第11讲_PT网和着色网(2)

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Construction
b1
b2
b4
b3
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Review Place/Transition System Colored Petri Net


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According to the above definition, give out the complements of the b1,b2,b3,b4, and b5
b1 e1 e2 b3 e4 b4 b5
b2
6
e3
e5
e6
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Construction
Marked place/transition nets generalize EN System Terminology
EN System Condition Event Case SCG
Marked P/T Net Place Transition Marking Marking Graph
Move3
Ready for insertion
Insert coin
Move2
Dispense item
Reject coin
Holding coin
Move1
=4
Ready to dispense
Accept coin
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Place/Transition Net
Ready for insertion
Insert coin Reject coin Holding coin
Dispense item
Accept coin
Ready to dispense
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A Vending Machine with Capacity 4
s S, m’(s) = m(s) – w(s,t) if s t - t m’(s) = m(s) + w(t,s) if s t - t m’(s) = m(s) – w(s,t) + w(t,s)if s t t m’(s) = m(s) if s t t
Let N =(B,E,F,Cin) be EN system Let B be a set disjoint with BE, and let :BB be a bijection The S-complementation of N is the EN system N ‘=(B’,E’,F’,Cin’) such that B’=BB, E’=E, F’=F {(e, (b)):eE&(b,e) F} {((b),e) :eE&(e,b) F}, Cin’=Cin (B-Cin)
S, set of Places; T, set of Transitions
(2) (3) (4)
k: S{1,2,3,…} {} w:F{1,2,3,…} M0:S{0,1,2,3,…}
<Capacity restriction <Weight function <Initial marking
2
t1/t2 t1/t2
t8
t5/t6 s3/s4 t3/t4 s9 t5/t6 t5/t6/t8 t8 s3/s4 s3/s4
s5/s6
22
2
t7
s7/s8 s7/s8/s9
t3/t4 t3/t4
s9
22
t7 t7
s7/s8
2
22
s5/s6 s5/s6
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Behavior of the Example
s0/s1/s2
(0,1,0,1)
2
t1/t2
(0,0,1,2) (1,0,0,2) (1,0,2,0)
(3,0,0,0)
(2,0,0,1)
t5/t6/t8 s3/s4 t3/t4 s7/s8/s9
(1,1,0,0) (0,0,0,3) (1,0,1,1)
Sequential Case Graph
e1
1 2
e2 1 4
e6
e3 5 2
3 2
e5
e4 1 6
e2 e1 3 4
e2 e3 5 4
e4 e1 3 6
e4 e3
5 6
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Case Graph
345 e5 6 e3 {e5,e3} e6 e4 2 3 4 156 e1 e2 e3 e5 6 {e2,e3} {e1,e5} 634 126 5 7 8 e2 e1 e5 e3 {e1,e2} 168 7 2 e1 e2
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Definition of Marked Place/Transition Net
A marked place/transition net is a tuple N=(S, T, F, k, w, M0) where (1) (S, T, F) is a net
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Folding a Net
s1/s2 s1 s0 s2
t1/t2 t1
t2
t5/t6 t5
s3 s3/s4
t3/t4 t3
t8
s4
t6 t4
s9 s5 s5/s6
10
2
t7
s6 s8
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s7 s7/s8
2
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Omit some Details
s1/s2 s0 s1/s2 s0 s1/s2 s0/t8/s9
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Example of P/T Net
s1/s2 s0
S={s0, s1/s2, s3/s4, s5/s6, s7/s8, s9} T={t1/t2, t3/t4, t5/t6, t7, t8}
t1/t2
t8
t5/t6 s3/s4 t3/t4 s9
t1/t2
t8
t5/t6
t8
s3/s4
t3/t4 s9
t5/t6
t1/t2/ s3/s4 t3/t4
t5/t6
s9
t1/t2/ s3/s4 t3/t4
s5/s6 s7/s8 11
2
t7 s7/s8
s5/s6
2
t7 s7/s8
s5/s6
2
t7
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2
2
2
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Control structure of Vending M
(0,0,2,1)
2
t7
23
2
s5/s6
(2,0,1,0)
(0,1,1,0)
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Adding Constraints
s0/s1/s2
<x>+<s0>
<y>
t5/t6/t8 t1/t2
<x, s0>
s3/s4
<y>
s7/s8/s9
<u, s0>
Declarations Type S12 = S1 | S2 Type S123 = S0| S1 | S2 Var x, u: S12 Var y: S123
e5
e4
5
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Complements
Condition b1, b2 are complements of each other, b1=b2 iff e E, b1 e iff b2 e and e E, b1 e iff b2 e
<s0> t3/t4
<s1>+<s2> <u> s5/s6 t7 <s1>+<s2>
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Marking with Multi Set
s0/s1/s2
Sets and Multi Set
<x>+<s0>
t1/t2
<y>
t5/t6/t8
<x, s0>
s3/s4
s1/s2 p1 s0/t8/s9 p2
(2,1,0,0)
t1
t5/t6 t3
t1/t2/ s3/s4 t1 t3/t4
t3
(1,1,1,0) (1,1,0,1) t1 t3 t2 (0,1,2,0) (0,1,0,2)
2
t7 t2
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s5/s6 p3
20
s7/s8 p4
7 8
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State Space Similar
e1
1 2
e2
345 e56 e3
e6 2 3 4 e6 1 5 6 3 2 1 4 e1 6 e2 e3 e5 e4 e3 e2e1 634 12 5 7 3 a 4 pair of bijections 1 6 8 5 Can 2 we find e2 6 e1 e5 ::v1v1’, and : ’? e3 e4 e1 e2 e3 such that 168 7 2 3 6 5 =v1’ 4 e1 e2 (v1) e4 e3 (c1, A, c2)Y1 iff ( (c1), (A), (c2)) Y1’ 7 8 5 6
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Main Contents

Review Place/Transition System Colored Petri Net


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Folding the Net Further
s1/s2 s0 s1/s2 s0/s1/s2 t1/t2 s0
DONG HUA UNIVERSITY
工作流建模技术
Petri Net(c2)
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Main Contents

Review Place/Transition System Colored Petri Net


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- M[t>
Are the transition t is enabled?
s1
2
t
s2
M(s1)
17
K(s2)
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Question 2
Let P/T Net , (S, T, F, k, w, M0 ) A transition t T is enabled. The occurrence of t leads to the successor marking m’:
A set has unordered elements, and only one occurrence of an elements A multi set, or a bag, allows more than one occurrence of an elements Tokens in s0/s1/s2 2‘<s0>+1’<s1>+1’<s2> 2‘<s0>+<s1>+<s2>
F={(s0,t1/t2), …}
k: sS, s
2
s5/s6
16
w= (s5/s6, t7)2, (t7, s7/s8) 2, others f 1
M0=(1, 2, 0, 0, 0, 0)
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s7/s8
2
t7
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Question 1
Let P/T Net , (S, T, F, k, w, M0 ) A transition t T is enabled at a marking m iff s t: m(s)w(s,t) s t : m(s)+w(t,s)k(s) Notation
M0 M Y initial vertex initial marking vertices set of reachable markings labeled-edges set of triples(m,t,m’), m[t>m’
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Example of Marking Graph
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Marki百度文库g Graph
Let P/T Net , (S, T, F, k, w, M0 ) The marking graph of a marked P/T Net is an edge-labeled graph with initial vertex, ((M, Y), M0)
<y>
s7/s8/s9
<u, s0>
<s0> t3/t4
<s1>+<s2> <u> s5/s6 t7 <s1>+<s2>
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