Abstract
We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ω-labeled input and output arcs, that is well-suited to analyse parametric concurrent systems with dynamic thread creation. Most techniques (such as the Karp and Miller tree or the Rackoff technique) that have been proposed in the setting of plain Petri nets do not apply directly to ωPN because ωPN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of ωPN. We show that an ωPN can be turned into a plain Petri net that allows to recover the reachability set of the ωPN, but that does not preserve termination. This yields complexity bounds for the reachability, (place) boundedness and coverability problems on ωPN. We provide a practical algorithm to compute a coverability set of the ωPN and to decide termination by adapting the classical Karp and Miller tree construction. We also adapt the Rackoff technique to ωPN, to obtain the exact complexity of the termination problem. Finally, we consider the extension of ωPN with reset and transfer arcs, and show how this extension impacts the decidability and complexity of the aforementioned problems.
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References
Abdulla, P.A., Cerans, K., Jonsson, B., Tsay, Y.-K.: General Decidability Theorems for Infinite-state Systems. In: LICS 1996. IEEE (1996)
Borosh, I., Treybig, L.: Bounds on positive integral solutions of linear diophantine equations. Proceedings of the American Mathematical Society 55(2), 299–304 (1976)
Brázdil, T., Jančar, P., Kučera, A.: Reachability games on extended vector addition systems with states. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 478–489. Springer, Heidelberg (2010)
Delzanno, G., Raskin, J.-F., Van Begin, L.: Towards the Automated Verification of Multithreaded Java Programs. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 173–187. Springer, Heidelberg (2002)
Delzano, G.: Constraint-Based Verification of Parameterized Cache Coherence Protocols. FMSD 23(3) (2003)
Dufourd, C.: Réseaux de Petri avec reset/transfert: Décidabilité et indécidabilité. PhD thesis, ENS de Cachan (1998)
Dufourd, C., Finkel, A., Schnoebelen, P.: Reset Nets Between Decidability and Undecidability. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 103–115. Springer, Heidelberg (1998)
Dufourd, C., Jančar, P., Schnoebelen, P.: Boundedness of reset P/T nets. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 301–310. Springer, Heidelberg (1999)
Esparza, J., Finkel, A., Mayr, R.: On the Verification of Broadcast Protocols. In: LICS 1999. IEEE (1999)
Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! TCS 256(1-2), 63–92 (2001)
Finkel, A., McKenzie, P., Picaronny, C.: A well-structured framework for analysing Petri net extensions. Inf. Comput. 195(1-2), 1–29 (2004)
Geeraerts, G., Heußner, A., Raskin, J.F.: Queue-Dispatch Asynchronous Systems. In: To appear in the Proceedings of ACSD. IEEE (2013), http://arxiv.org/abs/1201.4871
Geeraerts, G., Heußner, A., Praveen, M., Raskin, J.F.: ω-Petri nets ArXiV.org CoRR abs/1301.6572 (2013), http://arxiv.org/abs/1301.6572
Geeraerts, G., Raskin, J.-F., Van Begin, L.: Expand, enlarge and check: New algorithms for the coverability problem of wsts. J. Comput. Syst. Sci. 72(1) (2006)
German, S., Sistla, A.: Reasoning about systems with many processes. J. ACM 39(3), 675–735 (1992)
Karp, R.M., Miller, R.E.: Parallel Program Schemata. JCSS 3, 147–195 (1969)
Lipton, R.: The reachability problem requires exponential space. Tech. Report. Yale University (1963)
Mayr, E.W.: An algorithm for the general Petri net reachability problem. SIAM J. of Computing 3(13), 441–460 (1984)
Petri, C.A.: Kommunikation mit Automaten. PhD thesis, Institut fur Instrumentelle Mathematik, Bonn (1962)
Rackoff, C.: The covering and boundedness problems for vector addition systems. TCS 6, 223–231 (1978)
Reisig, W.: Petri Nets: An Introduction. Springer (1985)
Schnoebelen, P.: Revisiting Ackermann-hardness for lossy counter machines and reset Petri nets. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 616–628. Springer, Heidelberg (2010)
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Geeraerts, G., Heussner, A., Praveen, M., Raskin, JF. (2013). ω-Petri Nets. In: Colom, JM., Desel, J. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2013. Lecture Notes in Computer Science, vol 7927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38697-8_4
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DOI: https://doi.org/10.1007/978-3-642-38697-8_4
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