Undecidability of LTL for Timed Petri Nets (original) (raw)

Decidability of Properties of Timed-Arc Petri Nets

Lecture Notes in Computer Science, 2000

Timed-arc Petri nets (TAPN's) are not Turing powerful, because, in particular, they cannot simulate a counter with zero testing. Thus, we could think that this model does not increase significantly the expressiveness of untimed Petri nets. But this is not true; in a previous paper we have shown that the differences between them are big enough to make the reachability problem undecidable. On the other hand, coverability and boundedness are proved now to be decidable. This fact is a consequence of the close interrelationship between TAPN's and transfer nets, for which similar results have been recently proved. Finally, we see that if dead tokens are defined as those that cannot be used for firing any transition in the future, we can detect these kind of tokens in an effective way.

Comparison of the Expressiveness of Timed Automata and Time Petri Nets

Lecture Notes in Computer Science, 2005

In this paper we consider the model of Time Petri Nets (TPN) where time is associated with transitions. We also consider Timed Automata (TA) as defined by Alur & Dill, and compare the expressiveness of the two models w.r.t. timed language acceptance and (weak) timed bisimilarity. We first prove that there exists a TA A s.t. there is no TPN (even unbounded) that is (weakly) timed bisimilar to A. We then propose a structural translation from TA to (1-safe) TPNs preserving timed language acceptance. Further on, we prove that the previous (slightly extended) translation also preserves weak timed bisimilarity for a syntactical subclass T Asyn(≤, ≥) of TA. For the theory of TPNs, the consequences are: 1) TA, bounded TPNs and 1-safe TPNs are equally expressive w.r.t. timed language acceptance; 2) TA are strictly more expressive than bounded TPNs w.r.t. timed bisimilarity; 3) The subclass T Asyn(≤, ≥), bounded and 1-safe TPNs "à la Merlin" are equally expressive w.r.t. timed bisimilarity.

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

Lecture Notes in Computer Science

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 semiformally introduce these models, discuss their strengths and weaknesses, and provide an overview of the known results about the relationships among the models.

Using Forward Reachability Analysis for Verification of Timed Petri Nets

Nord. J. Comput., 2007

We consider verification of safety properties for concurrent real-timed systems modelled as timed Petri nets by performing symbolic forward reachability analysis. We introduce a formalism, called region generators, for representing sets of markings of timed Petri nets. Region generators characterize downward closed sets of regions and provide exact abstractions of sets of reachable states with respect to safety properties. We show that the standard operations needed for performing symbolic reachability analysis are computable for region generators. Since forward reachability analysis is necessarily incomplete, we introduce an acceleration technique to make the procedure terminate more often on practical examples. We have implemented a prototype for analyzing timed Petri nets and used it to verify a parameterized version of Fischer's protocol, Lynch and Shavit's mutual exclusion protocol and a producer-consumer protocol. We also used the tool to extract finite-state abstracti...

Undecidability of coverability and boundedness for timed-arc Petri nets with invariants

Proc. of MEMICS, 2009

Timed-Arc Petri Nets (TAPN) is a well studied extension of the classical Petri net model where tokens are decorated with real numbers that represent their age. Unlike reachability, which is known to be undecidable for TAPN, boundedness and coverability remain decidable. The model is supported by a recent tool called TAPAAL which, among others, further extends TAPN with invariants on places in order to model urgency. The decidability of boundedness and coverability for this extended model has not yet been considered. We present a reduction from two-counter Minsky machines to TAPN with invariants to show that both the boundedness and coverability problems are undecidable.

Comparison of Expressiveness for Timed Automata and Time Petri Nets

Combinatorial Optimization and Theoretical Computer Science, 2008

In this paper we consider the model of Time Petri Nets (TPN) "à la Merlin" where a time interval is associated with the firing of a transition, but we extend it with open intervals. We also consider Timed Automata (TA) as defined by Alur & Dill. We investigate some questions related to expressiveness for these models : we study the impact of slight variations of semantics for TPN and we compare the expressive power of TA and TPN, with respect to both time language acceptance and weak time bisimilarity. We prove that TA and bounded TPNs (enlarged with strict constraints) are equivalent w.r.t. timed language equivalence, providing an efficient construction of a TPN equivalent to a TA. We then exhibit a TA A such that no TPN (even unbounded) is weakly bisimilar to A. Because of this last result, it is natural to try and identify the (strict) subclass of TA that is equivalent to TPN w.r.t. weak timed bisimilarity. Thus we give some further results: 1) we characterize the subclass TA − of TA that is equivalent to the original model of TPN as defined by Merlin, i.e. restricted to closed intervals, 2) we show that the associated membership problem for TA − is P SP ACE-complete and 3) we prove that the reachability problem for TA − is also P SP ACE-complete.

A theory of implementation and refinement in timed Petri nets

Theoretical Computer Science, 1998

We define formally the notion of implementation for time critical systems in terms of provability of properties described abstractly at the specification level. We characterize this notion in terms of formulas of the temporal logic TRIO and operational models of timed Petri nets, and provide a method to prove that two given nets are in the implementation relation. Refinement steps are often used as a means to derive in a systematic way the system design starting from its abstract specification. We present a method to formally prove the correctness of refinement rules for timed Petri nets and apply it to a few simple cases. We show how the possibility to retain properties of the specification in its implementation can simplify the verification of the designed systems by performing incremental analysis at various levels of the specification/implementation hierarchy.

Decidability results in firstorder hybrid petri nets. Discrete Event Dynamic Systems

2001

In this paper we tackle the decidability of marking reachability for a hybrid formalism based on Petri nets. The model we consider is the untimed version of First–Order Hybrid Petri Nets: it combines a discrete Petri net and a continuous Petri net, the latter being a fluid version of a usual discrete Petri net. It is suggested that the decidability results should be pursued exploiting a hierarchy of models as it has been done in the framework of Hybrid Automata. In this paper we define the class of Single–Rate Hybrid Petri Nets: the continuous dynamics of these nets is such that the vector of the marking derivatives of the continuous places is constant but for a scalar factor. This class of nets can be seen as the counterpart of timed automata with skewed clocks. We prove that the reachability problem for this class can be reduced to the reachability problem of an equivalent discrete net and thus it is decidable. 1

Towards a Notion of Distributed Time for Petri Nets

Lecture Notes in Computer Science, 2001

We set the ground for research on a timed extension of Petri nets where time parameters are associated with tokens and arcs carry constraints that qualify the age of tokens required for enabling. The novelty is that, rather than a single global clock, we use a set of unrelated clocks-possibly one per place-allowing a local timing as well as distributed time synchronisation. We give a formal definition of the model and investigate properties of local versus global timing, including decidability issues and notions of processes of the respective models.