A Logical Product Approach to Zonotope Intersection

  • Khalil Ghorbal
  • Eric Goubault
  • Sylvie Putot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6174)


We define and study a new abstract domain which is a fine-grained combination of zonotopes with (sub-)polyhedric domains such as the interval, octagon, linear template or polyhedron domains. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid systems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection. We describe some examples and an implementation of our method in the APRON library, and discuss some further interesting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra.


Equality Test Abstract Interpretation Reachability Analysis Abstract Domain Logical Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adje, A., Gaubert, S., Goubault, E.: Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis. In: Proceedings of European Symposium on Programming (to appear, 2010)Google Scholar
  2. 2.
    Allamigeon, X., Gaubert, S., Goubault, E.: Inferring Min and Max Invariants Using Max-plus Polyhedra. In: Alpuente, M., Vidal, G. (eds.) SAS 2008. LNCS, vol. 5079, pp. 189–204. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Bouissou, O., Conquet, E., Cousot, P., Cousot, R., Feret, J., Ghorbal, K., Goubault, E., Lesens, D., Mauborgne, L., Mine, A., Putot, S., Rival, X.: Space software validation using abstract interpretation. In: Proceedings of the Int. Space System Engineering Conference, Data Systems in Aerospace DASIA’09 (2009)Google Scholar
  4. 4.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: Proceedings of SIBGRAPI (1993)Google Scholar
  5. 5.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proceedings of Principles Of Programming Languages, pp. 269–282. ACM Press, New York (1979)Google Scholar
  6. 6.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Proceedings of Principles of Programming Languages, pp. 84–96. ACM Press, New York (1978)Google Scholar
  7. 7.
    Ghorbal, K., Goubault, E., Putot, S.: The zonotope abstract domain Taylor1+. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 627–633. Springer, Heidelberg (2009)Google Scholar
  8. 8.
    Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Girard, A., Le Guernic, C.: Zonotope/hyperplane intersection for hybrid systems reachability analysis. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 215–228. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Goubault, E., Putot, S.: Weakly relational domains for floating-point computation analysis. Presented at the second international workshop on Numerical and Symbolic Abstract Domains (2005),
  11. 11.
    Goubault, E., Putot, S.: Static analysis of numerical algorithms. In: Yi, K. (ed.) SAS 2006. LNCS, vol. 4134, pp. 18–34. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Goubault, E., Putot, S.: Under-approximations of computations in real numbers based on generalized affine arithmetic. In: Riis Nielson, H., Filé, G. (eds.) SAS 2007. LNCS, vol. 4634, pp. 137–152. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Goubault, E., Putot, S.: Perturbed affine arithmetic for invariant computation in numerical program analysis. In: CoRR, abs/0807.2961 (2008),
  14. 14.
    Goubault, E., Putot, S.: A zonotopic framework for functional abstractions. In: CoRR, abs/0910.1763 (2009),
  15. 15.
    Gulwani, S., Tiwari, A.: Combining abstract interpreters. In: Proceedings of the ACM SIGPLAN conference on Programming language design and implementation, pp. 376–386. ACM Press, New York (2006)CrossRefGoogle Scholar
  16. 16.
    Keil, C.: Lurupa - rigorous error bounds in linear programming. In: Algebraic and Numerical Algorithms and Computer-assisted Proofs, Dagstuhl Seminar 5391 (2005)Google Scholar
  17. 17.
    Laviron, V., Logozzo, F.: Subpolyhedra: A (more) scalable approach to infer linear inequalities. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 229–244. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Manna, Z., Bradley, A.R.: The Calculus of Computation; Decision procedures with applications to verification. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  19. 19.
    Miné, A.: A new numerical abstract domain based on difference-bound matrices. In: Danvy, O., Filinski, A. (eds.) PADO 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    APRON Project. Numerical abstract domain library (2007),
  21. 21.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Khalil Ghorbal
    • 1
  • Eric Goubault
    • 1
  • Sylvie Putot
    • 1
  1. 1.Laboratory for the Modelling and Analysis of Interacting SystemsCEA, LISTGif-sur-YvetteFrance

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