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On Interval Semantics of Inhibitor and Activator Nets

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Application and Theory of Petri Nets and Concurrency (PETRI NETS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11522))

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Abstract

An interval operational semantics - in a form of interval sequences and step sequences - is introduced for elementary activator nets, and a relationship between inhibitor and activator nets is discussed. It is known that inhibitor and activator nets can simulate themselves for both standard firing sequence semantics and firing step sequence semantics. This paper shows that inhibitor and activator nets are not equivalent with respect to interval sequence and interval step sequence semantics, however, in some sense, they might be interpreted as equivalent with respect to pure interval order operational semantics.

Partially supported by Discovery NSERC Grant of Canada.

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Notes

  1. 1.

    When the inhibitor arc \((b,B_c)\) is removed from the net \(IN^1\) of Fig. 3, the interval sequence \(B_aE_aB_bB_cE_bE_c\) is a proper firing sequence of this new \(N^1\) and \(\blacktriangleleft _{B_aE_aB_bB_cE_bE_c}=\prec ^{strat}_{\lnot IN}\), but the stratified order \(\prec ^{strat}_{\lnot IN}\) is not expected to be a behaviour generated by the net IN. See [18] for details.

  2. 2.

    It can easily be shown that \(m[\![\{B_{a_1},\ldots ,B_{a_k}\}\rangle \!\rangle _{IN}m' \iff m[\![B_{a_{i_1}}\rangle \!\rangle _{IN}\ldots \) \([\![B_{a_{i_k}}\rangle \!\rangle _{IN} m'\) for any permutation \(l_1,\ldots ,l_k\) of \(1,\ldots ,k\), which is not true for \([\![...\rangle \!\rangle _{AN}\).

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Acknowledgment

The author gratefully acknowledges three anonymous referees for their helpful comments.

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Authors and Affiliations

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

    Ryszard Janicki

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  1. Ryszard Janicki
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Correspondence to Ryszard Janicki .

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Editors and Affiliations

  1. Università di Torino, Turin, Italy

    Susanna Donatelli

  2. Inria and LSV, CNRS & ENS Paris-Saclay, Cachan Cedex, France

    Stefan Haar

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Janicki, R. (2019). On Interval Semantics of Inhibitor and Activator Nets. In: Donatelli, S., Haar, S. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2019. Lecture Notes in Computer Science(), vol 11522. Springer, Cham. https://doi.org/10.1007/978-3-030-21571-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-21571-2_12

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