-of-the-art solvers have demonstrated high efficiency in handling conjunctive normal form (CNF) formulas, numerous applications require non-CNF (hybrid) constraints, such as XOR, cardinality, and Not-All-Equal constraints. Recent work leverages polynomial representations to represent such hybrid constraints, but it relies on box constraints that can limit the use of powerful unconstrained optimizers. In this paper, we propose unconstrained continuous optimization formulations for hybrid SAT solving by penalty terms. We provide theoretical insights into when these penalty terms are necessary and demonstrate empirically that unconstrained optimizers (e.g., Adam) can enhance SAT solving on hybrid benchmarks. Our results highlight the potential of combining continuous optimization and machine-learning-based methods for effective hybrid SAT solving.

Problem

• Solving hybrid SAT problems without box constraints
• Enhancing SAT solvers for non-CNF constraints
• Using unconstrained optimizers for hybrid benchmarks

Innovation

• Unconstrained continuous optimization for hybrid SAT
• Penalty terms replace box constraints
• Adam optimizer enhances hybrid SAT solving