Time dependant models have been intensively studied for many reasons, among others because of their applications in software verification and due to the development of embedded platforms where reliability and safety depend to a large extent on the time features. Many of the time dependant models were suggested as real-time extensions of several well-known untimed models. The most studied formalisms include Networks of Timed Automata which extend the model of communicating finite-state machines with a finite number of real-valued clocks, and timed extensions of Petri nets where the added time constructs include e.g. time intervals that are assigned to the transitions (Time Petri Nets) or to the arcs (Timed-Arc Petri Nets). In this paper, we shall semi-formally introduce these models, discuss their strengths and weaknesses, and provide an overview of the known results about the relationships among the models.

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Comparing the Expressiveness of Timed Automata and Timed Extensions of Petri Nets

We consider (unbounded) Timed Petri Nets (TPNs) where each token is equipped with a real-valued clock representing the "age" of the token. Each arc in the net is provided with a subinterval of the natural numbers, restricting the ages of the tokens travelling the arc. We apply a methodology developed in [AN00], based on the theory of better quasi orderings (BQOs), to derive an efficient constraint system for automatic verification of safety properties for TPNs. We have implemented a prototype based on our method and applied it for verification of a parametrized version of Fischer's protocol.

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Timed Petri Nets and BQOs

This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability of specific properties , various behavioural equivalences and finally the model checking problem for temporal logics .

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Decidability Issues for Petri Nets

A brief survey and synthesis of the roles of time in petri nets

Timed-arc Petri nets are not Turing powerful, because, in particular, they cannot simulate a counter with test on zero. Thus, we could think that this model does not extend significantly the expressiveness of untimed Petri nets. But this is not true; in this paper we show that the differences between them are big enough to make the reachability problem undecidable. We also define dead tokens as those that cannot be used for firing any transitions in the future and we present some particular cases where we can identify them on this kind of timed nets.

On non-decidability of reachability for timed-arc Petri nets

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. >

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

The authors consider timed Petri nets, for which a timed step semantics generalizing the ordinary step semantics is presented. A codification of these, with regard to such semantics, by ordinary Petri nets is presented. This codification is presented in a gradual way, beginning with that corresponding to a more restricted and thus simpler step semantics. This semantics does not allow the overlapped firing of several instances of each transition. The authors consider a step semantics which allows such overlapped executions. In restricted cases the construction is based on an automaton which controls the system evolution. The construction for the most general semantics is rather more complicated, that is why it is presented gradually. They consider three versions of it. >

V. Valero Ruiz

Papers

On non-decidability of reachability for timed-arc Petri nets

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

Simulation of timed Petri nets by ordinary Petri nets and applications to decidability of the timed reachability problem and other related problems