The Theory of WSTS: The Case of Complete WSTS

  • Alain Finkel
  • Jean Goubault-Larrecq
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7347)


We describe a simple, conceptual forward analysis procedure for ∞-complete WSTS \(\mathfrak S\). This computes the so-called clover of a state. When \(\mathfrak S\) is the completion of a WSTS \(\mathfrak X\), the clover in \(\mathfrak S\) is a finite description of the downward closure of the reachability set. We show that such completions are ∞-complete exactly when \(\mathfrak X\) is an ω 2 -WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized Karp-Miller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are clover-flattable. Finally, we apply this to well-structured Presburger counter systems.


Transition System Forward Analysis Downward Closure Forward Procedure Directed Subset 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AJ94]
    Abdulla, P.A., Jonsson, B.: Undecidable Verification Problems for Programs with Unreliable Channels. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 327–346. Springer, Heidelberg (1994)Google Scholar
  2. [ABJ98]
    Abdulla, P., Bouajjani, A., Jonsson, B.: On-The-Fly Analysis of Systems With Unbounded, Lossy Fifo Channels. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 305–318. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. [ADB07]
    Abdulla, P.A., Delzanno, G., Van Begin, L.: Comparing the Expressive Power of Well-Structured Transition Systems. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 99–114. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. [AN00]
    Abdulla, P., Nylén, A.: Better is Better than Well: On Efficient Verification of Infinite-State Systems. In: 14th LICS, pp. 132–140 (2000)Google Scholar
  5. [ACABJ04]
    Abdulla, P.A., Collomb-Annichini, A., Bouajjani, A., Jonsson, B.: Using forward reachability analysis for verification of lossy channel systems. Formal Methods in System Design 25(1), 39–65 (2004)zbMATHCrossRefGoogle Scholar
  6. [AČJT00]
    Abdulla, P.A., Čerāns, K., Jonsson, B., Tsay, Y.-K.: Algorithmic analysis of programs with well quasi-ordered domains. Information and Computation 160(1-2), 109–127 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [ADMN04]
    Abdulla, P.A., Deneux, J., Mahata, P., Nylén, A.: Forward Reachability Analysis of Timed Petri Nets. In: Lakhnech, Y., Yovine, S. (eds.) FORMATS/FTRTFT 2004. LNCS, vol. 3253, pp. 343–362. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. [AJ94]
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford University Press (1994)Google Scholar
  9. [BF12]
    Bonnet, R., Finkel, A.: Forward Analysis for WSTS: Beyond Regular Accelerations (February 2012) (submitted)Google Scholar
  10. [BFLS05]
    Bardin, S., Finkel, A., Leroux, J., Schnoebelen, P.: Flat Acceleration in Symbolic Model Checking. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 474–488. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. [BFHRV11]
    Bonnet, R., Finkel, A., Haddad, S., Rosa-Velardo, F.: Ordinal Theory for Expressiveness of Well Structured Transition Systems. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 153–167. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. [BG11]
    Bozzelli, L., Ganty, P.: Complexity Analysis of the Backward Coverability Algorithm for VASS. In: Delzanno, G., Potapov, I. (eds.) RP 2011. LNCS, vol. 6945, pp. 96–109. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. [CFP96]
    Cécé, G., Finkel, A., Purushothaman Iyer, S.: Unreliable channels are easier to verify than perfect channels. Information and Computation 124(1), 20–31 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [CFS11]
    Chambart, P., Finkel, A., Schmitz, S.: Forward Analysis and Model Checking for Trace Bounded WSTS. In: Kristensen, L.M., Petrucci, L. (eds.) PETRI NETS 2011. LNCS, vol. 6709, pp. 49–68. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. [CFS11]
    Chambart, P., Finkel, A., Schmitz, S.: Forward Analysis and Model Checking for Trace Bounded WSTS. In: Kristensen, L.M., Petrucci, L. (eds.) PETRI NETS 2011. LNCS, vol. 6709, pp. 49–68. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. [DFGvD06]
    Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Towards a Model-Checker for Counter Systems. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 493–507. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. [DFS98]
    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
  18. [EFM99]
    Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: 14th LICS, pp. 352–359 (1999)Google Scholar
  19. [EN98]
    Allen Emerson, E., Namjoshi, K.S.: On model-checking for non-deterministic infinite-state systems. In: 13th LICS, pp. 70–80 (1998)Google Scholar
  20. [FFSS11]
    Figueira, D., Figueira, S., Schmitz, S., Schnoebelen, P.: Ackermannian and Primitive-Recursive Bounds with Dickson’s Lemma. In: LICS, pp. 269–278 (2011)Google Scholar
  21. [Fin87]
    Finkel, A.: A Generalization of the Procedure of Karp and Miller to Well Structured Transition Systems. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 499–508. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  22. [Fin90]
    Finkel, A.: Reduction and covering of infinite reachability trees. Information and Computation 89(2), 144–179 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [Fin93]
    Finkel, A.: The Minimal Coverability Graph for Petri Nets. In: Rozenberg, G. (ed.) APN 1993. LNCS, vol. 674, pp. 210–243. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  24. [FMP04]
    Finkel, A., McKenzie, P., Picaronny, C.: A well-structured framework for analysing Petri net extensions. Information and Computation 195(1-2), 1–29 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [FS01]
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere? Theoretical Computer Science 256(1-2), 63–92 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [FG09a]
    Finkel, A., Goubault-Larrecq, J.: Forward analysis for WSTS, part I: Completions. In: Albers, S., Marion, J.-Y. (eds.) Proceedings of the 26th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2009). Leibniz International Proceedings in Informatics, vol. 3, pp. 433–444. Leibniz-Zentrum für Informatik, Freiburg (2009)Google Scholar
  27. [FG09b]
    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. LNCS, vol. 5556, pp. 188–199. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  28. [FG12a]
    Finkel, A., Goubault-Larrecq, J.: Forward analysis for WSTS, part I: Completions (in preparation, 2012); Journal version of [FG09a]Google Scholar
  29. [FG12b]
    Finkel, A., Goubault-Larrecq, J.: Forward analysis for WSTS, Part II: Complete WSTS (in preparation, 2012); Journal version of [FG09b]Google Scholar
  30. [GRVB07]
    Geeraerts, G., Raskin, J.-F., Van Begin, L.: Well-structured languages. Acta Inf. 44(3-4), 249–288 (2007)zbMATHCrossRefGoogle Scholar
  31. [GHK+03]
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press (2003)Google Scholar
  32. [GRvB06a]
    Ganty, P., Raskin, J.-F., Van Begin, L.: A Complete Abstract Interpretation Framework for Coverability Properties of WSTS. In: Allen Emerson, E., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 49–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  33. [GRvB06b]
    Geeraerts, G., Raskin, J.-F., van Begin, L.: Expand, enlarge and check: New algorithms for the coverability problem of WSTS. J. Comp. and System Sciences 72(1), 180–203 (2006)zbMATHCrossRefGoogle Scholar
  34. [GRvB07]
    Geeraerts, G., Raskin, J.-F., Van Begin, L.: On the Efficient Computation of the Minimal Coverability Set for Petri Nets. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 98–113. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  35. [GS64]
    Ginsburg, S., Spanier, E.H.: Bounded Algol-like languages. Trans. American Mathematical Society 113(2), 333–368 (1964)MathSciNetzbMATHGoogle Scholar
  36. [HP79]
    Hopcroft, J., Pansiot, J.-J.: On the reachability problem for 5-dimensional vector addition systems. Theoretical Computer Science 8, 135–159 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Jan99]
    Jančar, P.: A note on well quasi-orderings for powersets. Information Processing Letters 72(5-6), 155–160 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  38. [KM69]
    Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comp. and System Sciences 3(2), 147–195 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  39. [KS96]
    Kouchnarenko, O., Schnoebelen, P.: A model for recursive-parallel programs. Electr. Notes Theor. Comput. Sci. 5, 30 pages (1996)CrossRefGoogle Scholar
  40. [LNORW07]
    Lazić, R.S., Newcomb, T., Ouaknine, J., Roscoe, A.W., Worrell, J.B.: Nets with Tokens Which Carry Data. In: Kleijn, J., Yakovlev, A. (eds.) ICATPN 2007. LNCS, vol. 4546, pp. 301–320. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  41. [Mar94]
    Marcone, A.: Foundations of BQO theory. Trans. Amer. Math. Soc. 345(2), 641–660 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  42. [May03a]
    Mayr, R.: Undecidable problems in unreliable computations. Theor. Comput. Sci. 297(1-3), 337–354 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  43. [May03b]
    Mayr, R.: Undecidable problems in unreliable computations. Theoretical Computer Science 297(1-3), 337–354 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  44. [MM81]
    Mayr, E.W., Meyer, A.R.: The complexity of the finite containment problem for petri nets. J. ACM 28(3), 561–576 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  45. [Rac78]
    Rackoff, C.: The covering and boundedness problems for vector addition systems. Theor. Comput. Sci. 6, 223–231 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  46. [RMF11]
    Rosa-Velardo, F., Martos-Salgado, M., de Frutos-Escrig, D.: Accelerations for the Coverability Set of Petri Nets with Names. Fundam. Inform. 113(3-4), 313–341 (2011)zbMATHGoogle Scholar
  47. [RS04]
    Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory, Series B 92(2), 325–357 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Sch01]
    Schnoebelen, P.: Bisimulation and Other Undecidable Equivalences for Lossy Channel Systems. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 385–399. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  49. [SS11]
    Schmitz, S., Schnoebelen, P.: Multiply-Recursive Upper Bounds with Higman’s Lemma. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 441–452. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  50. [Val78]
    Valk, R.: Self-Modidying Nets, a Natural Extension of Petri Nets. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978. LNCS, vol. 62, pp. 464–476. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  51. [VF07]
    Rosa-Velardo, F., de Frutos-Escrig, D.: Name Creation vs. Replication in Petri Net Systems. In: Kleijn, J., Yakovlev, A. (eds.) ICATPN 2007. LNCS, vol. 4546, pp. 402–422. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  52. [VG05]
    Verma, K.N., Goubault-Larrecq, J.: Karp-Miller trees for a branching extension of VASS. Discrete Mathematics & Theoretical Computer Science 7(1), 217–230 (2005)MathSciNetzbMATHGoogle Scholar
  53. [WZH10]
    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
  54. [ZWH12]
    Zufferey, D., Wies, T., Henzinger, T.A.: Ideal Abstractions for Well-Structured Transition Systems. In: Kuncak, V., Rybalchenko, A. (eds.) VMCAI 2012. LNCS, vol. 7148, pp. 445–460. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alain Finkel
    • 1
  • Jean Goubault-Larrecq
    • 1
  1. 1.ENS CachanFrance

Personalised recommendations