Skip to main content
SpringerLink
Log in

Modeling Operational Semantics with Interval Orders Represented by Sequences of Antichains

  • Conference paper
  • First Online:
Application and Theory of Petri Nets and Concurrency (PETRI NETS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10877))

Abstract

A representation of interval orders by sequences of antichains is discussed and its relationship to the Fishburn’s representation by sequences of beginnings and endings is analyzed in detail. Using sequences of antichains to model operational semantics of elementary inhibitor nets is also discussed.

Partially supported by NSERC grant of Canada.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    ‘Operational semantics’ is not a generally agreed concept, in this paper this will be just a collection of all system runs (i.e. executions, observations) [3, 11, 15, 19]. A different meaning is used in for example [4].

  2. 2.

    For uncountable X it is additionally required that the equivalence relation \(\sim _<\) defined as \(a \sim _<~\!b \iff \forall c\in X. (c<a \Leftrightarrow c<b) \wedge (a<c \Leftrightarrow b<c)\) has countably many equivalence classes [6]. But in this paper we need only a simpler version for countable X, cf. [11].

  3. 3.

    Inhibitor arcs allow a transition to check for an absence of a token. In principle they allow ‘test for zero’, an operator the standard Petri nets do not have. They were introduced in [1].

References

  1. Agerwala, T., Flynn, M.: Comments on capabilities, limitations and “correctness” of Petri nets. Comput. Architect. News 4(2), 81–86 (1973)

    Article  Google Scholar 

  2. Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26, 832–843 (1983)

    Article  Google Scholar 

  3. Baldan, P., Busi, N., Corradini, A., Pinna, G.M.: Domain and event structure semantics for Petri nets with read and inhibitor arcs. Theor. Comput. Sci. 323, 129–189 (2004)

    Article  MathSciNet  Google Scholar 

  4. Bidoit, M., Hennicker, R., Wirsing, M.: Characterizing behavioural semantics and abstractor semantics. In: Sannella, D. (ed.) ESOP 1994. LNCS, vol. 788, pp. 105–119. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-57880-3_7

    Chapter  Google Scholar 

  5. Cormen, T.H., Leiserson, C.E., Rivest, D.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  6. Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  MathSciNet  Google Scholar 

  7. Fishburn, P.C.: Interval Orders and Interval Graphs. Wiley, New York (1985)

    MATH  Google Scholar 

  8. Grabowski, J.: On partial languages. Fundam. Inform. 4(2), 427–498 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Hopcroft, J.E., Motwani, R., Ullman, J.D.: Automata Theory, Languages, and Computation. Addison-Wesley, Boston (2001)

    MATH  Google Scholar 

  10. Janicki, R., Kleijn, J., Koutny, M., Mikulski, Ł.: Step traces. Acta Inform. 53, 35–65 (2016)

    Article  MathSciNet  Google Scholar 

  11. Janicki, R., Koutny, M.: Structure of concurrency. Theor. Comput. Sci. 112, 5–52 (1993)

    Article  MathSciNet  Google Scholar 

  12. Janicki, R., Koutny, M.: Semantics of inhibitor nets. Inf. Comput. 123(1), 1–16 (1995)

    Article  MathSciNet  Google Scholar 

  13. Janicki, R., Yin, X.: Modeling concurrency with interval orders. Inf. Comput. 253, 78–108 (2017)

    Article  Google Scholar 

  14. Kleijn, J., Koutny, M.: Process semantics of general inhibitor nets. Inf. Comput. 190, 18–69 (2004)

    Article  MathSciNet  Google Scholar 

  15. Kleijn, J., Koutny, M.: Formal languages and concurrent behaviours. In: Bel-Enguix, G., Jiménez-López, M.D., Martín-Vide, C. (eds.) New Developments in Formal Languages and Applications. SCI, vol. 113, pp. 125–182. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78291-9_5

    Chapter  MATH  Google Scholar 

  16. Nielsen, M., Rozenberg, G., Thiagarajan, P.S.: Behavioural notions for elementary net systems. Distrib. Comput. 4, 45–57 (1990)

    Article  MathSciNet  Google Scholar 

  17. Rozenberg, G., Engelfriet, J.: Elementary net systems. In: Reisig, W., Rozenberg, G. (eds.) ACPN 1996. LNCS, vol. 1491, pp. 12–121. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-65306-6_14

    Chapter  Google Scholar 

  18. van Glabbeek, R., Vaandrager, F.: Petri net models for algebraic theories of concurrency. In: de Bakker, J.W., Nijman, A.J., Treleaven, P.C. (eds.) PARLE 1987. LNCS, vol. 259, pp. 224–242. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-17945-3_13

    Chapter  Google Scholar 

  19. Vogler, W.: Partial order semantics and read arcs. Theor. Comput. Sci. 286(1), 33–63 (2002)

    Article  MathSciNet  Google Scholar 

  20. Wiener, N.: A contribution to the theory of relative position. In: Proceedings of the Cambridge Philosophical Society, vol. 17, pp. 441–449 (1914)

    Google Scholar 

  21. Zuberek, W.M.: Timed Petri nets and preliminary performance evaluation. In: Proceedings of the 7th Annual Symposium on Computer Architecture, La Baule, France, pp. 89–96 (1980)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing and Software, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Ryszard Janicki

Authors
  1. Ryszard Janicki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ryszard Janicki .

Editor information

Editors and Affiliations

  1. University of Newcastle, Newcastle upon Tyne, United Kingdom

    Victor Khomenko

  2. École Centrale de Nantes, Nantes Cedex 3, France

    Olivier H. Roux

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Janicki, R. (2018). Modeling Operational Semantics with Interval Orders Represented by Sequences of Antichains. In: Khomenko, V., Roux, O. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2018. Lecture Notes in Computer Science(), vol 10877. Springer, Cham. https://doi.org/10.1007/978-3-319-91268-4_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91268-4_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91267-7

  • Online ISBN: 978-3-319-91268-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation