On Sets with Cardinality Constraints in Satisfiability Modulo Theories

Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifier-free BAPA (QFBAPA). Deciding the satisfiability of QFBAPA formulas has been shown to be NP-complete using an eager reduction to quantifier-free Presburger arithmetic that exploits a sparse-solution property. In contrast to many other NP-complete problems (such as quantifier-free first-order logic or linear arithmetic), the applications of QFBAPA to a broader set of problems has so far been hindered by the lack of an efficient implementation that can be used alongside other efficient decision procedures. We overcome these limitations by extending the efficient SMT solver Z3 with the ability to reason about cardinality constraints. Our implementation uses the DPLL(T) mechanism of Z3 to reason about the top-level propositional structure of a QFBAPA formula, improving the efficiency compared to previous implementations. Moreover, we present a new algorithm for automated decomposition of QFBAPA formulas. Our algorithm alleviates the exponential explosion of considering all Venn regions, significantly improving the tractability of formulas with many set variables. Because it is implemented as a theory plugin, our implementation enables Z3 to prove formulas that use QFBAPA constructs alongside constructs from other theories that Z3 supports (e.g. linear arithmetic, uninterpreted function symbols, algebraic data types), as well as in formulas with quantifiers. We have applied our implementation to verification of functional programs; we show it can automatically prove formulas that no automated approach was reported to be able to prove before.