How to Compose Presburger-Accelerations: Applications to Broadcast Protocols

  • Alain Finkel
  • Jérôme Leroux
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2556)


Finite linear systems are finite sets of linear functions whose guards are defined by Presburger formulas, and whose the squares matrices associated generate a finite multiplicative monoid. We prove that for finite linear systems, the accelerations of sequences of transitions always produce an effective Presburger-definable relation. We then show how to choose the good sequences of length n whose number is polynomial in n although the total number of sequences of length n is exponential in n. We implement these theoretical results in the tool FAST [FAS] (Fast Acceleration of Symbolic Transition systems). FAST computes in few seconds the minimal deterministic finite automata that represent the reachability sets of 8 well-known broadcast protocols.


Presburger model checking verification infinite-state systems reachability set acceleration 


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  1. [AAB99]
    P. Abdulla, A. Annichini, and A. Bouajjani. Symbolic verification of lossy channel systems: Application to the bounded retransmission protocol. In (TACAS’99), volume 1579 of LNCS, pages 208–222. Springer-Verlag, 1999.Google Scholar
  2. [AAB00]
    A. Annichini, E. Asarin, and A. Bouajjani. Symbolic techniques for parametric reasoning about counter and clock systems. In (CAV’00), volume 1855 of LNCS, pages 419–434. Springer-Verlag, 2000.Google Scholar
  3. [AJ96]
    P. A. Abdulla and B. Jonsson. Verifying programs with unreliable channels. Information and Computation, 127(2):91–101, june 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BEM97]
    A. Bouajjani, J. Esparza, and O. Maler. Reachability analysis of pushdown automata: Application to model checking. In (CONCUR’97), volume 1243 of LNCS, pages 135–150. Springer-Verlag, June 1997.Google Scholar
  5. [BF99]
    B. Bérard and L. Fribourg. Reachability analysis of (timed) Petri nets using real arithmetic. In (CONCUR’99), volume 1664 of LNCS, pages 178–193. Springer-Verlag, 1999.Google Scholar
  6. [BF00]
    J.-P. Bodeveix and M. Filali. Fmona: a tool for expressing validation techniques over infinite state systems. In (TACAS’00), volume 1785 of LNCS, pages 204–219. Springer-Verlag, march 2000.Google Scholar
  7. [BGP97]
    T. Bultan, R. Gerber, and W. Pugh. Symbolic model checking of infinite state systems using presburger arithmetic. In (CAV’97), volume 1254 of LNCS, pages 400–411. Springer-Verlag, 1997.Google Scholar
  8. [BGWW97]
    B. Boigelot, P. Godefroid, B. Willems, and P. Wolper. The power of QDDs. In (SAS’97), volume 1302 of LNCS, pages 172–186. Springer-Verlag, 1997.Google Scholar
  9. [BH99]
    A. Bouajjani and P. Habermehl. Symbolic reachability analysis of FIFOchannel systems with nonregular sets of configurations. Theoretical Computer Science, 221(1–2):211–250, June 1999.Google Scholar
  10. [BJNT00]
    A. Bouajjani, B. Jonsson, M. Nilsson, and T. Touili. Regular model checking. In (CAV’00), volume 1855 of LNCS, pages 403–418. Springer-Verlag, July 2000.Google Scholar
  11. [Boi98]
    B. Boigelot. Symbolic Methods for Exploring Infinite State Spaces. PhD thesis, Université de Liège, 1998.Google Scholar
  12. [BW94]
    B. Boigelot and P. Wolper. Symbolic verification with periodic sets. In (CAV’94), volume 818 of LNCS, pages 55–67. Springer-Verlag, 1994.Google Scholar
  13. [CJ98]
    H. Comon and Y. Jurski. Multiple counters automata, safety analysis and presburger arithmetic. In (CAV’98), volume 1427 of LNCS, pages 268–279. Springer-Verlag, 1998.Google Scholar
  14. [Del00]
    Giorgio Delzann. Automatic verification of parameterized cache coherence protocols (extended and revised version). In (CAV’00), volume 1855 of LNCS, pages 53–68. Springer-Verlag, 2000.Google Scholar
  15. [DFS98]
    C. Dufourd, A. Finkel, and P. Schnoebelen. Reset nets between decidability and undecidability. In (ICALP’98), volume 1443 of LNCS, pages 103–115. Springer-Verlag, July 1998.Google Scholar
  16. [EFM99]
    J. Esparza, A. Finkel, and R. Mayr. On the verification of broadcast protocols. In (LICS’99), pages 352–359. IEEE Computer Society, July 1999.Google Scholar
  17. [EN98]
    A. Emerson and K. Namjoshi. On model checking for non-deterministic infinite-state systems. In (LICS’98), pages 70–80, 1998.Google Scholar
  18. [FB95]
    R. Floyd and R. W Beigel. Le langage des machines: introduction a la calculabilite et aux langages formels, chapter 7, page 433. (ITP 1995), 1995.Google Scholar
  19. [FL02]
    A. Finkel and J. Leroux. How to compose Presburger-accelerations: Applications to broadcast protocols. Technical report, Laboratoire Spécification et Vérification, CNRS UMR 8643 amp; ENS de Cachan, 2002.Google Scholar
  20. [FO97a]
    L. Fribourg and H. Olsén. A decompositional approach for computing least fixed-points of Datalog programs with Z-counters. Constraints, 2(3/4):305–335, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [FO97b]
    L. Fribourg and H. Olsén. Proving safety properties of infinite state systems by compilation into Presburger arithmetic. In (CONCUR’97), volume 1243 of LNCS, pages 213–227. Springer-Verlag, 1997.Google Scholar
  22. [FPS00]
    A. Finkel, S. Purushothaman Iyer, and G. Sutre. Well-abstracted transition systems. In (CONCUR’2000), volume 1877 of LNCS, pages 566–580. Springer-Verlag, 2000.Google Scholar
  23. [FS00a]
    A. Finkel and G. Sutre. An algorithm constructing the semilinear post* for 2-dim reset/transfer vass. In (MFCS’2000), volume 1893 of LNCS, pages 353–362. Springer-Verlag, 2000.Google Scholar
  24. [FS00b]
    A. Finkel and G. Sutre. Decidability of reachability problems for classes of two counters automata. In (STACS’2000), volume 1770 of LNCS, pages 346–357. Springer-Verlag, Feb 2000.Google Scholar
  25. [FWW97]
    A. Finkel, B. Willems, and P. Wolper. A direct symbolic approach to model checking pushdown systems (extended abstract). In (INFINITY’ 97), volume 9 of Electronical Notes in Theoretical Computer Science. Elsevier Science, July 1997.Google Scholar
  26. [GS66]
    S. Ginsburg and E. H. Spanier. Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics, 16(2):285–296, 1966.zbMATHMathSciNetGoogle Scholar
  27. [Jac78]
    G. Jacob. La finitude des representations lineaires des semi-groupes est decidable. J. Algebra, 52:437–459, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [MS77]
    A. Mandel and I. Simon. On finite semigroups of matrices. Theoretical Computer Science, 5(2):101–111, October 1977.CrossRefMathSciNetGoogle Scholar
  29. [MZ75]
    R. McNaughton and Y. Zalcstein. The burnside theorem for semi-groups. J. Algebra, 34:292–299, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [PS00]
    A. Pnueli and E. Shahar. Liveness and acceleration in parameterized verication. In (CAV’00), volume 1855 of LNCS, pages 328–343. Springer-Verlag, July 2000.Google Scholar
  31. [Rev90]
    P. Z. Revesz. A closed form for Datalog queries with integer order. In ICDT’90, Third International Conference on Database Theory, volume 470 of LNCS, pages 187–201. Springer-Verlag, December 1990.Google Scholar
  32. [SFRC99]
    G. Sutre, A. Finkel, O. Roux, and F. Cassez. Effective recognizability and model checking of reactive fiffo automata. In (AMAST’98), volume 1548 of LNCS, pages 106–123. Springer-Verlag, 1999.Google Scholar
  33. [Sut00]
    G. Sutre. Abstraction et accélération de systèmes infinis. PhD thesis, Ecole Normale Supérieure de Cachan, Laboratoire Spécification et Vérification. CNRS UMR 8643, october 2000.Google Scholar
  34. [WB98]
    P. Wolper and B. Boigelot. Verifying systems with infinite but regular state spaces. In (CAV’98), volume 1427 of LNCS, pages 88–97, June 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alain Finkel
    • 1
  • Jérôme Leroux
    • 1
  1. 1.Laboratoire Spécification et VérificationCNRS UMR 8643 amp; ENS de CachanCachan cedexFrance

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