Decidability of the strict reachability problem for TPN´s with rational and real durations

Timed Petri nets with rational and real durations are introduced. A timed version of the reachability problem, the strict reachability problem, which imposes a time limit to reach the given marking, is presented. It is shown that both in the case of rational durations, and in that of real durations, this strict reachability problem is decidable. In the first case, the result is obtained by scaling the time of the net, in such a way that all the durations of the transitions become integer. A previous result of the authors, showing that the strict reachability problem is decidable for this kind of net, is applied in order to conclude the desired result. Nevertheless, it is important to note that this application is not immediate, since, even though after scaling one can suppose that all the durations are integer, time is no more discrete, and thus the firing of a transition can happen at any (rational) moment. In the second case, a completely different approach is needed, since in general real durations cannot be normalized into integer durations. This consists of the counting of the passing of time in a symbolic way, and the simulation of the evolution of the original net without preserving the temporal order between the firing of the transitions, but only the causal order between them