Cooperation among constraint solvers is difficult because different solving paradigms have different theoretical foundations. Recent works have shown that abstract interpretation can provide a unifying theory for various constraint solvers. In particular, it relies on abstract domains which capture constraint languages as ordered structures. The key insight of this paper is viewing cooperation schemes as abstract domains combinations. We propose a modular framework in which solvers and cooperation schemes can be seamlessly added and combined. This differs from existing approaches such as SMT where the cooperation scheme is usually fixed (e.g., Nelson-Oppen). We contribute to two new cooperation schemes: (i) interval propagators completion that allows abstract domains to exchange bound constraints, and (ii) delayed product which exchanges over-approximations of constraints between two abstract domains. Moreover, the delayed product is based on delayed goal of logic programming, and it shows that abstract domains can also capture control aspects of constraint solving. Finally, to achieve modularity, we propose the shared product to combine abstract domains and cooperation schemes. Our approach has been fully implemented, and we provide various examples on the flexible job shop scheduling problem.

中文翻译：

---

通过抽象解释的模块化约束求解器合作

约束求解器之间的合作很困难，因为不同的求解范式有不同的理论基础。最近的工作表明，抽象解释可以为各种约束求解器提供统一的理论。特别是，它依赖于将约束语言捕获为有序结构的抽象域。本文的主要观点是将合作方案视为抽象领域的组合。我们提出了一个模块化框架，其中求解器和合作方案可以无缝添加和组合。这不同于现有的方法，例如 SMT，其中合作方案通常是固定的（例如，纳尔逊-奥本）。我们为两项新的合作计划做出了贡献：(i)间隔传播者完成允许抽象域交换绑定约束，以及 (ii)延迟产品它在两个抽象域之间交换约束的过度近似。此外，延迟乘积基于逻辑编程的延迟目标，表明抽象域也可以捕获约束求解的控制方面。最后，为了实现模块化，我们提出了共享产品结合抽象领域和合作方案。我们的方法已经完全实施，我们提供了关于灵活作业车间调度问题的各种示例。