Net theory is a theory of systems organization which had its origins, about 20 years ago, in the dissertation of C. A. Petri [1]. Since this seminal paper, nets have been applied in various areas, at the same time being modified and theoretically investigated. In recent time, computer scientists are taking a broader interest in net theory. The main concern of this book is the presentation of those parts of net theory which can serve as a basis for practical application. It introduces the basic net theoretical concepts and ways of thinking, motivates them by means of examples and derives relations between them. Some extended examples il­ lustrate the method of application of nets. A major emphasis is devoted to those aspect which distinguish nets from other system models. These are for instance, the role of concurrency, an awareness of the finiteness of resources, and the pos­ sibility of using the same representation technique of different levels of ab­ straction. On completing this book the reader should have achieved a system­ atic grounding in the subject allowing him access to the net literature [25]. These objectives determined the subjects treated here. The presentation of the material here is rather more axiomatic than in­ ductive. We start with the basic notions of 'condition' and 'event' and the con­ cept of the change of states by (concurrently) occurring events. By generali­ zation of these notions a part of the theory of nets is presented.

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1. Introductory Examples and Basic Definitions.- 1.1 Examples from Different Areas.- 1.2 Examples from Logic Circuits and Operating Systems.- 1.3 Non-Sequential Programs.- 1.4 An Example for Systems Analysis.- 1.5 Some Basic Definitions.- 1.6 Summary and Overview.- Exercises for Chapter 1.- 1. Condition/Event-Systems.- 2. Nets Consisting of Conditions and Events.- 2.1 Cases and Steps.- 2.2 Condition/Event-Systems.- 2.3 Cyclic and Live Systems.- 2.4 Equivalence.- 2.5 Contact-Free C/E-Systems.- 2.6 Case Graphs.- Exercises for Chapter 2.- 3. Processes of Condition/Event-Systems.- 3.1 Partially Ordered Sets.- 3.2 Occurrence Nets.- 3.3 Processes.- 3.4 The Composition of Processes.- 3.5 Processes and Case Graphs.- Exercises for Chapter 3.- 4. Properties of Systems.- 4.1 Synchronic Distances.- 4.2 Some Quantitative Properties of Synchronic Distances.- 4.3 Synchronic Distances in Sequential Systems.- 4.4 Synchronic Distances in Cyclic Systems.- 4.5 Facts.- Exercises for Chapter 4.- 2. Place/Transition-Nets.- 5. Nets Consisting of Places and Transitions.- 5.1 Place/Transition-Nets.- 5.2 Linear Algebraic Representation.- 5.3 Coverability Graphs.- 5.4 Decision Procedures for Some Net Properties.- 5.5 Liveness.- Exercises for Chapter 5.- 6. Net Invariants.- 6.1 S-Invariants.- 6.2 Nets Covered by S-Invariants.- 6.3 The Verification of System Properties Using S-Invariants.- 6.4 Properties of a Sender-Receiver Model.- 6.5 A Seat-Reservation System.- 6.6 The Verification of Facts in C/E-Systems by Means of S-Invariants.- 6.7 T-Invariants.- Exercises for Chapter 6.- 7. Liveness Criteria for Special Classes of Nets.- 7.1 Marked Nets, Deadlocks and Traps.- 7.2 Free Choice Nets.- 7.3 Marked Graphs.- Exercises for Chapter 7.- 3. Nets with Individual Tokens.- 8. Predicate/Event-Nets.- 8.1 An Introductory Example.- 8.2 Predicate/Event-Nets.- 8.3 An Organization Scheme for Distributed Databases.- 8.4 Facts in P/E-Nets.- 8.5 A Normal Form for P/E-Nets.- Exercises for Chapter 8.- 9. Relation Nets.- 9.1 Introductory Examples.- 9.2 Relation Nets.- 9.3 The Translation of P/E-Nets into Relation Nets.- 9.4 Calculation with Multirelations.- 9.5 A Matrix Representation for Relation Nets.- 9.6 S-Invariants for Relation Nets.- 9.7 An Example for Applying S-Invariants: The Verification of Facts.- 9.8 Relation Net Schemes.- Appendix. Mathematical Notions and Notation.- I. Sets.- II. Relations.- III. Mappings, Functions.- IV. Partial Orders.- VII. Vectors and Matrices.- Further Reading.- 1. Some Landmarks in the Development of Net Theory.- 2. Conferences on Petri Nets.- 3. Text Books.- 4. Bibliographies.- 5. References to Chapter 2.- 6. References to Chapter 3.- 7. References to Chapter 4.- 8. References to Chapter 5.- 9. References to Chapter 6.- 10. References to Chapter 7.- 11. References to Chapter 8.- 12. References to Chapter 9.- 13. Modifications and Generalizations of Place/Transition-Nets.- 14. Applications.- 15. Implementation and Automatic Analysis of Nets.- 16. Related System Models.