Exploring TCP Axioms and Chatting Philosophers in Process Algebra for Probabilistic Systems

1. Process Algebra (2IF45)Working with Probabilistic systems Dr. Suzana Andova

2. 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) Process Algebra (2IF45)

3. Chatting Philosophers example • 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. • 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. Switcher (2) Philosopher deall2 deall1 deall Think S 1 1-pi 2 ro all1 pi all2 all Chat 1-ro Process Algebra (2IF45)

4. Chatting Philosophers example • 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. • 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. • We consider a system of two philosophers and one switcher • First, we compute Phil1 || Phil2, where Phili = Thinki Process Algebra (2IF45)

5. Chatting Philosophers example tick (1-)(1-) (1-) (1-) S,T1,T2 tick tick deall2 deall1  all1 all2   S1,C1,T2 S2,T1,C2 all1 all2 1- 1- tick tick Process Algebra (2IF45)

6. Chatting Philosophers example tick (1-)(1-) (1-) (1-) S,T1,T2 tick tick deall2 deall1  all1 all2   S1,C1,T2 S2,T1,C2 all1 all2 1- 1- tick tick max\min Process Algebra (2IF45)

7. Chatting Philosophers example all2 all1  all1 all2 tick tick all2 all1 tick all1 all2 tick tick tick Process Algebra (2IF45)

8. Chatting Philosophers example all2 all1  all1 all2 tick tick tick all1 all2 tick tick tick all1 all2 Process Algebra (2IF45)

9. Chatting Philosophers example (small change) tick (1-)(1-) (1-) (1-) S,T1,T2 tick tick deall2 deall1  all1 all2   S1,C1,T2 S2,T1,C2 all1 all2 1- 1- tick tick max\min Process Algebra (2IF45)

10. Schedulers • Resolves nondeterminism • Allows for analysis (min/max) • Needed to define equivalence relations (with silent transitions) Process Algebra (2IF45)

11. Chatting Philosophers example (cont.) tick (1-)(1-) (1-) (1-) S,T1,T2 tick tick deall2 deall1  all1 all2   S1,C1,T2 S2,T1,C2 all1 all2 1- 1- tick tick Process Algebra (2IF45)

12. Chatting Philosophers example (cont) 2 1 3 S R S = s1(x).Sx Sx = i.s2(x).1 + i.s2(err).Sx R = r2(x).r3(x).1 + r2(err).R Sys 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) s1(x) i i c2(err) c2(x) s3(x) Process Algebra (2IF45)

13. ABP with unreliable channels 2 3 1 K 4 S R 5 6 L 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 Process Algebra (2IF45)