1. Process Algebra (2IF45) Probabilistic extension: semantics Parallel composition Dr. Suzana Andova

2. 1 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  ts  t  s  ts  t a s  ts  t

3. 2 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. 3 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. 4 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. 5 Probabilistic LTS Process Algebra (2IF45) 1 2/3 b 3 2 5 1/3 a 9 c 1 10 c c b 7 6 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. 6 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. 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 0

9. 8 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.x 1  a.x  x a

10. 9 SOS rules for PBPA(A) Process Algebra (2IF45) 1 a b 1 1/2 a b  1/2 = a.0b.0 a.0  1/2 b.0

11. 10 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  x a 1 a b 1 1/2 a b  1/2 = a.0b.0 a.0  1/2 b.0

12. 11 10 January 2008 1/2 a b + = 1/3 c d 2/3 1/3 ab 1/6 a 1/3 d c c d b SOS rules for PBPA(A)

13. 12 1/2 a b + = 1/3 c d 2/3 1/3 ab 1/6 a 1/3 d c c d b 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  x a

14. 13 SOS for action transitions Process Algebra (2IF45) Deduction rules for action transitions and termination 1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. 14 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. 15 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 ||= 1/3 c b 1/6 a a c d bd e 1 a a ddb 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 bc 111 1

17. 16 Deduction rules for action transitions and termination x  x’ x || y  x’ || y a a x  y  x || y  y  y’ x || y  x || y’ a a x  y  x | 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’, a  H  H (x)   H (x’) a a SOS semantics of PTCP(A,  )

18. 17 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. 18 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. 19 Unreliable communication – nondeterministic spec Process Algebra (2IF45) SR 2 S = s1(x).S x S x = i.s2(x).1 + i.s2(err).S x R = r2(x).r3(x).1 + r2(err).R Sys =  H (S || R) Sys =s1(x).  H (S x || R)  H (S x || R) = i.c2(x).s3(x).1 + i. c2(err).  H (S x || R) 13 Sys s1(x) c2(x) s3(x) i i c2(err)

21. 20 Unreliable communication – probabilistic spec Process Algebra (2IF45) SR 2 Specification of components: PS = s1(x).PS x PS x = s2(x).1  9/10 s2(err).PS x 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 (PS x || R)  H (PS x || R) = c2(x).s3(x).1  9/10 c2(err).  H (PS x || R) 13 PSys s1(x) c2(x) s3(x) 1/10 c2(err) 1 9/10 1

22. 21 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. 22 ABP with unreliable channels Process Algebra (2IF45) S K 2 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 3 R L 6 5 4