Programming by multiset transformation
Communications of the ACM
On the complexity of integer programming
Journal of the ACM (JACM)
Introduction to set constraint-based program analysis
Science of Computer Programming
Mechanizing a theory of program composition for UNITY
ACM Transactions on Programming Languages and Systems (TOPLAS)
Minimal Solutions of Linear Diophantine Systems: Bounds and Algorithms
RTA '91 Proceedings of the 4th International Conference on Rewriting Techniques and Applications
Combining Multisets with Integers
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
Deciding Boolean Algebra with Presburger Arithmetic
Journal of Automated Reasoning
Subtractive reductions and complete problems for counting complexity classes
Theoretical Computer Science - Mathematical foundations of computer science 2000
Theoretical Computer Science - Foundations of software science and computation structures
Modular data structure verification
Modular data structure verification
Automated verification of shape and size properties via separation logic
VMCAI'07 Proceedings of the 8th international conference on Verification, model checking, and abstract interpretation
Carathéodory bounds for integer cones
Operations Research Letters
CAV '08 Proceedings of the 20th international conference on Computer Aided Verification
Fractional Collections with Cardinality Bounds, and Mixed Linear Arithmetic with Stars
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
On Bounded Reachability of Programs with Set Comprehensions
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
Deciding Extensions of the Theories of Vectors and Bags
VMCAI '09 Proceedings of the 10th International Conference on Verification, Model Checking, and Abstract Interpretation
Decision procedures for algebraic data types with abstractions
Proceedings of the 37th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
PLDI '10 Proceedings of the 2010 ACM SIGPLAN conference on Programming language design and implementation
Combining theories with shared set operations
FroCoS'09 Proceedings of the 7th international conference on Frontiers of combining systems
Ordered sets in the calculus of data structures
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
What's decidable about sequences?
ATVA'10 Proceedings of the 8th international conference on Automated technology for verification and analysis
Decision procedures for automating termination proofs
VMCAI'11 Proceedings of the 12th international conference on Verification, model checking, and abstract interpretation
Building a calculus of data structures
VMCAI'10 Proceedings of the 11th international conference on Verification, Model Checking, and Abstract Interpretation
Collections, cardinalities, and relations
VMCAI'10 Proceedings of the 11th international conference on Verification, Model Checking, and Abstract Interpretation
MUNCH - automated reasoner for sets and multisets
IJCAR'10 Proceedings of the 5th international conference on Automated Reasoning
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Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these scenarios. Multisets arise 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 polynomial-space 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.