A Decision Procedure for a Theory of Finite Sets with Finite Integer Intervals
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In this paper we extend a decision procedure for the Boolean algebra of finite sets with cardinality constraints (ℒ_|·|) to a decision procedure for ℒ_|·| extended with set terms denoting finite integer intervals (ℒ_[ ]). In 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 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.
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