process algebra (2if45) probabilistic extension: semantics parallel composition dr. suzana andova
TRANSCRIPT
Process Algebra (2IF45)
Probabilistic extension: semanticsParallel composition
Dr. Suzana Andova
2
Probabilistic LTS
Process Algebra (2IF45)
Basic ingredients of a PLTS: • states
• non-detereministic states set N• probabilistic states set P
• transitions• action transitions labelled with actions and t P• probabilistic transitions labelled with probabilities
and t N• For a probabilistic state s,
= 1
s t
s t
as t
3 Process Algebra (2IF45)
Equational theory. Language
• Specify processes that can execute certain actions from a given set A
• The language of the Probabilistic Basic Process Algebra, namely, the operators in the signature• 0 deadlock constant (inaction)• 1 successful termination • a._ action prefix for a in A• + non-deterministic choice
probabilistic choice for (0,1)
4 Process Algebra (2IF45)
Axioms of PBPA(A)
PBPA(A)
Signature: 0,
a._ ,
_+_ ,
(A1) x+ y = y+x
(A2) (x+y) + z = x+ (y + z)
(A3) x + x = x
but (AA3) a.x+a.x = a.x
(A4) x+ 0 = x
5 Process Algebra (2IF45)
Axioms of PBPA(A)
PBPA(A)
Signature: 0,
a._ ,
_+_ ,
(PA1) x y = y 1- x
(PA2) x (y z) = (x y) z
where = /( + - ) and = + -
(PA3) x x = x
(PA4) (x y) + z = (x + z) (y + z)
6
Probabilistic LTS
Process Algebra (2IF45)
1 2/3
b32
5
1/3
a
9
c
1
10
c c
b
76
11
1/3
12
c c
a
8
1
13
c
b
1
c
2/3
4 1
b
1
a
1
c
1
7
SOS rules for PBPA(A)
Process Algebra (2IF45)
1
a
Process terms in the language of
the Probabilistic Basic Process Algebra,
• 0 deadlock constant (inaction)
• 1 successful termination
• a._ action prefix for a in A
• + non-deterministic choice
probabilistic choice for (0,1)
a.0
a.0?
0?
8
SOS rules for PBPA(A)
Process Algebra (2IF45)
1
a
Process terms in the language of
the Probabilistic Basic Process Algebra,
• 0 deadlock constant (inaction)
• 1 successful termination
• a._ action prefix for a in A
• + non-deterministic choice
probabilistic choice for (0,1)
a.0
a.0
0
9 Process Algebra (2IF45)
SOS rules for PBPA(A)
Signature: 0, a._ , _+_,
Set of closed terms C(PBPA(A))
Behaviour expressed by
• action transitions _ _ for a in A
• probabilistic transitions _ _ for (0,1]
• Behavioural equivalence is bisimilarity
a
Deduction rules
a.x a.x1
a.x xa
10
SOS rules for PBPA(A)
Process Algebra (2IF45)
1
a b
1 1/2
a b
1/21/2 =
a.0 b.0 a.0 1/2 b.0
11 Process Algebra (2IF45)
SOS rules for PBPA(A)
Deduction rules x x’
x y x’ a.x a.x
1 y y’
x y y’ (1-)
a.x xa
1
a b
1 1/2
a b
1/21/2 =
a.0 b.0 a.0 1/2 b.0
12 10 January 2008
1/2
a b
1/2+ =
1/3
c d
2/3 1/3
a b
1/61/6
a
1/3
dc cdb
SOS rules for PBPA(A)
13
1/2
a b
1/2+ =
1/3
c d
2/3 1/3
a b
1/61/6
a
1/3
dc cdb
SOS rules for PBPA(A)
Deduction rules x x’
x y x’ a.x a.x
1 y y’
x y y’ (1-)
x x’, y y’
x +y x’ + y’
a.x xa
14
SOS for action transitions
Process Algebra (2IF45)
• Deduction rules for action transitions and termination
1
x x’
x + y x’ a.x x
a
a
x (x + y)
a y y’
x + y y’
a
a
y (x + y)
15 Process Algebra (2IF45)
Extending the language with parallel composition – Probabilistic TCP(A, )
• Specify processes that can execute certain actions from a given set A
• The language of the Probabilistic Theory of Communicating Processes, namely, the operators in the signature• 0 deadlock constant (inaction)• 1 successful termination • a._ action prefix for a in A• + non-deterministic choice
probabilistic choice for (0,1)
• communication function (_,_)• parallel composition _ || _• communication composition _ | _
16 10 January 2008
SOS semantics of PTCP(A, )
where a and c communicate in e, and no other communication is defined (in this examples)
1/3
a b
2/3 1/2
c d
1/2|| =
1/3
c b
1/31/6
a
1/6
ac
db de
1
a a dd b
Deduction rules x x’
H(x) H(x’)
x x’, y y’
x || y x’|| y’
x x’, y y’
x | y x’ | y’
c
11
b c
1 11 1
17
• Deduction rules for action transitions and termination
x x’
x || y x’ || y
a
a
x yx || y
y y’
x || y x || y’
a
a
x yx | y
x x’ y y’, (a,b) = c
x || y x’ || y’
a
c
b x x’ y y’, (a,b) = c
x | y x’ || y’
a
c
b
x x’ , aH
H(x) H(x’) a
a
SOS semantics of PTCP(A, )
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
Exercises
Process Algebra (2IF45)
1. Consider process terms p = a.0 + a.0, q = a.0 1/3 b.0, r = c.(d.0 1/2 b.0). Draw the PLTSs of p, q and r using the SOS semantic rules. Use the rules
compute the PLTS of H(p || q || r) if (b,c) = e and H={b,c}Using the axioms derive a PBPA(A) process term t such that
PTCP(A, )├ H(p || q || r) = t, if (b,c) = e and H={b,c}. Draw the PLTS of t and establish a probabilistic bisimulation relation between
PLTS of t and PLTS of H(p || q || r).
20
Unreliable communication – nondeterministic spec
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)
21
Unreliable communication – probabilistic spec
Process Algebra (2IF45)
S R2
Specification of components:
PS = s1(x).PSx
PSx = s2(x).1 9/10 s2(err).PSx
R = r2(x).r3(x).1 + r2(err).R
Specification of the whole system,
derived from spec. above
PSys = H(PS || R)
PSys =s1(x). H(PSx || R)
H(PSx || R) = c2(x).s3(x).1 9/10 c2(err). H(PSx || R)
1 3
PSys
s1(x)
c2(x)
s3(x)
1/10
c2(err)
1
9/10
1
22
Unreliable communication – probabilistic spec
Process Algebra (2IF45)
Benefits of probabilistic wrt nondeterministic specification:
- no fairness assumption needed
- performance analysis is possible , for instance for this example we can compute the average number of the message x needs to be sent by S in order to be received by R; This number, of course, depends on the probability by which the message is correctly sent. Thus, for exaple, we compute, using probability theory techniques, that :
- for 1/10 vs. 9/10 in average a message needs to be sent 1.2 times
- for ½ vs. ½ in average a message needs to be sent 2 time
PSys
s1(x)
c2(x)
s3(x)
1/10
c2(err)
1
9/10
1
23
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