process algebra (2if45) working with probabilistic systems
DESCRIPTION
Process Algebra (2IF45) Working with Probabilistic systems. Dr. Suzana Andova. Axioms (not seen yet) of TCP(A, ). x|| y = x ╙ y + y ╙ x + x | y, only if x= x+x and y= y+y x || (y z) = (x || y) (x || z) (x y) || z = (x || z) (y || z) - PowerPoint PPT PresentationTRANSCRIPT
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Process Algebra (2IF45)
Working with Probabilistic systems
Dr. Suzana Andova
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18 Process Algebra (2IF45)
Axioms (not seen yet) of TCP(A, )
x|| y = x ╙ y + y ╙ x + x | y, only if x=x+x and y=y+y
x || (y z) = (x || y) (x || z)
(x y) || z = (x || z) (y || z)
x | (y z) = (x | y) (x | z)
(x y) | z = (x | z) (y | z)
H(x y) = H(x) H(y)
x ╙ (y z) = (x ╙ y) (x ╙ z)
(x y) ╙ z = (x ╙ z) (y ╙ z)
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19 Process Algebra (2IF45)
1. A chatting philosopher is a person dedicated to two activities: thinking and chatting. A philosopher uses his phone for chatting. He can decide to pick up the phone with probability pi, or stay thinking with probability 1-pi. Once he starts chatting, he end the call with probability ro, or keep chatting with probability 1-ro.
2. There is a switch which allocates connection to a philosopher, and also deallocating a connection. Our switcher is capable of handling only one connection at time.
Chatting Philosophers example
Think
Chat
pi
1-pi
1-ro
ro
all
deall
Philosopher
S1
all1
deall2
Switcher (2)
deall1
2
all2
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20 Process Algebra (2IF45)
1. A chatting philosopher is a person dedicated to two activities: thinking and chatting. A philosopher uses his phone for chatting. He can decide to pick up the phone with probability pi, or stay thinking with probability 1-pi. Once he starts chatting, he end the call with probability ro, or keep chatting with probability 1-ro.
2. There is a switch which allocates connection to a philosopher, and also deallocating a connection. Our switcher is capable of handling only one connection at time.
3. We consider a system of two philosophers and one switcher
4. First, we compute Phil1 || Phil2, where Phili = Thinki
Chatting Philosophers example
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21 Process Algebra (2IF45)
Chatting Philosophers example
S,T1,T2
S1,C1,T2
deall2
all1
tick
S2,T1,C2all2
(1-)(1-)
(1-)(1-)
1-
1-
deall1
tick
tick
ticktick
all1 all2
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22 Process Algebra (2IF45)
Chatting Philosophers example
S,T1,T2
S1,C1,T2
deall2
all1
tick
S2,T1,C2all2
(1-)(1-)
(1-)(1-)
1-
1-
deall1
tick
tick
ticktick
all1 all2
max\min
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23 Process Algebra (2IF45)
Chatting Philosophers example
tick tick
all1 all2
all1 all2
tick
tick tick
all1 all2
all1 all2
tick
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24 Process Algebra (2IF45)
Chatting Philosophers example
tick tick
all1 all2
all1 all2
tick
ticktick
all1 all2
all1
all2
tick
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25 Process Algebra (2IF45)
Chatting Philosophers example (small change)
S,T1,T2
S1,C1,T2
deall2
all1
tick
S2,T1,C2all2
(1-)(1-)
(1-)(1-)
1-
1-
deall1
tick
tick
ticktick
all1 all2
max\min
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26 Process Algebra (2IF45)
1. Resolves nondeterminism
2. Allows for analysis (min/max)3. Needed to define equivalence relations (with silent transitions)
Schedulers
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27 Process Algebra (2IF45)
Chatting Philosophers example (cont.)
S,T1,T2
S1,C1,T2
deall2
all1
tick
S2,T1,C2all2
(1-)(1-)
(1-)(1-)
1-
1-
deall1
tick
tick
ticktick
all1 all2
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28
Chatting Philosophers example (cont)
Process Algebra (2IF45)
S R2
S = s1(x).Sx
Sx = i.s2(x).1 + i.s2(err).Sx
R = r2(x).r3(x).1 + r2(err).R
Sys = H(S || R)
Sys =s1(x). H(Sx || R)
H(Sx || R) = i.c2(x).s3(x).1 + i. c2(err). H(Sx || R)
1 3
Sys
s1(x)
c2(x)
s3(x)
i i
c2(err)
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32
ABP with unreliable channels
Process Algebra (2IF45)
SK2
S = S0 S1 S
Sn = d r1(d).Snd
Snd = s2(dn). Tnd
Tnd = r6(1-n).Snd + s6(err).Snd + r6(n).1
R = R1 R0 R
Rn = r3(err).s5(n).Rn
+ d,n r3(dn).s5(n).Rn + d,n r3(d(1-n)).s4(d).s5(1-n).1
K = d,n r2(dn).(i.s3(dn).K + i.s3(err).K)
L = n r5(n).(i.s6(n).K + i.s6(err).L)
Specify K and L with probabilistic choice operator.
Derive the spec. of the whole system
1 3R
L6 5
4