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On the Structure of Cycloids Introduced by Carl Adam Petri

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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

Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri’s general systems theory and have very different interpretations, ranging from Einstein’s relativity theory to elementary information processing gates. Despite their simple definitions, their properties are still not completely understood. This contribution provides for the first time a formal definition together with new results concerning their structure. For instance, it is shown that the minimal length of a cycle is the length of a local basic circuit, possibly decreased by an integer multiple of the number of semi-active transitions.

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Notes

  1. 1.

    In the net, the gaps are also ordinary black tokens, but represented here by a cross to distinguish them from the cars.

  2. 2.

    In his Hamburg lecture 2004 [8] and in [9] Petri introduced the denomination natural coordinates whereas in an earlier publication [7] the term Minkowski coordinates has been used. This shows that he also saw the necessity to use a different name.

  3. 3.

    See more about the notion of slowness in Sect. 4.

  4. 4.

    The cycloid model generator cyclogen written by Fenske, in combination with the RENEW tool.

  5. 5.

    The net is known as “oscillator net” or “four seasons net”. See also Fig. 9.

  6. 6.

    \(F^{-1}[x]\) is the relational image of the element x with respect to the inverse of the relation F.

  7. 7.

    denotes the set of positive integers.

  8. 8.

    \(\mathord {\equiv }[A]\) is the relational image of the set A with respect to the relation \(\mathord {\equiv }\).

    \([\![x]\!]_{\equiv }\) denotes the equivalence class to which x belongs in the quotient \(X_1/_\equiv \).

  9. 9.

    These sets are not disjoint.

  10. 10.

    Reported by Fenske.

  11. 11.

    They are in the concurrency relation co.

  12. 12.

    We will use the term initial marking, for short.

  13. 13.

    For the definitions of safe and secure see [7], whereas live was used in the usual form (e.g. see [2], p. 59). For a cycloid to be secure \(\alpha ,\beta ,\gamma ,\delta \ge 2\) is required.

  14. 14.

    First and last with respect to the space dimension.

  15. 15.

    It may be helpful for the reader to consider Fig. 7, which illustrates the proof and is explained afterwards.

  16. 16.

    Such unfoldings of the fundamental parallelogram have been frequently used by Petri, see “Nets, Time and Space” [7], Fig. 11, for instance.

References

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  9. Petri, C.A., Valk, R.: On the Physical Basics of Information Flow - Results obtained in cooperation Konrad Zuse. http://www.informatik.uni-hamburg.de/TGI/mitarbeiter/profs/petri_eng.html (2008)

  10. Smith, E., Reisig, W.: The semantics of a net is a net - an exercise in general net theory. In: Voss, K., Genrich, J., Rozenberg, G. (eds.) Concurrency and Nets, pp. 461–479. Springer, Berlin (1987). https://doi.org/10.1007/978-3-642-72822-8_29

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Hamburg, Hamburg, Germany

    Rüdiger Valk

Authors
  1. Rüdiger Valk
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Correspondence to Rüdiger Valk .

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

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Valk, R. (2018). On the Structure of Cycloids Introduced by Carl Adam Petri. 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_15

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  • DOI: https://doi.org/10.1007/978-3-319-91268-4_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91267-7

  • Online ISBN: 978-3-319-91268-4

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