Bruno Marnette, Viktor Kuncak, and Martin Rinard. Polynomial constraints for sets with cardinality bounds. In Foundations of Software Science and Computation Structures (FOSSACS), volume 4423 of LNCS, March 2007.

Logics that can reason about sets and their cardinality bounds are useful in program analysis, program verification, databases, and knowledge bases. This paper presents a class of constraints on sets and their cardinalities for which the satisfiability and the entailment problems are computable in polynomial time. Our class of constraints, based on tree-shaped formulas, is unique in being simultaneously tractable and able to express 1) that a set is a union of other sets, 2) that sets are disjoint, and 3) that a set has cardinality within a given range. As the main result we present a polynomial-time algorithm for checking entailment of our constraints.

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