DESCRIPTION OF THE ALGORITHM FOR FINDING THE SHORTEST SEQUENCE OF TRANSITIONS IN A PETRI NET (original) (raw)

The Possibilities of Modeling Petri Nets and Their Extensions

Computer Simulation [Working Title], 2019

This chapter is dedicated to several structure features of Petri nets. There is detailed description of appropriate access in Petri nets and reachable tree mechanism construction. There is an algorithm that describes the minimum sequence of possible transitions. The algorithm developed by us finds the shortest possible sequence for the network promotion state, which transfers the mentioned network state to the coverage state. The corresponding theorem is proven, which states that due to the describing algorithm, the number of transitions in the covering state is minimal. This chapter studies the interrelation of languages of colored Petri nets and traditional formal languages. The Venn diagram, modified by the author, is presented, which shows the relationship between the languages of the colored Petri nets and some traditional languages. As a result, it is shown that the language class of colored Petri nets includes an entire class of context-free languages and some other classes. The results obtained show that it is not possible to model the Patil problem using the well-known semaphores P and V or classical Petri nets, so the mentioned systems have limited properties.

Analysis of Petri Nets by Partitioning: Splitting Transitions

Theoretical Computer Science, 1990

In this paper, a method of analysis of large Petri nets by partitioning is proposed. This method permits a great saving of computation time and storage. Useless efforts spent in the analysis of large Petri nets are spared by a look to the partitions of interest. It is possible to study the characteristics of !he required p!aces by involving them in a partitition. It was sho#~r that partitioning preserves the characteristics of the main Petri net. The reachability tree method or the matrix equations approach, which were untractable at the whole net level, may be used at the subnet level to get the needed analysis criteria. 0304-3975!90/$03.50 @ 199~Elsevier Science Publishers B.Y. (North-iioiiand)

Deriving Petri nets from finite transition systems

IEEE Transactions on Computers, 1998

This paper presents a novel method to derive a Petri Net from any specification model that can be mapped into a statebased representation with arcs labeled with symbols from an alphabet of events (a Transition System, TS). The method is based on the theory of regions for Elementary Transition Systems (ETS). Previous work has shown that, for any ETS, there exists a Petri Net with minimum transition count (one transition for each label) with a reachability graph isomorphic to the original Transition System. Our method extends and implements that theory by using the following three mechanisms that provide a framework for synthesis of safe Petri Nets from arbitrary TSs. First, the requirement of isomorphism is relaxed to bisimulation of TSs, thus extending the class of synthesizable TSs to a new class called Excitation-Closed Transition Systems (ECTS). Second, for the first time, we propose a method of PN synthesis for an arbitrary TS based on mapping a TS event into a set of transition labels in a PN. Third, the notion of irredundant region set is exploited, to minimize the number of places in the net without affecting its behavior. The synthesis method can derive different classes of place-irredundant Petri Nets (e.g., pure, free choice, unique choice) from the same TS, depending on the constraints imposed on the synthesis algorithm. This method has been implemented and applied in different frameworks. The results obtained from the experiments have demonstrated the wide applicability of the method.

A Petri Net Modeling Approach Based on Boolean Function Transition

2012 International Symposium on Computer, Consumer and Control, 2012

A Petri Net is a mathematical modeling language for the description of distributed systems. A Petri net is a directed bipartite graph, in which the elements consist of place, transition, and arc. A Petri Net offers a kind of graphical notation for stepwise processes which include choice, iteration, and concurrent execution. However, when the system is getting large, the complexity of Petri Nets will be increased more quickly. In this paper, we propose a new Petri Net modeling language which is based on the Boolean Function Transition (BFT-PN, for short). Via combining the function with Boolean mathematic, it can describe both of PTP (Place-Transition-Place) table and the state transition of system more easily. BFT-PN can also reduce the complexity of transition of system on when the system is getting large and having a lot of events. In the other words, it can be dealt with very complex transition states by stepwise process not only more easily but also very clearly.

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

Mathematical programming approach to the Petri nets reachability problem

European Journal of Operational Research, 2007

This paper focuses on the resolution of the reachability problem in Petri nets, using the mathematical programming paradigm. The proposed approach is based on an implicit traversal of the Petri net reachability graph. This is done by constructing a unique sequence of Steps that represents exactly the total behaviour of the net. We propose several formulations based on integer and/or binary linear programming, and the corresponding sets of adjustments to the particular class of problem considered. Our models are validated on a set of benchmarks and compared with standard approaches from IA and Petri nets community.

A Symbolic Algorithm for the Synthesis of Bounded Petri Nets

2008

This paper presents an algorithm for the synthesis of bounded Petri nets from transition systems. A bounded Petri net is always provided in case it exists. Otherwise, the events are split into several transitions to guarantee the synthesis of a Petri net with bisimilar behavior. The algorithm uses symbolic representations of multisets of states to efficiently generate all the minimal regions. The algorithm has been implemented in a tool. Experimental results show a significant net reduction when compared with approaches for the synthesis of safe Petri nets.

ω-Petri Nets: Algorithms and Complexity

Fundamenta Informaticae

We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ω-labeled input and output arcs, that is well-suited to analyse parametric concurrent systems with dynamic thread creation. Most techniques (such as the Karp and Miller tree or the Rackoff technique) that have been proposed in the setting of plain Petri nets do not apply directly to ωPN because ωPN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of ωPN. We show that an ωPN can be turned into a plain Petri net that allows us to recover the reachability set of the ωPN, but that does not preserve termination (an ωPN terminates iff it admits no infinitely long execution). This yields complexity bounds for the reachability, (place) boundedness and coverability problems on ωPN. We provide a practical algorithm to compute a coverability set of the ωPN and to decide termination by adapting the classical Karp and Miller tree construction. We also adapt the Rackoff technique to ωPN, to obtain the exact complexity of the termination problem. Finally, we consider the extension of ωPN with reset and transfer arcs, and show how this extension impacts the decidability and complexity of the aforementioned problems.

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.

Petri Net Reachability Checking Is Polynomial with Optimal Abstraction Hierarchies

Lecture Notes in Computer Science, 2005

The Petri net model is a powerful state transition oriented model to analyse, model and evaluate asynchronous and concurrent systems. However, like other state transition models, it encounters the state explosion problem. The size of the state space increases exponentially with the system complexity. This paper is concerned with a method of abstracting automatically Petri nets to simpler representations, which are ordered with respect to their size. Thus it becomes possible to check Petri net reachability incrementally. With incremental approach we can overcome the exponential nature of Petri net reachability checking. We show that by using the incremental approach, the upper computational complexity bound for Petri net reachability checking with optimal abstraction hierarchies is polynomial. The method we propose considers structural properties of a Petri net as well an initial and a final marking. In addition to Petri net abstraction irrelevant transitions for a given reachability problem are determined. By removing these transitions from a net, impact of the state explosion problem is reduced even more.