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Extensional Petri net


Petri nets form a concurrent model for distributed and asynchronous systems. They are capable of modeling information flow in a closed system, but are generally not suitable for the study of compositionality. We address the issue of Petri net compositionality by introducing extensional Petri nets. In an extensional Petri net some places are external while others are internal. Every external place is labeled by a distinguished interface name. When composing two extensional Petri nets two places with a same label are coerced. An external place can be turned into an internal place by applying localization operator. The paper takes a look at bisimulation semantics and observational properties of the extensional Petri nets.

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  1. Baldan P, Bonchi F, Gadducci F (2009) Encoding asynchronous interactions using open Petri nets. In: CONCUR'09, LNCS, vol 5710, pp 99–114

  2. Baldan P, Corradini A, Ehrig H, Heckel R (2001) Compositional modeling of reactive systems using extensional nets. In: CONCUR'01, LNCS, vol 2154, pp 502–518

  3. Baldan, P., Corradini, A., Ehrig, H., Heckel, R.: Compositional semantics for open Petri nets based on deterministic processes. Math Struct Comput Sci 15(1), 1–35 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  4. Baldan P, Corradini A, Ehrig H, Heckel R, König B (2007) Bisimilarity and behaviour-preserving reconfigurations of open Petri nets. In: CALCO'07, LNCS, vol 4624, pp 126–142

  5. Baldan P, Corradini A, Ehrig H, König B (2008) Open Petri nets: non-deterministic processes and compositionality. In: ICGT'08, LNCS, vol 5214, pp 257–273

  6. Best, E., Devillers, R., Hall, J.G.: The Petri box calculus: a new causal algebra with multi-label communication. Advances in Petri nets, LNCS 609, 21–69 (1992)

    Google Scholar 

  7. Best E, Devillers R, Koutny M (2001) A unified model for nets and process algebras. In: Bergstra JA, Ponse A, Smolka SA (eds) Handbook of process algebra. Elsevier Science, Amsterdam, pp 875–944

  8. Best E, Devillers R, Koutny M (2001) Petri net algebra. In: Monographs in theoretical computer science, an EATCS series. Springer, Berlin

  9. Bergstra JA, Klop JW (1984) The algebra of recursively defined processes and the algebra of regular processes. In: ICALP'84, LNCS, vol 172, pp 82-95

  10. Busi N, Gorrieri R (1995) A Petri net semantics for pi-calculus. In: CONCUR' 95, LNCS, vol 962, pp 145–159

  11. Busi, N., Gorrieri, R.: Distributed semantics for the \(\pi \)-calculus based on Petri nets with inhibitor arcs. J Log Algebr Program 78(3), 138–162 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  12. Cao, M., Wu, Z., Yang, G.: Pi net–a new modular higher Petri net. J Shanghai Jiaotong Univ 38(1), 52–58 (2004)

    MathSciNet  Google Scholar 

  13. Devillers R, Klaudel H, Koutny M (2004) Petri net semantics of the finite pi calculus. In: FORTE'04, LNCS, vol 3235, no. 2, pp 309–325

  14. Degano, P., Nicola, R.D., Montanari, U.: A distributed operational semantics for CCS based on C/e Systems. Acta Inform 26, 59–91 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  15. Fu Y (2013) The value-passing calculus. In: Theories of programming and formal methods. Lecture Notes in Computer Science, vol 8051, pp 166–195

  16. Fu, Y.: Nondeterministic structure of computation. Math Struct Comput Sci 25, 1295–1338 (2015)

    Article  MATH  Google Scholar 

  17. Fu, Y.: Theory of interaction. Theor Comput Sci 611, 1–49 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  18. Fu Y (2017) On the power of name-passing communication. In: CONCUR 2017

  19. Fu, Y.: The universal process. Log Methods Comput Sci 13, 1–23 (2017)

    MathSciNet  Google Scholar 

  20. Fu, Y., Lv, H.: On the expressiveness of interaction. Theor Comput Sci 411, 1387–1451 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  21. Fu Y, Zhu H (2015) The name-passing calculus. arXiv:1508.00093

  22. Guo X, Hao K, Hou H, Ding J The representation of Petri nets with prohibition arcs by Pi+ calculus. J Syst Simul S2:9–12

  23. Goltz, U.: CCS and Petri nets. Chapter, semantics of systems of concurrent processes, LNCS 469, 334–357 (1990)

    MathSciNet  Article  Google Scholar 

  24. Hao, K.: Open nets–a model for interative concurrent systems. J Northwest Univ 27(6), 461–466 (1997)

    MathSciNet  Google Scholar 

  25. Hoare, C.A.R.: Communicating sequential processes. Commun ACM 21(8), 666–677 (1978)

    Article  MATH  Google Scholar 

  26. Karp, R., Miller, R.: Parallel program schemata. J Comput Syst Sci 3, 147–195 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  27. Koutny M, Esparza J, Best E (1994) Operational semantics for the Petri box calculus. In: CONCUR'94, LNCS, vol 836, pp 210–225

  28. Kindler, E.: A compositional partial order semantics for Petri net components. Application and theory of Petri nets, LNCS 1248, 235–252 (1997)

    MathSciNet  Google Scholar 

  29. Kosaraju S (1982) Decidability of reachability in vector addition systems. In: STOC, pp 267–281

  30. Koutny, M., Best, E.: Operational and denotational semantics for the box algebra. Theor Comput Sci 211(1–2), 1–83 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  31. Lambert, J.: A structure to decide reachability in Petri nets. Theor Comput Sci 99, 79–104 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  32. Leroux J, Schmitz S (2015) Demystifying reachabiity in vector addition systems. In: LICS 2015

  33. Liu, G., Jiang, C., Zhou, M.: Interactive Petri nets. IEEE Trans Syst Man Cybern Syst 43(2), 291–302 (2013)

    Article  Google Scholar 

  34. Mayr E (1981) An algorithm for the general Petri net reachability problem. In: STOC, pp 238–246

  35. Meseguer, J., Montanari, U.: Petri nets are monoids. Inf Comput 88, 105–155 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  36. Milner, R.: Communication and concurrency. Prentice Hall, Upper Saddle River (1989)

    MATH  Google Scholar 

  37. Milner R, Sangiorgi D (1992) Barbed bisimulation. In: ICALP'92, LNCS, pp 685-695

  38. Milner R, Parrow J, Walker D (1992) A calculus of mobile processes. Inf Comput 1–40 (Part I), 41–77 (Part II)

  39. Nielsen M (1987) CCS and its relationship to net theory. In: Petri nets: applications and relationships to other models of concurrency. LNCS, vol 255, pp 393–415

  40. Nielsen M, Priese L, Sassone V (1995) Characterizing behavioural congruences for Petri nets. In: CONCUR 955, vol 962, pp 175–189

  41. Park, D.: Concurrency and automata on infinite sequences, theoretical computer Science. LNCS 104, 167–183 (1981)

    Google Scholar 

  42. Peterson, J.: Petri net theory and the modelling of systems. Prentice Hall, Upper Saddle River (1981)

    MATH  Google Scholar 

  43. Priese, L., Wimmel, H.: A uniform approach to true-concurrency and interleaving semantics for Petri nets. Theor Comput Sci 206, 219–256 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  44. Priese, L.: On the concept of simulation in asynchronous, concurrent systems. Prog Cybern Syst Res 7, 85–92 (1978)

    Google Scholar 

  45. Rackoff, C.: The covering and boundedness problems for vector addition systems. Theor Comput Sci 6, 223–231 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  46. Sacerdote G, Tenney R (1977) The decidability of the reachability problem for vector addition systems. In: STOC, pp 61–76

  47. van der Aalst, W.: Pi calculus versus Petri nets: let us eat "humble pie" rather than further inflate the "Pi hype". BPTrends 3(5), 1–5 (2005)

    Google Scholar 

  48. van Glabbeek R, Weijland W (1989) Branching time and abstraction in bisimulation semantics. In: Information processing'89, pp 613–618

  49. Winskel, G.: Event structures. Petri nets: applications and relationships to other models of concurrency, LNCS 255, 325–392 (1987)

    MathSciNet  MATH  Google Scholar 

  50. Yu, Z., Cai, Y., Xu, H.: Petri net semantics for Pi Calculus. Control Decis 22(8), 864–868 (2007)

    MathSciNet  Google Scholar 

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The work is supported by National Natural Science Foundation of China (61472239, 61772336). The authors would like to thank the anonymous reviewers for the comments.

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Correspondence to Yuxi Fu.

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Naijun Zhan, Heike Wehrheim, Martin Fränzle, and Deepak Kapur

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Dong, X., Fu, Y. & Varacca, D. Extensional Petri net. Form Asp Comp 31, 47–58 (2019).

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  • Petri net
  • Compositionality
  • Bisimulation