Canonizable Partial Order Generators

  • Mateus de Oliveira Oliveira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7183)


In a previous work we introduced slice graphs as a way to specify both infinite languages of directed acyclic graphs (DAGs) and infinite languages of partial orders. Therein we focused on the study of Hasse diagram generators, i.e., slice graphs that generate only transitive reduced DAGs. In the present work we show that any slice graph can be transitive reduced into a Hasse diagram generator representing the same set of partial orders. This result allow us to establish unknown connections between the true concurrent behavior of bounded p/t-nets and traditional approaches for representing infinite families of partial orders, such as Mazurkiewicz trace languages and Message Sequence Chart (MSC) languages. Going further, we identify the family of weakly saturated slice graphs. The class of partial order languages which can be represented by weakly saturated slice graphs is closed under union, intersection and a suitable notion of complementation (bounded cut-width complementation). The partial order languages in this class also admit canonical representatives in terms of Hasse diagram generators, and have decidable inclusion and emptiness of intersection. Our transitive reduction algorithm plays a fundamental role in these decidability results.


Partial Orders Automata Canonization 


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  1. 1.
    Alur, R., Yannakakis, M.: Model Checking of Message Sequence Charts. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 114–129. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Badouel, E., Darondeau, P.: Theory of Regions. In: Reisig, W., Rozenberg, G. (eds.) APN 1998. LNCS, vol. 1491, pp. 529–586. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Mathematical Systems Theory 20(2-3), 83–127 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bergenthum, R., Desel, J., Lorenz, R., Mauser, S.: Synthesis of Petri nets from finite partial languages. Fundamenta Informaticae 88(4), 437–468 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bergenthum, R., Desel, J., Mauser, S., Lorenz, R.: Synthesis of Petri nets from term based representations of infinite partial languages. Fundamenta Informaticae 95(1), 187–217 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Best, E., Wimmel, H.: Reducing k-Safe Petri Nets to Pomset-Equivalent 1-Safe Petri Nets. In: Nielsen, M., Simpson, D. (eds.) ICATPN 2000. LNCS, vol. 1825, pp. 63–82. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Bossut, F., Dauchet, M., Warin, B.: A Kleene theorem for a class of planar acyclic graphs. Inform. and Comput. 117(2), 251–265 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bozapalidis, Kalampakas: Recognizability of graph and pattern languages. Acta Informatica 42 (2006)Google Scholar
  9. 9.
    Courcelle, B.: Graph expressions and graph rewritings. Mathematical Systems Theory 20, 83–127 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Droste, M.: Concurrent automata and domains. International Journal of Foundations of Computer Science 3(4), 389–418 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ehrenfeucht, A., Rozenberg, G.: Partial (set) 2-structures. Part I: Basic notions and the representation problem. Acta Informatica 27(4), 315–342 (1989)CrossRefzbMATHGoogle Scholar
  12. 12.
    Engelfriet, J., Vereijken, J.J.: Context-free graph grammars and concatenation of graphs. Acta Informatica 34 (1997)Google Scholar
  13. 13.
    Esparza, J., Römer, S., Vogler, W.: An improvement of McMillan’s unfolding algorithm. Formal Methods in System Design 20(3), 285–310 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gaifman, H., Pratt, V.R.: Partial order models of concurrency and the computation of functions. In: Proc. of LICS 1987, pp. 72–85 (1987)Google Scholar
  15. 15.
    Giammarresi, D., Restivo, A.: Two-dimensional finite state recognizability. Fundamenta Informaticae 25(3), 399–422 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gischer, J.L.: The equational theory of pomsets. Theoret. Computer Science 61, 199–224 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grabowski, J.: On partial languages. Fundamenta Informaticae 4(2), 427 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hayman, J., Winskel, G.: The unfolding of general Petri nets. In: Proc. of FTTCS 2008. LIPIcs, vol. 2, pp. 223–234 (2008)Google Scholar
  19. 19.
    Henriksen, J.G., Mukund, M., Kumar, K.N., Sohoni, M.A., Thiagarajan, P.S.: A theory of regular MSC languages. Inform. and Comput. 202(1), 1–38 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hoogers, P., Kleijn, H., Thiagarajan, P.: A trace semantics for Petri nets. Inform. and Comput. 117(1), 98–114 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hoogers, P.W., Kleijn, H.C.M., Thiagarajan, P.S.: An event structure semantics for general Petri nets. Theoret. Computer Science 153(1-2), 129–170 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Husson, J.-F., Morin, R.: On Recognizable Stable Trace Languages. In: Tiuryn, J. (ed.) FOSSACS 2000. LNCS, vol. 1784, pp. 177–191. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  23. 23.
    Jategaonkar, L., Meyer, A.R.: Deciding true concurrency equivalences on safe, finite nets. Theoret. Computer Science 154(1), 107–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kupferman, O., Lustig, Y., Vardi, M.Y., Yannakakis, M.: Temporal synthesis for bounded systems and environments. In: STACS 2011, pp. 615–626 (2011)Google Scholar
  25. 25.
    Kuske, D., Morin, R.: Pomsets for local trace languages. Journal of Automata, Languages and Combinatorics 7(2), 187–224 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lodaya, K., Weil, P.: Series-parallel languages and the bounded-width property. Theoret. Computer Science 237(1-2), 347–380 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lorenz, R., Juhás, G., Bergenthum, R., Desel, J., Mauser, S.: Executability of scenarios in Petri nets. Theor. Comput. Sci. 410(12-13), 1190–1216 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mazurkiewicz, A.W.: Trace Theory. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) APN 1986. LNCS, vol. 255, pp. 279–324. Springer, Heidelberg (1987)Google Scholar
  29. 29.
    Montanari, U., Pistore, M.: Minimal Transition Systems for History-Preserving Bisimulation. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 413–425. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  30. 30.
    Morin, R.: On Regular Message Sequence Chart Languages and Relationships to Mazurkiewicz Trace Theory. In: Honsell, F., Miculan, M. (eds.) FOSSACS 2001. LNCS, vol. 2030, pp. 332–346. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  31. 31.
    Muscholl, A., Peled, D., Su, Z.: Deciding Properties for Message Sequence Charts. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 226–242. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  32. 32.
    de Oliveira Oliveira, M.: Hasse diagram generators and Petri nets. Fundamenta Informaticae 105(3), 263–289 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Thomas, W.: Finite-state recognizability of graph properties. Theorie des Automates et Applications 172, 147–159 (1992)Google Scholar
  34. 34.
    Vogler, W.: Modular Construction and Partial Order Semantics of Petri Nets. LNCS, vol. 625. Springer, Heidelberg (1992)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mateus de Oliveira Oliveira
    • 1
  1. 1.School of Computer Science and CommunicationKTH Royal Institute of TechnologyStockholmSweden

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