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.
翻译:在本文中,我们扩展了对具有基本限制限制的定型(mathcal{L ⁇ lvert\cdot\rvert}$)的布列恩代数的确定程序,将其扩展至对美元(mathcal{L ⁇ lvert\cd\rvert}$)的确定程序,以固定条件注明一定的整数间隔(mathcal{L ⁇ {L ⁇ [\\\\]}$)。在$\mathcal{L ⁇ }[\,}}$的间隔限制可以是整线性条件,包括\emph{未受约束的变量}。这些间隔是一个有用的扩展。因为它们允许表达非三角设置的操作员,如一组操作员的最小和上限, 仍然在量化符的逻辑中。 因此, 通过为$\mathcal{L\\\\\\\\\\\\\\\\}$提供一个决定程序, 可以自动解释一个新的量化无定型公式类别。 决定程序可以作为 $ ⁇ _} 工具的一部分实施。 。 。文件包括基于电梯算算法的案例研究研究, 显示$\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\