System schemata

1
Chapter 5

• System schemata
• and Place Invariants

2
reminder the biscuit vending machine

• two kinds of biscuits
• and ?

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
3
The empty can return machine
interpret the symbols differently empty can
1 coins
E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
you get 2 Euros for one empty can!
no signal
G F
slot is free
D
coin back
b
coin store
4
What is the net really?

• a model of a biscuit vending machine?
• a model of a can return machine?
• it is a schema with symbols that deserve
  interpretation

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
5
many different interpretations

• you may interpret the net as a biscuit vending
  machine
• you may interpret the net as a can return machine
• you may interpret the schema in different ways

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
6
We can analyze the schema!

• M0(A) M0(D) M0(H) 6
• ?( 1 lt M0(A) M0(D) M0(H) )

in ? in ?
E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
7
marking M1

• M1(A) M1(D) M1(H) 6
• ?( 1 lt M1(A) M1(D) M1(H) )

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
1
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
8
some properties hold at some markings

• M (A) M(D) M(H) 6
• holds for M M0 and M M1
• but not for M M3

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
9
some properties hold at all reachable markings
A D E 2 holds in the schema is a valid
equation

• M (A) M(D) M(E) 2
• holds for all reachable M

E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
10
valid equations

• A D 1
• A D1 1
• E 1
• B G ?

H 2G 2F 7 H - 2B 2F 5
cont(E) 2F 5 H - 2F cont(E) 2B
E
H
? ?
5
A
. a
y,z
outbox
C
d
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
11
Equation at a branching transition
f(A) g(A) B C f(u), f(v), g(u),
g(v)
12
Equation at a branching transition
f(A) g(A) B C f(u), f(v), g(u),
g(v)
13
Equation at a branching place
A f-1(B) g-1(C) u,v
14
Equation at a loop
A f(A) u,u
15
Equations of the philosophers schema
A C a, b, c B l(C) r(C) a, b,
c l(A) r(A) B a, b, c
16
Matrix and place invariant of a trans.
f(A) g(A) B C f(u), f(v), g(u),
g(v)
17
Matrix and place invariant of two trans.
A f-1(B) g-1(C) u,v
18
Matrix and place invariant of a loop
A f(A) u,u
19
... of the philosophers schema
A C a, b, c B l(C) r(C) a, b,
c l(A) r(A) B a, b, c
20
End of Chapter 5

• System schemas
• and Place Invariants

21
4 th Advanced Course on Petri Nets The end of
High Level Petri Nets

• Wolfgang Reisig
• Humboldt-Universität zu Berlin

thank you very much
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The philosophers Schema
27
marking M1

• 1st case For each arc (p,t) holds pt??.
• let M enable t. Then for each place p
• M(p) M(p) pt tp
• M0 ?? M1 with M1(D) , M1(A) 1

E
H
? ?
5
A
. a
y,z
outbox
C
d
signal
B
y,z
y
1
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
b
coin store
28
marking M2

• M1 enables b with valuation x 5
• but not with x 4
• M1 ????? M2

b, x 5
? ?
4
y,z
C
d
B
y,z
y
?
b
c
G F
D
b
1
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? ? ? Math1 ? ? ? ? ? ? ? Math4 ?? ? ? ?
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? Math5 ? ? ? ? ? Symbol ? ? ? ? ? ? ? ?
??? ?? ? ? ? ? ? ? ? . ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .
?? ? def M? M M? WFv(A) SFv(A) ??Nexta?vars?
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