Abstract
We investigate the problem of parameter synthesis for time Petri nets with a cost variable that evolves both continuously with time, and discretely when firing transitions. More precisely, parameters are rational symbolic constants used for time constraints on the firing of transitions and we want to synthesise all their values such that the cost variable stays within a given budget.
We first prove that the mere existence of such values for the parameters is undecidable. We nonetheless provide a symbolic semi-algorithm that is proved both sound and complete when it terminates. We also show how to modify it for the case when parameters values are integers. Finally, we prove that this modified version terminates if parameters are bounded. While this is to be expected since there are now only a finite number of possible parameter values, this is interesting because the computation is symbolic and thus avoids an explicit enumeration of all those values. Furthermore, the result is a symbolic constraint representing a finite union of convex polyhedra that is easily amenable to further analysis through linear programming.
We finally report on the implementation of the approach in Romeo, a software tool for the analysis of hybrid extensions of time Petri nets.
This work is partially supported by the ANR national research program PACS (ANR-14-CE28-0002).
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Notes
- 1.
Every 200 t.u., since T1 is executed twice as often as T2, T1 is running during \((22+a)*2=44+2a\) t.u. whereas T2 is running during \(28+(76-2a)=104-2a\) t.u.
- 2.
To ensure a correct access to the cores, we could have added one place for each core and some arcs on each task to capture and release them but the resulting net would have been quite unreadable. Instead, we chose to add 2 integer variables C0 and C1 (both initialised to 0); a variable equal to 0 (resp. 1) obviously means the corresponding core is idle (resp. busy).
- 3.
The last term ensures that such cases where an instance of a task is activated while a previous one is running are heavily penalised.
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Lime, D., Roux, O.H., Seidner, C. (2019). Parameter Synthesis for Bounded Cost Reachability in Time Petri 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_22
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