Section: New Results

Fundamental results and algorithms: timed models

Participants : Claude Jard, Aurore Junier, Sundararaman Akshay, Loïc Hélouët.

Our work on that subject mainly concerns Time Petri Nets (TPNs) and their robustness. Robustness of timed systems aims at studying whether infinitesimal perturbations in clock values can result in new discrete behaviors. A model is robust if the set of discrete behaviors is preserved under arbitrarily small (but positive) perturbations. We have tackled this problem for Time Petri Nets (TPNs for short) by considering the model of parametric guard enlargement which allows time-intervals constraining the firing of transitions in TPNs to be enlarged by a (positive) parameter.

We have shown that TPNs are not robust in general and that checking if they are robust with respect to standard properties (such as boundedness, safety) is undecidable. We have also provided two decidable robustly bounded subclasses of TPNs, and shown that one can effectively build a timed automaton which is timed bisimilar even in presence of perturbations. This allowed us to apply existing results for timed automata to these TPNs and show further robustness properties. This work was published in [20] .

In a second work, we have considered robustness issues in Time Petri Nets (TPN) under constraints imposed by an external architecture. Our main objective was to check whether a timed specification, given as a TPN behaves as expected when subject to additional time and scheduling constraints. These constraints are given by another TPN that constrains the specification via read arcs. Our robustness property says that the constrained net does not exhibit new timed or untimed behaviors. We show that this property is not always guaranteed but that checking for it is always decidable in 1-safe TPNs. We further show that checking if the set of untimed behaviors of the constrained and specification nets are the same is also decidable. Next we turn to the more powerful case of labeled 1-safe TPNs with silent transitions. We show that checking for the robustness property is undecidable even when restricted to 1-safe TPNs with injective labeling, and exhibit a sub-class of 1-safe TPNs (with silent transitions) for which robustness is guaranteed by construction. This sub-class already lies close to the frontiers of intractability. This work was published in  [19] .

Finally, in cooperation with IRCCyN in Nantes, we defined a more general model, called “clock transition systems”, which generalizes both TPNs and networks of timed automata  [32] . This model will allow us to transfer new results on TPNs to the timed automata community.