A Decision Procedure for a Theory of Finite Sets with Finite Integer Intervals

Abstract: 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.