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Rewriting Petri Nets As Directed Graphs INTERNATIONAL JOURNAL OF COMPUTERS Issue 2, Volume 5, 2011 Rewriting Petri Nets as Directed Graphs A. Spiteri Staines properties are i) reachability, ii) boundedness, iii) safeness, iv) Abstract— This work attempts to understand some of the basic conservativeness, v) liveness, vi) reversiblility, vii) properties of Petri nets and their relationships to directed graphs. repetitiveness, viii) home states. Sometimes these properties Different forms of directed graphs are widely used in computer are also called structural properties by some authors [6], [7]. science. Normally various names are given to these structures. E.g. Other properties like persistence and synchronic distance directed acyclical graphs (DAGs), control flow graphs (CFGs), task graphs, generalized task graphs (GTGs), state transition diagrams could be included. From these main properties others like (STDs), state machines, etc. Some structures might exhibit bi- partially conservative, structurally bounded, partially similarity. The justification for this work is that Petri nets are based repetitive, etc. can be defined. These properties can be found on graphs and have some similarities to them. Transforming Petri using reachability methods such as the marking graph or place nets into graphs opens up a whole set of new interesting possible and transition invariants or the analysis of the Petri net experimentations. Normally this is overlooked. Directed Graphs have incidence matrix [6],[7]. Other analysis is based on the siphon a lot of theory and research associated with them. This work could be further developed and used for Petri net evaluation. The related and trap method. Most of these forms of analysis are works justifies the reasoning how and why Petri nets are obtained or applicable to structurally bounded Petri net structures with the supported using graphs. The transformation approach can be formal exception of reachability which can be solved to provide for or informal. The main problem tackled is how graphs can be obtained some unbounded states. Simulation is another method by from Petri nets. Possible solutions that use reduction methods to which the Petri net can be tested and functionally verified. simplify the Petri net are presented. Different methods to extract This should normally be done after the behavioral properties graphs from the basic or fundamental Petri net classes are explained. Some examples are given and the findings are briefly discussed. have been checked and verified. In general for the purpose of analysis, Petri net structures can be classified into two Keywords— Directed Graphs, Graphs, Petri nets, categories: i) unsolvable and ii) solvable. The structurally Transformation, Reduction limited Petri nets are normally solvable whilst those that are not structurally limited and have state space explosion I. INTRODUCTION problems are not simple to solve. One possible solution is to ETRI nets are expressive graphical formalisms that serve to reduce the structure. P model discrete event behavior that takes place in different One of the salient points for using Petri nets is precisely the systems [12]-[15]. They are designed to model system ability to transform them or obtain them from other behavior like: sequential behavior, concurrency, mutual formalisms or notations. Petri nets are classified as directed exclusion, non-determinism, choice and conflict. Petri nets bi-partite graphs, definitely sharing some common properties are classified into different classes ranging from elementary with graphs. This means that they could be transformed into nets to higher order nets, colored Petri nets and object oriented graphs and analyzed from this point of view. This could serve nets. All these classes can be converted to time Petri nets. to generate many new ideas. E.g. the static structure or Ordinary Petri nets have a ‘dual identity’ they can be topological features are examinable. Structure is easier to represented graphically or by using equations. These can be control and understand than behavior. This is because analyzed using mathematical models. Petri nets have at least normally the structure should remain fixed in relation to time, three decades of use. Normally speaking, the analysis of Petri whist behavior can be modified or applied differently, being nets is based on i) structural properties and ii) behavioral dynamic. Another important aspect is the reduction of the properties [6]. The structural properties of Petri nets are Petri net model. Even though normally reduction implies suitable to understand the basic underlying structure. If the fusion/augmentation of places, transitions etc.; in the wider Petri net is viewed, basic structural features can be seen. E.g. sense a higher order net can be reduced to a simple place the Petri net can be cycle free (acyclical) [9]. It could have transition net by keeping the graphical outline structure and bounded places, etc. On the other hand behavioral properties removing other information. explain the behavior of the Petri net. These properties cannot The work in this paper is restricted to the basic or be applied to all types of Petri nets especially if the net is fundamental classes of Petri nets. unbounded or improperly designed. Some basic behavioral II. BACKGROUND A normal Petri net is basically defined as directed bipartite Manuscript received Feb 23, 2011. Anthony (Tony) Spiteri Staines, is with the Department of Information Systems, Faculty of ICT, University of Malta, graph or bipartite digraph that can be basically represented as (corresponding phone: 00356-21373402,e-mail: [email protected]) 289 INTERNATIONAL JOURNAL OF COMPUTERS Issue 2, Volume 5, 2011 a five tuple (P, T, I,O,M0) where P is a finite set of places, T is associated with graphs and graph theory. Most of the work a finite set of transitions, I ⊆ (P x T) Input arcs, and O ⊆ (T about Petri nets does not usually consider them from the graph point of view. This opens up a lot of new exciting possibilities x P) Output arcs, P ∪ T ≠ φ and P ∩ T =φ , M0 represents for analyzing Petri nets. If Petri nets are transformed into the initial marking. graphs, then they can be analysed using graph theory. The Normal Petri nets are very simple and convenient to use for graphs can also serve for visualization. It can be shown that a variety of purposes. Similar to them are elementary nets and some notations like control flow graphs (CFGs), state augmented marked graphs which are a special subclass of transition diagrams (STDs), etc. can be obtained directly from Petri nets. Normal Petri nets have a reduced or limited state certain Petri net types. space. There are problems to find simple ways for understanding IV. PROBLEM FORMULATION and analyzing Petri nets. Another aspect is that certain Petri net models that are created are just too complex to analyze and The main problem that is dealt with in this paper is to try to verify using the traditional approaches. Other fundamental examine how Petri nets can be converted into graphs for the properties of Petri nets are normally not applicable to certain purpose of analysis. It is possible to transform Petri nets into classes of Petri nets like higher order nets. An interesting graphs. There are different ways how to obtain graphs from idea, that is often overlooked, is to analyze the structural Petri nets. To obtain graphs from the Petri net the Petri net properties of Petri nets by representing the Petri net as a should have a reduced structure and has to be bounded. A graph. The graph although static can be used for different possible solution it to reduce the class and structure of the objectives such as visual inspection, etc. Petri net before applying analysis methods and transformation Some works are listed below. These just confirm the of the Petri net structure into a graph. Sometimes there can be importance of Petri nets for supporting other forms of various issues, especially if the Petri net is too complex. It has graphical structures and the possible transformation or support to be reduced. I.e. it can be reduced structurally to a simpler of Petri nets using graphs. A vast amount of literature is model or class reduction could be performed. E.g. a more available in this respect. complex class can be reduced to a lower class by replacing or Obviously the transformation or mapping is done eliminating some properties or information. informally or formally or it just happens. In [6] it is shown how a Petri net having exactly one incoming arc and exactly V. PROBLEM SOLUTION one outgoing arc with unit weight can be directly represented There are two aspects of the solution i.e. i) reduction of the as directed marked graph where directed edges correspond to Petri net and ii) explaining the possible transformations that places and nodes to transitions. The augmented marked graphs can be done. Reduction might imply i) class reduction and ii) presented in [4] seem to share similar properties to this. The structural reduction. same Petri net can also be represented as a state machine. Once the Petri net is reduced it is possible to transform the Augmented Marked Graphs are a special sub-class of Petri Petri net to a graph by simply replacing the nodes and edges in nets [4]. These are structurally bounded Petri nets that the Petri net. preserve certain properties. In [10] Petri nets are obtained Another simple way of obtaining a graph from a Petri net is from transfer resource graphs (TRGs). Here Petri nets are used by generating the marking graph or the reachability graph. to model a system at a higher level of abstraction. In [1] Petri Other possible methods could include replacing the Petri net net elements are defined as TGG rules from project or object node and edges. elements. This is a form of formal mapping. In [2] it is A.
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