Process Algebra (2IF45) Probabilistic Branching Bisimulation: Exercises
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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
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Combining theminto
Probabilistic Branching Bisimulation
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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
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Exercise 1.
Process Algebra (2IF45)
Figure 1
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0 0 0
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0
s
n
0 t
m
0
p
k
9 Process Algebra (2IF45)
Figure 2
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0 00
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0 0
10 Process Algebra (2IF45)
Figure 3
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2 3 4
0 0 0
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0 0
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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
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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)
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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)
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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)
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A simple modeling language
Process Algebra (2IF45)
• Lego Mindstorms
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Language environment: Language semantics
SLE'11: Frank Stappers
SLCOPA-like
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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
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