Abstract
One-counter automata are a fundamental and widely-studied class of infinite-state systems. In this paper we consider one-counter automata with counter updates encoded in binary—which we refer to as the succinct encoding. It is easily seen that the reachability problem for this class of machines is in PSpace and is NP-hard. One of the main results of this paper is to show that this problem is in fact in NP, and is thus NP-complete.
We also consider parametric one-counter automata, in which counter updates be integer-valued parameters. The reachability problem asks whether there are values for the parameters such that a final state can be reached from an initial state. Our second main result shows decidability of the reachability problem for parametric one-counter automata by reduction to existential Presburger arithmetic with divisibility.
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References
Abdulla, P.A., Cerans, K.: Simulation is decidable for one-counter nets. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 253–268. Springer, Heidelberg (1998)
Alur, R., Henzinger, T.A., Vardi, M.Y.: Parametric real-time reasoning. In: Proc. STOC 1993, pp. 592–601. ACM, New York (1993)
Bouajjani, A., Bozga, M., Habermehl, P., Iosif, R., Moro, P., Vojnar, T.: Programs with lists are counter automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 517–531. Springer, Heidelberg (2006)
Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 577–588. Springer, Heidelberg (2006)
Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and presburger arithmetic. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427. Springer, Heidelberg (1998)
Cormen, T., Leiserson, C., Rivest, R.: Introduction to algorithms. MIT Press and McGraw-Hill (1990)
Demri, S.: Logiques pour la spécification et vérification. Mémoire d’habilitation, Université Paris 7 (2007)
Demri, S., Gascon, R.: The effects of bounding syntactic resources on Presburger LTL. In: Proc. TIME 2007. IEEE Computer Society Press, Los Alamitos (2007)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Ibarra, O.H., Dang, Z.: On two-way finite automata with monotonic counters and quadratic diophantine equations. Theor. Comput. Sci. 312(2-3), 359–378 (2004)
Ibarra, O.H., Dang, Z.: On the solvability of a class of diophantine equations and applications. Theor. Comput. Sci. 352(1), 342–346 (2006)
Ibarra, O.H., Jiang, T., Trân, N., Wang, H.: New decidability results concerning two-way counter machines and applications. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 313–324. Springer, Heidelberg (1993)
Jančar, P., Kučera, A., Moller, F., Sawa, Z.: DP lower bounds for equivalence-checking and model-checking of one-counter automata. Information Computation 188(1), 1–19 (2004)
Kučera, A.: Efficient verification algorithms for one-counter processes. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, p. 317. Springer, Heidelberg (2000)
Lafourcade, P., Lugiez, D., Treinen, R.: Intruder deduction for AC-like equational theories with homomorphisms. In: Research Report LSV-04-16. LSV, ENS de Cachan (2004)
Leroux, J., Sutre, G.: Flat counter automata almost everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)
Lipshitz, L.: The diophantine problem for addition and divisibility. Transaction of the American Mathematical Society 235, 271–283 (1976)
Mayr, E.W.: An algorithm for the general petri net reachability problem. In: Proc. STOC 1981, pp. 238–246. ACM, New York (1981)
Minsky, M.: Recursive unsolvability of post’s problem of ”tag” and other topics in theory of turing machines. Annals of Math. 74(3) (1961)
Xie, G., Dang, Z., Ibarra, O.H.: A solvable class of quadratic diophantine equations with applications to verification of infinite-state systems. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 668–680. Springer, Heidelberg (2003)
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Haase, C., Kreutzer, S., Ouaknine, J., Worrell, J. (2009). Reachability in Succinct and Parametric One-Counter Automata. In: Bravetti, M., Zavattaro, G. (eds) CONCUR 2009 - Concurrency Theory. CONCUR 2009. Lecture Notes in Computer Science, vol 5710. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04081-8_25
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DOI: https://doi.org/10.1007/978-3-642-04081-8_25
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