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 (${\mathcal{L}}_{|\cdot |}$) to a decision procedure for ${\mathcal{L}}_{|\cdot |}$ 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.

arXiv admin note: text overlap with arXiv:2102.05422