CSL 2008: Computer Science Logic pp 124-138

# Fractional Collections with Cardinality Bounds, and Mixed Linear Arithmetic with Stars

• Ruzica Piskac
• Viktor Kuncak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5213)

## Abstract

We present decision procedures for logical constraints involving collections such as sets, multisets, and fuzzy sets. Element membership in our collections is given by characteristic functions from a finite universe (of unknown size) to a user-defined subset of rational numbers. Our logic supports standard operators such as union, intersection, difference, or any operation defined pointwise using mixed linear integer-rational arithmetic. Moreover, it supports the notion of cardinality of the collection, defined as the sum of occurrences of all elements. Deciding formulas in such logic has applications in software verification.

Our decision procedure reduces satisfiability of formulas with collections to satisfiability of formulas in an extension of mixed linear integer-rational arithmetic with a “star” operator. The star operator computes the integer cone (closure under vector addition) of the solution set of a given formula. We give an algorithm for eliminating the star operator, which reduces the problem to mixed linear integer-rational arithmetic. Star elimination combines naturally with quantifier elimination for mixed integer-rational arithmetic. Our decidability result subsumes previous special cases for sets and multisets. The extension with star is interesting in its own right because it can encode reachability problems for a simple class of transition systems.

## Keywords

verification and program analysis sets multisets fuzzy sets cardinality operator mixed linear integer-rational arithmetic

## References

1. 1.
Berezin, S., Ganesh, V., Dill, D.L.: An online proof-producing decision procedure for mixed-integer linear arithmetic. In: TACAS (2003)Google Scholar
2. 2.
Bradley, A.R., Manna, Z.: The Calculus of Computation. Springer, Heidelberg (2007)
3. 3.
Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)
4. 4.
Dutertre, B., de Moura, L.: Integrating Simplex with DPLL(T). Technical Report SRI-CSL-06-01, SRI International (2006)Google Scholar
5. 5.
Eisenbrand, F., Shmonin, G.: Carathéodory bounds for integer cones. Operations Research Letters 34(5), 564–568 (2006), http://dx.doi.org/10.1016/j.orl.2005.09.008
6. 6.
Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)
7. 7.
Ginsburg, S., Spanier, E.: Semigroups, Pressburger formulas and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)
8. 8.
Kuncak, V., Nguyen, H.H., Rinard, M.: Deciding Boolean Algebra with Presburger Arithmetic. J. of Automated Reasoning (2006), http://dx.doi.org/10.1007/s10817-006-9042-1
9. 9.
Kuncak, V., Rinard, M.: Towards efficient satisfiability checking for Boolean Algebra with Presburger Arithmetic. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 215–230. Springer, Heidelberg (2007)
10. 10.
Lugiez, D.: Multitree automata that count. Theor. Comput. Sci. 333(1-2), 225–263 (2005)
11. 11.
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
12. 12.
Piskac, R., Kuncak, V.: Decision procedures for multisets with cardinality constraints. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 218–232. Springer, Heidelberg (2008)
13. 13.
Piskac, R., Kuncak, V.: Linear arithmetic with stars. In: CAV (2008)Google Scholar
14. 14.
Piskac, R., Kuncak, V.: On linear arithmetic with stars. Technical Report LARA-REPORT-2008-005, EPFL (2008)Google Scholar
15. 15.
Pottier, L.: Minimal solutions of linear diophantine systems: Bounds and algorithms. In: Book, R.V. (ed.) RTA 1991. LNCS, vol. 488. Springer, Heidelberg (1991)Google Scholar
16. 16.
Pugh, W.: The Omega test: a fast and practical integer programming algorithm for dependence analysis. In: ACM/IEEE conf. Supercomputing (1991)Google Scholar
17. 17.
Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1998)
18. 18.
Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: ISSAC, pp. 129–136 (1999)Google Scholar
19. 19.
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
20. 20.
Zarba, C.G.: Combining multisets with integers. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392. Springer, Heidelberg (2002)Google Scholar