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Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises

Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises. Dr. Suzana Andova. Example 1 (cont.). x. Property1: A path has a trace c*a. n. p. 1/3. 1/6. 1/2. p. k. s. s. c. b. a. 0. p. 1/3. 1/6. k. s. 1/2. k. s. s. 0. c. b. a. 0. p.

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Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises

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

  2. Example 1 (cont.) x Property1: A path has a trace c*a n p 1/3 1/6 1/2 p k s s c b a 0 p 1/3 1/6 k s 1/2 k s s 0 c b a 0 p . . . 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 Process Algebra (2IF45)

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

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

  5. 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 Process Algebra (2IF45)

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

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

  8. Figure 2 4 1 5 2 3 0 0 0 0 0 Process Algebra (2IF45)

  9. Figure 3 8 1 5 9 2 3 4 6 7 0 0 0 0 0 0 0 0 Process Algebra (2IF45)

  10. Figure 4 Process Algebra (2IF45)

  11. Process Algebra (2IF45)

  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? 1 5 2 6 0 3 4 0 Process Algebra (2IF45)

  13. Philosophers example - revised The system consist of the following components; • Philosopher 1 is specified as: T1 = a1.C1 pithink1.T1 C1 = talk1.(d1.T1 ro C1) • Philosopher 2 is specified as: T2 = a2.C2 pthink2.T2 C2 = talk2.(d2.T2 q C2) • Server is specified as: S = a1.d1.S + a2.d2.S Process Algebra (2IF45)

  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)

  15. 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)

  16. tau

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

  18. 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? Process Algebra (2IF45)

  19. 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? verification (model checking) In (model checking) tools manipulating the state space (LTSs): UPPAAL, Prism, MRMC manipulating the specification (language): mCRL2, Chi, CADP, FDR, PEPA, MRMC updated IMC MODELING LANGUAGE simulation testing visualization execution performance analysis reduction on specification components’ specifications Language SEMANTICS reduction on specification … Language transformation the whole system specification composition by axiom reduction on LTSs No! the state space simulation traces SS generation by the SOS rules executable code verification model checking (P)LTSs property specification visualizal rep. PLTS Yes! Process Algebra (2IF45)

  20. A simple modeling language • Lego Mindstorms Process Algebra (2IF45)

  21. Language environment: Language semantics SLCO PA-like semantic based semantic based specification semantic based execution code simulation analysis semantic based Interchange format SLE'11: Frank Stappers

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

  23. Related courses the educational program 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

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