Flat Counter Automata Almost Everywhere!

  • Jérôme Leroux
  • Grégoire Sutre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3707)


This paper argues that flatness appears as a central notion in the verification of counter automata. A counter automaton is called flat when its control graph can be “replaced”, equivalently w.r.t. reachability, by another one with no nested loops. From a practical view point, we show that flatness is a necessary and sufficient condition for termination of accelerated symbolic model checking, a generic semi-algorithmic technique implemented in successful tools like Fast, Lash or TReX. From a theoretical view point, we prove that many known semilinear subclasses of counter automata are flat: reversal bounded counter machines, lossy vector addition systems with states, reversible Petri nets, persistent and conflict-free Petri nets, etc. Hence, for these subclasses, the semilinear reachability set can be computed using a uniform accelerated symbolic procedure (whereas previous algorithms were specifically designed for each subclass).


Regular Language Reachability Problem Counter Machine Reachability Property Presburger Arithmetic 
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. [ABS01]
    Annichini, A., Bouajjani, A., Sighireanu, M.: TReX: A tool for reachability analysis of complex systems. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 368–372. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. [AK77]
    Araki, T., Kasami, T.: Decidable problems on the strong connectivity of Petri net reachability sets. Theoretical Computer Science 4(1), 99–119 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BF97]
    Bouziane, Z., Finkel, A.: Cyclic petri net reachability sets are semi-linear effectively constructible. In: Proc. 2nd Int. Workshop on Verification of Infinite State Systems (INFINITY 1997), Bologna, Italy, July 1997. Electronic Notes in Theor. Comp. Sci, vol. 9. Elsevier, Amsterdam (1997)Google Scholar
  4. [BFLP03]
    Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: Fast Acceleration of Symbolic Transition systems. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. [BGWW97]
    Boigelot, B., Godefroid, P., Willems, B., Wolper, P.: The power of QDDs. In: Van Hentenryck, P. (ed.) SAS 1997. LNCS, vol. 1302, pp. 172–186. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  6. [BH99]
    Bouajjani, A., Habermehl, P.: Symbolic reachability analysis of FIFO-channel systems with nonregular sets of configurations. Theoretical Computer Science 221(1–2), 211–250 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BM99]
    Bouajjani, A., Mayr, R.: Model checking lossy vector addition systems. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 323–333. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. [Boi03]
    Boigelot, B.: On iterating linear transformations over recognizable sets of integers. Theoretical Computer Science 309(2), 413–468 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [BW94]
    Boigelot, B., Wolper, P.: Symbolic verification with periodic sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994)Google Scholar
  10. [CJ98]
    Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. [Dic13]
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with r distinct prime factors. Amer. Journal Math. 35, 413–422 (1913)CrossRefzbMATHGoogle Scholar
  12. [DJS99]
    Dufourd, C., Jančar, P., Schnoebelen, P.: Boundedness of Reset P/T nets. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 301–310. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. [Esp97]
    Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae 31(1), 13–25 (1997)zbMATHMathSciNetGoogle Scholar
  14. [FIS03]
    Finkel, A., Iyer, S.P., Sutre, G.: Well-abstracted transition systems: Application to FIFO automata. Information and Computation 181(1), 1–31 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [FL02]
    Finkel, A., Leroux, J.: How to compose Presburger-accelerations: Applications to broadcast protocols. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. [FO97]
    Fribourg, L., Olsén, H.: A decompositional approach for computing least fixed-points of Datalog programs with Z-counters. Constraints 2(3/4), 305–335 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [FS00a]
    Finkel, A., Sutre, G.: An algorithm constructing the semilinear post * for 2-dim reset/transfer vass. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 353–362. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. [FS00b]
    Finkel, A., Sutre, G.: Decidability of reachability problems for classes of two counters automata. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 346–357. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. [GS66]
    Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas and languages. Pacific J. Math. 16(2), 285–296 (1966)zbMATHMathSciNetGoogle Scholar
  20. [Hir94]
    Hirshfeld, Y.: Congruences in commutative semigroups. Research report ECS-LFCS-94-291, Laboratory for Foundations of Computer Science, University of Edinburgh, UK (1994)Google Scholar
  21. [HP79]
    Hopcroft, J.E., Pansiot, J.-J.: On the reachability problem for 5-dimensional vector addition systems. Theoretical Computer Science 8(2), 135–159 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [Iba78]
    Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. Journal of the ACM 25(1), 116–133 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Kos82]
    Kosaraju, S.R.: Decidability of reachability in vector addition systems. In: Proc. 14th ACM Symp. Theory of Computing (STOC 1982), San Francisco, CA, May 1982, pp. 267–281 (1982)Google Scholar
  24. [Las]
  25. [Lat04]
    Latour, L.: From automata to formulas: Convex integer polyhedra. In: Proc. 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004), Turku, Finland 2004, pp. 120–129. IEEE Comp. Soc. Press, Los Alamitos (2004)CrossRefGoogle Scholar
  26. [LR78]
    Landweber, L.H., Robertson, E.L.: Properties of conflict-free and persistent petri nets. Journal of the ACM 25(3), 352–364 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  27. [LS04]
    Leroux, J., Sutre, G.: On flatness for 2-dimensional vector addition systems with states. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 402–416. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  28. [May81]
    Mayr, E.W.: Persistence of vector replacement systems is decidable. Acta Informatica 15, 309–318 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  29. [May84]
    Mayr, E.W.: An algorithm for the general Petri net reachability problem. SIAM J. Comput. 13(3), 441–460 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  30. [Min67]
    Minsky, M.L.: Computation: Finite and Infinite Machines, 1st edn. Prentice-Hall, London (1967)zbMATHGoogle Scholar
  31. [Tai68]
    Taiclin, M.A.: Algorithmic problems for commutative semigroups. Soviet Math. Doklady 9(1), 201–204 (1968)MathSciNetGoogle Scholar
  32. [VVN81]
    Valk, R., Vidal-Naquet, G.: Petri nets and regular languages. Journal of Computer and System Sciences 23(3), 299–325 (1981)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jérôme Leroux
    • 1
  • Grégoire Sutre
    • 2
  1. 1.IRISA, Vertecs ProjectRennesFrance
  2. 2.LaBRI, CNRS UMR 5800TalenceFrance

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