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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.

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction

Abstract
Petri nets, the subject of this book, are a model for procedures, organizations and devices where regulated flows, in particular information flows, play a role.
Wolfgang Reisig

Introductory Examples and Basic Definitions

Chapter 1. Introductory Examples and Basic Definitions

Abstract
In the preface and the introduction, we have already used the terms “system organization”, “system model”, “condition”, “event” and “information transformation” without explaining them. These notions are of fundamental importance in net theory. However, as they are concepts from the real world, we shall not try to give precise definitions of them but rather appeal to the intuition and general understanding of the reader. But, we shall have to consider properties of objects of this kind, and also the relationships between such objects. We shall say, for instance, that “system models” represent real systems more or less adequately, that “events” occur and that “conditions” do or do not hold.
Wolfgang Reisig

Condition/Event-Systems

Frontmatter

Chapter 2. Nets Consisting of Conditions and Events

Abstract
First, for nets consisting of conditions and events, we must make precise the meaning of “occurrence of a single event or several independent events”. For this, the notion of a step is introduced. A notion of equivalence for condition/ event systems (C/E-systems) is then introduced, and we show how each system can be transformed to an equivalent contact-free normal form. Finally we discuss the case graph of a C/E-system. This graph provides an overview of all cases and steps of the system.
Wolfgang Reisig

Chapter 3. Processes of Condition/Event-Systems

Abstract
This chapter deals with processes which can run on C/E-systems. One may be tempted to define a process of a C/E-system as a path of its case graph. But what we mean intuitively when speaking of processes is not adequately described by such a path: the total ordering of its elements does not give any information as to whether the events actually occur one after the other or whether they are independent of each other. The partial order in which events occur is only indirectly represented in the case graph by the set of all possibilities of occurrences as successions of steps.
Wolfgang Reisig

Chapter 4. Properties of Systems

Abstract
In the previous chapter we saw how to describe C/E-systems and how to define and analyse their dynamic behaviour. We shall now concern ourselves with some properties of C/E-systems. We shall see that some of those properties can again be described by means of the net calculus.
Wolfgang Reisig

Place/Transition-Nets

Frontmatter

Chapter 5. Nets Consisting of Places and Transistions

Abstract
As a first example in this chapter we consider a system consisting of one producer and two consumers. We have already seen this in Fig. 5. In this modified version
(1)
the buffer may contain at most five tokens,
 
(2)
the producer generates three tokens in each step,
 
(3)
at most one consumer is able to access the buffer in each configuration of the system,
 
(4)
each consumer removes two tokens when accessing the buffer,
 
(5)
the production steps of the producer are counted.
 
Wolfgang Reisig

Chapter 6. Net Invariants

Abstract
In this chapter, we are first concerned with sets of places of P/T-nets which do not change their token count during transition firings. Knowledge about any such sets of places not only helps in analysing liveness but also allows us to investigate other properties of systems (for instance, facts in C/E-systems). Such sets of places will be called S-invariants.Since invariants are characterized by solutions of linear equation systems of the form N x = 0, (N denotes the transpose of N, cf. Appendix VII) it is possible to compute them by the well-known methods of linear algebra.
Wolfgang Reisig

Chapter 7. Liveness Criteria for Special Classes of Nets

Abstract
In this chapter, we investigate marked nets; these are special P/T-nets which are suitable for many applications. The liveness analysis for such nets is not much simpler than for P/T-nets in general, but there are special classes of marked nets for which criteria for liveness or safeness are known. These criteria are the main topic of this chapter.
Wolfgang Reisig

Nets with Individual Tokens

Frontmatter

Chapter 8. Predicate/Event-Nets

Abstract
We consider an example which is well known as “The Dining Philosophers Problem”. To start with, we represent it as a C/E-system.
Wolfgang Reisig

Chapter 9. Relation Nets

Abstract
After introducing P/E-nets, we now present a further net model using individuals as tokens. This new model, in particular, supports a calculus of invariants.
Wolfgang Reisig

Backmatter

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