Méthodes probabilistes pour la vérification des systèmes distribués

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HAL Id: tel-00136083

https://tel.archives-ouvertes.fr/tel-00136083

Submitted on 12 Mar 2007

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Méthodes probabilistes pour la vérification des systèmes

distribués

Stéphane Messika

To cite this version:

Stéphane Messika. Méthodes probabilistes pour la vérification des systèmes distribués. Réseaux et

télécommunications [cs.NI]. École normale supérieure de Cachan - ENS Cachan, 2004. Français.

�tel-00136083�

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T =

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s

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; ; # 7 : ! 7 8&

(25)

7 )* 8& & . & )* 1 2 ! & : # / % %$ ! &

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:

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Ω

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, ρ(s

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s

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t

s

i

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(26)

s

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; ! )* 8& ( /)0 1 &

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(0 &

E(X) =

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k∈

∪∞

k

×

(X

−1

(

{k}))

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X

−1

(

{∞})

1

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2 # (

(#$&++'+ )& )2$3+3#6 ./(#-3&6+

$ 3 6: ; #7 8& 7 8& ! #6 1 2 : 5#7 7 8& ! 8& ! #; :

Acts

"# 3 8& 333 ! $

(S, A, p)

$ "

S

%$

s

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A(s)

_{⊆ Acts}

%$ ! %$ $

s

s, t

_{∈ S}

a

_{∈ A(s)}

p

st

(a)

$ % %$ $

s

!

t

$

a

$

s, t

_{∈ S}

a

_{∈ A(s)}

0 _{≤ p}

st

(a)

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P

t∈S

p

st

(a) = 1

6 8& 3 ! ) 3 # 4 33 "$

421

$ ! ! $ % % $$ % ! $ % "

421

111

611

666

511

555

(30)

411

444

311

333

211

222

654

543

432

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! $

665

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$ % $ $$

1

$ ! $ $ ! $

2

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3

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U

1 U

2 U

3

$

D

12 D

13 D

23

$ $

T

$ $ $ $ $

A

! $ ! $ $ $$ $ $ $$ % $ ! "

Acts =

{A, U

1 , U

2 , U

3 , D

12 , D

13 , D

23 , T

}

S =

_{{(abc, n)|a ≥ b ≥ c ∧ (a, b, c) ∈ {1, 2, 3, 4, 5, 6}}

3 _{∧ n ∈ {1, 2, 3}}}

abc

$

n

$ %

s = (abc, 3)

$

A(s) =

_{A}

A(s) =

_{{A, U}

1 , U

2 , U

3 , D

12 , D

13 , D

23 , T

}

$ % %$ %$ $ $

(211, 1)

$ $ $ $

U

1

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t

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≥ 0

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s

Ω

s

$%$

s

X

i

Y

i

$ ! %$ $ $

i

!

(31)

>

(211, 1)

(111, 2)

(211, 2)

(311, 2)

(411, 2)

(511, 2)

(611, 2)

1/6

(211, 1)

(211, 2)

(221, 2)

(321, 2)

(421, 2)

(521, 2)

(621, 2)

1/6

# & : 4 33 $$ !

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! % $ $ $ ! %$ $ ! $ $ %$ "

532, D

12 , 442, U

1 , 432, U

2 , ...

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etat

_{− action}

& 6:# 33!

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etat

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etat

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s, a

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s

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a

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χ

Π

=

{(s, a)|s ∈ S ∧ a ∈ A(s)}

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´

etat

_{− action}

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etat

_{− action s, a}

Succ(s, a) = t

|p

st

(a) > 0

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s

$

a

6 3 ( 7 8& 3 ! : # 4 ;

s

_{∈ S}

B

s

⊆ 2

Ω

s

*

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)

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s

|X

0 = s

∧ Y

0 = a

0 ∧ ... ∧ X

n

= s

n

∧ Y

n

= a

n

}

n

≥ 0

s

0 , ...s

n

∈ S

a

0 ∈ A(s

0 )

###

a

n

∈ A(s

n

)

# *

σ

* 5 : & #