**1. Introduction**

Prof. Carl Adam Petri wrote: "In order to apply net theory with success, a user of net theory can just rely on the fact that every net which he can specify explicitly (draw on paper) can be connected by a short (≤ 4) chain of net morphisms to the physical real word; your net is, in a very precise sense, physically implementable." (Status Report On Net Theory, 1989, a forward for my book Petri Nets in Chinese[1]).

Why a net is physically implementable? The reason is, every concept in net theory is carefully chosen based on nature laws, and well defined in terms of precise mathematics and logic. For example, the concept of global time does not belong to net theory. Time measured with real numbers exists only in theories like theoretical physics. Logical time does not exist in the real world. For net theory, time is just "clock reading", a measurement of physical changes. Global time is not realistic for systems in which a shared clock is not available.

On the other hand however, it is easy to find in the literature, that many an author introduces new concepts into his or her Petri net with implementation totally forgotten. "Timed Petri Net" is just one of such examples.

As one of the chapters in this book on Petri nets, implementable concepts and only implementable concepts will be introduced.

We start with the definition of a directed net, which is the most fundamental concept in net theory. The next two sections serve to keep this chapter self-reliant.

This chapter is organized as below:

Sections 2 and 3 recall basic definitions of Petri Nets: The concept of directed net deserves a separate section since it is the foundation of the whole net theory. Section 3 is mainly about Place/Transition-systems, based on which workflow models are to be constructed.

Section 4 is an introduction of synchrony, a branch in net theory on transition synchronization that provides theoretical support to workflow modelling.

Workflow Modelling Based on Synchrony 109

x={y|(y,x)∈ F} is the pre-set of x,

﹒ is the extension of t.◆

A triple N=(S,T;F) is a directed net if S∪T≠∅∧S∩T=∅ ∧ F ⊑ (S×T∪T×S) ∧ dom(F) ∪

A directed net used to be called "Petri net". We keep the term "Petri net" to mean "net theory", a term Carl Adam Petri used in *Status Report*. "Net" is often used to mean "directed

S∪T≠∅∧S∩T=∅ demands that a net consists of at least one element, and its elements are

F ⊑ S × T ∪ T × S indicates that direct dependence does not exist between elements in the

A directed net has a graphical presentation as shown in Figure 1, in which elements in S and T appear as circles and boxes respectively while elements in F appear as arrows (arcs). The

Petri net (net theory) consists of Special Net Theory and General Net Theory. Special Net Theory focuses on system modelling and General Net Theory focuses on theories supporting system modelling. The concept of directed nets is their common foundation.

Special Net Theory (SNT for short) and General Net Theory (GNT for short) are derived

The S-complementation operation on a net leads to the removal of contact. The Tcomplementation operation on a net leads to the removal of differences between forward

from Directed Net based on pre-sets and post-sets of elements as described below.

The concept of extensions of transitions leads to the principle of local determinism.

Let N=(S,T;F) be a directed net and X=S∪T, For x in X, ﹒

x﹒={y|(x,y)∈ F} is the post-set of x. For t∈ T, ﹒t∪t

and backward flow of tokens (SNT).

cod(F) = S ∪ T Where dom(F)={x∣∃y: (x, y) ∈ F} and cod (F)={y∣∃x: (x, y) ∈ F}.◆

same class. dom(F) ∪ cod(F) = S ∪ T excludes isolated elements from a net.

**2. Directed net** 

clearly classified.

**Figure 1.** A Directed Net

Definition 2

net" if it causes no ambiguity.

arrow from x to y represents (x,y) in F.

Definition 1

Section 5 talks about business processes, the subject of workflow research. A full understanding of the concept of business processes makes a good start.

Section 6 proposes the concepts of synchronizers and workflow logic. A synchronizer connects transitions in two consecutive steps in a business process, and workflow logic is obtained when all transitions in a business process are so connected. Properties and analysis methods of workflow logic are defined and proposed. A transition in workflow logic represents a business task while a synchronizer represents a task in workflow management. Workflow logic specifies all possible routes a business case may take when it is processed. An individual business case corresponds to a unique route among all routes given by workflow logic. This route is considered as the semantics of that case. Section 7 defines the concept of case semantics.

Section 8 is about business process management, or automatic management. The dual net of workflow logic is exactly the logic of management, based on which workflow engine conducts the process of individual cases.

Section 9 concludes this chapter with acknowledgement, and the last section is a list of references.

This chapter is about Petri nets and workflow modelling.

There are many ways in the literature to define nets and net systems. For example, the concept of flow relation, namely F, has been made implicit by many researchers. It is often combined with weight function W. Without the flow relation, the concept of directed nets would disappear; and without the concept of directed nets, the whole theoretical part of Petri nets would be without a foundation. Thus, this chapter starts from the definition of directed nets given by C. A. Petri himself.

Petri net systems have been considered as one of the adequate and promising candidates for workflow modelling. The concept of WF-nets proposed by Prof. Aalst from Holland has become popular in the last 10 to 20 years. A team in the Software Development Company of Peking University tried to use WF-nets as a formal model to develop software for a government organization at a time around the year 2000. The WF-nets didn't work, and they didn't know why. The author joint them, and we found problems of WF-nets, leading to failure. The concept of WF-nets was proposed without theoretical foundation. These problems were discussed in our paper titled "A Three Layer Model for Business process" [7]. The concepts of synchronizers, workflow logic and case semantics etc were defined in this paper for the first time. After so many years since then, people interested in workflow modelling remain sticking to WF-nets. Many people do not even know our work, it seems. The reason is, the author guess, General Net Theory (theoretical part of Petri nets) is not popular yet. This chapter shows how important "Synchrony" is to a successful application of Petri nets in the area of workflow modelling.
