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 ( \({\mathcal {L}_{\vert {\cdot }\vert }} \) ) to a decision procedure for \({\mathcal {L}_{\vert {\cdot }\vert }} \) extended with set terms denoting finite integer intervals ( \({\mathcal {L}_{[\,]}} \) ). In \({\mathcal {L}_{[\,]}} \) interval limits can be integer linear terms including 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 } (‘setlog’) 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.