Lazy Constrained Monotonic Abstraction

  • Zeinab GanjeiEmail author
  • Ahmed Rezine
  • Petru Eles
  • Zebo Peng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9583)


We introduce Lazy Constrained Monotonic Abstraction (lazy CMA for short) for lazily and soundly exploring well structured abstractions of infinite state non-monotonic systems. CMA makes use of infinite state and well structured abstractions by forcing monotonicity wrt. refinable orderings. The new orderings can be refined based on obtained false positives in a CEGAR like fashion. This allows for the verification of systems that are not monotonic and are hence inherently beyond the reach of classical analysis based on the theory of well structured systems. In this paper, we consistently improve on the existing approach by localizing refinements and by avoiding to trash the explored state space each time a refinement step is required for the ordering. To this end, we adapt ideas from classical lazy predicate abstraction and explain how we address the fact that the number of control points (i.e., minimal elements to be visited) is a priori unbounded. This is unlike the case of plain lazy abstraction which relies on the fact that the number of control locations is finite. We propose several heuristics and report on our experiments using our open source prototype. We consider both backward and forward explorations on non-monotonic systems automatically derived from concurrent programs. Intuitively, the approach could be regarded as using refinable upward closure operators as localized widening operators for an a priori arbitrary number of control points.


Constrained monotonic abstraction Lazy exploration Well structured systems Safety properties Counter machines reachability 


  1. 1.
    Abdulla, P.A., Delzanno, G., Rezine, A.: Parameterized verification of infinite-state processes with global conditions. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 145–157. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Abdulla, P.A., Čerāns, K., Jonsson, B., Tsay, Y.-K.: General decidability theorems for infinite-state systems. In: Proceedigs of the LICS 1996, \(11^{th}\) IEEE International Symposium on Logic in Computer Science, pp. 313–321 (1996)Google Scholar
  3. 3.
    Abdulla, P.A., Chen, Y.-F., Delzanno, G., Haziza, F., Hong, C.-D., Rezine, A.: Constrained monotonic abstraction: a CEGAR for parameterized verification. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 86–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Abdulla, P.A., Delzanno, G., Ben Henda, N., Rezine, A.: Regular model checking without transducers (on efficient verification of parameterized systems). In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 721–736. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Ball, T., Rajamani, S.K.: The SLAM Project: debugging system software via static analysis. In: Proceedings of the 29th ACM SIGPLAN-SIGACT, POPL 2002, pp. 1–3. ACM, New York (2002)Google Scholar
  6. 6.
    Bardin, S., Finkel, A., Leroux, J.: FASTer acceleration of counter automata in practice. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 576–590. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Bardin, S., Leroux, J., Point, G.: FAST extended release. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 63–66. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Bonnet, R., Finkel, A., Leroux, J., Zeitoun, M.: Model checking vector addition systems with one zero-test (2012). arXiv preprint arXiv:1205.4458
  9. 9.
    Bouajjani, A., Bozga, M., Habermehl, P., Iosif, R., Moro, P., Vojnar, T.: Programs with lists are counter automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 517–531. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Clarke, E., Kroening, D., Sharygina, N., Yorav, K.: SATABS: SAT-based predicate abstraction for ANSI-C. In: TACAS, pp. 570–574. Springer (2005)Google Scholar
  11. 11.
    Cogumbreiro, T., Hu, R., Martins, F., Yoshida, N.: Dynamic deadlock verification for general barrier synchronisation. In: Proceeding of the 20th ACM SIGPLAN PPoPP Symposium, pp. 150–160. ACM (2015)Google Scholar
  12. 12.
    de Moura, L., Bjørner, N.S.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Delzanno, G.: Automatic verification of parameterized cache coherence protocols. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 53–68. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Donaldson, A.F., Kaiser, A., Kroening, D., Tautschnig, M., Wahl, T.: Counterexample-guided abstraction refinement for symmetric concurrent programs. Formal Meth. Syst. Des. 41(1), 25–44 (2012)zbMATHCrossRefGoogle Scholar
  15. 15.
    Downey, A.: The Little Book of SEMAPHORES (2nd Edition): The Ins and Outs of Concurrency Control and Common Mistakes. Createspace Ind, Pub (2009).
  16. 16.
    Farzan, A., Kincaid, Z., Podelski, A.: Proofs that count. In: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2014, pp. 151–164. ACM, New York (2014)Google Scholar
  17. 17.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere!. Theor. Comput. Sci. 256(1–2), 63–92 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Ganjei, Z., Rezine, A., Eles, P., Peng, Z.: Abstracting and counting synchronizing processes. In: D’Souza, D., Lal, A., Larsen, K.G. (eds.) VMCAI 2015. LNCS, vol. 8931, pp. 227–244. Springer, Heidelberg (2015)Google Scholar
  19. 19.
    Geeraerts, G., Raskin, J.-F., Van Begin, L.: Expand, enlarge and check.. made efficient. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 394–407. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Ghilardi, S., Ranise, S.: MCMT: a model checker modulo theories. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 22–29. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Henzinger, T.A., Jhala, R., Majumdar, R., Sutre, G.: Lazy abstraction. In: Proceedings of the 29th ACM SIGPLAN-SIGACT, POPL 2002, pp. 58–70. ACM, New York (2002)Google Scholar
  22. 22.
    Higman, G.; Ordering by divisibility in abstract algebras. In: Proceedings of the London Mathematical Society, pp. 326–336 (1952)Google Scholar
  23. 23.
    Kaiser, A., Kroening, D., Wahl, T.: Efficient coverability analysis by proof minimization. In: Koutny, M., Ulidowski, I. (eds.) CONCUR 2012. LNCS, vol. 7454, pp. 500–515. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Kloos, J., Majumdar, R., Niksic, F., Piskac, R.: Incremental, inductive coverability. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 158–173. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  25. 25.
    Liu, P., Wahl, T.: Infinite-state backward exploration of boolean broadcast programs. In: Proceedings of the 14th Conference on Formal Methods in Computer-Aided Design, pp. 155–162. FMCAD Inc (2014)Google Scholar
  26. 26.
    McMillan, K.L.: Lazy abstraction with interpolants. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 123–136. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Weissenbacher, G., Kroening, D., Malik, S.: Wolverine: battling bugs with interpolants. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 556–558. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Zeinab Ganjei
    • 1
    Email author
  • Ahmed Rezine
    • 1
  • Petru Eles
    • 1
  • Zebo Peng
    • 1
  1. 1.Linköping UniversityLinköpingSweden

Personalised recommendations