On persistent reachability in Petri nets (original) (raw)
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Electronic Notes in Theoretical Computer Science, 2005
Persistence of information is common in modern computer systems. This paper describes how to extend Petri nets, a traditional model of concurrent and distributed computations, to take account of conditions that are persistent. We found use for this kind of nets in modelling untrustworthy networks on which messages are exchanged according to a security protocol. The paper explains a construction where persistent conditions are unfolded and a basic net is recovered. Conditions are given under which the unfolded net exhibits the same finite behaviours as the original net with persistence.
Linear time algorithms for liveness and boundedness in conflict-free Petri nets
LATIN'92, 1992
We introduce the notion of a T -path within Petri nets, and propose to adopt the model of directed hypergraphs in order to determine properties of nets; in particular, we study the relationships between T -paths and firable sequences of transitions. Let us consider a Petri net P = ⟨P, T , A, M 0 ⟩ and the set of places with a positive marking in M 0 , i.e., P 0 = {p | M 0 (p) > 0}. If we regard the net as a directed graph, the existence of a simple path from any place in P 0 to a transition t is, of course, a necessary condition for the potential firability of t. This is sufficient only if the net is a state machine, where | • t| = |t • | = 1 for all t ∈ T . In this paper we show that the existence of a T -path from any subset of P 0 to a transition t is a more restrictive condition and is, again, a necessary condition for the potential firability of t. But, in this case: (a) if P is a conflict-free Petri net, this is also a sufficient condition, (b) if P is a general Petri net, t is potentially firable by increasing the number of tokens in P 0 . For conflict-free nets (CFPN) we consider the following problems: (a) determining the set of firable transitions, (b) determining the set of coverable places, (c) determining the set of live transitions, (d) deciding the boundedness of the net. For all these problems we provide algorithms requiring linear space and time, i.e., O(|P| + |T | + |A|), for a net P = ⟨P, T , A, M 0 ⟩. Previous results for this class of networks are given by Howell et al. (1987) [20], providing algorithms for solving problems in conflict-free nets in O(|P| × |T |) time and space. Given a Petri net and a marking M, the well-known coverability problem consists in finding a reachable marking M ′ such that M ′ ≥ M; this problem is known to be EXPSPACE hard (Rackoff (1978) [33]). For general Petri nets we provide a partial answer to this problem. M is coverable by augmentation if it is coverable from an augmented marking M ′ 0 of the initial marking M 0 : M ′ 0 ≥ M 0 and, for all p ∈ P, M ′ 0 (p) = 0 if M 0 (p) = 0. We solve this problem in linear time.
Linear time analysis of properties of conflict-free and general Petri nets
Theoretical Computer Science, 2011
We introduce the notion of T -path within Petri nets, and propose a simple approach, based on previous work developed for directed hypergraphs, in order to determine structural properties of nets; in particular, we study the relationships between T -paths in a Petri net and firable sequences of transitions.
Decidability Problems in Petri Nets with Names and Replication
Fundamenta Informaticae
In this paper we study decidability of several extensions of P/T nets with name creation and/or replication. In particular, we study how to restrict the models of RN systems (P/T nets extended with replication, for which reachability is undecidable) and ν-RN systems (RN extended with name creation, which are Turing-complete, so that coverability is undecidable), in order to obtain decidability of reachability and coverability, respectively. We prove that if we forbid synchronizations between the different components in a RN system, then reachability is still decidable. Similarly, if we forbid name communication between the different components in a ν-RN system, or restrict communication so that it is allowed only for a given finite set of names, we obtain decidability of coverability. Finally, we consider a polyadic version of ν-PN (P/T nets extended with name creation), that we call pν-PN, in which tokens are tuples of names. We prove that pν-PN are Turing complete, and discuss how the results obtained for ν-RN systems can be translated to them.
Continuous Reachability for Unordered Data Petri Nets is in PTime
Lecture Notes in Computer Science, 2019
Unordered data Petri nets (UDPN) are an extension of classical Petri nets with tokens that carry data from an infinite domain and where transitions may check equality and disequality of tokens. UDPN are well-structured, so the coverability and termination problems are decidable, but with higher complexity than for Petri nets. On the other hand, the problem of reachability for UDPN is surprisingly complex, and its decidability status remains open. In this paper, we consider the continuous reachability problem for UDPN, which can be seen as an over-approximation of the reachability problem. Our main result is a characterization of continuous reachability for UDPN and polynomial time algorithm for solving it. This is a consequence of a combinatorial argument, which shows that if continuous reachability holds then there exists a run using only polynomially many data values.
A Reduced Reachability Tree for a Class of Unbounded Petri Nets
As a powerful analysis tool of Petri nets, reachability trees are fundamental for systematically investigating many characteristics such as boundedness, liveness and reversibility. This work proposes a method to generate a reachability tree, called ωRT for short, for a class of unbounded generalized nets called ω-independent nets based on new modified reachability trees (NMRTs). ωRT can effectively decrease the number of nodes by removing duplicate and ω-duplicate nodes in the tree, and verify properties such as reachability, liveness and deadlocks. Two examples are provided to show its superiority over NMRTs in terms of tree size. Citation: Shouguang Wang, Mengdi Gan, Mengchu Zhou, Dan You. A reduced reachability tree for a class of unbounded Petri nets. IEEE/CAA Journal of Automatica Sinica, 2015, 2(4): 345-352
Decidability Results for Restricted Models of Petri Nets with Name Creation and Replication
Lecture Notes in Computer Science, 2009
In previous works we defined ν-APNs, an extension of P/T nets with the capability of creating and managing pure names. We proved that, though reachability is undecidable, coverability remains decidable for them. We also extended P/T nets with the capability of nets to replicate themselves, creating a new component, initially marked in some fixed way, obtaining g-RN systems. We proved that these two extensions of P/T nets are equivalent, so that g-RN systems have undecidable reachability and decidable coverability. Finally, for the class of the so called ν-RN systems, P/T nets with both name creation and replication, we proved that they are Turing complete, so that also coverability turns out to be undecidable. In this paper we study how can we restrict the models of ν-APNs (and, therefore, g-RN systems) and ν-RN systems in order to keep decidability of reachability and coverability, respectively. We prove that if we forbid synchronizations between the different components in a g-RN system, then reachability is still decidable. The proof is done by reducing it to reachability in a class of multiset rewriting systems, similar to Recursive Petri Nets. Analogously, if we forbid name communication between the different components in a ν-RN system, or restrict communication to happen only for a given finite set of names, we obtain decidability of coverability.
On Nets with Structured Concurrency
An extension of Petri nets with a statechart-like AND/OR state hierarchy is defined and studied. The resulting net variant, state- chart nets, is shown to coincide, under certain conditions, with a strict subclass of safe nets in which concurrency is structured. Next, syntac- tic constraints on statechart nets are defined that guarantee absence of deadlocks and livelocks. Such syntactic constraints are hard to give for ordinary Petri nets. Together, these results give more insight into the expressive power and usefulness of AND/OR state hierarchies, and into the differences between statecharts and Petri nets.