Process Algebra (2IF45)

Working with Probabilistic systems

Dr. Suzana Andova

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)

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

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

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

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

23 Process Algebra (2IF45)

Chatting Philosophers example

tick tick

all1 all2

all1 all2

tick

tick tick

all1 all2

all1 all2

tick

24 Process Algebra (2IF45)

Chatting Philosophers example

tick tick

all1 all2

all1 all2

tick

ticktick

all1 all2

all1

all2

tick

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

26 Process Algebra (2IF45)

1. Resolves nondeterminism

2. Allows for analysis (min/max)3. Needed to define equivalence relations (with silent transitions)

Schedulers

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

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)

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