We consider the problem of deciding the satisfiability of quantifier-free formulas in the theory of finite sets with cardinality constraints. Sets are a common high-level data structure used in programming; thus, such a theory is useful for modeling program constructs directly. More importantly, sets are a basic construct of mathematics and thus natural to use when formalizing the properties of computational systems. We develop a calculus describing a modular combination of a procedure for reasoning about membership constraints with a procedure for reasoning about cardinality constraints. Cardinality reasoning involves tracking how different sets overlap. For efficiency, we avoid considering Venn regions directly, as done in previous work. Instead, we develop a novel technique wherein potentially overlapping regions are considered incrementally as needed, using a graph to track the interaction among the different regions. The calculus has been designed to facilitate its implementation within SMT solvers based on the DPLL($T$) architecture. Our experimental results demonstrate that the new techniques are competitive with previous techniques and can scale much better on certain classes of problems.

@article{paperbot359,
  title = {Reasoning with Finite Sets and Cardinality Constraints in SMT},
  author = {Kshitij Bansal and Clark Barrett and Andrew Reynolds and Cesare Tinelli},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 14, Issue 4},
  year = {2018},
  doi = {10.23638/lmcs-14(4:12)2018}
}