Skip to main content
SpringerLink
Log in
SpringerLink
Log in

How to Compose Presburger-Accelerations: Applications to Broadcast Protocols

  • Conference paper
  • First Online:
FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2556))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 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. P. A. Abdulla and B. Jonsson. Verifying programs with unreliable channels. Information and Computation, 127(2):91–101, june 1996.

    Article  MATH  MathSciNet  Google Scholar 

  4. 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. 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. 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. 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. 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. 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. 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. B. Boigelot. Symbolic Methods for Exploring Infinite State Spaces. PhD thesis, Université de Liège, 1998.

    Google Scholar 

  12. 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. 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. 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. 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. 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. A. Emerson and K. Namjoshi. On model checking for non-deterministic infinite-state systems. In (LICS’98), pages 70–80, 1998.

    Google Scholar 

  18. 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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  21. 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. 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. 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. 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. 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. S. Ginsburg and E. H. Spanier. Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics, 16(2):285–296, 1966.

    MATH  MathSciNet  Google Scholar 

  27. G. Jacob. La finitude des representations lineaires des semi-groupes est decidable. J. Algebra, 52:437–459, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Mandel and I. Simon. On finite semigroups of matrices. Theoretical Computer Science, 5(2):101–111, October 1977.

    Article  MathSciNet  Google Scholar 

  29. R. McNaughton and Y. Zalcstein. The burnside theorem for semi-groups. J. Algebra, 34:292–299, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  30. 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. 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. 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. 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. 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 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Spécification et Vérification, CNRS UMR 8643 amp; ENS de Cachan, 61 av. du Président Wilson, 94235, Cachan cedex, France

    Alain Finkel & Jérôme Leroux

Authors
  1. Alain Finkel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jérôme Leroux
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology, 208016, Kanpur, India

    Manindra Agrawal  & Anil Seth  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Finkel, A., Leroux, J. (2002). How to Compose Presburger-Accelerations: Applications to Broadcast Protocols. In: Agrawal, M., Seth, A. (eds) FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2002. Lecture Notes in Computer Science, vol 2556. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36206-1_14

Download citation

  • DOI: https://doi.org/10.1007/3-540-36206-1_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00225-3

  • Online ISBN: 978-3-540-36206-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation