In constraint programming (CP), a combinatorial problem is modeled declaratively as a conjunction of constraints, each of which captures some of the combinatorial substructure of the problem. Constraints are more than a modeling convenience: every constraint is partially implemented by an inference algorithm, called a propagator, that rules out some but not necessarily all infeasible candidate values of one or more unknowns in the scope of the constraint. Interleaving propagation with systematic search leads to a powerful and complete solution method, combining a high degree of re-usability with natural, high-level modeling.A propagator can be characterized as a sound approximation of a constraint on an abstraction of sets of candidate values; propagators that share an abstraction are similar in the strength of the inference they perform when identifying infeasible candidate values. In this thesis, we consider abstractions of sets of candidate values that may be described by an elegant mathematical formalism, the Galois connection. We develop a theoretical framework from the correspondence between Galois connections and propagators, unifying two disparate views of the abstraction-propagation connection, namely the oft-overlooked distinction between representational and computational over-approximations. Our framework yields compact definitions of propagator strength, even in complicated cases (i.e., involving several types, or unknowns with internal structure); it also yields a method for the principled derivation of propagators from constraint definitions.We apply this framework to the extension of an existing CP solver to constraints over strings, that is, words of finite length. We define, via a Galois connection, an over-approximation for bounded-length strings, and demonstrate two different methods for implementing this over-approximation in a CP solver. First we use the Galois connection to derive a bounded-length string representation as an aggregation of existing scalar types; propagators for this representation are obtained by manual derivation, or automated synthesis, or a combination. Then we implement a string variable type, motivating design choices with knowledge gained from the construction of the over-approximation. The resulting CP solver extension not only substantially eases modeling for combinatorial string problems, but also leads to substantial efficiency improvements over prior CP methods.

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除数字外的其他内容：抽象，约束传播和字符串变量类型

在约束编程（CP）中，将组合问题声明为约束的结合，每个约束都捕获了该问题的某些组合子结构。约束不仅提供建模方面的便利：每个约束都由称为传播器的推理算法部分实现，该推理算法排除了约束范围内一个或多个未知数的一些但不一定是所有不可行的候选值。传播与系统搜索的交织导致一种强大而完整的解决方案方法，将高度可重用性与自然的高级建模相结合。传播子可以表征为对候选值集的抽象约束的声音近似; 共享抽象的传播者在识别不可行的候选值时执行的推理强度相似。在本文中，我们考虑了一组候选值的抽象，这些候选值可以用一种优雅的数学形式主义-伽罗瓦联系来描述。我们从伽罗瓦联系与传播者之间的对应关系中建立了一个理论框架，统一了抽象-传播联系的两种不同观点，即表示和计算过度逼近之间经常被忽视的区别。即使在复杂的情况下（即涉及多种类型或内部结构未知的情况），我们的框架也可以得出传播强度的紧凑定义。它还提供了一种从约束定义中原理性推导传播子的方法。我们将此框架应用于现有CP解算器的扩展，以扩展对字符串（即长度有限的单词）的约束。我们通过Galois连接定义有界字符串的过度逼近，并演示了两种不同的方法来在CP解算器中实现这种过度逼近。首先，我们使用Galois连接来导出有限长度的字符串表示形式，作为现有标量类型的集合；通过手动推导，自动合成或组合获得用于此表示的传播器。然后，我们实现一个字符串变量类型，利用从过逼近的构造中获得的知识来激励设计选择。由此产生的CP求解器扩展不仅大大简化了组合字符串问题的建模，