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Flat Parametric Counter Automata

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Automata, Languages and Programming (ICALP 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4052))

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Abstract

In this paper we study the reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic. The equivalence between non-parametric flat counter automata and Presburger arithmetic has been established previously by Comon and Jurski [5]. We simplify their proof by introducing finite state automata defined over alphabets of a special kind of graphs (zigzags). This framework allows one to express also the reachability problem for parametric automata with one control loop as the existence of solutions of a 1-parametric linear Diophantine systems. The latter problem is shown to be decidable, using a number-theoretic argument. Finally, the general reachability problem for parametric flat counter automata with more than one loops is shown to be undecidable, by reduction from Hilbert’s Tenth Problem [9].

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References

  1. 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)

    Chapter  Google Scholar 

  2. Finkel, A., Leroux, J., Petrucci, L., Bardin, S.: FAST: Fast Acceleration of Symbolic Transition Systems. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, Springer, Heidelberg (2003)

    Google Scholar 

  3. Boigelot, B.: On iterating linear transformations over recognizable sets of integers. TCS 309(2), 413–468 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Comon, H., Cortier, V.: Flatness Is Not a Weakness. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 262–276. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. 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)

    Chapter  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173 (1931)

    Article  Google Scholar 

  9. Hilbert, D.: Mathematische probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris. In: Nachrichten von der Königliche Gesellschaft der Wissenschaften zu Göttingen, pp. 253–297 (1900)

    Google Scholar 

  10. Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. Journal of the Association for Computing Machinery 25(1), 116–133 (1978)

    MATH  MathSciNet  Google Scholar 

  11. Ibarra, O.H., Dang, Z.: On the solvability of a class of diophantine equations and applications (submitted, 2005)

    Google Scholar 

  12. Matiyasevich, Y.: Enumerable sets are diophantine. Journal of Sovietic Mathematics 11, 354–358 (1970)

    MATH  Google Scholar 

  13. Matiyasevich, Y.: Personal communication (2005)

    Google Scholar 

  14. Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  15. Pheidas, T., Zahidi, K.: Undecidability of existential theories of rings and fields: A survey. Contemporary Mathematics 270, 49–106 (2000)

    MathSciNet  Google Scholar 

  16. Pottier, L.: Solutions minimales des systemes diophantiens lineaires: bornes et algorithmes. Technical Report 1292, INRIA Sophia Antipolis (1990)

    Google Scholar 

  17. Presburger, M.: Uber die Vollstandigkeit eines gewissen Systems der Arithmetik. In: Comptes rendus du I Congrés des Pays Slaves, Warsaw (1929)

    Google Scholar 

  18. Wolper, P., Boigelot, B.: Verifying systems with infinite but regular state spaces. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 88–97. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. VERIMAG, 2 Avenue de Vignate, 38610, Gières, France

    Marius Bozga, Radu Iosif & Yassine Lakhnech

Authors
  1. Marius Bozga
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  2. Radu Iosif
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  3. Yassine Lakhnech
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università Ca’ Foscari, Venice,  

    Michele Bugliesi

  2. Interdisciplinary Institute for BroadBand Technology (IBBT), Belgium

    Bart Preneel

  3. ECS, University of Southampton, UK

    Vladimiro Sassone

  4. FB Informatik, LS2, Univ. Dortmund, 44221, Dortmund, Germany

    Ingo Wegener

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© 2006 Springer-Verlag Berlin Heidelberg

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Bozga, M., Iosif, R., Lakhnech, Y. (2006). Flat Parametric Counter Automata. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_49

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  • DOI: https://doi.org/10.1007/11787006_49

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35907-4

  • Online ISBN: 978-3-540-35908-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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