GRAFCET and Petri Nets Outline Introduction GRAFCET - EPFL

Outline 

GRAFCET and Petri Nets 

Prof. J.-D. Decotignie 

CSEM Centre Suisse d’Electronique et de Microtechnique SA 

Jaquet-Droz 1, 2007 Neuchâtel 

jean-dominique.decotignie@epfl.ch 

introduction 

GRAFCET 

GEMMA 

Petri nets 

Real-Time Programming GRAFCET and Petri nets 2 

© J.-D. Decotignie, 2007 

Introduction 


Description of process evolution 

Description of process interaction 


Functional specifications 

Requirements for automata 


 

 

 

 

 

Graphe de Commande Etape Transition (Step Transition Control 

Graph) 

2 levels 

Functional specification 

Operational specification 

Described in the international standard IEC 848 under the name of 

function charts 

[Dav90] R. David, A. Alla, « Petri nets and Grafcet", Prentice 

Hall, 1992. 

R. David, Grafcet: a powerful tool for specification of logic 

controllers, IEEE Trans. on Control Systems Technology, vol. 

3, Issue 3, Sept. 1995, pp.253 - 268 




© J.-D. Decotignie, 2007


GRAFCET - definition 

Definition 

Evolution rules 

Defining actions 

Taking time into account 

Defining transition conditions 

Execution algorithm 

Macrostep and macroactions 

Directed graph derived from PN 

Quadruple C = {S, TR, A, M 0 } 

N steps s i ∈ S; each step s i may be 

active (X i = true) or not (X i =false). 

M 0 denotes the set of steps active at startup 

L transitions tr i ∈ TR; to each transition is 

associated a boolean condition (receptivity) 

Steps and transitions are linked by arcs a i ∈ A 

+ EVOLUTION CONDITIONS 

1 L=1 

b1 

2 L=0 

/b1 





Exercise - syntax 

Exercise - syntax (2) 

a 

a 

a 

a 

a 

b 

a 

a 

A 

B 

C 

D 

E 

K 

L 

M 

a 

a 

b 

a 

b 

a 

b 

a 

F 

G 

H 

I 

N 

O 

P 





Exercise - syntax (3) 

GRAFCET – an example 

request 

Loading place 

m 

OP 

A 

B 

a 

G 

D 

b 

a 

b 

Q 

a 

b 

R 

a 

Q 

F 

O 

N 

C 

T 

I 

N 

A 

L 

1 Load used / waiting in A 

2 

3 

4 

Load requested 

Go to right 

Arrived at right 

loading 

Loading terminated 

Go to left 

Arrived at left 

m 

Control 

part 

a b p D G OP 

Operative part 

(wagon, door,... 

O 

P 

E 

R 

A 

T 

I 

O 

N 

A 

L 

1 

2 

3 

4 

m 

b 

p 

a 

p 

D 

OP 

G 





Evolution Conditions 

Evolution Conditions (2) 

 

Evolution is performed from the initial state defined as M 0 by 

clearing transitions according to 5 rules: 

all steps immediately preceding the transition must be active, the 

transition is then enabled 

then, if receptivity is true, the transition may be cleared 

all transitions that may be cleared are cleared simultaneously 

clearing a transition leads to the activation of all immediately following 

steps and deactivation of all immediately preceding steps 

a step that is activated and deactivated remains active 

1 

(1) a (2) a 

2 

1 

a=1 

clearing 

3 

4 

3 

(1) a (2) a 

5 

(3) b (4) b 

6 

b=1 

clearing 

5 

7 

(3) b (4) b 

6 

(34) 

5 

b 

b=1 

clearing 

(34) 

5 

b 

7 

2 

4 

6 

7 

6 

7 





Exercise - transitions 

A. enabled transitions; B. firable transitions if a=1 and b=0; 

C. state after clearing the transitions when possible 

Exercise - transitions (2) 

A. enabled transitions; B. firable transitions if a=1 and b=0; 

C. state after transition when possible 

a 

A 

a 

B 

/a 

C 

/a 

D 

a./b 

E 

a.b 

a 

a./b 

a./b 

a./b 

/b 

/b 

a 

I 

J 

K 

L 

F 

G 

H 

M 

b 

N 





Conflicts 

Exercise: is there a conflict ? 

5 

(3) a (4) b 

6 7 

1 

a.b 

/a.c 

4 

a.b 

a.c 

a 

7 

/a.b 

/a./b 

5 

(3) a./b (4) b 

a./b 

5 

ab 

/a.b 

2 

3 

A 

5 

6 

B 

8 

9 

10 

C 

6 

7 

6 

7 





Divider by 2 

Examples of logical graphs 

(1) 

(2) 

(3) 

(4) 

1 

2 

3 

4 

a 

a' 

a 

a' 

S 

S 

a 

S 

↑a 

1 

(1) ↑a 

2 S 

(2) ↑a 

a 

Controlling 

system 

S 

a 

m 

G 

(A) 

(1) 

(2) 

(3) 

1 

2 

3 

↑m 

D 

b 

G 

a 

D 

(1) m pressed 

b 

2 Go to right 

(2) Arrived on right 

(B) 

3 Go to left 

1 

(3) Arrived on left 

(1) ↑m 

2 V 

1 

(2) b 

(1) ↑m.a 

3 

2 V 

(3) a 

(2) b 

1 





Example 

Increment a counter 

reservoir 

m 

 

 

 

Initial conditions 

Tanks empty, valves closed 

Sensors and actuators 

V i , W i = 1 if open 

V 1 

h i , b i = 1 if level above sensor tank 1 

b 1 

operations: 

W 1 

Fill each tank until above h i , 

close valve V i and open W i until 

level below b i . Procees cannot 

be repeated before both tanks 

are empty 

h 1 

h 2 

b 2 

W 2 

tank 2 

(1) 

(2) 

(3) 

1 

2 

3 

(C←0)* 

=1 

↑a 

(C←C+1)* 

V 2 

GRAFCET and Petri nets 20 

↑a 

a 

Stable 

situation 

(C←0)* 

(C←C+1)* 

{2} {3} {2} {3} 



Real-Time Programming © J.-D. Decotignie, 2007

Example (cont) 

(B) 

4 7 

(A) 1 

(1) ↑m 

(1) ↑m 

(2) 

(3) 

2 V 1 5 

h 1 (4) 

3 W 1 6 

/b 1 (5) /b 2 

4 7 

(6) =1 

V 2 

h 2 

W 2 

2 

(2) 

3 

(3) 

V 1 

h 1 

W 1 

5 

(4) 

V 2 

h 2 

W 2 

6 

/b 1 (5) /b 2 

reservoir 

m 

V1 

V2 

h1 

b1 

tank 1 

W1 

h2 

b2 

tank 2 

W2 

(C) 

4 7 

(1') ↑m .X 7 (1") ↑m .X 4 

(2) 

(3) 

2 

3 

V 1 

h 1 

W 1 

/b 1 

(1) 

1 

(4) 

(5) 

↑m 

5 

6 

2 V 1 5 

(2) h 1 (4) 

34 W 1if b 1 67 

(6) /b 1./b 2 

(D) 

V 2 

h 2 

W 2 

h 2 

/b 2 

V 2 

W 2if b 2 

Taking time into account 

5 

(2) a 

b 

6 T=10s 

(3) b 

a 

X 5 

X 6 

10s 

t/6/10s 

20s 

t/6/20s 





Timer behavior 

Continuous actions (à niveau) 

X i 

t/i/Δ 

↑t/i/Δ 

Δ 

Δ Δ Δ Δ 

(2) 

(3) 

5 

i 

a 

b 

T=Δ 

1 

(1) a 

2 Action A 

(2) b 

5 

(5) 

(6) 

6 

a 

b 

X2 

action A 

a 

X6 

T=Δ; ActionA 

t/6/Δ 

if t/6/Δ 

b 

action A 

a 

b 

3 

(3) a 

4 Action A 

if C 

(4) b 

a 

b 

X4 

c 

action A 

7 

b 

(7) a 

X8 

Δ 8 T=Δ; ActionA 

t/8/Δ 

if (t/8/Δ)' 

(8) b 

action A 

a 

Δ 





Duration of a continuous action 

Continuous actions are only defined for stable situations 

a 

5 

b 

(2) a 

X5 

6 Action A X6 

(3) b 

action A 

Pulse shaped actions 

c1 

3 

(2) c1 

c2 

c3 

4 open P* 

(3) c2 

c4 

X3 

5 

(4) 

6 

c3 

close P* 

X4 

X5 

X6 

(5) c4 

open P* 

close P* 

P open 





Conditional Pulse Shaped Actions 

Actions types 

b 

(3) a 

A* if b=1 when ↑X 

3 Action A* if b 

3 

A* when X 3 =1 as soon as b=1 

(4) c 

A* when ↑(X 3 .b) 

A* if b is ambiguous. Unsufficient notation !! 

 

We shall not keep this capability 

X 3 

Continuous actions (à niveau) 

unconditional 

conditional 

Condition is a boolean variable 

Condition is based on a timer 

Pulse shaped actions 

Always unconditional 





Exercise: drilling machine 

Actions and outputs 

h 

b 1 

b 2 

b 3 

h: upper position 

b1: end of approach 

b3: end of drilling 

High 

speed 

Working 

speed 

High 

speed up 

b1: end of approach 

b2: end of deburring 

High 

speed 

Working 

speed 

h: upper position 

High 

speed up 

b3: end of drilling 

Working 

speed 

High 

speed up 

E 

D 

m 

Operator or 

supervising system 

Controlling part 

Operative part of the 

controlling part 

(calculations, timers,..) 

C* b 

Control part of the 

controlling part 

(described by the graph) 

a A B* 

Operative part 

(processus to control) 

inputs: a,b,m 

actions: A, B*, D, C* 

outputs: A, B*, D, E 





Transition conditions 

Events 

variables 

External boolean variables 

Coming from the process or from the external world 

relative to time (t/i/Δ) 

Internal boolean variables 

Relative to the state of a step (Xi) 

State of the operative part (of the controlling part) 

receptivity R i = C i E i 

condition C i = boolean combination of variables 

event E i = rising (falling) edge of an external variable (e always 

occurring event) 

∀a ∈ {0,1} si a(t)=1 when t ∈ [t1, t2[ et t ∈ [t3, t4[ 

definition 1: ↑a occurs in t1 and in t3 

definition 2: ↓a occurs in t2 and in t4 

definition 3: ↑a.b occurs at the same instant as ↑a each time 

b=1 at this instant 

definition 4: ↑a.↑b occurs when ↑a and ↑b occur at the same 

instant 

Show that 

↑a = ↓a' ↑a.a = ↑a ↑a.a' = 0 ↓a.a = 0 

↑a.↑a = ↑a ↑a. ↑a' = 0 ↑a.e = ↑a 





Graph interpretation 

2 persons in front of the same graph and assuming the 

same sequence of inputs should deduce the same 

sequence of outputs. 

Asumptions: 

2 uncorelated events may not occur at the same time 

The graph has enough time to reach a stable state before the 

occurrence of the next event 

Interpretation Algorithm (without 

stability search) 

1. Read the state of the inputs 

2. evolve (one or more 

simultaneous clearings) 

3. Execute actions 

4. goto 1 

a 

c 

b 

(1) ↑b 

7 G 

(4) c 

d 

1 

(5) 

(7) 

5 

H D 

(2) 

↑b 

c 

G 

3 

↑a 

D 

(6) 

(8) 

(3) 

6 

2 

4 

=1 

d 

D 

m.a' 

G 

H 

D 





Iterated Clearing 

Interpretation Algorithm (with 

stability search) 

4 

(1) 

5 

(2) 

6 

(3) 

a 

/b 

↑c 

4 

(1) 

5 

(2) 

6 

(3) 

↑a 

/b 

↑c 

a 

b 

c 

Stable 

situation 

{4} 

{6} 

1. initialization: activation initial step(s), execution of associated 

pulse shaped actions; go to 5 

2. If an external event Ei occurs, determine the set T 1 of transitions 

that can be cleared on Ei; if T 1 ≠∅ goto 3, else modify 

conditional actions and goto 2 

3. Clear transitions that can be. If, after clearing, the state is 

unchanged goto 6 

4. Execute the pulse shaped actions that are associated to the step 

that were activated at step 3 (incl. timers) 

5. Determine the set T2 of transitions that can be cleared on 

occurrence of e. If T 2 ≠∅, goto 3 






stability search)(2) 

6. A stable situation has been reached 

1. Determine the set A 0 of continuous actions that must be 

deactivated (incl. conditional actions) 

2. Determine the set A 1 of continuous actions that must be 

activated (incl. conditional actions) 

3. Set to 0 all the actions that belong to A 0 and not to A 1 . Set to 

1 all the actions that belong to A 1 

4. go to 2 


stability search)(3) 

stage 1: situation {1,2} 

stage 5: (1) (2) et (3) enabled, 

cannot be cleared on e, T 2 =∅ 

stage 6: stable situation {1,2}, 

A 0 =∅, A 1 = {H,D} 

stage 2: m=1, on e, T 1 = {3} 

stage 3: clearing (3) 

stage 4: no pulse shaped action 

stage 5: T 2 =∅ 

stage 6: A 0 =∅=A 1 

stage 2: on ↑a, T 1 = {2} 


(1) 

↑b 

1 

7 G 

(4) c 

(5) 

H D 

(2) 

↑b 


stage 5: T 2 = {(6)} 



stage 5: T 2 =∅ 

stage 6: A 0 = {H,D}, A 1 = {D} 

...... 

3 

5 G 

(7) c 

↑a 

D 

(6) 

(8) 

(3) 

6 

2 

4 

=1 

d 

D 

m.a' 





Non stable cycle 

Exercise [Dav90] 

1 

a' 

2 

(3) a 

(1) 

(4) 

5 A 

4 

(5) a' 

(6) 

b 

b' 

B 

b' 

(2) 

12 

Let H1 and H2 be two wagons carrying goods from loading points C1 and C2 

respectively to an unloaded point D. Variables c1,c2 and d correspond to end 

of track sensors. They turn to 1 when a wagon is present at the given point. 

Variable a1 turns to 1 when the front wheels of wagon H1 are on the tracks 

between A1 and D (same for a2 if wagon H2 is between A2 and D). 

If wagon H1 is in C1 and if button m1 is pressed, a cycle C1 → D → C1 starts 

with a possible wait in A1 until the track common to both wagons is free, then a 

wait in D during 100s. Wagon H2 performs the same on C2 → D → C2. The 

path C1-D is set when V equals 1, otherwise path C2-D is set. 

(3) a 

(1) 

(4) b 

5 A 

4 B 

(5) a' 

(6) b' 



m1 

m2 

c1 

C1 

c2 

C2 

G1 

H1 

D1 



A1 

A2 

V 

G2 

H2 

D2 

d 

D

(4) 

(5) 

4 

5 

M30 

6 

Macro-step 

A 

a.d' 

b 

G 

↑a 

32 V1 

c 

34 D 

c'.d 

E30 B 

S30 

b 

31 V1 

↑a' 

33 B 

c 

(4) 

4 

↑a 

32 V1 

c 

34 D 

c'.d 

6 

A 

a.d' 

30 B 

30' 

(5) b 

b 

31 V1 

↑a' 

33 B 

c 

G 

Pseudo macro-step 

Represents a set of steps 

All arcs do not necessarily go to the same step (no need 

for a unique entry step) 

All arcs do not necessarily go to the same step (no need 

for a unique exit step) 

All arcs entering or leaving the pseudo macro-step need 

be represented 





Pseudo macro-step 

1 

5 

P1 

start 

end 

System stopped 

Start of units 

2 Normal operations 

Stop request 

P2 5 

end 

Stop units 

P1 

P2 



1 

3 

4 

start 

end 

System stopped 

Start unit 1 

2 Normal operations 

Stop request 

5 

6 

end 

end 

end 

Start unit 2 

Stop unit 1 

Stop unit 2 

restart 

restart 

Macro actions 

Globaly act on a GRAFCET 

Do not increase the expression power 

But ease specification 

Exhibit same properties as actions 

Continuous 

Forçage(forcing), figeage(freezing), masquage(masking) 

Pulse shaped 

Forcer (force), masquer(mask), démasquer(unmask), figer(freeze), 

libérer(release) 



Macroaction: Forcing (Forçage) 

Macroaction: Freezing (Figeage) 

11 

11 

11 

11 

5 

d 

forçage 

G2:{12} 

d' 

G 1 G 2 

a 

12 A 

b 

13 B* 

/a 

⇔ 

d 

5 

d' 

G 12 

12 A 

13 

a 

b./X 5 

/a 

B* 

X 5 

X 5 

X 5 


6 

d 

d' 

figeage G2 

G 1 G 2 

12 A 

13 

a 

b 

/a 

B* 

⇔ 

d 

6 

d' 

G 12 

a./X 6 

12 A 

b./X 6 

13 B* 

/a./X 6 



Real-Time Programming © J.-D. Decotignie, 2007 

Macroaction: Masking (Masquage) 

Macrocaction : Force (Forcer) 

11 

11 

11 

11 

a 

a 

a 

↑X 4 

a 

↑X 4 

7 

d 

Masquage 

G2:{A} 

d' 

G 1 G 2 

12 A 

b 

13 B* 

/a 

⇔ 

d 

7 

d' 

G 12 

12 A if /X 7 

b 

13 B* 

/a 

4 

d 

forcer 

G2:{12} 

d' 

G 1 G 2 

12 A 

b 

13 B* 

/a 

⇔ 

d 

4 

d' 

G 12 

12 A 

b 

13 B* 

/a 

↑X 4 






GEMMA 

introduction 


GEMMA 


Guide d'Etude des Modes de Marche et d'Arrêt 

Problems with GRAFCET 

Objectives 

Concepts 

How to use GEMMA 

Advantages 

Drawbacks 





Problems with GRAFCET 

Objectives of GEMMA inventors 

GRAFCET is good to describe normal operation but: 

Not well suited to describe 

Emergency cases 

Manual modes 

Degraded modes 

Does not favor a good separation of operating modes 

Does not give a clear vocabulary concerning modes 

Does not include any security aspect 

Create a graphical tool that can deal with: 

Emergency cases 

Manual modes 

Degraded modes 

Estblish a precise vocabulary 

Tool show present itself as a guide (checklist) 





GEMMA concepts 

Assumes that controlling part is operational 

Separate production states from other states 

Distinguishes 3 categories of operating modes 

Working procedures 

Stopping procedures 

Failure procedures 

control 

part 

power 

off 

control 

part 

power 

off 

PC 

power 

on 

CP 

power 

off 

PC 

power 

on 

CP 

power 

off 

A 

A6 

D1 

 

A7 

D2 

 

A1 

D3 

A4 

 

A5 

F 

PRODUCTION 

F4 

 

F5 

 

F6 

WORKING PROCEDURES 

 





How to use GEMMA 

Example – joint test bed 

Selection of modes 

Showing relationships 

Search for evolution conditions 

 

 

 

3 tested values V, C, β 

Long tests on a typical run 

Periodic request for withdraw 

and check 

ß 

cardan 

V C β 

Frein couple C 

on off ack AU 

moteur 

vitesse V 

Verrin pour réglage 

angle ß 

man. 

auto 





partie 

commande 

hors 

énergie 

partie 


hors 

énergie 

mise en 

énergie 

de PC 

mise hors 

énergie 

de PC 

mise en 

énergie 

de PC 

mise hors 

énergie 

de PC 

A 

PROCEDURES D'ARRET DE LA PARTIE OPERATIVE (PO) 

A6 

all setpoints to 0 

 

power on OP, 

clear default 

D2 

 

en fin de dans état 

clear setpoints 

cycle> déterminé> 

end of tests 

or 

end for check 

stop 

stop 

²V/V or 

²C/C or 

²ß/ß>10% 

ack 

from any 

state 

F1 

F 

F2 

time 

setup 

auto.marche 

PROCEDURES DE FONCTIONNEMENT 

< production normale> 

TEST according to 

program 

 

PRODUCTION 

Em.Stop or Vmax or Cmax exceeded 

Détection 

défaillances 

F PROCEDURES DE FONCTIONNEMENT 

F4 

 

check 

feedback 

loops for V, C 

and ß by 

displaying min 

and max 

performances 

Tests with 

display of 

setpoints 

V,C and ß 

[duration 

test] 

F6 

 

Example – valve test 

C 

C 

E 

z y 

D 

z y 

D 

A 

x V 

Before feeding 

B 

x 

V 

After feeding 

B 

E 





GUIDE DES MODES DE MARCHE ET D'ARRET (GEMMA) 


partie 


hors 

énergie 

mise en 

énergie 

de PC 

A 


A6 

par boutons poussoirs 

avec les sécurités 

b+ si e- et a+ si /z /y 

D2 

A7 

 

CI 

A1 

 

 

A4 

demande 

de marche 

F 

F2 


F3 

 

F4 

 

partie 


hors 

énergie 

mise en 

énergie 

de PC 

A 


A6 




A7 

 

A1 

 

 

A4 

/m2 

m2 


de marche 

F 

F2 


F3 

 

F4 

par boutons 

poussoirs avec 

les sécurités 

b+ si e- et 

a+ si /y /z 

mise hors 

énergie 

de PC 

 

PRODUCTION 


d'arrêt 

F1 


F5 

 

F5 

 

D3 

 

PRODUCTION 

D PROCEDURES EN DEFAILLANCE DE LA PARTIE OPERATIVE 

Détection 

défaillances 

F 

PRODUCTION 

F6 

cycle de test 

du dispositif 

de contrôle 

d'étanchéité 


partie 


hors 

énergie 

mise en 

énergie 

de PC 

mise hors 

énergie 

de PC 

D2



partie 


hors 

énergie 

mise en 

énergie 

de PC 

A 


A6 




A7 

 

CI 

A1 

 

 

A4 

m1 


de marche 

F 

F2 


F3 

 

F4 

 

partie 


hors 

énergie 

mise en 

énergie 

de PC 

A 


A6 

 




A7 

D2 

 

CI 

A1 

 

 

A4 

T 

m2 

/m2 

m1 


de marche 

F 

F2 

 


F3 

 

F4 

par boutons 

poussoirs avec 

les sécurités 

b+ si e- et 

A1 si /z /y 

mise hors 

énergie 

de PC 

Fcyraz 

 

PRODUCTION 


d'arrêt 

F1 


cycle de contrôle de 

l'étanchéité des soupapes 

F5 

 

cycle de contrôle de 

l'étanchéité des soupapes 

F5 

 

D3 

coupure de l'alimentation de la partie 

opérative PO 

 

PRODUCTION 

D PROCEDURES EN DEFAILLANCE DE LA PARTIE OPERATIVE 

Détection 

défaillances 

Système de test d'étanchéité de soupapes: arrêt d'urgence 

PRODUCTION 

F6 

F PROCEDURES DE FONCTIONNEMENT 



AU 

partie 


hors 

énergie 

mise en 

énergie 

de PC 

mise hors 

énergie 

D2

How to manage an automation 

project 

Phases according to GEMMA tenants 

Define requirements 

Use GEMMA for start/stop operations 

Create GRAFCET 

Select control technology 

Technological realization 

Trial and commissioning 


introduction 


GEMMA 







Petri Nets 

 

 

 

 

 

Invented by C. Petri in 1962 in his PhD thesis ("Kommunikation 

mit Automaten) 

J. Peterson, "Petri Net Theory and the Modelling of Systems", 

Prentice Hall, 1981. 

R. David, A. Alla, "Du Grafcet aux réseaux de Petri", Hermes, 

1990. 

T. Murata, Petri nets: Properties, analysis and applications, Proc. 

of the IEEE, Vol. 77 (4), April 1989, pp.541 - 580 

2 view points 

Theoretical 

Definitions and evolution rules 

Properties 

Application to functional specifications 



Definition 

Marking 

PN types 

Modelling with Petri nets 

Modelling synchronization and cooperation between 

tasks 

Design 

Analysis and validation 



Petri nets - definition 

Petri net example 

Directed graph 

tuple C={P, T, I, O} 

N places p i ∈ P 

L transitions t i ∈ T 

Input matrix I [LxN] 

places → transitions 

Output matrix O [LxN] 

transitions → places 

p 1 p 2 p 3 

t 1 

p 4 p 5 

I= 

O= 

P = {p 1 , p 2 , p 3 , p 4 , p 5 } 

T = {t 1 , t 2 } 

p 1 p 2 p 3 p 4 p 5 

1 1 2 0 0 t 1 

0 0 0 0 1 t 2 

p 1 p 2 p 3 p 4 p 5 

0 0 0 1 2 t 1 

0 0 1 0 0 t 2 

p 1 p 2 p 3 

2 

t 1 t 2 

2 

p 4 p 5 

#(p i , I(t j )) weight of arc p i → t j 

#(p i , O(t j )) weight of arc t j → p i 





Petri nets - marking 

PN – evolution rules 

marking 

M = {m(p 1 ),....., m(p i ), ....., m(p N )} 

with m(p i ) = number of tokens 

in place p i 

Initial marking M 0 , 

example M 0 ={2,1,3,0,1} 

Evolution by firing transitions 

Can then represent behavior (dynamic) 

p 1 p 2 p 3 



t 1 

t 2 

2 

p 4 p 5 

2 

a transition t i may be fired iif for all i 

m(p i ) ≥ #(p i , I(t j )) 

Only one transition may fired at a time 

A transition is fired instantaneously by: 

Removing in each place that immediately preceeds the transition 

a number of tokens equal the weight of arc that links them 

Adding in each place that immediately follows the transition a 

number of tokens equal the weight of arc that links them 

m(p i ) := m(p i ) - #(p i , I(t j )) + #(p i , O(t j )) 

EVOLUTION IS NOT DETERMINISTIC 



Evolution rules 

p 1 p 2 p 3 

2 

Exercise - syntax 

For each of the graphs below, answer the following questions: 

1. Is it a PN ? 2. What are the validated transitions ? 

3. What will be the marking after firing ? 4. What are the validated transitions 

t 1 

t 2 

2 

p 4 p 5 

p 1 p 2 p 3 

p 1 p 2 p 3 

A B C D E 

2 

2 

t 1 

t 2 

t 1 

t 2 

2 

2 

p 4 p 5 

p 4 p 5 

F G H I 





Exercise – syntax (2) 

PN types 

For each of the graphs below, answer the following questions: 

1. Is it a PN ? 2. What are the validated transitions ? 

3. What will be the marking after firing ? 4. What are the validated transitions 

Generalized PNs 

There a weight on each arc 

Can be transformed into ordinary PN 

PNs with predicates 

Cannot be transformed into ordinary PN 

J K L 

Marc 

X 

Y father of X t 1 

Y 

X 

t 1 

Y 

Luc 





PN types (2) 

PN types (3) 

PNs with capacity 

Can be 

transformed 

into ordinary PN 

t 1 

p 2 

t 2 

p 2 

t 1 fired t 1 fired 

t 1 

t 1 

t 1 

cap(p 2 )=2 

t 2 

t 2 

t 2 

t 1 fired t 1 fired 

t 1 

t 1 

p 2 ' p 2 ' p 2 ' 

t 2 

t 2 

Colored PNs 

Tokens have colors 

Any colored PN with a finite number of colors may be 

transformed into an ordinary PN 

PNs with priorities 


Continuous PNs 

The number of tokens can be a real number 






PN types (4) 

Exercise 

PNs with inhibiting arcs 

Cannot be transformed into 

ordinary PNs 

No autononous PNs 

Several types 

Synchronized PNs 

Time PNs 

Stochastic PNs 

Stochastic Time PNs 

.... 

Cannot be transformed into ordinary PNs 

p 1 p 2 p 3 

t 1 

p 4 p 5 

2 

2 

t 2 

Transform this PN into an ordinary PN 

t 1 p 1 

p 2 

t 2 

cap(p 3 )=3 

t 3 

p 3 t 5 

t 4 

p 4 





Modelling with PNs 

Conditions « and" & « or" 

actions associated to places 

Activated by token presence 

actions associated to 

transitions (short duration) 

Activated by transition 

firing 

Task 

waiting 

Start of 

execution 

executing p 3 

End of 

execution 

Tasked 

ended 

p 1 p 2 

p 4 

Processor 

free 

type "AND" type "OR" 

p 1 p 2 p 3 

p 1 p 2 p 3 

x 1 x 2 x 3 

x 1 x 2 x 3 

y 

p 4 

t 1 

t 1 

p 4 

t 

t 3 

2 

y 





Introduction of external conditions 

When synchronisation is required 

controlled system → 

controlling system 

Label on the transition 

(if condition, action) 

Transition may be fired iif 

external condition satisfied 

Timers (extern or 

minimal sejourn duration in a place) 

Task 

waiting 

(.,begin_exec) 

executing 

p 3 

(end_exec,.) 

Task 

terminated 

p 1 p 2 

p 4 

Processor 

free 


(2) 

Task 

waiting 

(.,begin_exec) 

executing 

p 3 

(end_exec,.) 

Task 

terminated 

p 1 p 2 

p 4 

Processor 

free 

p 13 

p 14 

t 11 

p 17 

(.,end_exec) 

p 11 

t 12 t 13 

(begin_exec, .) 

p 15 

t 14 

t 15 

p 16 






(3) 

p 1 

Modelling task synchronisation 

mechanisms 

Mutual synchronisation (rendez-vous) 

t 1 

(.,start_timer) 

A 

A 1 

A 2 

t 2 

p 3 

(end_timer,.) 

timer 

A 1 

t 2 t 3 

t 1 

B 1 

B B 1 

A 

A 1 

OR 

B 2 

A 2 

A 2 B 2 

p 4 

B B 1 

B 2 






mechanisms(2) 


mechanisms(3) 

mailbox 

A 1 

t 2 t 3 

t 1 

A 2 B 2 

B 1 

A 

A 1 

B B 1 

A 2 

B 2 

Interrupt driven task 

As an external 

condition 

on a transition 

t 

Interrupt handler 1 

modeled using 

2 places 

A 

p 3 

t 5 

t 2 

B 

p 4 

(interrupt,...) t 3 (interrupt,...) 

p 2 

B p 1 

t 4 

t 6 






mechanisms (4) 

Mutual exclusion through semaphore 

Mutual exclusion 

Modeled by a single 

token 

P(S): AND condition on 

a transition 

V(S): OR condition on 

a transition 

Critical section 

synchronized by an auxiliary 

place 

P(S) 

V(S) 

t 11 

t 1 

S 

S 

p 1 

p 1 

Task 

waiting 

t 1 

Critical 

section 

t 3 

p 0 

p 3 

p 5 

S 

p 1 

p 2 

t 2 

p 4 

t 4 

Task 

waiting 

p 6 

Critical 

section 





Readers – writers problem 

Producers - consumers 

READERS 

WRITERS 

PRODUCERS 

CONSUMERS 

Task 

waiting 

p 0 

Task 

waiting 

p 3 

p 5 

s 

t 1 

p 1 

t 2 

t 3 

messages 

t 2 

p 2 

reading p 3 S s p 4 writing 

t 3 

t 

s 4 

p 5 p 6 

production p 4 

p 1 

t 1 s p 2 

positions 

p 6 

t 4 

use 





Dining Philosophers 

t 7 

Exercise 

t 8 

eat p 11 

t 4 

p 4 

p 5 

p 10 p 6 

t 1 

p 7 

p 1 

meditate 

p 8 

t 2 

p 3 

p 2 

t 3 

p 12 

t 6 

 

A consulate let its clients enter then the entrance door is closed 

and the client may be served. After being served, each client 

leaves the consulate through another door. The entrance door 

cannot be opened again before all the clients have left. 

Design a PN with inhibitor arcs that represents this specification 

Modify the first solution so that only one client may enter at a time 

Describe the same case using an ordinary PN 

A maximum of 4 clients may enter before the door is closed. Model this 

case using an ordinary PN 

p 13 

p 5 

t 5 





Design and validation 

Petri nets properties 

Petri nets properties 

Design by successive refinements 

Analysis by reduction 

Structural properties 

Are only dependent on the topology 

Behavioral properties 

Depend on the initial marking 





Structural properties 

Structural properties (2) 

State graph 

Event graph 

Without conflict 

Free choice 

Extended free choice 

simple 

pure 

Without loop 

State graph 

iif each transition 

has exactly one input 

and one output place 

Event graph 

iif each place 

has exactly one input 

and one output 

transition 

p k 

p 1 

p 1 

t i 

t i 

p 2 p 3 p 2 

t i 

t 1 t 3 

p 1 

t j 

t 2 







Without conflit 

Each place has at most 

one output transition 

simple 

Each transition is 

involved in at most 

one conflict 

p 3 

p 3 p 2 

t 1 

t 2 

p 4 

p 3 

t 2 

p 3 

t 1 t 2 

t 1 

p 4 

t 2 

Free choice 

In case of conflict, 

 

none of the transitions 

has another input place 

than p j 

Extended free choice 

In case of conflict, all 

the involved transitions 

have the same input places 

p 3 

t 1 

p 3 

t 1 

t 2 

p 3 

p 4 

t 2 

t 1 

p 4 

t 2 






Exercise - properties 

Pure PN 

p 3 

There is no transition 

that has an input 

place that is also an 

output place 

Without loop 

If there exists t j and p i that 

is at the same time input 

and output place for t j 

then t j has at least another 

input place 

t 1 

p 3 

p 3 t 1 

p 4 

t 1 ' 

p 4 p 0 

t 1 '' 

For each PN below, respond to the following questions: 

1. Is it a state graph ? 2. Is it an event graph ? 

3. Is it without conflict ? 4. It is simple ? 

5. Is it pure ? 6. Is it without loop ? 

p 1 

t 1 

p 2 

t 2 

p 1 t 1 

p 2 





Behavioral properties 

Reachability 

Reachability 

Boundedness 

Conservativeness 

Liveness 

Reversibility, home state 

Coverability 

Persistence 

Synchronic distance 

Fairness 

Can the system reach a given state or exhibit a given 

behavior (correct or not) ? 

To answer this question, one has to find a sequence of firing 

that brings from M 0 to M i 

A marking M i is said reachable from M 0 , if there exists a 

sequence of firings that transforms M 0 in M i 

M i is said immediately reachable from M 0 , if there is a unique 

firing that transforms M 0 en M i 

R(M 0 ): set of all reachable markings from M 0 

L(M 0 ): set of all firing sequences from M 0 





Bounded and safe 

Bounded and safe (2) 

Places are often used to represent: 

Information storage in communication systems and computers 

Products and tools in production systems 

The boundedness property is useful to detect overflows 

in resource usage 

A PN is said k-bounded if the number of tokens 

whichever the place does not exceed k for all markings 

in R(M 0 ) 

A PN is said safe if k=1 (1-bounded) 

p 1 

t 1 

p 3 p 2 

t 2 

p 4 

t 3 

p 1 

t 1 

p 3 p 2 

t 2 

p 4 

t 3 





Exercise 

Conservativeness 

2 tasks T1 and T2 execute on the same processeur in 

time sharing (part of T1 is executed, then part of T2 is, 

then part of T1, …) 

Model the same behavior with 4 tasks 

Is the resulting PN bounded ? 

In a real system, the number of resources is limited. If 

tokens represent resources, it may be a desired properties 

that the number of tokens remains constant whichever 

the marking in R(M 0 ) 

A PN is said strictly conservative if the number of tokens is 

constant 

A PN is said conservative if there exists a weighting vector 

w=[w 1 ,....., w i , ....., w N ] such that for each marking in R(M 0 ), 

∑w i m(p i ) remains constant. 





Conservativeness - example 

Liveness 

p 1 

t 1 

t 3 

p 3 p 2 

p 3 

t 2 

p 4 

PRODUCERS 

t 3 messages 

production p 4 

p 1 

t 1 s p 2 

positions 

CONSUMERS 

p 5 

t 2 

p 6 use 

t 4 

The idea of liveness is closely related to blocking and 

interblocking cases 

Necessary conditions (all 4 together) 

Limited access 

Resources can only be shared by a finite number of tasks 

No preemption 

Once allocated, a resource cannot be withdrawn from the task 

Multiple requests 

Tasks already own resources when they additional resource requests 

The request graph has circular paths 





Liveness (2) 

Liveness - example 

A transition t of a PN is said : 

L0-live (dead) if t does not belong to any sequence of L(M 0 ) 

L1- live (potentialy firable) if t may fired at least once in a 

sequence of L(M 0 ) 

L2- live if t may be fired k times (k integer >1) in a sequence 

of L(M 0 ) 

L3- live if t may be fired an infinite number of times in a 

sequence of L(M 0 ) 

L4- live (live) if t is L1- live for any marking in R(M 0 ) 

live 

p 1 

t 1 

t 3 

p 2 

t 2 , t 3 are L1, L2 et L3 

live 

t 1 is L1 live 

t 2 

p 3 

p 1 

t 3 

p 2 

t 2 

t 1 

p 3 





Exercise 

Reversibility and home state 

2 computers share a common memory. It is assumed that 

each computer may be in one of the following states: 

Does not need the shared memory 

Needs the shared memory, but it has not yet been assigned 

Uses the shared memory 

Your assignment 

Model this behavior 

Is the result conservative 

Show that the resulting PN is live 

For a real-time system, it is of prime importance to 

recover in case of error and then return to a correct state 

A PN is said reversible if the initial marking M 0 may be 

reached from any marking in R(M 0 ) 

A given state M i of a PN is said « home state » if for any 

marking M of R(M 0 ), M i can be reached from M 





Reversibility and home state - 

example 

Exercise - properties 

 

reversible 

p 1 

t 1 

t 3 

 

Home state 

For each PN below, please indicate if it is: 

1. bounded ? 2. live ? 3. Without blocking? 4. conservative ? 

p 2 

p 3 

p 0 

p 1 

t 2 

t 2 

t 0 

t 1 

 

Not 

reversible 

p 1 

t 3 

p 2 

t 1 

t 1 

p 3 p 2 

p 4 

t 3 

p 1 

t 1 

p 1 p 1 

t 1 

p 1 

t 1 p1 

t 1 

A B C p 2 

D t 2 

E 

t 2 

p 3 





PN Properties 

PN Properties (2) 

Reachable marking 

A marking that can reached from M 0 

K-bounded (if K=1, safe) 

Maximum number of tokens ∀place =K 

reversible 

M 0 may be reached from any marking 

conservative (same weights, strictly conservative) 

Weighted sum of tokens in places = Constant 

live 

All transitions are live 

Well shaped 

live + bounded + reversible 





Design by refinement 

Design by refinement (2) 

p 1 

p 1 

First step 

Simple net 

Complex tasks associated to transitions 

validation 

Second step 

Transitions are replace by a block (a PN that begins with an 

initial transition and ends with a final transition) t 11 

p 

validation 

13 

t 12 t 13 

Following steps 

p 14 

Repeat step 2 

t 14 

p 2 

p 3 

t 1 

t 2 

t 3 

1st STEP 

BLOCK 

p 15 

p 16 

p 1 

t 1 

p 2 

t 2 

p 3 

t 3 

1st STEP 

p 13 

p 14 

t 11 

t 12 t 13 

t 14 

BLOCK 

p 15 

p 16 

p 13 

p 14 

t 1 

p 2 

t 11 

t 12 t 13 

t 14 

p 3 

t 3 

p 15 

p 16 





Structure approach 

Analysis by reduction 

We only use blocks that have good properties 

p k 

t i 

Sequence simplification 

Eliminate sequences of places 

Concatenate parallel paths 

Suppress identity transitions 

t j 

SEQUENCE 

IF THEN ELSE 

WHILE DO 

ORIGINAL PROPERTIES MUST BE 

PRESERVED 





Analysis by reduction (2) 

Reduction techniques 

p 1 

Preserving boundedness and liveness properties 

p 1 

R1: place substitution 

p 1 

R2: implicit place 

p 2 

p 2 

R3: neutral transition 

R4: identical transitions 

Preserving invariants 

p 1 

Ra: non pure transition for ordinary PN 

p 1 

p 1 

Rb: pure transition for ordinary PN 





Reduction R1: place substitution 

Reduction R1- example 

Place Pi can be removed if it complies with the 3 

following conditions 

Pi output transitions have no other input place than Pi 

There is no transition Tj that is at the same time input and 

output transition of Pi 

At least one output transition of Pi is not a sink transition 

P 1 

P 2 

T 1 T 2 

P 3 

T 3 T 4 

P 4 P 5 

P 1 

P 2 

T 23 

T 13 T 24 

P 4 P 5 

T 14 

Keeps: bounded, safe, live, without blocking, home state, 

conservative. However, bound and home state are not 

always known 





Reduction R2: implicit place 

Reduction R2- example 

Place Pi is implicit if it fulfils the following 2 conditions 

Marking of this place may never block the firing of its output 

transitions 

Its marking may be deducted from the marking of other places 

according to 

M 

( P ) = ∑α M ( P ) + β 

i 

k≠i 

k 

k 

P 1 

P 1 

T 1 P 3 T 1 

P 2 

p 2 

p 3 

Keeps: bound, live, without blocking, home state, 

conservative. May be safe after reduction even if original 

is not. It is not always possible to know the home state 

and the bound. 

t 1 

t 2 

t 3 

p 1 

p 2 

p 3 

t 1 

t 2 

t 3 





Reduction R3: neutral transition 

Reduction R4 – identical transitions 

Transition T is neutral iif the set of all its input places is 

identical to the set of all its output places 

2 transtions are identical if they have the same set of 

input places and the same set of output places 

T 1 

T 3 

T 1 

T 3 

p 3 

p 4 

p 3 

p 4 

P 1 

T 5 

P 2 

P 1 

P 2 

t 1 

T 2 

T 4 


conservative. It is not always possible to know the home 

state and the bound. 

T 2 

T 4 


t 2 

t 1 

p 3 


conservative. It is not always possible to find the home 

state and the bound. 

p 3 



Real-Time Programming © J.-D. Decotignie, 2007 

Reduction Ra – non pure transition 

Reduction Rb – pure transition 

T 1 

P 1 

Transition Tj with place Pi and arcs Tj->Pi and Pi->Tj 

Reduction 

Suppress arcs Tj->Pi and Pi->Tj 

Suppress transition Tj if it is isolated 

P 2 T 1 

P 2 

T 3 P 1 

T 3 

p 1 

The transition must possess at least one input and one 

output place 

Reduction 

p 1 

Suppress transition 

Each couple of places Pi,Pk such that Pi 

p 2 

is an input place and Pk is an output place, 

is replaced by a place Pi+Pk (union of places) 

T 2 P 2 

T 2 

P 2 

p 1 

p 1 





Reduction Rb- example 

Exercise 

Reduce the following PN 

p 1 

t 1 

p 2 

T 1 

T 2 

T 1 

T 2 

t 11 

P 1 

P 2 

T 5 

P 1+4 

P 2+4 

P 1+3 

P 2+3 

p 13 

t 12 t 13 

p 15 

P 4 P 3 

p 14 

p 16 

t 14 

T 3 

T 4 

T 3 

T 4 

p 3 

t 3 





Exercise 

PN Formal analysis 

Reduce the PN below and find its place invariants 

t 1 

P 1 

t 2 

P 2 

P 3 

t 3 

Systematic evolution from the initial marking 

Gives the set of reachable markings 

From which the properties are derived 

3 techniques 

Reachability graph 

Reachability tree 

Matrix analysis 

Place invariant 

Transition invariant 





Reachability graph 

Reachability tree 

We create a graph of 

reachable markings 

Exhaustive search 

Risk of combinatorial 

explosion 

t 1 

p 1 

t 2 t 3 

t 1 

p 2 p 3 

t 3 t 2 t 1 

[2,0,0] [1,0,1] [0,1,1] [1,0,0] 

 

If we want to show that 

the PN is not bounded, 

it might be useless to continue 

p 1 

This is the case if M 1 

that has been reached from M, 

M 1 ≥M (∀i: m 1 (p i ) ≥m(p i )) 

if M 1 =M, we are back to the 

same marking 

t 2 

if M 1 >M, it is possible to repeat 

[1,0,1] 

the same firing sequence and 

increment again the number of tokens 

t 2 t 3 

t 1 

p 2 p 3 

[1,0,0] 

t 1 

[0,1,1] 

t 3 

[1,1,0] 





Analyis by reachability tree 

Matrix analysis 

The tree is always finite 

The tree permits to check the following properties 

Not bounded if there exsists M1>M 

Conservative if the number of tokens is constant 

L1-Live is each transitions appears at least once in the tree 

Reversibility and home state 

Complex to use 

Uses I and O matrices 

Look for a non trivial solution to 

| O-I |.x T =0 avec x= | x 1 , ...., x i , ...., x N | (1) 

Let M et M 1 be two successive markings 

M 1 = M + O[t j ] – I[t j ] after firing t j (2) 

Let ρ = | ρ 1 , ...., ρ i , ...., ρ N | be a solution of (1) 

Let multiply each term of (2) by ρ i and sum up on i 

N 

N 

N 

∑m 

pi 

i = ∑ im 

pi 

+ ∑ i O t j pi 

− I t j pi 

i= 

1 1 ρ ρ ρ , , 

i= 

1 i= 

1 

( ) ( ) [ ( ) ( )] 





Place invariant 

result: 

If all ρ i have the same sign and are non zero 

The PN is conservative (strictly is ρ i are all equal) 

The PN is bounded 

N 

∑m 

= N 

i i i 

i= 

1 1 ρ 

i= 

1 

( p ) ρ ∑ m( p ) 

If all ρ i have the same sign and some are zero 

A subset of the PN is conservative 

i 

Place invariant - example 

I= 

O= 

⏐O-I⏐.x T = 

0 1 0 1 0 0 

1 0 0 0 1 0 

0 0 1 0 0 0 

0 0 0 0 0 1 

1 0 1 0 0 0 

0 1 0 0 0 1 

0 0 0 1 0 0 

0 0 0 0 1 0 

1 -1 1 -1 0 0 

-1 1 0 0 -1 1 

0 0 -1 1 0 0 

0 0 0 0 1 -1 

PRODUCTEURS 

p 3 

messages 

Possible solution x i =1 , invariant = s+2; bounded, 

strictly conservative 

t 3 

production 

. x T =0 

p 4 

p 1 

t 1 s p 2 

positions 

CONSOMMATEURS 

p 5 

t 2 

t 4 

p 6 

utilisation 





Exercise – readers-writers 

Transition invariant 

Task 

waiting 

READERS 

t 1 

p 0 

reading p 3 

S 

p 1 

s 

s 

WRITERS 

p 2 

Task 

waiting 

t 2 

p 4 

writing 

Look for a non trivial solution to 

| O-I | T .x T =0 avec x= | x 1 , ...., x i , ...., x M | (1) 

M is the number of transitions 

Gives an indication of liveness 

Sequence of transitions that can be fired repeatedly 

t 3 

s 

t 4 





Interesting information 

http://www.informatik.uni-hamburg.de/TGI/PetriNets/ 

http://home.arcor.de/henryk.a/petrinet/e/hpsim_e.htm

GRAFCET and Petri Nets Outline Introduction GRAFCET - EPFL

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