**2. Directed net**

#### Definition 1

108 Petri Nets – Manufacturing and Computer Science

concept of case semantics.

references.

conducts the process of individual cases.

directed nets given by C. A. Petri himself.

of Petri nets in the area of workflow modelling.

This chapter is about Petri nets and workflow modelling.

Section 4 is an introduction of synchrony, a branch in net theory on transition

Section 5 talks about business processes, the subject of workflow research. A full

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

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

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

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

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

synchronization that provides theoretical support to workflow modelling.

understanding of the concept of business processes makes a good start.

A triple N=(S,T;F) is a directed net if S∪T≠∅∧S∩T=∅ ∧ F ⊑ (S×T∪T×S) ∧ dom(F) ∪ cod(F) = S ∪ T Where dom(F)={x∣∃y: (x, y) ∈ F} and cod (F)={y∣∃x: (x, y) ∈ 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 net" if it causes no ambiguity.

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

F ⊑ S × T ∪ T × S indicates that direct dependence does not exist between elements in the same class. dom(F) ∪ cod(F) = S ∪ T excludes isolated elements from a net.

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

**Figure 1.** A Directed Net

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.

## Definition 2

Let N=(S,T;F) be a directed net and X=S∪T, For x in X, ﹒ x={y|(y,x)∈ F} is the pre-set of x, x﹒={y|(x,y)∈ F} is the post-set of x. For t∈ T, ﹒t∪t ﹒ is the extension of t.◆

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

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 and backward flow of tokens (SNT).

The S-completion operation on a net leads to Synchrony while the T-completion operation on a net leads to Enlogy(GNT).

Workflow Modelling Based on Synchrony 111

complete marking. Many system models take global states as a means of system control. But this is not always implementable, since a global state is not always instantly known. It takes time to know the current global state, and this delay may be significant to an effective real

Figure 2 illustrates how to represent a net system graphically. The black dot inside place s is a token, denoting M0(s) = 1. Empty places have no token (resource). Conventionally, M(s)=0,

Let Σ=(S,T;F,K,W,M0) be a net system, M is a marking and t1, t2 are transitions.

+ W(t2,s)≤ K(s),then t1 and t2 are in concurrent relation at M, denoted by M[{t1,t2}>.

4. If there is a transition t and a place b such that ∀s ∈﹒t1 : M(s)≥W(s,t)∧M(b) +

The marking reached by concurrently fired transitions can also be reached by the transitions

This theorem guarantees that the next definition includes markings reached by concurrent

The set of markings reachable from M0 by consecutive transition firings is usually denoted

3. If M[t1>∧M[t2>∧¬M[{t1, t2}>, then t1,t2 are in conflict at M, denoted by cf(t1,t2,M).

Two concurrently enabled transitions can fire either concurrently or one after another.

W(t,b)>K(b), then t leads to a contact in b at M, denoted by ct(t,b,M). ◆

2. If M[t1>∧M[t2>, and ∀s ∈﹒t1 ∩﹒t2 : M(s) ≥W(s,t1) + W(s,t2) and ∀s ∈ t1

[t2>∧¬M[t2>, then t1 and t2 are in sequential relation at M, denoted by

﹒ ∩ t2

﹒: M(s) + W(t1,s)

time control.

**Figure 2.** A Net System

∧M'

Definition 5

1. If M[t1>M'

Theorem 1

fired one after another.◆

transition firings.

Definition 6

by [M0>.◆

M[t1,t2>.

W(x,y) =1 and K(s) = ∞ are shown by default.

A directed net implies a unique undirected net that leads to Net Topology (GNT).

A special class of directed nets is the occurrence nets that lead to Concurrency (GNT).

Synchrony will be briefly introduced in Section 3, since it provides guidance to workflow modelling. It is impossible in this chapter to go any further on SNT and GNT. The point is, successful applications of Petri nets rely on both SNT and GNT, not only SNT.
