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Flat Petri Nets (Invited Talk)

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Book cover Application and Theory of Petri Nets and Concurrency (PETRI NETS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12734))

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Abstract

Vector addition systems with states (VASS for short), or equivalently Petri nets are one of the most popular formal methods for the representation and the analysis of parallel processes. The central algorithmic problem is reachability: whether from a given initial configuration there exists a sequence of valid execution steps that reaches a given final configuration. This paper provides an overview of results about the reachability problem for VASS related to Presburger arithmetic, by presenting 1) a simple algorithm for deciding the reachability problem based on invariants definable in Presburger arithmetic, 2) the class of flat VASS for computing reachability sets in Presburger arithmetic, and 3) complexity results about the reachability problem for flat VASS.

The author is supported by the grant ANR-17-CE40-0028 of the French National Research Agency ANR (project BRAVAS).

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References

  1. Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. In: Chaudhuri, S., Kutten, S. (eds.) Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, PODC 2004, St. John’s, Newfoundland, Canada, July 25–28, 2004, pp. 290–299. ACM (2004). https://doi.org/10.1145/1011767.1011810

  2. Annichini, A., Asarin, E., Bouajjani, A.: Symbolic techniques for parametric reasoning about counter and clock systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 419–434. Springer, Heidelberg (2000). https://doi.org/10.1007/10722167_32

    Chapter  Google Scholar 

  3. 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). https://doi.org/10.1007/3-540-44585-4_34

    Chapter  Google Scholar 

  4. Bardin, S., Finkel, A., Leroux, J.: FASTer acceleration of counter automata in practice. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 576–590. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24730-2_42

    Chapter  MATH  Google Scholar 

  5. Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: fast acceleration of symbolic transition systems. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45069-6_12

    Chapter  Google Scholar 

  6. Bardin, S., Leroux, J., Point, G.: FAST extended release. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 63–66. Springer, Heidelberg (2006). https://doi.org/10.1007/11817963_9

    Chapter  Google Scholar 

  7. Bardin, S., Finkel, A., Leroux, J., Schnoebelen, P.: Flat acceleration in symbolic model checking. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 474–488. Springer, Heidelberg (2005). https://doi.org/10.1007/11562948_35

    Chapter  Google Scholar 

  8. Boigelot, B.: On iterating linear transformations over recognizable sets of integers. Theor. Comput. Sci. 309(1–3), 413–468 (2003). https://doi.org/10.1016/S0304-3975(03)00314-1

  9. Boigelot, B.: Domain-specific regular acceleration. Int. J. Softw. Tools Technol. Transf. 14(2), 193–206 (2012). https://doi.org/10.1007/s10009-011-0206-x

  10. Boigelot, B., Godefroid, P.: Symbolic verification of communication protocols with infinite state spaces using qdds. Formal Methods Syst. Des. 14(3), 237–255 (1999). https://doi.org/10.1023/A:1008719024240

  11. Boigelot, B., Godefroid, P., Willems, B., Wolper, P.: The power of QDDs (extended abstract). In: Van Hentenryck, P. (ed.) SAS 1997. LNCS, vol. 1302, pp. 172–186. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0032741

    Chapter  Google Scholar 

  12. Boigelot, B., Herbreteau, F.: The power of hybrid acceleration. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 438–451. Springer, Heidelberg (2006). https://doi.org/10.1007/11817963_40

    Chapter  Google Scholar 

  13. Boigelot, B., Herbreteau, F., Mainz, I.: Acceleration of affine hybrid transformations. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 31–46. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11936-6_4

    Chapter  Google Scholar 

  14. Boigelot, B., Wolper, P.: Symbolic verification with periodic sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58179-0_43

    Chapter  Google Scholar 

  15. Bouajjani, A., Habermehl, P.: Symbolic reachability analysis of FIFO-channel systems with nonregular sets of configurations. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 560–570. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63165-8_211

    Chapter  Google Scholar 

  16. Bouajjani, A., Habermehl, P.: Symbolic reachability analysis of fifo-channel systems with nonregular sets of configurations. Theor. Comput. Sci. 221(1–2), 211–250 (1999). https://doi.org/10.1016/S0304-3975(99)00033-X

  17. Bozga, M., Iosif, R.: On flat programs with lists. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 122–136. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-69738-1_9

    Chapter  Google Scholar 

  18. 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). https://doi.org/10.1007/11787006_49

    Chapter  MATH  Google Scholar 

  19. Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. Fundam. Informaticae 91(2), 275–303 (2009). https://doi.org/10.3233/FI-2009-0044

  20. Czerwiński, W., Lasota, S., Lazić, R., Leroux, J., Mazowiecki, F.: The reachability problem for petri nets is not elementary. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23–26, 2019, pp. 24–33. ACM (2019). https://doi.org/10.1145/3313276.3316369

  21. Czerwinski, W., Lasota, S., Lazic, R., Leroux, J., Mazowiecki, F.: Reachability in fixed dimension vector addition systems with states. In: Konnov, I., Kovács, L. (eds.) 31st International Conference on Concurrency Theory, CONCUR 2020, September 1–4, 2020, Vienna, Austria (Virtual Conference). LIPIcs, vol. 171, pp. 48:1–48:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.CONCUR.2020.48

  22. Englert, M., Hofman, P., Lasota, S., Lazic, R., Leroux, J., Straszynski, J.: A lower bound for the coverability problem in acyclic pushdown VAS. Inf. Process. Lett. 167, 106079 (2021). https://doi.org/10.1016/j.ipl.2020.106079

  23. Englert, M., Lazic, R., Totzke, P.: Reachability in two-dimensional unary vector addition systems with states is nl-complete. In: Grohe, M., Koskinen, E., Shankar, N. (eds.) Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016, New York, NY, USA, July 5–8, 2016, pp. 477–484. ACM (2016). https://doi.org/10.1145/2933575.2933577

  24. Esparza, J., Nielsen, M.: Decidability issues for petri nets - a survey. Bull. Eur. Assoc. Theor. Comput. Sci. 52, 245–262 (1994)

    MATH  Google Scholar 

  25. Esparza, J., Ganty, P., Leroux, J., Majumdar, R.: Verification of population protocols. In: Aceto, L., de Frutos-Escrig, D. (eds.) 26th International Conference on Concurrency Theory, CONCUR 2015, Madrid, Spain, September 1.4, 2015. LIPIcs, vol. 42, pp. 470–482. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015). https://doi.org/10.4230/LIPIcs.CONCUR.2015.470

  26. Finkel, A., Leroux, J.: How to compose presburger-accelerations: applications to broadcast protocols. In: Agrawal, M., Seth, A. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36206-1_14

    Chapter  Google Scholar 

  27. Fribourg, L., Olsén, H.: Proving safety properties of infinite state systems by compilation into Presburger arithmetic. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 213–227. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63141-0_15

    Chapter  Google Scholar 

  28. Hack, M.H.T.: Decidability questions for Petri nets. Ph.D. thesis, MIT (1975). http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-161.pdf

  29. Hauschildt, D.: Semilinearity of the Reachability Set is Decidable for Petri Nets. Ph.D. thesis, University of Hamburg (1990)

    Google Scholar 

  30. Hopcroft, J.E., Pansiot, J.J.: On the reachability problem for 5-dimensional vector addition systems. Theor. Comput. Sci. 8, 135–159 (1979)

    Article  MathSciNet  Google Scholar 

  31. Karp, R.M., Miller, R.E.: Parallel program schemata. J. Comput. Syst. Sci. 3(2), 147–195 (1969). https://doi.org/10.1016/S0022-0000(69)80011-5

    Article  MathSciNet  MATH  Google Scholar 

  32. Kosaraju, S.R.: Decidability of reachability in vector addition systems (preliminary version). In: STOC, pp. 267–281. ACM (1982). https://doi.org/10.1145/800070.802201

  33. Kosaraju, S.R., Sullivan, G.F.: Detecting cycles in dynamic graphs in polynomial time (preliminary version). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2–4, 1988, Chicago, Illinois, USA, pp. 398–406. ACM (1988). https://doi.org/10.1145/62212.62251

  34. Lambert, J.: A structure to decide reachability in Petri nets. Theor. Comput. Sci. 99(1), 79–104 (1992). https://doi.org/10.1016/0304-3975(92)90173-D

    Article  MathSciNet  MATH  Google Scholar 

  35. Leroux, J.: The general vector addition system reachability problem by Presburger inductive invariants. In: Logical Methods in Computer Science, vol. 6, no. 3 (2010). https://doi.org/10.2168/LMCS-6(3:22)2010

  36. Leroux, J.: Vector addition system reachability problem: a short self-contained proof. In: POPL, pp. 307–316. ACM (2011). https://doi.org/10.1145/1926385.1926421

  37. Leroux, J.: Vector addition systems reachability problem (A simpler solution). In: Turing-100. EPiC Series in Computing, vol. 10, pp. 214–228. EasyChair (2012). http://www.easychair.org/publications/paper/106497

  38. Leroux, J.: Presburger vector addition systems. In: 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, pp. 23–32. IEEE Computer Society (2013). https://doi.org/10.1109/LICS.2013.7

  39. Leroux, J., Schmitz, S.: Demystifying reachability in vector addition systems. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6–10, 2015, pp. 56–67. IEEE Computer Society (2015). https://doi.org/10.1109/LICS.2015.16

  40. Leroux, J., Schmitz, S.: Reachability in vector addition systems is primitive-recursive in fixed dimension. In: 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24–27, 2019, pp. 1–13. IEEE (2019). https://doi.org/10.1109/LICS.2019.8785796

  41. Leroux, J., Sutre, G.: On flatness for 2-dimensional vector addition systems with states. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 402–416. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28644-8_26

    Chapter  Google Scholar 

  42. Leroux, J., Sutre, G.: Flat counter automata almost everywhere! In: Software Verification: Infinite-State Model Checking and Static Program Analysis, 19.02. - 24.02.2006. Dagstuhl Seminar Proceedings, vol. 06081. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany (2006). http://drops.dagstuhl.de/opus/volltexte/2006/729

  43. Leroux, J.: When reachability meets grzegorczyk. In: Hermanns, H., Zhang, L., Kobayashi, N., Miller, D. (eds.) LICS 2020: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8–11, 2020, pp. 1–6. ACM (2020). https://doi.org/10.1145/3373718.3394732

  44. Mayr, E.W.: An algorithm for the general petri net reachability problem. In: Proceedings of the 13th Annual ACM Symposium on Theory of Computing, May 11–13, 1981, Milwaukee, Wisconsin, USA, pp. 238–246. ACM (1981). https://doi.org/10.1145/800076.802477

  45. Mayr, E.W.: An algorithm for the general Petri net reachability problem. SIAM J. Comput. 13(3), 441–460 (1984). https://doi.org/10.1137/0213029

    Article  MathSciNet  MATH  Google Scholar 

  46. Mayr, E.W., Meyer, A.R.: The complexity of the finite containment problem for petri nets. J. ACM 28(3), 561–576 (1981). https://doi.org/10.1145/322261.322271

    Article  MathSciNet  MATH  Google Scholar 

  47. McAloon, K.: Petri nets and large finite sets. Theor. Comput. Sci. 32, 173–183 (1984). https://doi.org/10.1016/0304-3975(84)90029-X

    Article  MathSciNet  MATH  Google Scholar 

  48. Sacerdote, G.S., Tenney, R.L.: The decidability of the reachability problem for vector addition systems (preliminary version). In: Proceedings of the 9th Annual ACM Symposium on Theory of Computing, May 4–6, 1977, Boulder, Colorado, USA, pp. 61–76. ACM (1977). https://doi.org/10.1145/800105.803396

  49. Schmitz, S.: The complexity of reachability in vector addition systems. SIGLOG News 3(1), 4–21 (2016). https://dl.acm.org/citation.cfm?id=2893585

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Authors and Affiliations

  1. University Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, Talence, 33405, France

    Jérôme Leroux

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  1. Jérôme Leroux
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Correspondence to Jérôme Leroux .

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  1. Université de Genève, Carouge, Switzerland

    Didier Buchs

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep Carmona

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Leroux, J. (2021). Flat Petri Nets (Invited Talk). In: Buchs, D., Carmona, J. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2021. Lecture Notes in Computer Science(), vol 12734. Springer, Cham. https://doi.org/10.1007/978-3-030-76983-3_2

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  • DOI: https://doi.org/10.1007/978-3-030-76983-3_2

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