The octagon abstract domain is a widely used numeric abstract domain expressing relational information between variables whilst being both computationally efficient and simple to implement. Each element of the domain is a system of constraints where each constraint takes the restricted form $$\pm \, x_i \pm x_j \le c$$±xi±xj≤c. A key family of operations for the octagon domain are closure algorithms, which check satisfiability and provide a normal form for octagonal constraint systems. We present new quadratic incremental algorithms for closure, strong closure and integer closure and proofs of their correctness. We highlight the benefits and measure the performance of these new algorithms.

中文翻译：

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逐渐闭合的八边形

八边形抽象域是一种广泛使用的数值抽象域，它表示变量之间的关系信息，同时计算效率高且易于实现。域的每个元素都是一个约束系统，其中每个约束采用受限形式 $$\pm \, x_i \pm x_j \le c$$±xi±xj≤c。八边形域的一个关键操作族是闭包算法，它检查可满足性并为八边形约束系统提供标准形式。我们提出了用于闭包、强闭包和整数闭包的新的二次增量算法，并证明了它们的正确性。我们强调了这些新算法的好处并衡量了它们的性能。