Expressiveness of Petri Nets with Stopwatches. Discrete-time Part
With this contribution, we aim to draw a comprehensive classification of Petri nets with stopwatches w.r.t. expressiveness and decidability issues. This topic is too ambitious to be summarized in a single paper. That is why we present our results in two different parts. In the first part of our work, we established new results regarding to both dense-time and discrete-time semantics. We now focus on the discrete-time specificities. We address the class of bounded Petri nets with stopwatches and reset arcs (rSwPNs), which is an extension of T-time Petri nets (TPNs) where time is associated with transitions. Stopwatches can be reset, stopped and started. We recall the formal dense-time and discrete-time semantics of these models in terms of Transition Systems. We study the expressiveness of rSwPNs and its subclasses w.r.t. (weak) bisimilarity (behavioral semantics). The main results are following: 1) Discrete-time bounded TPNs, discrete-time bounded rSwPNs and untimed Petri nets are equally expressive; 2) The resulting (final) classification of models is given by a set of relations explained in Fig. 7. While investigating expressiveness, we exhibit proofs that can be easily extended to the resolution of decidability issues. Among other results, we prove that, for bounded rSwPNs, the state and marking reachability problems - undecidable with dense-time semantics - are decidable when discrete-time is considered. Table 1 gives a synthesis of the main decidability results for these models. For the sake of simplicity, our results are explained on a model such that the stopwatches behaviors are expressed using inhibitor arcs. Our conclusions can however be easily extended to the general class of Stopwatch Petri nets.
expressiveness petri nets with stopwatches decidability discrete-time semantics real-time systems time petri nets