Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises

Process Algebra (2IF45)

Probabilistic Branching Bisimulation: Exercises

Dr. Suzana Andova

2

Example 1 (cont.)

Process Algebra (2IF45)

Property1: A path has a trace c*a

n

p

k s

0

x p

k s s

0p

1/31/2 1/6

a b c

k s s

0p

1/2 1/6

a b c

1/3

.

.

.Prob(SetPaths1) = 1/3 + 1/6x1/3 + (1/6)^2x1/3 + ….

= k0 1/3x(1/6)^k = (1/3)/ (1-1/6) = 2/5

3

Towards probabilistic branching bisimulation

Process Algebra (2IF45)

Recall Branching bisimulation on LTss

st

s’

t s

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 )

11

22

Combining theminto

Probabilistic Branching Bisimulation

4

Missing ingredients

Process Algebra (2IF45)

s

0

a

u

1

k

0

a

r1

n

p

1

m

q

1

Relate probabilistic and non-deter.states!

unob

serv

able

pa

th

Prob(s, {s}) = 1

6 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

Examples: Probabilistic Branching Bisimulation

Process Algebra (2IF45)

Distributed pages, also on

http://www.win.tue.nl/~andova/education/2IF45/ExBB.pdf

8

Exercise 1.

Process Algebra (2IF45)

Figure 1

1

2

3

4

0

0 0 0

5

6

7

8 9

0

s

n

0 t

m

0

p

k

9 Process Algebra (2IF45)

Figure 2

1

2 3

0 00

4

5

0 0

10 Process Algebra (2IF45)

Figure 3

1

2 3 4

0 0 0

6

8

7

0

5

00

0 0

9

11 Process Algebra (2IF45)

Figure 4

12 Process Algebra (2IF45)

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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

4 0

5

6 0

14

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

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

17

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

tau

Process Algebra (2IF45)

Closing: Relating / positioning / applying the knowledge from this course

Dr. Suzana Andova

20 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 (P)L

TSs

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

verificationmodel checking

reductionon specification

reductionon specification

reductionon LTSs

composition by axiom

SS generation by the SOS rules

property specification

Yes!

No!

…

MODELING LANGUAGE

Language SEMANTICS

execution

simulation visualizationtesting

PLTS

simulation traces

executable code

performance analysis

verification (model checking)

22

A simple modeling language

Process Algebra (2IF45)

• Lego Mindstorms

23

Language environment: Language semantics

SLE'11: Frank Stappers

SLCOPA-like

24

Language technology in practice

We have currently three ongoing projects with ASML on language development related topics!

Process Algebra (2IF45)

25

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 interoperability1-2 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