Decision Procedures for Multisets with Cardinality Constraints

Abstract

Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of linked data structure with duplicate elements leads to multisets. Interactive theorem provers such as Isabelle specify theories of multisets and prove a number of theorems about them to enable their use in interactive verification. However, the decidability and complexity of constraints on multisets is much less understood than for constraints on sets. The first contribution of this paper is a polynomial-space algorithm for deciding expressive quantifier-free constraints on multisets with cardinality operators. Our decision procedure reduces in polynomial time constraints on multisets to constraints in an extension of quantifier-free Presburger arithmetic with certain "unbounded sum" expressions. We prove bounds on solutions of resulting constraints and describe a polynomialspace decision procedure for these constraints. The second contribution of this paper is a proof that adding quantifiers to a constraint language containing subset and cardinality operators yields undecidable constraints. The result follows by reduction from Hilbert's 10th problem.

Collections, Cardinalities, and Relations

“…It then constrains the cardinality of the multiset obtained through the image to be equal to the cardinality of the original set. This final formula is a formula in the NP-complete logic for reasoning about multisets and cardinality constraints [22,23].…”

“…Apply the algorithm in Figure 7 to translate F into an equisatisfiable multiset formula F ′ in the syntax given in Figure 1 in [22]. In this step we eliminate function symbols in a way similar to that described in Section 3.…”

“…One such logic is QFBAPA (quantifier-free Boolean Algebra with Presburger Arithmetic), which we recently proved to be in NP [16], an improvement over the previous NEXPTIME algorithms based on quantifier elimination [9,14]. We subsequently generalized this result to quantifier-free constraints on multisets (bags), collections in which an element can occur multiple times [22,23]. The usefulness of collections and cardinality measures on them has been established through a number of examples from software analysis and verification, including not only decision procedures [14,16,29] but also static analyses that operate directly on the set abstraction or the cardinality abstraction [11,13,21].…”

“…We show that the sparse model solution phenomenon of QFBAPA continues to apply in the presence of such relations, and use it to prove that the logic remains inside NP. -We show NEXPTIME completeness (by reduction to [22]) of a logic that allows computing multisets instead of sets as function images, preserving the multiplicity of elements that occur in the range of the function multiple times. This is a natural definition of the notion of multiset comprehension and arises e.g.…”

Building a Calculus of Data Structures

“…The motivation for multisets comes from verification of data structures with possibly repeated elements, where, in addition to knowing whether an element occurs in the data structure, we are also interested how many times it occurs. A detailed description of decision procedures for satisfiability of multisets with cardinality constraints is in [PK08a,PK08c,PK08b].…”