DOCSLIB.ORG

Modeling Games with Petri Nets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modeling Games with Petri Nets Modeling Games with Petri Nets Manuel Araújo Licínio Roque Departamento de Engenharia Informática Departamento de Engenharia Informática Faculdade de Ciências e Tecnologia Faculdade de Ciências e Tecnologia Universidade de Coimbra Universidade de Coimbra Coimbra, Portugal Coimbra, Portugal [email protected] [email protected] ABSTRACT game systems. We will then discuss some of their This paper describes an alternate approach to the modeling advantages and disadvantages when compared to the other of game systems and game flow with Petri nets. Modeling modeling languages. languages usually used in this area are of limited efficiency MODELING IN GAME DESIGN when it comes to validating the underlying game systems. Natural language is easy to read, but it is not easy to review We provide a case study to show that Petri Nets can be used a large natural language specification [15]. For that reason with advantages over other modeling languages. Their we use charts and diagrams to explain things visually and graphical notation is simple, yet it can be used to model synthetically. This kind of communication can be as complex game systems. Their mathematically defined expressive as and easier to understand than the verbal structure enables the modeled system to be formally descriptions used for the same information [15]. People analyzed and its behavior’s simulation offers the possibility doing game design make intensive use of visual of detecting unwanted behaviors, loop-holes or balancing communication [9, 10, 12, 13, 19]. The flow of the game is issues while still in the game design stage. usually depicted visually instead of a pure textual approach Keywords [10, 13, 19]. We will now take a look at some of the most Game design, Petri Nets, game flow modeling, simulation popular diagrams in this branch of software design. INTRODUCTION Unified Modeling Language (UML) is a modeling language Visual communication is an essential part of software widely used in software specification and design. As of design. In game development projects, this sentence version 2.0, it comprises 13 diagrams, divided into 6 remains true. Be it for identifying and constructing use categories [21]. A number of these can be of great use to cases, charting the flow between menus, modeling a game design [9]. Use cases are used to represent actors sequence of events or showing the relationship between (player, external systems, etc.) and their actions or actors in the game, diagrams and charts are extensively interactions with the system. They can be used to identify used throughout the conceptual and design phases of game and collect requirements in the early stages of conception. development [9, 10]. The representations used for this Other diagrams, like sequence, class and activity diagrams, purpose, however, present a number of limitations when it can help game designers further document and structure comes to the verification, validation and simulation of the ideas and later help the programmers structure their code in underlying system [11, 17, 18]. This prevents designers the development phase. from using these diagrams in a more effective way, such as As proposed in [9], although with a bigger focus on use for finding balancing issues or problems within the game's case diagrams, UML makes the game designer divide flow. his/her project in a number of different kinds of diagrams, In this paper, we will explore Petri Nets for modeling a each one serving its purpose. Those who are going to read game’s systems and flow. We will try to show that this those diagrams have to be able to understand all of them approach can be as simple and yet more powerful than other and cope with possible semantic inconsistencies [18], since graphical tools used in this domain. they define a range of possible interpretations instead of conveying an exact meaning [18, 20]. UML diagrams, We will begin by analyzing modeling languages used in especially when used at the conceptual level, lack formal game design. Next we will describe Petri Nets structure, semantics that prevent them from being used in rigorous how they work and some useful extension to their basic model analysis [11, 17, 18]. In large, complex models this structure. In the following section we will develop the case can present a problem since validation can be study to show how Petri Nets can be used in describing compromised. Although there are tools for analysis and Breaking New Ground: Innovation in Games, Play, Practice and Theory. Proceedings of DiGRA 2009 © 2009Authors & Digital Games Research Association (DiGRA). Personal and educational classroom use of this paper is allowed, commercial use requires specific permission from the author. simulation, these use intermediate models like graphs or amongst events or the existence of appropriate Petri nets [11] or focus more on syntax and consistency synchronization [1]. They can also be used to measure the checks [17]. There are also a number of proposals for the performance of the underlying system [1, 7]. definition of formal semantics [11, 17, 20], but none of Petri nets can be described both mathematical and these are yet part of the UML standard. graphically. For the sake of simplicity, this paper will not Other popular diagrams are flowcharts. As the name states, feature the mathematical description. Most of the Petri net it is a chart where one can model the flow of events in a diagrams present in this document were made using Yasper system. They are made up processing steps, conditional [5]. To illustrate a difference in notation, another diagram branching nodes and arrows connecting them. Flowcharts was made using Woped [6]. can be used to explain the steps needed for a certain action Graphically, Petri nets are described as a diagram with to occur or simply to document the flow of the game [10, circles (places), bars or squares (transitions) and arrows 12, 13, 16]. Decision trees are a type of flowchart that (arcs) connecting them. Depending on the interpretation the branch on every situation where the player can make a designer wishes to give them, places can represent decision, effectively mapping all the possible decisions and conditions, input/output data or resources. Transitions can outcomes in a game [16, 19]. be interpreted as events, tasks or clauses, among others [2]. Flowcharts are limited to the modeling of sequential, non- Places can have multiple arcs from and to transitions and concurrent systems, which is something most games are not. transitions can have multiple arcs from and to places. A Simulation can be done only in a case-by-case fashion, as transition can have arcs going back to its input places, there is no way to actively insert, for instance, conflict or symbolized in figure 1 by a double arrow. A place can hold concurrency for resources. While they can be used to chart one or more tokens, symbolized by one or more dots. simple decision based systems, when decision depends of Depending on the interpretation given to places, a token can external processes running in parallel their capabilities are represent resources or whether a condition is true or false limited. [1]. In its basic incarnation, a transition is enabled and can Petri nets are both a mathematical and a graphical tool for fire when all the places that are connecting to it hold at least modeling, analyzing and designing discrete event systems one token. When the transition fires, it removes (consumes) used in many different industries [1]. When compared with a token from all the incoming places and adds (produces) behavior modeling based on Statecharts, e.g., [15], Petri another token in all the outgoing places. Nets tend to be, in general, a more economic representation In the example given in figure 1, T1 has one input place when state-space complexity increases. (P1) and two output places (P1 and P2) and it is enabled In [22], Brom and Abonyi describe a technique for because all of its input places have tokens. In figure 2, T1 "authoring a nonlinear plot and managing a story" has consumed a token from P1 and produced another in P1 according to this plot using Petri nets. This technique makes and P2. The transition can fire again indefinitely because use of both the formal and graphical natures of Petri nets P1 will always have a token. for drafting, simulating and building plots for a story-driven More complex behaviors can be modeled by adding places virtual reality application. It was later applied successfully and transitions to the diagram. In figure 3 we have a to the story manager of serious games Europe 2045 [24] situation where a F1 car is making a pit stop to change all and Karo [23]. We believe that Petri nets can be applied to of its tires. When the Stop transition fires, the places other areas of game design, not just plot description or story managing. OVERVIEW OF PETRI NETS Petri Nets were created by German mathematician Carl Adam Petri for the purpose of describing chemical processes [3]. They can be applied to many different areas, Figure 1: Simple Petri Net with two places and one such as the modeling of production lines, distributed- transition database systems and communication networks or the design and analysis of workflows and business processes, among many others [1, 2]. They can also simulate such processes with the help of a number of computer applications [4, 5, 6]. Figure 2: Result after transition has fired Due to the fact that Petri nets can be described as a set of algebraic equations [3], they are considered as a powerful representing the tires – front and rear, left and right – analysis tool [1, 2, 7]. Petri nets can be used to check for receive a token.
Load more

Recommended publications
  • Stochastic-Petri Net Modeling and Optimization for Outdoor Patients in Building Sustainable Healthcare System Considering Staff Absenteeism
    Article Stochastic-Petri Net Modeling and Optimization for Outdoor Patients in Building Sustainable Healthcare System Considering Staff Absenteeism Chang Wook Kang 1, Muhammad Imran 2, Muhammad Omair 3, Waqas Ahmed 2, Misbah Ullah 4 and Biswajit Sarkar 1,* 1 Department of Industrial & Management Engineering, Hanyang University, Ansan, Gyeonggido 155 88, Korea; [email protected] 2 Department of Management & HR’, NUST Business School, National University of Sciences & Technology, Islamabad 44000, Pakistan; [email protected] (M.I.); [email protected] (W.A.) 3 Department of Industrial Engineering, Jalozai Campus, University of Engineering & Technology, Peshawar 25000, Pakistan; [email protected] 4 Department of Industrial Engineering, University of Engineering & Technology, Peshawar 25000, Pakistan; [email protected] * Correspondence: [email protected]; Tel.: +82-31-400-5259; Fax: +82-31-436-8146 Received: 13 April 2019; Accepted: 29 May 2019; Published: 1 June 2019 Abstract: Sustainable healthcare systems are gaining more importance in the era of globalization. The efficient planning with sustainable resources in healthcare systems is necessary for the patient’s satisfaction. The proposed research considers performance improvement along with future sustainability. The main objective of this study is to minimize the queue of patients and required resources in a healthcare unit with the consideration of staff absenteeism. It is a resource-planning model with staff absenteeism and operational utilization. Petri nets have been integrated with a mixed integer nonlinear programming model (MINLP) to form a new approach that is used as a solution method to the problem. The Petri net is the combination of graphical, mathematical technique, and simulation for visualizing and optimization of a system having both continuous and discrete characteristics.
    [Show full text]
  • Petri Nets for Dynamic Event-Driven System Modeling
    24 Petri Nets for Dynamic Event-Driven System Modeling Jiacun Wang 24.1 Introduction .................................................................24-1 Monmouth University 24.2 Petri Net Definition .....................................................24-1 24.3 Transition Firing ..........................................................24-3 24.4 Modeling Power ...........................................................24-4 24.5 Petri Net Properties ......................................................24-5 Reachability • Safeness • Liveness 24.6 Analysis of Petri Nets ...................................................24-7 Reachability Analysis • Incidence Matrix and State Equation • Invariant Analysis • Simulation 24.7 Colored Petri Nets ........................................................24-10 24.8 Timed Petri Nets ..........................................................24-12 Deterministic Timed Petri Nets • Stochastic Timed Petri Nets 24.9 Concluding Remark .....................................................24-16 24.1 Introduction Petri nets were introduced in 1962 by Dr. Carl Adam Petri (Petri, 1962). Petri nets are a powerful modeling formalism in computer science, system engineering, and many other disciplines. Petri nets combine a well-defined mathematical theory with a graphical representation of the dynamic behavior of systems. The theoretical aspect of Petri nets allows precise modeling and analysis of system behavior, while the graphical representation of Petri nets enables visualization of the modeled system state
    [Show full text]
  • Petri Nets: Properties, Analysis and Applications
    Petri Nets: Properties, Analysis and Appl kations TADAO MURATA, FELLOW, IEEE Invited Paper This is an invited tutorial-review paper on Petri nets-a graphical to the faculty of Mathematics and Physics at the Technical and mathematical modeling tool. Petri nets are a promising tool University of Darmstadt, West Germany. The dissertation for describing and studying information processing systems that was prepared while C. A. Petri worked as a scientist at the are characterized as being concurrent, asynchronous, distributed, parallel, nondeterministic, and/or stochastic. Universityof Bonn. Petri’swork[l], [2]came totheattention The paper starts with a brief review of the history and the appli- of A. W. Holt, who later led the Information System Theory cation areas considered in the literature. It then proceeds with Project of Applied Data Research, Inc., in the United States. introductory modeling examples, behavioral and structural prop- The early developments and applications of Petri nets (or erties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs- their predecessor)arefound in the reports [3]-[8] associated the concurrent system model most amenable to analysis. In addi- with this project, and in the Record [9] of the 1970 Project tion, the paper presents introductory discussions on stochastic nets MAC Conference on Concurrent Systems and Parallel with their application to performance modeling, and on high-level Computation. From 1970 to 1975, the Computation Struc- nets with their application to logic programming. Also included are recent results on reachability criteria. Suggestions are provided ture Group at MIT was most active in conducting Petri-net for further reading on many subject areas of Petri nets.
    [Show full text]
  • Introduction to Petri Net Theory *
    Introduction to Petri Net Theory ? Hsu-Chun Yen Dept. of Electrical Engineering, National Taiwan University Taipei, Taiwan, R.O.C. ([email protected]) Abstract. This paper gives an overview of Petri net theory from an algorithmic viewpoint. We survey a number of analytical techniques as well as decidability/complexity results for various Petri net problems. 1 Introduction Petri nets, introduced by C. A Petri in 1962 [54], provide an elegant and useful mathematical formalism for modelling concurrent systems and their behaviors. In many applications, however, modelling by itself is of limited practical use if one cannot analyze the modelled system. As a means of gaining a better under- standing of the Petri net model, the decidability and computational complexity of typical automata theoretic problems concerning Petri nets have been exten- sively investigated in the literature in the past four decades. In this paper, we first give an overview of a number of analytical techniques known to be useful for reasoning about either structural or behavioral proper- ties of Petri nets. However, due to the intricate nature of Petri nets, none of the available analytical techniques is a panacea. To understand the limitations and capabilities of analyzing Petri nets from an algorithmic viewpoint, we also summarize a variety of decidability/complexity results reported in the literature for various Petri net problems including boundedness, reachability, containment, equivalence, and more. Among them, Lipton [40] and Rackoff [56] have shown exponential space lower and upper bounds, respectively, for the boundedness problem. As for the containment and the equivalence problems, Rabin [2] and Hack [19], respectively, have shown these two problems to be undecidable.
    [Show full text]
  • Abstract Petri Net Notation
    Abstract Petri Net Notation FalkoBause Peter Kemp er Pieter Kritzinger Forschungsb ericht der Universitat Dortmund Lehrstuhl Informatik IV Universitat Dortmund D Dortmund Germany email fbausekemp erglsinformatikunidortmundde Pieter Kritzinger is from Data Network Architecture Lab oratory Department of Computer Science UniversityofCapeTown Private Bag RONDEBOSCH South Africa pskcsuctacza Abstract The need to interchange the description of Petri Net mo dels amongst re searchers and users was recognised as long ago as This do cument prop oses an Abstract Petri Net Notation APNN in whichvarious nets can b e describ ed using a common notation The use of the notation in the context of Petri Net software to ols is shown and the general requirements for such a notation to b e generally acceptable are suggested Keywords in a the notation are similar to L T Xcommands in order to supp ort readabil E ity By employing an appropriate stylele a net description can thus b e a included directly into a L T X source do cument The notation is given for E untimed PlaceTransition Nets and Coloured Petri Nets as well as Gener alised Sto chastic Petri Nets and Hierarchical Petri Nets So eine Arbeit wird eigentlich nie fertig man muss sie fur fertig erklaren wenn man nach Zeit und Umstanden das Moglichste getan hat JW Go ethe Italienische Reise Intro duction Petri Netshave b ecome widely accepted as a formalism for describing concurrent systems and an active community of researchers exist in the eld As the theory of Petri Nets develop ed so were software
    [Show full text]
  • PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010
    UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Facultat de Lletres Departament de Filologies Romàniques Petri Net Controlled Grammars PhD Dissertation Presented by Sherzod Turaev Supervised by Jürgen Dassow Tarragona, 2010 UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Supervisor: Prof. Dr. Jürgen Dassow Fakultät für Informatik Institut für Wissens- und Sprachverarbeitung Otto-von-Guericke-Universität Magdeburg D-39106 Magdeburg, Germany Tutor: Dr. Gemma Bel Enguix Grup de Recerca en Lingüística Matemàtica Facultat de Lletres Universitat Rovira i Virgili Av. Catalunya, 35 43002 Tarragona, Spain UNIVERSITAT ROVIRA I VIRGILI PETRI NET CONTROLLED GRAMMARS Sherzod Turaev ISBN:978-84-693-1536-1/DL:T-644-2010 Acknowledgements It is a pleasure to thank many people who helped me to made this thesis possible. First of all I would like to express my sincere gratitude to my super- visor, Professor Jürgen Dassow. He helped me with his enthusiasm, his inspiration, and his efforts. During the course of this work, he provided with many helpful suggestions, constant encouragement, important advice, good teaching, good company, and lots of good ideas. It has been an honor to work with him. I would like to thank the head of our research group, Professor Car- los Martín-Vide, for his great effort in establishing the wonderful circum- stances of PhD School on Formal Languages and Applications, and for his help in receiving financial support throughout my PhD. This work was made thanks to the financial support of research grant 2006FI-01030 from AGAUR of Catalonia Government.
    [Show full text]
  • Coding Mobile Synchronizing Petri Nets Into Rewriting Logic⋆
    RULE 2006 Coding Mobile Synchronizing Petri Nets into Rewriting Logic ? Fernando Rosa-Velardo Departamento de Sistemas Inform´aticos y Programaci´on Universidad Complutense de Madrid, Spain e-mail: [email protected] Abstract Mobile Synchronizing Petri Nets (MSPN’s) are a model for mobility and coordina- tion based on coloured Petri Nets, in which systems are composed of a collection of (possibly mobile) hardware devices and mobile agents, both modelled homoge- nously. In this paper we approach their verification, for which we have chosen to code MSPN’s into rewriting logic. In order to obtain a representation of MSPN systems by means of a rewrite theory, we develop a class of them, that we call ν-Abstract Petri nets (ν-APN’s), which are easily representable in that framework. Moreover, the obtained representation provides a local mechanism for fresh name generation. Then we prove that, even if ν-APN’s are a particular class of MSPN systems, they are strong enough to capture the behaviour of any MSPN system. We have chosen Maude to implement ν-APN’s, as well as the translation from MSPN’s to ν-APN’s, for which we make intensive use of its reflective features. Key words: Mobility, Petri nets, rewriting, security, specifications 1 Introduction In several previous papers [14,15] we have presented a model for concurrent, mobile and ubiquitous systems [5,12], based on Petri Nets, that we call Mo- bile Synchronizing Petri Nets (MSPN). Petri nets provide friendly graphical representations of systems, and we can profit from their solid theoretical back- ground.
    [Show full text]
  • A PROPOSAL of USING DEVS MODEL for PROCESS MINING Yan Wang, Grégory Zacharewicz, David Chen, Mamadou Kaba Traoré
    A PROPOSAL OF USING DEVS MODEL FOR PROCESS MINING Yan Wang, Grégory Zacharewicz, David Chen, Mamadou Kaba Traoré To cite this version: Yan Wang, Grégory Zacharewicz, David Chen, Mamadou Kaba Traoré. A PROPOSAL OF US- ING DEVS MODEL FOR PROCESS MINING. 27th European Modeling & Simulation Symposium (Simulation in Industry), Sep 2015, Bergeggi, Italy. pp.403-409. hal-01536390 HAL Id: hal-01536390 https://hal.archives-ouvertes.fr/hal-01536390 Submitted on 11 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright A PROPOSAL OF USING DEVS MODEL FOR PROCESS MINING Yan Wang(a), Grégory Zacharewicz(b), David Chen(c), Mamadou Kaba Traoré(d) (a),(b),(c) IMS, University of Bordeaux, 33405 Talence Cedex, France (d) LIMOS, Université Blaise Pascal, 63173 Aubiere Cedex, France (a)[email protected], (b)[email protected], (c)[email protected], (d)[email protected] ABSTRACT many models can be selected as the resulting model for Process mining is a relative young research area which business process for example workflow nets and consists of process modeling and data mining. Process BPMN, whereas Petri Nets [Peterson, 1977] is discovery as a part of the process mining focuses on frequently identified as the direct resulting process converting event logs into process models.
    [Show full text]
  • Petri Nets in Software Engineering Von Prof. Dr. Robert Gold
    Arbeitsberichte Working Papers Kompetenz schafft Zukunft Creating competence for the future Petri Nets in Software Engineering von Prof. Dr. Robert Gold Fachhochschule Ingolstadt University of Applied Sciences Arbeitsberichte Working Papers Petri Nets in Software Engineering von Prof. Dr. Robert Gold Heft Nr. 5 aus der Reihe "Arbeitsberichte - Working Papers" ISSN 1612-6483 Ingolstadt, im Juni 2004 Abstract In this paper we investigate the use of Petri nets in software engineering extending the classical software development process with simulation and mathematical analysis based on place/transition nets. The advantage is that requirements can be validated earlier and fault detection and correction is less expensive. We show how to construct nets from basic patterns and demonstrate this for an application in automotive electronics, the cruise control with distance warning. The resulting nets can be simulated and analysed using Petri net tools and embedded into an object-oriented framework, where transitions are triggered by messages. 0 Introduction Almost every process model for software development is build around the phases requirements analysis, preliminary and detailed design, implementation, integration and test. The V-model (Figure 1), an extension of the well-known waterfall model, depicts the development phases in the form of the letter V with the verification activities module test, integration test and system test between them [Bal98], [Bal00], [Car02], [Ger97], [Jal97]. System test Requirements analysis Integration test Integration Preliminary design Module test Detailed design Implementation Fig. 1: V-model One inherent problem of such process models is that, since the first executable software emerges not before the end of the integration, the requirements cannot be validated earlier.
    [Show full text]
  • A Petri Net Based Methodology for Business Process Modeling and Simulation
    A Petri Net Based Methodology for Business Process Modeling and Simulation Joseph Barjis, Han Reichgelt Georgia Southern University P.O. Box 8150 Statesboro, GA 30460, U.S.A. Abstract. In this paper we introduce a modeling methodology for the study of business processes, including their design, redesign, modeling, simulation and analysis. The methodology, called TOP, is based on the theoretical concept of a transaction which we derive from the Language Action Perspective (LAP). In particular, we regard business processes in an organization as patterns of communication between different actors to represent the underlying actions. TOP is supported by a set of extensions of basic Petri nets notations, resulting in a tool that can be used to build models of business processes and to analyze these models. The result is a more practical methodology for business process modeling and simulation. 1 Introduction Research into business processes has recently seen a re-emergence as evidenced by the increasing number of publications and corporate research initiatives. Research into business processes encompasses a broad spectrum of topics, including business process design, redesign, modeling, simulation and analysis. The reason for this resurgence in the study of business processes are not hard to fathom. Although organizations have invested heavily in new IT applications, and the software engineering community has worked diligently on producing new methodologies and tools for constructing IT applications, the quality of many IT applications remains poor (see, e.g., the articles in [7]). As organizations struggle to ensure that their IT investments do indeed provide value, their focus is shifting increasingly to the improvement, transformation and management of business processes [9].
    [Show full text]
  • Petri Nets(PDF)
    Petri Nets* JAMES L. PETERSON Department of Computer Sciences, The Unwers~ty of Texas, Austin, Texas 78712 Over the last decade, the Petri net has gamed increased usage and acceptance as a basic model of systems of asynchronous concurrent computation. This paper surveys the basic concepts and uses of Petm nets. The structure of Petri nets, their markings and execution, several examples of Petm net models of computer hardware and software, and research into the analysis of Petm nets are presented, as are the use of the reachability tree and the decidability and complexity of some Petri net problems. Petri net languages, models of computation related to Petm nets, and some extensions and subclasses of the Petri net model are also bmefly discussed Keywords and Phrases: Petn nets, system models, asynchronous concurrent events. CR Categories. 1.3, 5.29, 8.1 INTRODUCTION tems of parallel or concurrent activities. Section 3 presents a more formal definition A Petri net is an abstract, formal model of and discussion of the fundamental con- information flow. The properties, con- cepts and notations of Petri nets. Section 4 cepts, and techniques of Petri nets are considers the extensive body of research being developed in a search for natural, dealing with the analysis of Petri nets, simple, and powerful methods for describ- their advantages, and their limitations. ing and analyzing the flow of information Petri net languages are presented in Sec- and control in systems, particularly sys- tion 5. Finally, in Section 6, we consider tems that may exhibit asynchronous and some of the many variations of Petri nets concurrent activities.
    [Show full text]
  • Petri Nets Petri Nets
    Petri nets Petri nets Petri nets are a basic model of parallel and distributed systems (named after Carl Adam Petri). The basic idea is to describe state changes in a system with transitions. s1 s2 @ @R t @ @R s3 s4 Petri nets contain places and transitions that may be connected by directed arcs. Places symbolise states, conditions, or resources that need to be met/be available before an action can be carried out. Transitions symbolise actions. 2 Behaviour of Petri nets Places may contain tokens that may move to other places by executing (“firing”) actions. A token on a place means that the corresponding condition is fulfilled or that a resource is available: s1 s2 }@ @R t @ @R s3 s4 } In the example, transition t may “fire” if there are tokens on places s1 and s3. Firing t will remove those tokens and place new tokens on s2 and s4. 3 Why Petri Nets? low-level model for concurrent systems expressly models concurrency, conflict, causality, . finite-state or infinite-state models Content: Semantics of Petri nets Modelling with Petri nets Analysis methods: finite/infinite-state case, structural analysis Remark: Many variants of Petri nets exist in the literature; we regard a special simple case also called P/T nets. 4 Petri Net A Petri net is a tuple N = hP; T ; F; W ; m0i, where • P is a finite set of places, • T is a finite set of transitions, • the places P and transitions T are disjoint (P \ T = ;), • F ⊆ (P × T ) [ (T × P) is the flow relation, • W : ((P × T ) [ (T × P)) ! IN is the arc weight mapping (where W (f ) = 0 for all f 2= F, and W (f ) > 0 for all f 2 F), and • m0 : P ! IN is the initial marking representing the initial distribution of tokens.
    [Show full text]