In this paper we extend a decision procedure for the Boolean algebra of finite sets with cardinality constraints ($\mathcal{L}_{\lvert\cdot\rvert}$) to a decision procedure for $\mathcal{L}_{\lvert\cdot\rvert}$ extended with set terms denoting finite integer intervals ($\mathcal{L}_{[\,]}$). In $\mathcal{L}_{[\,]}$ interval limits can be integer linear terms including \emph{unbounded variables}. These intervals are a useful extension because they allow to express non-trivial set operators such as the minimum and maximum of a set, still in a quantifier-free logic. Hence, by providing a decision procedure for $\mathcal{L}_{[\,]}$ it is possible to automatically reason about a new class of quantifier-free formulas. The decision procedure is implemented as part of the $\{log\}$ tool. The paper includes a case study based on the elevator algorithm showing that $\{log\}$ can automatically discharge all its invariance lemmas some of which involve intervals.

中文翻译：

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具有有限整数区间的有限集理论的决策程序

在本文中，我们将具有基数约束（$ \ mathcal {L} _ {\ lvert \ cdot \ rvert} $）的有限集布尔代数的决策过程扩展为$ \ mathcal {L} _ {\ lvert \ cdot \ rvert} $使用表示有限整数间隔（$ \ mathcal {L} _ {[\，]} $）的设置项扩展。在$ \ mathcal {L} _ {[\，]} $中，区间限制可以是整数线性项，包括\ emph {unbounded variables}。这些间隔是有用的扩展，因为它们允许仍使用无量词逻辑来表示非平凡的集合运算符，例如集合的最小值和最大值。因此，通过提供$ \ mathcal {L} _ {{\，]} $的决策程序，可以自动推理出一类新的无量纲公式。决策过程是$ \ {log \} $工具的一部分。