1. Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises Dr. Suzana Andova

2. 1 Example 1 (cont.) Process Algebra (2IF45) Property1: A path has a trace c*a n p ks 0 x p kss 0 p 1/3 1/2 1/6 a b c kss 0 p 1/2 1/6 a b c 1/3...... Prob(SetPaths1) = 1/3 + 1/6x1/3 + (1/6)^2x1/3 + …. =  k  0 1/3x(1/6)^k = (1/3)/ (1-1/6) = 2/5

3. 2 Towards probabilistic branching bisimulation Process Algebra (2IF45) Recall Branching bisimulation on LTss s t s’  ts t’s’ a t’’ a  Recall Strong Probabilistic bisimulation on PLTss s t C1 (eq. class ) s s’ a t’ a t C2 (eq. class ) 11 11 22 22 Combining them into Probabilistic Branching Bisimulation

4. 3 Missing ingredients Process Algebra (2IF45) s 0 a u 1 k 0 a r 1 n p 1  m q 1  Relate probabilistic and non-deter. states! unobservable path Prob(s, {s}) = 1

5. 4 Probabilistic Branching Bisimulation Process Algebra (2IF45) Definition : An equivalence relation R ⊆ S × S is a probabilistic branching bisimulation iff for every (s, t) ∈ R the following two conditions hold: (i)if s –-> s′ for a ∈ A or a= , then there exist states t0,..., tn, t′ such that t = t0 -------> t1 ------> … tn –-> t’ and (s, ti) ∈ R for all 0 ≤ i ≤ n, and (s′, t′) ∈ R, (ii) for all equivalence classes of states M ∈ S/R, Prob(s,M) = Prob(t,M). States s and t are branching bisimilar, denoted by s ∼ pbb t, if (s, t) ∈ R for some branching bisimulation relation R. a  or  a

6. 5 Process Algebra (2IF45) An equivalence relation R  S x S is probabilistic branching bisimulation iff for every (s, t)  R the following conditions hold: - - -> is either probabilistic or  transition - - -> is either probabilistic or  transition

7. 6 Examples: Probabilistic Branching Bisimulation Process Algebra (2IF45) Distributed pages, also on http://www.win.tue.nl/~andova/education/2IF45/ExBB.pdf

8. 7 Exercise 1. Process Algebra (2IF45) Figure 1 1 2 3 4 0 000 5 6 7 8 9 0 s n 0t m 0 p k

9. 8 Process Algebra (2IF45) Figure 2 1 23 0 0 0 4 5 00

10. 9 Process Algebra (2IF45) Figure 3 1 2 34 000 6 8 7 0 5 0 0 00 9

11. 10 Process Algebra (2IF45) Figure 4

12. 11 Process Algebra (2IF45)

13. 12 1. Is PLTS a. probabilistically branching bisimilar to the PLTS in b? Why? 2. What does your intuition tells you? 3. If a. is counterintuitive (goes against b.) can you foresee what may be the reason that this solution is chosen? Process Algebra (2IF45) 1 2 3 40 5 60

14. 13 Philosophers example - revised The system consist of the following components; Philosopher 1 is specified as: T1 = a1.C1  pi think1.T1 C1 = talk1.(d1.T1  ro C1) Philosopher 2 is specified as: T2 = a2.C2  p think2.T2 C2 = talk2.(d2.T2  q C2) Server is specified as: S = a1.d1.S + a2.d2.S Process Algebra (2IF45)

15. 14 Philosophers example - revised The PLTS specifying the behaviour of the system  H (T1 || T2 || S) is given on the next slide. Note that the system is a bit simplified on state R1’’’’: think1 and think2 are forced to synchronize in think_both, while they shall be also allowed to interleave. Process Algebra (2IF45)

16. 15

17. 16 Philosophers example - revised On the next slide some of the actions are hidden, i.e. renamed into . Exercise: Minimize this PLTS using the probabilistic branching bisimilarity. Process Algebra (2IF45)

18. 17 tau

19. Process Algebra (2IF45) Closing: Relating / positioning / applying the knowledge from this course Dr. Suzana Andova

20. 19 Process Algebra (2IF45) Questions from our first lecture When modeling a system, for verification purposes, is an LTS a representation (model) to start with or it is to be obtained as a final or side product? What ingredient do we need to have predefined, to be able to produce / work with LTSs?

21. 20 (P)LTSs visualizal rep. Process Algebra (2IF45) More opening questions When modeling a system, is an LTS a model to start with or is it something to be obtained as a final or side product? In (model checking) tools manipulating the state space (LTSs): UPPAAL, Prism, MRMC manipulating the specification (language): mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMC components’ specifications the whole system specification the state space verification model checking reduction on specification reduction on specification reduction on LTSs composition by axiom SS generation by the SOS rules property specification Yes! No! … MODELING LANGUAGE Language SEMANTICS execution simulation visualization testing PLTS simulation traces executable code performance analysis verification (model checking)

22. 21 A simple modeling language Process Algebra (2IF45) Lego Mindstorms

23. 22 Language environment: Language semantics SLE'11: Frank Stappers SLCO PA-like

24. 23 Language technology in practice We have currently three ongoing projects with ASML on language development related topics! Process Algebra (2IF45)

25. 24 1-2 2IS15 Generic language technology 1-2 2IS95 Seminar software engineering and technology 3-4 2IP45 Software project management ! 3-4 2IS55 Software evolution 1-2 2IF85 Formal verification techniques 1-2 2IW26 System validation 1-2 2IW55 Algorithms for model checking 3-4 2IF75 Quantitative formal methods 1 2II65 Metamodeling and interoperability 1-2 2II70 Constraint programming 3-4 2IF65 Proving with computer assistance 3-4 2IW15 Automated reasoning 1-2 2II45 Architecture of distributed systems Related courses the educational program