PowerPointPrsentation

1
Chapter 2
Modeling with High Level Petri Nets
2
A variant of the biscuit vending machine

• two kinds of biscuits
• and ?

3
A variant of 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
d
coin store
4
The components of a HL net
Everything can be inscribed
5
The data structure of a HL Net Multisets

• we dont want to distinguish 1-Euro coins
• usually a place contains many copies of
  different sorts
• the sorts constitute the universe, U
• multiset m U ? ?
• we assume a sufficiently large universe
• Let U contain whatever sort we consider
• let ? denote the set of all multisets over U

6
some notations

• a multiset m U ? ? is finite if m(u) ? 0
  for finitely many u only
• notation for finite multisets
• m() 3, m(?) 2, m(u) 0, for all other u?U
  ,,,?,?
• emty multiset (u) 0 for all u?U

7
a little mathematics on multisets

• you can add multisets m,n ? ?
• ,? ? ,?,?
• for each u?U let (mn)(u) def m(u)n(u).
• you can compare multisets m,n ? ?
• ,? lt ,?,?
• let m ? n iff for each u?U m(u) ? n(u)
• you can subtract a multiset m from a multiset n
  if m ? n ,?,? - ? ,?
• for each u?U let (m - n)(u) def m(u) - n(u)

8
marking M P ? ?

• M0(A) M0(B) M0(C) M0(F)
• M0(D) M0(G) ?
• M0(E) 5
• M0(H) ?,?, , ,

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
d
coin store
9
a notation

• for an arc (x,y)
• xy denotes the inscription of (x,y).
• aA 1 Da ? Eb x bE x-2
• xy if (x,y) is no arc.
• aE

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
d
coin store
10
a marking M enables a transition t

• 1st case For each arc (p,t) holds pt ? ? .
• Def. M enables t iff for each (p,t) pt lt
  M(p) .
• M0 enables a

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
d
coin store
11
step M ?? M
t

• 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
2
signal
B
y,z
y
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
d
coin store
12
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
2
signal
B
y,z
y
1
xgt 2
b
c
remove biscuits
no signal
G F
slot is free
D
coin back
d
coin store
13
variables in arc inscriptions

• 2nd case some arc inscriptions pt and tp
  contain variables
• Let x1, ... xn be the variables in arc inscr.
  around t
• Let u1, ..., un? U. Then x1 u1, ... xn un is a
  valuation.
• Each valuation yields pt ? ?.
• Continue as above.

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
d
coin store
14
variables in arc inscriptions

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

b, x 5
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
d
coin store
15
marking M2

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

b, x 5
? ?
3
y,z
C
d
2
B
y,z
y
?
xgt 2
b
c
G F
D
d
1
16
four enabeling valuations for c

• M2 enables c with valuations
• y z ?
• y z
• y , z ?
• y ?, z

? ?
3
y,z
C
d
2
B
y,z
y
?
xgt 2
b
c
G F
D
d
1
17
Frequent special case

• U ?
• M(p) n ?
• xy n ?
• classical Petri Nets

18
End of Chapter 2
Modeling with High Level Petri Nets
19
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? M?
M M? ? ? def ? ? ? ? ?? ? ? ? ? ? ? ???????? ?
???????? ? ? ???????? ? ???????? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ??
WFv(A) SFv(A) ??Nexta?vars? var?var?var helveti
ca 24 ???????????
20
MathB ? ? ? ? ? ? ? ? MathC ? ? ? ? ? ? ? ? ? ?
? ? ? Math1 ? ? ? ? ? ? ? Math4 ?? ? ? ?
???????? ? ???????? ? ? ???????? ? ????????
? Math5 ? ? ? ? ? Symbol ? ? ? ? ? ? ? ?
??? ?? ? ? ? ? ? ? ? . ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? .
?? ? def M? M M? WFv(A) SFv(A) ??Nexta?vars?
helvetica 24