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## Über dieses Buch

The theory of Petri nets is a part of computer science whose importance is increasingly acknowledged. Many papers and anthologies, whose subject matter is net theory and its applications, have appeared to date. There exist at least seven introductory textbooks on the theory. The present monograph augments this literature by offering a mathematical treatment of one of the central aspects of net theory: the modelling of concur­ rency by partially ordered sets. Occurrence nets - which are special nets as well as special partial orders - are proposed by net theory for this purpose. We study both the general properties of occurrence nets and their use in describing the concurrent behaviour of systems. Occurrence nets may be contrasted with a more language-oriented approach to the modelling of concurrency known as arbitrary interleaving. We will dis­ cuss some connections between these' two approaches. Other approaches based on partially ordered sets - such as the theory of traces, the theory of event structures and the theory of semi words - are not considered in this book, in spite of the strong links between them and net theory.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The concept of concurrency plays a fundamental role in computer science. A sizeable share of present-day computer systems feature concurrency in an explicit way, being distributed systems or multi-processor systems or both. In the guise of co-existing variables and data structures, concurrency pervades the more conventional sequential systems as well.
Eike Best, César C. Fernández

### Chapter 2. Partially Ordered Sets

Abstract
The present chapter gives some mathematical theory of partially ordered sets. Referring to the appendix on terminology, we recall that a partially ordered set is a pair (X, ≺) where ≺ is an irreflexive and transitive relation on X. We shall not immediately give the interpretation of the elements of X. For the purpose of this chapter, it suffices to think of a poset (X, ≺) as describing a history or a process (of a concurrent system) and of an element xX as representing a basic occurrence, i.e., an item which has occurred once, and only once, in the history given by (X, ≺). The relation ≺ means ‘before’; that is, xy means that x has occurred earlier than y in the history given by (X, ≺).
Eike Best, César C. Fernández

### Chapter 3. Petri Nets

Abstract
We have motivated the study of partially ordered sets by the claim that they may be used to define concurrent behaviour faithfully. But we have not yet specified the systems the behaviours of which are to be described.
Eike Best, César C. Fernández

### Chapter 4. Connections Between Systems and Processes

Abstract
Having defined the processes of a system net via the process definition 3.3.3, or (as we know: equivalently for system nets of finite synchronisation) via the construction given in Definition 3.4.1, it is now possible to translate properties from a system to the set of its processes and vice versa. It is possible to ask, for example, what it would mean for a system that all of its processes enjoy some interesting property, or conversely.
Eike Best, César C. Fernández

### Backmatter

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