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
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)
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)
Boigelot, B.: On iterating linear transformations over recognizable sets of integers. TCS 309(2), 413–468 (2003)
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)
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)
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)
Ginsburg, S., Spanier, E.H.: Semigroups, presburger formulas and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173 (1931)
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)
Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. Journal of the Association for Computing Machinery 25(1), 116–133 (1978)
Ibarra, O.H., Dang, Z.: On the solvability of a class of diophantine equations and applications (submitted, 2005)
Matiyasevich, Y.: Enumerable sets are diophantine. Journal of Sovietic Mathematics 11, 354–358 (1970)
Matiyasevich, Y.: Personal communication (2005)
Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)
Pheidas, T., Zahidi, K.: Undecidability of existential theories of rings and fields: A survey. Contemporary Mathematics 270, 49–106 (2000)
Pottier, L.: Solutions minimales des systemes diophantiens lineaires: bornes et algorithmes. Technical Report 1292, INRIA Sophia Antipolis (1990)
Presburger, M.: Uber die Vollstandigkeit eines gewissen Systems der Arithmetik. In: Comptes rendus du I Congrés des Pays Slaves, Warsaw (1929)
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)
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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
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