Ideal Abstractions for Well-Structured Transition Systems

  • Damien Zufferey
  • Thomas Wies
  • Thomas A. Henzinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7148)


Many infinite state systems can be seen as well-structured transition systems (WSTS), i.e., systems equipped with a well-quasi-ordering on states that is also a simulation relation. WSTS are an attractive target for formal analysis because there exist generic algorithms that decide interesting verification problems for this class. Among the most popular algorithms are acceleration-based forward analyses for computing the covering set. Termination of these algorithms can only be guaranteed for flattable WSTS. Yet, many WSTS of practical interest are not flattable and the question whether any given WSTS is flattable is itself undecidable. We therefore propose an analysis that computes the covering set and captures the essence of acceleration-based algorithms, but sacrifices precision for guaranteed termination. Our analysis is an abstract interpretation whose abstract domain builds on the ideal completion of the well-quasi-ordered state space, and a widening operator that mimics acceleration and controls the loss of precision of the analysis. We present instances of our framework for various classes of WSTS. Our experience with a prototype implementation indicates that, despite the inherent precision loss, our analysis often computes the precise covering set of the analyzed system.


Coverability Problem Widening Operator Communication Graph Process Term Abstract Domain 
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.


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  1. 1.
    Abdulla, P.A., Cerans, K., Jonsson, B., Tsay, Y.-K.: General decidability theorems for infinite-state systems. In: LICS, pp. 313–321 (1996)Google Scholar
  2. 2.
    Abdulla, P.A., Collomb-Annichini, A., Bouajjani, A., Jonsson, B.: Using forward reachability analysis for verification of lossy channel systems. FMSD 25(1), 39–65 (2004)zbMATHGoogle Scholar
  3. 3.
    Abdulla, P.A., Jonsson, B.: Verifying programs with unreliable channels. In: LICS, pp. 160–170 (1993)Google Scholar
  4. 4.
    Azzopardi, T.: Generic compute server in Scala using remote actors (2008), (accessed November 2011)
  5. 5.
    Bagnara, R., Hill, P.M., Zaffanella, E.: Widening operators for powerset domains. Software Tools for Technology Transfer 8(4/5), 449–466 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Calcagno, C., Distefano, D., O’Hearn, P.W., Yang, H.: Compositional shape analysis by means of bi-abduction. In: POPL, pp. 289–300 (2009)Google Scholar
  7. 7.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252 (1977)Google Scholar
  8. 8.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: POPL, pp. 269–282. ACM (1979)Google Scholar
  9. 9.
    Cousot, P., Cousot, R.: Abstract interpretation frameworks. Journal of Logic and Computation 2(4), 511–547 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dufourd, C., Finkel, A., Schnoebelen, P.: Reset Nets Between Decidability and Undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Engelfriet, J., Gelsema, T.: Multisets and structural congruence of the pi-calculus with replication. Theor. Comput. Sci. 211(1-2), 311–337 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Finkel, A., Goubault-Larrecq, J.: Forward Analysis for WSTS, Part I: Completions. In: STACS. Dagstuhl Sem. Proc., vol. 09001, pp. 433–444 (2009)Google Scholar
  13. 13.
    Finkel, A., Goubault-Larrecq, J.: Forward Analysis for WSTS, Part II: Complete WSTS. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 188–199. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theor. Comput. Sci. 256(1-2), 63–92 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ganty, P., Raskin, J.-F., Van Begin, L.: A Complete Abstract Interpretation Framework for Coverability Properties of WSTS. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 49–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Geeraerts, G., Raskin, J.-F., Van Begin, L.: Expand, Enlarge and Check: New algorithms for the coverability problem of WSTS. J. Comput. Syst. Sci. 72(1), 180–203 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goubault-Larrecq, J.: On noetherian spaces. In: LICS, pp. 453–462. IEEE Computer Society (2007)Google Scholar
  18. 18.
    Haller, P., Odersky, M.: Scala actors: Unifying thread-based and event-based programming. Theor. Comput. Sci. 410(2-3), 202–220 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Joshi, S., König, B.: Applying the Graph Minor Theorem to the Verification of Graph Transformation Systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 214–226. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. Syst. Sci. 3(2), 147–195 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lift. Lift web framework,
  22. 22.
    Meyer, R.: On boundedness in depth in the pi-calculus. In: IFIP TCS. IFIP, vol. 273, pp. 477–489. Springer, Boston (2008)Google Scholar
  23. 23.
    Milner, E.C.: Basic wqo- and bqo-theory. Graphs and order (1985)Google Scholar
  24. 24.
    Milner, R.: The polyadic pi-calculus: A tutorial. In: Logic and Algebra of Specification. Computer and Systems Sciences. Springer, Heidelberg (1993)Google Scholar
  25. 25.
    Petri, C.A., Reisig, W.: Scholarpedia 3(4), 6477 (2008),
  26. 26.
    Rival, X., Mauborgne, L.: The trace partitioning abstract domain. ACM Trans. Program. Lang. Syst. 29(5) (2007)Google Scholar
  27. 27.
    Schnoebelen, P.: Revisiting Ackermann-Hardness for Lossy Counter Machines and Reset Petri Nets. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 616–628. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  28. 28.
    Wies, T., Zufferey, D., Henzinger, T.A.: Forward Analysis of Depth-Bounded Processes. In: Ong, L. (ed.) FOSSACS 2010. LNCS, vol. 6014, pp. 94–108. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  29. 29.
    Zufferey, D., Wies, T.: Picasso Analyzer,
  30. 30.
    Zufferey, D., Wies, T., Henzinger, T.A.: On ideal abstractions for well-structured transition systems. Technical Report IST-2011-10, IST Austria (November 2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Damien Zufferey
    • 1
  • Thomas Wies
    • 2
  • Thomas A. Henzinger
    • 1
  1. 1.ISTAustria
  2. 2.New York UniversityUSA

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