The Boolean SATisfiability problem (SAT) is of central importance in computer science. Although SAT is known to be NP-complete, progress on the engineering side—especially that of Conflict-Driven Clause Learning (CDCL) and Local Search SAT solvers—has been remarkable. Yet, while SAT solvers, aimed at solving industrial-scale benchmarks in Conjunctive Normal Form (CNF), have become quite mature, SAT solvers that are effective on other types of constraints (e.g., cardinality constraints and XORs) are less well-studied; a general approach to handling non-CNF constraints is still lacking.

To address the issue above, we design FourierSAT,¹ an incomplete SAT solver based on Fourier Analysis (also known as Walsh-Fourier Transform) of Boolean functions, a technique to represent Boolean functions by multilinear polynomials. By such a reduction to continuous optimization, we propose an algebraic framework for solving systems consisting of different types of constraints. The idea is to leverage gradient information to guide the search process in the direction of local improvements. We show this reduction enjoys interesting theoretical properties. Empirical results demonstrate that FourierSAT can be a useful complement to other solvers on certain classes of benchmarks.

布尔可满足性问题 (SAT) 在计算机科学中至关重要。尽管众所周知 SAT 是 NP 完全的，但工程方面的进展——尤其是冲突驱动子句学习 (CDCL) 和本地搜索 SAT 求解器——的进展非常显着。然而，虽然旨在解决联合范式 ( CNF ) 中的工业规模基准的SAT 求解器已经变得非常成熟，但对其他类型的约束（例如，基数约束和XOR s）有效的 SAT 求解器的研究较少; 仍然缺乏处理非CNF约束的通用方法。

为了解决上述问题，我们设计了FourierSAT，¹是一种基于布尔函数傅里叶分析（也称为 Walsh-Fourier 变换）的不完全 SAT 求解器，这是一种通过多重线性多项式表示布尔函数的技术。通过这种对连续优化的简化，我们提出了一个代数框架来解决由不同类型的约束组成的系统。这个想法是利用梯度信息来引导搜索过程朝着局部改进的方向发展。我们表明这种减少具有有趣的理论特性。实证结果表明，在某些类别的基准测试中，FourierSAT可以作为其他求解器的有用补充。