Expressiveness of Petri Nets with Stopwatches. Dense-time Part

Abstract

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. The scope of this first paper is to address the general results that apply for both dense-time and discrete-time semantics. We study 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 give 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) bounded rSw- PNs and 1-safe rSwPNs are equally expressive; 2) For all models, reset arcs add expressiveness. 3) The resulting partial classification of models is given by a set of relations explained in Fig. 7: in the forthcoming paper, we will complete these results by covering expressiveness and decidability issues when discrete-time nets are considered. 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.