**6.3. Throughness and reduction rules**

Let Σ = (S,T; F, K, W, M0) be a well-structured workflow logic, and M is a reachable marking.

Definition 17


The workflow logic in Figure 2 is through.

Theorem 5

If workflow logicΣis through, then for every transition t, there is a reachable marking M that enables t and there is a route to which t belongs.◆

From the fact that Σ is acyclic and s0 is unique, we know that there is a directed path from s0 to t. By mathematical reduction on the length of this path, this theorem is easy to prove.

Soundness of a WF-net is proved via computing T-invariant by adding an extra transition between its unique place o and its unique initial place i. This method applies to workflow logic as well. But we do it differently.

Keep in mind the principle of local determinism. Our attention focuses on local structures of workflow logic rather than global properties like T-invariants.

Characteristics of local structures bring up the following reduction rules for proving throughness of workflow logic.

Reduction Rule 1

Synchronizer p1 = (T1,T0,(a1, a2)) and p2 = (T0,T2,(b1,b2)) can be reduced to synchronizer p = (T1,T2,(a1,b2)) if a2 = b1. ◆

Figure 14 illustrates this rule.

Let Σ be the workflow logic to which rule 1 is applied and Σ' is resulted by replacing p1, T0, p2 with p in Σ. We have

Theorem 6

Σ is through if and only if Σ' is through.◆

Synchronizer p1 requires that a2 transitions from T0 to fire after the firings of a1 transitions from T1 while synchronizer p2 requires that b1 transitions from T0 to be fired before any transition firing from T2. Given a2 = b1, p1 and p2 are consistent with each other on T0: a1 transitions from T1 followed by b2 transitions from T2. Thus, the theorem is true. (p1,T0, p2) is in fact the detail of p. This reduction of omitting consistent detail is a net morphism in net topology. No wonder the property of being through is reserved. All reduction rules are in fact to conceal local details, and as such, they reserve throughness. But there will be no more theorems and proofs to be given below for simplicity.

**Figure 14.** Reduction Rule 1

Reduction Rule 2

126 Petri Nets – Manufacturing and Computer Science

soundness are covered by being through.

**6.3. Throughness and reduction rules** 

also a termination marking. ◆

logic as well. But we do it differently.

throughness of workflow logic.

Reduction Rule 1

(T1,T2,(a1,b2)) if a2 = b1. ◆

p2 with p in Σ. We have

Theorem 6

Figure 14 illustrates this rule.

Σ is through if and only if Σ' is through.◆

The workflow logic in Figure 2 is through.

that enables t and there is a route to which t belongs.◆

workflow logic rather than global properties like T-invariants.

marking.

Definition 17

Theorem 5

In what follows, workflow logic is always assumed well-structured.

What we propose is throughness: the token in s0 (an abstract case) will be passed over by transition firings (tasks) via the engine, to a unique end-place. All properties required by

Let Σ = (S,T; F, K, W, M0) be a well-structured workflow logic, and M is a reachable

1. M is a termination marking if it enables no transition; M is an end-marking if there is a

2. Σ is through if every termination marking is an end-marking and every end-marking is

If workflow logicΣis through, then for every transition t, there is a reachable marking M

From the fact that Σ is acyclic and s0 is unique, we know that there is a directed path from s0 to t. By mathematical reduction on the length of this path, this theorem is easy to prove.

Soundness of a WF-net is proved via computing T-invariant by adding an extra transition between its unique place o and its unique initial place i. This method applies to workflow

Keep in mind the principle of local determinism. Our attention focuses on local structures of

Characteristics of local structures bring up the following reduction rules for proving

Synchronizer p1 = (T1,T0,(a1, a2)) and p2 = (T0,T2,(b1,b2)) can be reduced to synchronizer p =

Let Σ be the workflow logic to which rule 1 is applied and Σ' is resulted by replacing p1, T0,

unique end-place e such that M(e) = 1 and M(s) = 0 for all places s other than e.

For synchronizer p = ({t1}, {t2}, (1,1)), if t1. ∩. t2 = {p}, then (t1, p, t2) can be replaced by a single transition t with . t = . t1∪. t2– {p} and t. = t1. ∪t2. – {p}. All weights on remaining arcs remain.◆

Figure 15 illustrates this rule.

**Figure 15.** Reduction Rule 2

It is easy to see that the combined effect of t1 and t2 on all places other than p is the same as the effect of t. The dotted circles and arcs may exist, but not necessarily.

Reduction Rule 3

For transition t and places p1, p2, if p1. ∩. p2 ={t} and W(p1, t) = W(t, p2) = 1, then (p1,t,p2) can be replaced by p, . p = . p1∪. p2– {t} and p. = p1. ∪p2. –{t}. All weights on remaining arcs remain. ◆

Note that places p1 and p2 are not necessarily synchronizers, i.e. it is possible . p1 = ∅ and/or p2. = ∅. 

Figure 16 (a) illustrates this rule.

Workflow Modelling Based on Synchrony 129

We will see the significance of this rule and the rule next when management logic is discussed, since separated managements before the rule is applied become centralized

If pi = ({t}, {ti}, (1, 1)) is a synchronizer for every i, i = 1, 2, …, a, then these synchronizers can

If pj = ({tj}, {t}, (1, 1)) is a synchronizer for every j, j = 1, 2, …, b, then these synchronizers can

The next reduction rule reserves throughness, but a place element rather than a

In this rule, i = 1, 2, …, a and j = 1, 2, …, b. For transition sets T = {t1, t2,…, ta} and Ti = {ti1, ti2, …, tib}, if pi = ({ti}, Ti, (1, b)) is a synchronizer for every i, then these synchronizers can be

{ti}, (b, 1)) is a synchronizer for every i, then these synchronizers can be reduced to the place

A noticeable fact is: places p and q are different from a synchronizer at two aspects. Firstly, their capacities are both ab instead of the product of the input weight and the output weight. Secondly, transitions in Ti and Tj , i ≠ j, maybe enabled at different times before reduction, but they will be enabled at the same time by p; similarly, transitions in T will be enabled at the same time by q, but maybe not before reduction. The important point is, the number of transitions to be selected for a route remain unchanged (real selection is to be

��� , K(p) = ab, W(ti, p) = b and W(p, tij) = 1, If pi

.' = (Ti,

be reduced to a single synchronizer p = ({t}, {t1, t2,…, ta}, (1, a)).

be reduced to a single synchronizer p = ({t1, t2,…, tb}, {t}, (b, 1)).◆

Figure 18 illustrates this rule, where a = b = 3. .

management after it.

**Figure 18.** Reduction Rule 5

Reduction Rule 6

q = ⋃ T� � ��� , q.

q: .

reduced to the place p:.

synchronizer is used for reduction.

p = T, p.

Figure 19 illustrates this rule, where a = 3 and b = 2.

= ⋃ T� �

= T, K(q) = ab, W(tij, q) = 1, W(q, ti) = b.◆

Reduction Rule 5

**Figure 16.** Reduction Rule 3

The workflow logic in Figure 16 (b) is not through since the termination marking would have two tokens at two different end-places. Rule 3 can be applied to reduce it in size, but the resulted place p turns out to be inconsistent between its pre-set and its post-set.

#### Reduction Rule 4

For T1 = {t1, t2,…, ta} and T2 = {t1', t2',…, tb'}, if pi = ({ti}, T2, (1, b)) is a synchronizer, i = 1, 2,…, a, then p1, p2,… pa can be reduced to a single synchronizer p = (T1,T2, (a, b)); If pi' = (T1, {ti ' }, (a,1)) is a synchronizer i = 1,2,…, b, then p1', p2',…pb' can be reduced to the same single synchronizer p = (T1,T2, (a, b)).◆

Figure 17 illustrates this rule where a = 3 and b = 2.

**Figure 17.** Redaction Rule 4

We will see the significance of this rule and the rule next when management logic is discussed, since separated managements before the rule is applied become centralized management after it.

#### Reduction Rule 5

128 Petri Nets – Manufacturing and Computer Science

**Figure 16.** Reduction Rule 3

synchronizer p = (T1,T2, (a, b)).◆

**Figure 17.** Redaction Rule 4

Figure 17 illustrates this rule where a = 3 and b = 2.

Reduction Rule 4

The workflow logic in Figure 16 (b) is not through since the termination marking would have two tokens at two different end-places. Rule 3 can be applied to reduce it in size, but

For T1 = {t1, t2,…, ta} and T2 = {t1', t2',…, tb'}, if pi = ({ti}, T2, (1, b)) is a synchronizer, i = 1, 2,…, a, then p1, p2,… pa can be reduced to a single synchronizer p = (T1,T2, (a, b)); If pi' = (T1, {ti

(a,1)) is a synchronizer i = 1,2,…, b, then p1', p2',…pb' can be reduced to the same single

the resulted place p turns out to be inconsistent between its pre-set and its post-set.

If pi = ({t}, {ti}, (1, 1)) is a synchronizer for every i, i = 1, 2, …, a, then these synchronizers can be reduced to a single synchronizer p = ({t}, {t1, t2,…, ta}, (1, a)).

If pj = ({tj}, {t}, (1, 1)) is a synchronizer for every j, j = 1, 2, …, b, then these synchronizers can be reduced to a single synchronizer p = ({t1, t2,…, tb}, {t}, (b, 1)).◆

Figure 18 illustrates this rule, where a = b = 3. .

**Figure 18.** Reduction Rule 5

The next reduction rule reserves throughness, but a place element rather than a synchronizer is used for reduction.

#### Reduction Rule 6

' },

> In this rule, i = 1, 2, …, a and j = 1, 2, …, b. For transition sets T = {t1, t2,…, ta} and Ti = {ti1, ti2, …, tib}, if pi = ({ti}, Ti, (1, b)) is a synchronizer for every i, then these synchronizers can be reduced to the place p:. p = T, p. = ⋃ T� � ��� , K(p) = ab, W(ti, p) = b and W(p, tij) = 1, If pi .' = (Ti, {ti}, (b, 1)) is a synchronizer for every i, then these synchronizers can be reduced to the place q: . q = ⋃ T� � ��� , q. = T, K(q) = ab, W(tij, q) = 1, W(q, ti) = b.◆

Figure 19 illustrates this rule, where a = 3 and b = 2.

A noticeable fact is: places p and q are different from a synchronizer at two aspects. Firstly, their capacities are both ab instead of the product of the input weight and the output weight. Secondly, transitions in Ti and Tj , i ≠ j, maybe enabled at different times before reduction, but they will be enabled at the same time by p; similarly, transitions in T will be enabled at the same time by q, but maybe not before reduction. The important point is, the number of transitions to be selected for a route remain unchanged (real selection is to be determined by concrete data from a practical case later by case semantics.) due to the fact that all synchronizers in question are either ALL-split or ALL-join. This means that the property of being through is reserved.

Workflow Modelling Based on Synchrony 131

tokens each. As shown in the figure, a virtual synchronizer p5 is obtained when rule 6 is applied. The resulted system after rule 6 remains being not through. If rule 1' is further applied to p5 and p4, we get a system that is through. Inconsistence between p1 and p4 is concealed. The virtual synchronizer p5 does not tell whether it is ALL-split or OR-split. It

inherits property of being ALL-split or OR-split from p1.

**Figure 20.** Reduction Rule 1'

applicable. Theorem 7

leads to Error

synchronizer. This time the property of being through is reserved.

Figure 21 illustrates Reduction Rule 1', in which the workflow logic on top is apparently through. After twice applications of rule 6, the resulted p6 and p7 are virtual synchronizers both. By reduction rule 1', these two virtual synchronizers are replaced by p, which is a

Reduction Rule 6 is suggested not to be used as long as there is a different reduction rule

A well-structured workflow logic has the property of being through if it can be reduced to a

single isolated place, i.e. a place whose pre-set and post-set are both empty.◆

**Figure 19.** Reduction Rule 6

Definition 18

A place p with . p = T1 and p. = T2, T1≠ ∅, T2≠ ∅, is called a virtual synchronizer if there exist integers a and b such that ∀t ∈ T1:W(t,p)=b,∀t ∈ T2:W(p, t)=a, K(p) ≠ ab, but K(p)/b = n, K(p)/a = m, where n, m are respectively the numbers of transitions in T1 and T2. For virtual synchronizer p, we also write p = (. p, p. , (a,b)).◆

The two places p and q in Figure 19 are virtual synchronizers. It is easy to check that the two places used for reduction by Rule 6 are virtual synchronizers in general.

For further reduction when a virtual synchronizer is involved, we have the revised version of Reduction Tule 1:

Reduction Rule 1':

Let each of p1 = (T1, T0, (a1, a2)) and p2 = (T0, T2, (b1, b2)) be a synchronizer or a virtual synchronizer. As long as K(p1)/a1 = K(p2)/b2, (p1, T0, p2) can be reduced to p = (T1, T2, (a, b)), where a = K(p1)/b1 and b = K(p2)/a2, and K(p) = ab, ∀t ∈ T2: W(t, p) = b and ∀t ∈ T2: W(t, p) = a. ◆

This rule is applicable when the involved virtual synchronizer is of ALL-join and ALL-split nature. Otherwise, local consistence should be checked since a virtual synchronizer is more flexible than a synchronizer. Figure 20 explains: the workflow logic on top is not through since just one of p2 and p3 may have 2 tokens while p4 requires the two of them to have two tokens each. As shown in the figure, a virtual synchronizer p5 is obtained when rule 6 is applied. The resulted system after rule 6 remains being not through. If rule 1' is further applied to p5 and p4, we get a system that is through. Inconsistence between p1 and p4 is concealed. The virtual synchronizer p5 does not tell whether it is ALL-split or OR-split. It inherits property of being ALL-split or OR-split from p1.

**Figure 20.** Reduction Rule 1' leads to Error

Figure 21 illustrates Reduction Rule 1', in which the workflow logic on top is apparently through. After twice applications of rule 6, the resulted p6 and p7 are virtual synchronizers both. By reduction rule 1', these two virtual synchronizers are replaced by p, which is a synchronizer. This time the property of being through is reserved.

Reduction Rule 6 is suggested not to be used as long as there is a different reduction rule applicable.

#### Theorem 7

130 Petri Nets – Manufacturing and Computer Science

property of being through is reserved.

**Figure 19.** Reduction Rule 6

p = T1 and p.

synchronizer p, we also write p = (.

Definition 18

A place p with .

of Reduction Tule 1:

Reduction Rule 1':

◆

determined by concrete data from a practical case later by case semantics.) due to the fact that all synchronizers in question are either ALL-split or ALL-join. This means that the

= T2, T1≠ ∅, T2≠ ∅, is called a virtual synchronizer if there exist

integers a and b such that ∀t ∈ T1:W(t,p)=b,∀t ∈ T2:W(p, t)=a, K(p) ≠ ab, but K(p)/b = n, K(p)/a = m, where n, m are respectively the numbers of transitions in T1 and T2. For virtual

The two places p and q in Figure 19 are virtual synchronizers. It is easy to check that the two

For further reduction when a virtual synchronizer is involved, we have the revised version

Let each of p1 = (T1, T0, (a1, a2)) and p2 = (T0, T2, (b1, b2)) be a synchronizer or a virtual synchronizer. As long as K(p1)/a1 = K(p2)/b2, (p1, T0, p2) can be reduced to p = (T1, T2, (a, b)), where a = K(p1)/b1 and b = K(p2)/a2, and K(p) = ab, ∀t ∈ T2: W(t, p) = b and ∀t ∈ T2: W(t, p) = a.

This rule is applicable when the involved virtual synchronizer is of ALL-join and ALL-split nature. Otherwise, local consistence should be checked since a virtual synchronizer is more flexible than a synchronizer. Figure 20 explains: the workflow logic on top is not through since just one of p2 and p3 may have 2 tokens while p4 requires the two of them to have two

, (a,b)).◆

p, p.

places used for reduction by Rule 6 are virtual synchronizers in general.

A well-structured workflow logic has the property of being through if it can be reduced to a single isolated place, i.e. a place whose pre-set and post-set are both empty.◆

Workflow Modelling Based on Synchrony 133

**Figure 22.** Reducing Workflow to Harmony

synchronizers.

Figure 23 is a more complicated net from Dr. Aalst during his visit to Tsinghua University (Beijing, China) in 2004, where (a) is the original net in which every place has the capacity 1 and all weights on arcs are also 1, i.e. a WF-net. The author would suggest to take (b) as the workflow logic for whatever business in mind. Note that (c) contains virtual

**Figure 21.** Reduction Rule 1'

This theorem is true since all reduction rules (including carefully used rule 1') reserve throughness . The isolated place is the start-place and the end place at the same time, thus the concealed detail has the property of being through.

This isolated place represents "harmony" as far as workflow logic is concerned. An interesting fact is, the isolated transition represents "contradiction" when it is resulted by applying the Resolution Rule and the Expansion Rule in Enlogy (a branch in GNT, net logic) for logical reasoning. A pair of dual elements, i.e. a place and a transition, lead to opposite concepts in philosophy, could this happen by chance? In fact as we will see soon, the dual of workflow logic is exactly the logic for workflow management.

We do not claim the completeness of this set of reduction rules. Whenever a user finds workflow logic not reducible to "harmony" though it is indeed through, please try to figure out new reduction rules and to let us know. Reduction rules would be enriched this way, we hope.

Figure 22 shows how to reduce to harmony the workflow logic for insurance claim where t0, t1, t2, t3 and t4 are respectively the tasks "accept", "check policy", "check claim", "send letter" and "pay" (See *Workflow Management*).

**Figure 22.** Reducing Workflow to Harmony

132 Petri Nets – Manufacturing and Computer Science

**Figure 21.** Reduction Rule 1'

the concealed detail has the property of being through.

workflow logic is exactly the logic for workflow management.

and "pay" (See *Workflow Management*).

This theorem is true since all reduction rules (including carefully used rule 1') reserve throughness . The isolated place is the start-place and the end place at the same time, thus

This isolated place represents "harmony" as far as workflow logic is concerned. An interesting fact is, the isolated transition represents "contradiction" when it is resulted by applying the Resolution Rule and the Expansion Rule in Enlogy (a branch in GNT, net logic) for logical reasoning. A pair of dual elements, i.e. a place and a transition, lead to opposite concepts in philosophy, could this happen by chance? In fact as we will see soon, the dual of

We do not claim the completeness of this set of reduction rules. Whenever a user finds workflow logic not reducible to "harmony" though it is indeed through, please try to figure out new reduction rules and to let us know. Reduction rules would be enriched this way, we hope.

Figure 22 shows how to reduce to harmony the workflow logic for insurance claim where t0, t1, t2, t3 and t4 are respectively the tasks "accept", "check policy", "check claim", "send letter" Figure 23 is a more complicated net from Dr. Aalst during his visit to Tsinghua University (Beijing, China) in 2004, where (a) is the original net in which every place has the capacity 1 and all weights on arcs are also 1, i.e. a WF-net. The author would suggest to take (b) as the workflow logic for whatever business in mind. Note that (c) contains virtual synchronizers.

Workflow Modelling Based on Synchrony 135

pi = {ti},

**6.4. Develop a workflow logic for a given business** 

.

M0(s0) = 1 and all other elements in S have no token.

the unique start task with arc weight 1.

24 shows the developed workflow logic.

**Figure 24.** A developed Workflow Logic

' ( 1, ai')).

more.

Σ.

synchronizer: pi = ({ti}, pi

them. E contains nothing else.

Let T = {t1, t2,…tn} and B = {b1, b2, …bm} be respectively the set of business tasks with < as its immediate successor relation, and the set of blanks on the B-form for a well-designed business. There should be a fixed correspondence between T and B to make clear responsibilities for each of the tasks in T. This correspondence will not be mentioned any

Let Σ= (S,T;F,K,W,M0) be the net system such that T is the task set and S = {p1, p2, …pk}∪{s0}

pi. is the set of immediate successors of ti. Let ai be the number of tasks in this immediate successor set, ai' is the intended number of tasks to be executed after ti, then K(pi) = ai', and W(ti, pi) = ai' for every immediate successor t of ti, W(pi, t) = 1. It is easy to see that pi is a

The place named s0 is the unique start place: it has an empty pre-set and its post-set contains

For every end-task in T, there is a unique end place e in E with 1 as the arc weight between

As a workflow logic of a business process, transition rules for RP/T-systems are assumed for

Σ may be not well-structured as shown by Figure 12 (a). In this case, measures must be

As an example, Let T = {t1, t2, t3, t4, t5} and <= {(t1, t2), (t1, t3), (t2, t4), (t2, t5), (t3, t4), (t3, t5)}. There are 3 non-end tasks in T, namely t1, t2 and t3. So there 3 synchronizers: p1, p2 and p3. Figure

There is redundancy in the developed system at p2 and p3. By Reduction Rule 3, these two synchronizers are reduced to a single synchronizer p = ({t2, t3}, {t4, t5}, (2, 1)). This is the workflow logic for insurance claim, as shown in Figure 22, there it was reduced to harmony.

Σ must be well-ordered since < is exactly its next relation among transitions.

taken as suggested from Figure 12 (a) to (b), then to Figure 13.

Redundant synchronizers may be removed by Reduction Rule 3.

∪E where for each i, i = 1, 2, …, k, k is the number of non-end tasks t1, t2, …tk in T, .

**Figure 23.** Reduction Process
