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Dynamic Analysis of Petri Net-Based Discrete Systems PDF

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Lecture Notes in Control and Information Sciences 356 Editors:M.Thoma,M.Morari Andrei Karatkevich Dynamic Analysis of Petri Net-Based Discrete Systems ABC SeriesAdvisoryBoard F.Allgöwer,P.Fleming,P.Kokotovic, A.B.Kurzhanski,H.Kwakernaak, A.Rantzer,J.N.Tsitsiklis Author Dr.AndreiKaratkevich InstituteofComputerEngineeringandElectronics FacultyofElectricalEngineering, ComputerScienceandTelecommunications UniversityofZielonaGóra Ul.Pogfórna50 65-246ZielonaGóra Poland Email:[email protected] LibraryofCongressControlNumber:2007923719 ISSNprintedition:0170-8643 ISSNelectronicedition:1610-7411 ISBN-10 3-540-71464-2 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-71464-4 SpringerBerlinHeidelbergNewYork This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationor partsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,in itscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliablefor prosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandSPSusingaSpringerLATEXmacropackage Printedonacid-freepaper SPIN:11903901 89/SPS 543210 Preface Design of modern digital hardware systems and of complex software systems is almost always connected with parallelism. For example, execution of an object-oriented pro- gram can be considered as parallel functioning of the co-operating objects; all modern operating systems are multitasking, and the software tends to be multithread; many complex calculation tasks are solved in distributed way. But designers of the control systems probably have to face parallelism in more evident and direct way. Controllers rarely deal with just one controlled object. Usually a system of several objects is to be controlled, and then the control algorithm naturally turns to be parallel. So, classical and very deeply investigated model of discrete device, Finite State Machine, is not expressive enough for the design of control devices and systems. Theoretically in most of cases behavior of a controller can be described by an FSM, but usually it is not convenient; such FSM description would be much more complex, than a parallel specification (even as a network of several communicating FSMs). The engineers and researchers became aware of practical necessity of developing of parallel discrete models about forty years ago. There were (and are) two main ap- proaches to such models. One is a direct development of the FSM, being enhanced by parallelism and hierarchy. Another one is based on the parallelism “from the very beginning”. The most famous and popular model of this second kind is Petri nets. A big family of more or less detailed behavioral specifications of parallel systems is based on this formalism. Design and, especially, analysis and verification of systems, which behavior is specified by the parallel models, is a remarkably more complex task, than design and analysis of strictly sequential systems. In this book, we are concerned about the formal analysis and verification of the parallel systems, specified by the Petri nets and the extended Petri net models. Be- sides, some results presented here are related to the FSM networks (but, again, we model them by means of Petri nets) and to the sequent automata (a kind of parallel descriptions other than Petri nets). To formulate some general affirmations, we use a general model of a parallel discrete system, which covers all specific models studied in the book. We have focused on the approach of reduced exploration of state spaces. This ap- proach is selected here as the basic one, because the state exploration provides the most detailed information about system behavior among other analysis approaches, and, on the other hand, such exploration does not have to be full to decide many im- portant properties of the systems (especially with restricted structure). The analysis methods of such kind are thoroughly developed; our work was in- spired by the results of many authors, first of all of A. Valmari and P. Godefroid. For inspiration of another kind (the interesting parallel models and the methodology of research) we are grateful to A. Zakrevskij. The original results presented in this book VI Preface are mostly connected to generalization of the known methods or, vice versa, to apply- ing them to specific subclasses of parallel systems, which sometimes allows to obtain more information than in general case. We intended this book to be useful to CAD researches and designers of parallel control systems. Content of the book is mostly theoretical, but it was written bearing in the mind possible practical applications. It may also be useful for the students of electrical engineering and computer science. March 2007 Andrei Karatkevich Zielona Góra Symbols Main symbols A FSM or parallel automaton C incidence matrix of a Petri net D siphon e event E set of edges of a graph F set of arcs of a Petri net G graph I set of input (external) events k elementary conjunction; implicant of a Boolean function L cycle in a graph M marking or global state M deadlock d M initial marking or initial global state 0 mp macroplace N FSM network O set of output events p place or local state p Boolean variable corresponding to place or local state p i i P set of places or set of local states Pin set of input places Pout set of output places Q path in a graph Qp code of local state p (an elementary conjunction) i i R partial order relation s sequent S sequent automaton t transition T set of transitions T persistent set P T stubborn set S VIII Preface u elementary disjunction V set of nodes of a graph w weight (of an arc) X set of input variables Y set of output variables Z set of internal variables or events Γ set of events ∆ step of concurrent simulation (cid:3) Boolean variable corresponding to event e i i µ initial label of a parallel automaton transition ν terminal label of a parallel automaton transition ρ priority relation σ firing sequence Σ Petri net ϕ left part of a sequent ψ right part of a sequent Main operators and functions enabled(M) set of transitions enabled in M M(p) number of tokens in place p or activity of local state p at M M(P) sum of tokens in places belonging to P at M [M(cid:1) set of markings or global states, reachable from M P(p) set of local states parallel to p V(G) set of nodes of graph G •x set of predecessors of node (place or transition) x of a Petri net x• set of successors of node (place or transition) x of a Petri net [x] cluster of Petri net containing node x |σ| length of firing sequence σ Main abbreviations BDD binary decision diagram CAD computer aided design CNF conjunctive normal form DNF disjunctive normal form EFC extended free choice FSM finite state machine HPN hierarchical Petri net IPN interpreted Petri net JPVM Java ParallelVirtual Machine LS live and safe OPN operational Petri net OPT ”optimal simulation” PN Petri net Preface IX PNSF Petri Net Specification Format PSS ”parallel selective search” PDG program dependency graph RRG reduced reachability graph SCC strongly connected component SFC Sequential Function Chart SM state machine TC terminal component UML Unified Modelling Language XML Extensible Markup Language Contents 1. Introduction................................................ 1 1.1 Analysis of Parallel Discrete Systems........................ 3 1.2 Preliminary Remarks ..................................... 5 1.3 The Scope of the Book .................................... 6 2. Main Notions, Problems and Methods...................... 9 2.1 Models and Specifications of Parallel Discrete Systems ........ 9 2.1.1 ParallelDiscrete Systems (General Definition) ......... 9 2.1.2 Petri Nets and Their Extensions...................... 10 2.1.3 Paralleland Sequent Automata ...................... 16 2.1.4 Sequential Function Charts .......................... 17 2.1.5 Statecharts ........................................ 17 2.1.6 Finite State Machines and FSM Networks ............. 20 2.2 The Tasks of Analysis..................................... 21 2.3 The Methods of Analysis .................................. 23 3. Reduced Reachability Graphs .............................. 27 3.1 Review of Known Methods ................................ 27 3.1.1 Persistent Set Methods.............................. 27 3.1.2 Other Methods..................................... 32 3.2 A Generalization of Stubborn Set Method ................... 33 3.3 Weak Persistent Sets ..................................... 34 3.4 On Combining the Persistent Set Approach and Concurrent Simulation............................................... 37 3.4.1 Concurrent Simulation and Persistent Sets............. 37 3.4.2 Comparison with the Janicki-Koutny’s Method......... 39 3.4.3 Conflicts, State Explosion and Decomposition.......... 41 3.5 Analysis of Special Classes of Petri Nets..................... 42 3.5.1 Properties and Analysis of α-Nets .................... 42 3.5.2 A Hypothesis on EFC-Nets .......................... 53 3.5.3 Analysis of s-Nets .................................. 53 3.6 Minimization of Space .................................... 56 XII Contents 3.6.1 Dynamic Reduction of Reachability Graphs............ 57 3.6.2 Reducing the Space for Various Analysis Tasks......... 59 3.6.3 Example .......................................... 60 3.6.4 Conclusive Notes on the Method ..................... 61 4. Decomposition for Analysis................................. 63 4.1 Block Decomposition ..................................... 63 4.1.1 Operational Petri Nets .............................. 63 4.1.2 Analysis of Operational Petri Nets.................... 64 4.1.3 Analysis of a Class of Cyclic Nets .................... 66 4.1.4 Example and Experimental Results ................... 67 4.2 HierarchicalDecomposition ................................ 69 4.2.1 A Conception of Hierarchical Decomposition of Petri Nets....................................... 70 4.2.2 Properties of P-Decomposition ....................... 74 4.2.3 Finding P-Blocks................................... 75 4.3 Decomposition and Persistent Sets.......................... 78 4.4 ParallelAnalysis ......................................... 79 4.5 Distributed Analysis ...................................... 82 4.5.1 A Method of Distributed Analysis .................... 82 4.5.2 Implementation of the Method ....................... 84 4.5.3 Experimental Results and Concluding Remarks ........ 85 5. Analysis by Solving Logical Equations – Calculation of Siphons and Traps.......................................... 87 5.1 Known Methods of Calculation of Siphons and Traps ......... 88 5.1.1 Calculation of Siphons and Traps by Means of Solving Logical Equations .................................. 88 5.1.2 Other Approaches to Calculation of Siphons and Traps ............................... 89 5.2 Algorithm to Find Siphons and Traps....................... 90 5.3 Example ................................................ 91 5.3.1 The ProposedMethod .............................. 91 5.3.2 Some Other Symbolic Methods....................... 92 5.3.3 The Linear Algebraic Method........................ 92 5.4 Concluding Remarks...................................... 93 6. Verification of Detailed System Descriptions................ 95 6.1 Application of the Described Approaches to Other Parallel Discrete Models .......................................... 95 6.1.1 Interpreted Petri Nets and Sequent Automata.......... 95 6.1.2 Statecharts ........................................ 102 6.1.3 FSM Networks..................................... 108 6.2 Verification of Parallel Automata Implementation ............ 113 6.2.1 Testing Approach .................................. 113 6.2.2 Analytical Approach................................ 117

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