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 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 quantifierfree 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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aiken, A.: Introduction to set constraint-based program analysis. Science of Computer Programming 35, 79–111 (1999)
Banâtre, J.-P., Le Métayer, D.: Programming by multiset transformation. Commun. ACM 36(1), 98–111 (1993)
Bès, A.: Definability and decidability results related to the elementary theory of ordinal multiplication. Fund. Math. 171, 197–211 (2002)
Domenjoud, E.: Solving systems of linear diophantine equations: An algebraic approach. In: MFCS, pp. 141–150 (1991)
Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. Theor. Comput. Sci. 340(3), 496–513 (2005)
Eisenbrand, F., Shmonin, G.: Carathéodory bounds for integer cones. Operations Research Letters 34(5), 564–568 (2006)
Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)
Ginsburg, S., Spanier, E.: Semigroups, Pressburger formulas and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)
Kuncak, V.: Modular Data Structure Verification. PhD thesis, EECS Department, Massachusetts Institute of Technology (February 2007)
Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean Algebra with Presburger Arithmetic. J. of Automated Reasoning (2006)
Kuncak, V., Rinard, M.: On the theory of structural subtyping. Technical Report 879, LCS, Massachusetts Institute of Technology (2003)
Kuncak, V., Rinard, M.: Towards efficient satisfiability checking for Boolean Algebra with Presburger Arithmetic. In: CADE-21 (2007)
Lugiez, D.: Multitree automata that count. Theor. Comput. Sci. 333(1-2), 225–263 (2005)
Lugiez, D., Zilio, S.D.: Multitrees Automata, Presburger’s Constraints and Tree Logics. Research report 08-2002, LIF, Marseille, France (June 2002), http://www.lif-sud.univ-mrs.fr/Rapports/08-2002.html
Marnette, B., Kuncak, V., Rinard, M.: On algorithms and complexity for sets with cardinality constraints. Technical report, MIT CSAIL (August 2005)
Matiyasevich, Y.V.: Enumerable sets are Diophantine. Soviet Math. Doklady 11(2), 354–357 (1970)
Misra, J.: A logic for concurrent programming (in two parts): Safety and progress. Journal of Computer and Software Engineering 3(2), 239–300 (1995)
Nguyen, H.H., et al.: Automated verification of shape, size and bag properties via separation logic. In: VMCAI (2007)
Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
Nipkow, T., et al.: Multiset theory version 1.30 (Isabelle distribution) (2005), http://isabelle.in.tum.de/dist/library/HOL/Library/Multiset.html
Christos, H., Papadimitriou, C.H.: On the complexity of integer programming. J. ACM 28(4), 765–768 (1981)
Paulson, L.C.: Mechanizing a theory of program composition for UNITY. ACM Trans. Program. Lang. Syst. 23(5), 626–656 (2001)
Pottier, L.: Minimal solutions of linear diophantine systems: Bounds and algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488, Springer, Heidelberg (1991)
Schwartz, J.T.: On programming: An interim report on the SETL project. Technical report, Courant Institute, New York (1973)
Zarba, C.G.: Combining multisets with integers. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, Springer, Heidelberg (2002)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Piskac, R., Kuncak, V. (2008). Decision Procedures for Multisets with Cardinality Constraints. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2008. Lecture Notes in Computer Science, vol 4905. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78163-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-78163-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78162-2
Online ISBN: 978-3-540-78163-9
eBook Packages: Computer ScienceComputer Science (R0)