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Quantitative evaluation of time-dependent Petri nets and applications to biochemical networks

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Abstract

Time Petri nets (TPN) are a well-known extension of standard Petri nets, where each transition gets a continuous time interval, specifying the range of the transition’s firing time. In contrast, Timed Petri nets are a different time-dependent extension where a time duration is associated with each transition. We sketch a locally defined transformation from a Timed into a Time Petri net. Additionally, we consider time-dependent Petri nets, where the firing of each transition lasts a certain time which is limited by both a lower and an upper bound. These nets can also be transformed locally into TPN and are used in this paper for modelling and analysing biochemical systems, and we present algorithms allowing their quantitative analyses. We consider algorithms which work for arbitrary systems, i.e., bounded as well as unbounded ones, and algorithms, which are suitable for bounded systems only. The crucial point is the state space reduction, which exploits basically two ideas: parametric state description and discretisation of the state space. Altogether, we introduce eight problems, characterised by their input/ output relation. A sketch of the solution idea as well as possible application scenarios to evaluate biochemical systems are given, too.

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Notes

  1. Each non-trivial non-negative integer solution y of the equality \( y^{ T} \cdot C_{{\mathcal{N}}} = 0\) is called a P-invariant of the Petri net \({\mathcal{N}}.\)

  2. Proved with TINA, Berthomieu (2008).

  3. The reachability graph was computed with INA (cf. Starke 2003).

  4. Each non-trivial non-negative integer solution x of the equality \(C_{{\mathcal{N}}}\cdot x = 0\) is called a T-invariant of the Petri net \({\mathcal{N}}\).

  5. Computed with INA (Starke 2003).

  6. Please remember that the places are ordered as follow: (aa, Y, pM, M, YP, C2, CP,…). The order of transitions is: (r 1, r 3, r ′4 , r 4, r 6, r 7, r 8, r 9, t 8, t 9, t 10, t 11).

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Acknowledgments

The author would like to thank Monika Heiner for many discussions in preparing this paper, Peter Starke for the implementation of time lengths for transition sequences in INA, Jan Richling and Bernard Berthomieu for their support in the computational experiments as well as Ben Collins for editing the paper.

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  1. Department of Computer Science, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Louchka Popova-Zeugmann

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  1. Louchka Popova-Zeugmann
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Popova-Zeugmann, L. Quantitative evaluation of time-dependent Petri nets and applications to biochemical networks. Nat Comput 10, 1017–1043 (2011). https://doi.org/10.1007/s11047-010-9211-3

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