Language Inclusion for Finite Prime Event Structures

  • Andreas Fellner
  • Thorsten Tarrach
  • Georg WeissenbacherEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11990)


We study the problem of language inclusion between finite, labeled prime event structures. Prime event structures are a formalism to compactly represent concurrent behavior of discrete systems. A labeled prime event structure induces a language of sequences of labels produced by the represented system. We study the problem of deciding inclusion and membership for languages encoded by finite prime event structures and provide complexity results for both problems. We provide a family of examples where prime event structures are exponentially more succinct than formalisms that do not take concurrency into account. We provide a decision algorithm for language inclusion that exploits this succinctness. Furthermore, we provide an implementation of the algorithm and an evaluation on a series of benchmarks. Finally, we demonstrate how our results can be applied to mutation-based test case generation.


Event structures Language inclusion Concurrency Mutation-based test case generation 


  1. 1.
    Abdulla, P.A., Chen, Y.-F., Holík, L., Mayr, R., Vojnar, T.: When simulation meets antichains. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 158–174. Springer, Heidelberg (2010). Scholar
  2. 2.
    Bertoni, A., Mauri, G., Sabadini, N.: Equivalence and membership problems for regular trace languages. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 61–71. Springer, Heidelberg (1982). Scholar
  3. 3.
    Bertoni, A., Mauri, G., Sabadini, N.: Membership problems for regular and context-free trace languages. Inf. Comput. 82(2), 135–150 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bouajjani, A., Enea, C., Lahiri, S.K.: Abstract semantic diffing of evolving concurrent programs. In: Ranzato, F. (ed.) Proceedings of the Static Analysis - 24th International Symposium, SAS 2017, New York, NY, USA, 30 August–1 September 2017, LNCS, vol. 10422, pp. 46–65. Springer (2017). Scholar
  5. 5.
    Boudol, G.: Flow event structures and flow nets. In: Guessarian, I. (ed.) LITP 1990. LNCS, vol. 469, pp. 62–95. Springer, Heidelberg (1990). Scholar
  6. 6.
    Černý, P., et al.: From non-preemptive to preemptive scheduling using synchronization synthesis. Formal Methods Syst. Des. 50(2), 97–139 (2017)CrossRefGoogle Scholar
  7. 7.
    Diekert, V., Métivier, Y.: Partial commutation and traces. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 457–533. Springer, Heidelberg (1997). Scholar
  8. 8.
    Dietsch, D., Jakobs, M.C.: VMCAI 2020 virtual machine, November 2019.
  9. 9.
    Clarke, J.E.M., Grumberg, O., Kroening, D., Peled, D., Veith, H.: Model Checking. 2nd edn. MIT Press (2018)Google Scholar
  10. 10.
    Esparza, J.: Decidability and complexity of Petri net problems — an introduction. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1491, pp. 374–428. Springer, Heidelberg (1998). Scholar
  11. 11.
    Esparza, J., Nielsen, M.: Decidability issues for Petri nets - a survey. Bull. EATCS 52, 244–262 (1994)zbMATHGoogle Scholar
  12. 12.
    Fatès, N.: A guided tour of asynchronous cellular automata. In: Kari, J., Kutrib, M., Malcher, A. (eds.) AUTOMATA 2013. LNCS, vol. 8155, pp. 15–30. Springer, Heidelberg (2013). Scholar
  13. 13.
    Fellner, A., Krenn, W., Schlick, R., Tarrach, T., Weissenbacher, G.: Model-based, mutation-driven test-case generation via heuristic-guided branching search. ACM Trans. Embed. Comput. Syst. 18(1), 4:1–4:28 (2019)CrossRefGoogle Scholar
  14. 14.
    Fellner, A., Tarrach, T., Weissenbacher, G.: Language inclusion for finite prime event structures (2019).
  15. 15.
    Fellner, A., Tarrach, T., Weissenbacher, G.: Language Inclusion for Finite Prime Event Structures. Artifact (2019).
  16. 16.
    Friedman, E.P.: The inclusion problem for simple languages. Theor. Comput. Sci. 1(4), 297–316 (1976)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gaubert, S., Giua, A.: Petri net languages and infinite subsets of Nm. J. Comput. Syst. Sci. 59(3), 373–391 (1999). Scholar
  18. 18.
    van Glabbeek, R., Goltz, U.: Refinement of actions in causality based models. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1989. LNCS, vol. 430, pp. 267–300. Springer, Heidelberg (1990). Scholar
  19. 19.
    van Glabbeek, R.J., Plotkin, G.D.: Configuration structures, event structures and Petri nets. Theor. Comput. Sci. 410(41), 4111–4159 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Godefroid, P.: Partial-order methods for the verification of concurrent systems (1996)Google Scholar
  21. 21.
    Grabowski, J.: The unsolvability of some Petri net language problems. Inf. Process. Lett. 9(2), 60–63 (1979). Scholar
  22. 22.
    Hack, M.: Decidability questions for Petri Nets. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (1976).
  23. 23.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. 3rd edn. Pearson (2013)Google Scholar
  24. 24.
    Jancar, P.: Nonprimitive recursive complexity and undecidability for Petri net equivalences. Theor. Comput. Sci. 256(1–2), 23–30 (2001). Scholar
  25. 25.
    Jones, N.D., Landweber, L.H., Lien, Y.E.: Complexity of some problems in Petri nets. Theor. Comput. Sci. 4(3), 277–299 (1977). Scholar
  26. 26.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). Scholar
  27. 27.
    Ko, K.I., Lin, C.L.: On the complexity of min-max optimization problems and their approximation. In: Du, D.Z., Pardalos, P.M. (eds.) Minimax and Applications. Nonconvex Optimization and Its Applications, vol. 4, pp. 219–239. Springer, Boston (1995). Scholar
  28. 28.
    Lamport, L., et al.: Paxos made simple. ACM SIGACT News 32(4), 18–25 (2001)Google Scholar
  29. 29.
    Madhusudan, P.: Model-checking trace event structures. In: Proceedings of the 18th IEEE Symposium on Logic in Computer Science (LICS 2003), Ottawa, Canada, 22–25 June 2003, pp. 371–380. IEEE Computer Society (2003).
  30. 30.
    Mayr, R., Clemente, L.: Advanced automata minimization. In: ACM SIGPLAN Notices, vol. 48, pp. 63–74. ACM (2013)Google Scholar
  31. 31.
    Mazurkiewicz, A.: Trace theory. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) ACPN 1986. LNCS, vol. 255, pp. 278–324. Springer, Heidelberg (1987). Scholar
  32. 32.
    Mazurkiewicz, A.: Introduction to trace theory. In: The Book of Traces, pp. 3–41 (1995)CrossRefGoogle Scholar
  33. 33.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: SWAT (FOCS), pp. 125–129 (1972)Google Scholar
  34. 34.
    Nguyen, H.T.T., Rodríguez, C., Sousa, M., Coti, C., Petrucci, L.: Quasi-optimal partial order reduction. In: Chockler, H., Weissenbacher, G. (eds.) CAV 2018. LNCS, vol. 10982, pp. 354–371. Springer, Cham (2018). Scholar
  35. 35.
    Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains, part I. Theor. Comput. Sci. 13(1), 85–108 (1981)CrossRefGoogle Scholar
  36. 36.
    Penczek, W.: Model-checking for a subclass of event structures. In: Brinksma, E. (ed.) TACAS 1997. LNCS, vol. 1217, pp. 145–164. Springer, Heidelberg (1997). Scholar
  37. 37.
    Peterson, J.: Petri Net Theory and the Modeling of Systems. Independently Published (2019).
  38. 38.
    Rodríguez, C., Sousa, M., Sharma, S., Kroening, D.: Unfolding-based partial order reduction. In: 26th International Conference on Concurrency Theory (CONCUR 2015), pp. 456–469 (2015)Google Scholar
  39. 39.
    Stearns, R.E., Hunt III, H.B.: On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata. SIAM J. Comput. 14(3), 598–611 (1985)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Valk, R., Vidal-Naquet, G.: Petri nets and regular languages. J. Comput. Syst. Sci. 23(3), 299–325 (1981). Scholar
  41. 41.
    Valmari, A.: Stubborn sets for reduced state space generation. In: Rozenberg, G. (ed.) ICATPN 1989. LNCS, vol. 483, pp. 491–515. Springer, Heidelberg (1991). Scholar
  42. 42.
    Van Glabbeek, R., Goltz, U.: Refinement of actions and equivalence notions for concurrent systems. Acta Informatica 37(4–5), 229–327 (2001)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Winskel, G.: An introduction to event structures. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1988. LNCS, vol. 354, pp. 364–397. Springer, Heidelberg (1989). Scholar
  44. 44.
    Winskel, G.: Event structures, stable families and concurrent games (2016)Google Scholar
  45. 45.
    Winskel, G., Nielsen, M.: Handbook of logic in computer science. In: Models for Concurrency, vol. 4, pp. 1–148. Oxford University Press Inc., New York (1995).
  46. 46.
    Wrathall, C.: Complete sets and the polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 23–33 (1976)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zielonka, W.: Notes on finite asynchronous automata. RAIRO-Theor. Inf. Appl. 21(2), 99–135 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Andreas Fellner
    • 1
    • 2
  • Thorsten Tarrach
    • 1
  • Georg Weissenbacher
    • 2
    Email author
  1. 1.AIT Austrian Institute of TechnologyViennaAustria
  2. 2.TU WienViennaAustria

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