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Texte intégral

(1)

Alexandre Duret-Lutz

mars 2009

(2)

¯

r1

¯

r2

¯

d

1

¯

d

2

¯

r

1

¯

r2 d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

¯

d1

¯

d2

¯

r1

¯

r2

d1d2

¯

r1

¯

r2

¯

d1

¯

d2

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r

1

¯

r2 d

1

¯

d

2

¯

r

1 r

2

¯

d

1

¯

d

2

r

1

¯

r2

¯ ¯

r1

r

2

¯

q0

q

1

q

2

q3 q4 q5

q6 q

7

q

8

q9

q

10

q

11

(3)

G

(

d1

→

Fr1

)

.

⊤

d

1

∧ ¬

r1

¬

r1

q

C

q

D

(4)

q0

,

qC

q1

,

q C

q2

,

q C

q3

,

qC q4

,

qC

q5

,

qC

q6

,

qC q7

,

q C

q8

,

q C

q9

,

qC

q10

,

qC q11

,

qC

q0

,

qD

q1

,

qD q2

,

qD

q3

,

qD

q4

,

qD q5

,

qD

q6

,

q D

q7

,

qD q8

,

qD

q9

,

qD

, ,

(5)

1

On-the-Fly EmptinessCheks for Generalized Bühi Automata

2

UsingPartialOrders to Reduethe State Spae

(6)

On-the-Fly Emptiness Cheks for GBA

Jean-Mihel Couvreur,Alexandre Duret-Lutz, Denis Poitrenaud

(7)

A(transition-based) Bühi automatonhas:

Aset of states, with a designated initial state,

Aset of transitions between states,

Aset of aepting transitions.

Aninnite runof this automatonis aepting ifit visits an aepting

transitioninnitely often.

s

4 s

5

(8)

A(transition-based) Bühi automatonhas:

Aset of states, with a designated initial state,

Aset of transitions between states,

Aset of aepting transitions.

Aninnite runof this automatonis aepting ifit visits an aepting

transitioninnitely often.

s

4 s

5

(9)

Emptiness Chek =Does an automaton have noaepting run?

= ⇒

Searh for an aepting yle reahable from the initial state.

(10)

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

(11)

s

1

s

2

s

3 s

4 s

5

(12)

s

1

s

2

s

3 s

4 s

5

(13)

s

1

s

2

s

3 s

4 s

5

(14)

s

1

s

2

s

3 s

4 s

5

(15)

s

1

s

2

s

3 s

4 s

5

(16)

s

1

s

2

s

3 s

4 s

5

(17)

s

1

s

2

s

3 s

4 s

5

(18)

s

1

s

2

s

3 s

4 s

5

(19)

s

1

s

2

s

3 s

4 s

5

(20)

s

1

s

2

s

3 s

4 s

5

(21)

s

1

s

2

s

3 s

4 s

5

(22)

s

1

s

2

s

3 s

4 s

5

(23)

s

1

s

2

s

3 s

4 s

5

(24)

s

1

s

2

s

3 s

4 s

5

(25)

s

1

s

2

s

3 s

4 s

5

(26)

s

1

s

2

s

3 s

4 s

5

(27)

s

1

s

2

s

3 s

4 s

5

(28)

s

1

s

2

s

3 s

4 s

5

(29)

s

1

s

2

s

3 s

4 s

5

(30)

s

1

s

2

s

3 s

4 s

5

(31)

s

1

s

2

s

3 s

4 s

5

(32)

s

1

s

2

s

3 s

4 s

5

Found!

(33)

s

1

s

2

s

3 s

4 s

5

Found!

(34)

AGeneralized (transition-based) Bühi automaton has:

Aset of states, with a designated initial state,

Aset of transitions between states,

Aset of aepting sets of transitions.

Aninnite runof this automatonis aepting ifit visits a transition

fromeah aepting set innitely often.

s

4 s

5

(35)

AGeneralized (transition-based) Bühi automaton has:

Aset of states, with a designated initial state,

Aset of transitions between states,

Aset of aepting sets of transitions.

Aninnite runof this automatonis aepting ifit visits a transition

fromeah aepting set innitely often.

s

4 s

5

(36)

Ageneralized automaton with

n states

m aeptane onditions

an be degeneralized into an automaton with

nm states at worst

1aeptane ondition

(37)

entries in hash tablesize searh stak states

hash table in bits depth traversed

n n

(

s

+

2

)

n 2n

n states,

s bits per state.

(38)

entries in hash tablesize searh stak states

hash table in bits depth traversed

nm nm

(

sd

+

2

)

nm 2nm

n states, m aeptane onditions,

s

d

bits per degeneralized state.

(39)

entries in hash tablesize searh stak states

hash table in bits depth traversed

nm nm

(

sd

+

2

)

nm 2nm

n n

(

sg

+

2m

)

nm 2nm

n states, m aeptane onditions,

s

d

bits perdegeneralized state, s

g

bits pergeneralized state (s

g

≤

sd).

(40)

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

(41)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

(42)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

(43)

s

1

s

2

s

3 s

4 s

5

(44)

s

1

s

2

s

3 s

4 s

5

(45)

s

1

s

2

s

3 s

4 s

5

(46)

s

1

s

2

s

3 s

4 s

5

(47)

s

1

s

2

s

3 s

4 s

5

(48)

s

1

s

2

s

3 s

4 s

5

(49)

s

1

s

2

s

3 s

4 s

5

(50)

s

1

s

2

s

3 s

4 s

5

(51)

s

1

s

2

s

3 s

4 s

5

(52)

s

1

s

2

s

3 s

4 s

5

(53)

s

1

s

2

s

3 s

4 s

5

(54)

s

1

s

2

s

3 s

4 s

5

(55)

s

1

s

2

s

3 s

4 s

5

(56)

s

1

s

2

s

3 s

4 s

5

(57)

s

1

s

2

s

3 s

4 s

5

(58)

s

1

s

2

s

3 s

4 s

5

(59)

s

1

s

2

s

3 s

4 s

5

(60)

s

1

s

2

s

3 s

4 s

5

(61)

s

1

s

2

s

3 s

4 s

5

(62)

s

1

s

2

s

3 s

4 s

5

(63)

s

1

s

2

s

3 s

4 s

5

(64)

s

1

s

2

s

3 s

4 s

5

(65)

s

1

s

2

s

3 s

4 s

5

(66)

s

1

s

2

s

3 s

4 s

5

(67)

s

1

s

2

s

3 s

4 s

5

(68)

s

1

s

2

s

3 s

4 s

5

(69)

s

1

s

2

s

3 s

4 s

5

(70)

s

1

s

2

s

3 s

4 s

5

(71)

s

1

s

2

s

3 s

4 s

5

(72)

s

1

s

2

s

3 s

4 s

5

(73)

s

1

s

2

s

3 s

4 s

5

Found!

(74)

s

1

s

2

s

3 s

4 s

5

Found!

(75)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

(76)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

(77)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(78)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(79)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(80)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(81)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(82)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(83)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

without

(84)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

(85)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

1

with

(86)

Mergemore reent optimizations of Gastin et al. ('04) and

Shwoon & Esparza ('04) into Taurainen's algorithm.

Introdue another optimization: weighted blue stak.

s

4 s

5

1

with

(87)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

(88)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

SCC Lihtenstein & Pnueli '83 Couvreur '99

(89)

s

4 s

5

s

1

s

2

s

3

(90)

s

4 s

5

s

1

s

2

s

3

(91)

s

4 s

5

s

1

s

2

s

3

(92)

s

4 s

5

s

1

s

2

s

3

(93)

s

4 s

5

s

1

s

2

s

3

(94)

s

4 s

5

s

1

s

2

s

3

(95)

s

4 s

5

s

1

s

2

s

3

(96)

s

4 s

5

s

1

s

2

s

3

(97)

s

4 s

5

s

1

s

2

s

3

(98)

s

4 s

5

s

1

s

2

s

3

(99)

s

4 s

5

s

1

s

2

s

3

(100)

s

4 s

5

s

1

s

2

s

3

(101)

s

4 s

5

s

1

s

2

s

3

(102)

s

4 s

5

s

1

s

2

s

3

(103)

s

4 s

5

s

1

s

2

s

3

Found!

(104)

s

4 s

5

s

1

s

2

s

3

Found!

entries in hash table size searh stak states

(105)

s

4 s

5

s

1

s

2

s

3

Found!

entries in hash table size searh stak states

(106)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

SCC Lihtenstein & Pnueli '83 Couvreur '99

(107)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

SCC Lihtenstein & Pnueli '83 Couvreur '99

(108)

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

SCC Lihtenstein & Pnueli '83 Couvreur '99

(109)

H1: visit transitions that goto visited states rst.

s

1

s

2

s

3

s

4

(110)

H1: visit transitions that goto visited states rst.

s

1

s

2

s

3

s

4

(111)

H1: visit transitions that goto visited states rst.

s

1

s

2

s

3

s

4

H2: H1 + onsider the DFS in term of SCC when hoosing a

(112)

H1: visit transitions that goto visited states rst.

s

1

s

2

s

3

s

4

H2: H1 + onsider the DFS in term of SCC when hoosing a

(113)

Upper bounds easyto have.

Objetive : evaluateall these algorithms on the average, on

non-empty automata.

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Tauriainen '03

(114)

Upper bounds easyto have.

Objetive : evaluateall these algorithms on the average, on

non-empty automata.

degeneralized generalized

nested DFS Couroubetis et al. '90

Godefroid & Holzmann'93

Holzmann et al. '96

Gastin et al. '04

Tauriainen '03

(115)

nested DFS

Shwoon & Esparza '04

Tauriainen '03

Couvreur et al. '05

SCC Geldenhuys & Valmari '04 Couvreur et al. '05

uniquestatesvisited

transitionsvisited

searhstaksize

(116)

generalized vs. non-generalized:

generalizedalgorithmsrequire lessmemory

generalizedalgorithmsprodue more meaningful

ounterexamples

weak fairnessexpressibleusinggeneralizedonditions

non-generalizedNDFSsprodue ounterexamplesdiretly

NDFS vs. SCC algorithms:

SCCalgorithms hekemptiness faster

SCCalgorithms saleto generalizedonditions andfairness

onditionseasily

(117)

Partial Order Methods

Mostly based onSetion 4of:

MarkoRauhamaa

Aomparative study of methods for eient reahability analysis.

(118)

¯

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

¯

d1

¯

d2

¯

r

1

¯

r2 d1d2

¯

r

1

¯

r2

¯

d1

¯

d2

r1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1r2

¯

d

1

¯

d

2

r

1

¯

r2

¯

d1d2

¯

r1 r

2

d1

¯

d2

(119)

Wewantto verify G

(

d1

→

Fr1

)

.

Onthis formula, the following twoexeutions areequivalent:

lient C

1

sends a request

lient C

2

sends a request

otherevents...

lient C

2

sends a request

lient C

1

sends a request

otherevents (in sameorder)

Theorder between the two requests does not make anydierene.

Some notes:

(120)

¯

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

¯

d1

¯

d2

¯

r

1

¯

r2 d1d2

¯

r

1

¯

r2

¯

d1

¯

d2

r1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1r2

¯

d

1

¯

d

2

r

1

¯

r2

¯

d1d2

¯

r1 r

2

d1

¯

d2

(121)

¯

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r

1

¯

r2 d1d2

¯

r

1

¯

r2

¯

d1

¯

d2

r1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1r2

¯

d

1

¯

d

2

r

1

¯

r2

¯

d1d2

¯

r1 r

2

d1

¯

d2

(122)

a

0 b

0

t

a t

b

a

1

b

1

00 10

01 11

t

a

t

b t

b

t

a

(123)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

001 101

t

a

t

b t

b

t

a t

t

t

t

b t

a

t

b

t

a

t

(124)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

001 101

t

a

t

b t

b

t

a t

t

t

t

b t

a

t

b

t

a

t

(125)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

t

a

t

b t

b

t

a

t

(126)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

110 t

a

t

b

t

(127)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

(128)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state?

(129)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state? Notall, obviously

(130)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state? Notall, obviously

Deadlok detetion?

(131)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state? Notall, obviously

Deadlok detetion? Yes

(132)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state? Notall, obviously

Deadlok detetion? Yes

(133)

Away to redue state spae with a priori knowledge.

Theanalyst manually supplies a partialordering of the events.

E.g. t

a

≺

tb

≺

t.

Thisorderingis used whenever thereis a hoie between events.

Theinformationis stati.

What kindof property does it preserves?

(assuming the partial ordering has been setproperly)

Reahability ofa state? Notall, obviously

Deadlok detetion? Yes

(134)

h

S

,

T

,

W

,

M0

i

where

S is the setof states,

T is the setof transitions,

W

: (

S

×

T

) ∪ (

T

×

S

) → N

is the ar weight funtion,

M

0

:

S

→ N

is the initial marking.

For x

∈

S

×

T wedenote

•

x

= {

y

|

W

(

y

,

x

) >

0

}

.

For x

∈

T

×

S wedenote x

• = {

y

|

W

(

x

,

y

) >

0

}

.

Amarking is a S

→ N

funtion.

Atransition t

∈

T is enabled at marking M (denoted M

− →

t ) if

(135)

Thereahability graph of a Petri net

h

S

,

T

,

W

,

M0

i

isa pair

h

V

,

E

i

where

V (verties) is a set of markings,

E

⊂

V

×

T

×

V (edges)

and the followinghold

M

0

∈

V,

ifM

∈

V and M

− →

t M

′

then M

′ ∈

V and

(

M

,

t

,

M

′ ) ∈

E,

(136)

For

σ =

t1t2

· · ·

tn

∈

T

⋆

wedenote M

− → σ

is there existsM1, M2,

...M

n

−

1 suh that M

t

− →

1 M1

− →

t2 M2

· · ·

Mn

−

1

− →

tn .

Similarly M

− σ

→

M

′

ifthere additionallyexists M

′

suh that

M t

− →

1 M1

− →

t2 M2

· · ·

Mn

−

1

− →

tn M

′

(137)

ThePetri net is split in two parts: a blakbox and an environment,

suh that the transitions of the two sets areindependent.

Inthe following we assume that t is a blak box transition while

σ

is

a sequene of transitions from the environment.

Priniple 1 If

¬

M

− →

T and M

− → σ

,then

¬

M

−→ σ

t .

In other words, ring transitions in the environment

annot enable a disabled transition ofthe blak box.

− →

T

− → σ −→ σ

t

−→

t

σ

(138)

P

1

If

¬

M

− →

T and M

− → σ

, then

¬

M

−→ σ

t .

P

2 If M

− →

T and M

− → σ

, then M

−→ σ

t and M

−→

t

σ

.

When looking for dead states, we an simplify the reahability graph

by ring anyblak box transitions (t) before environmenttransitions

(

σ

).

Letthere be an enabled transition r in the blakbox and a path

π

leading to a dead marking.

Then

π

must ontain some transitiont from the blakbox.

π

(139)

P

1

If

¬

M

− →

T and M

− → σ

, then

¬

M

−→ σ

t .

P

2 If M

− →

T and M

− → σ

, then M

−→ σ

t and M

−→

t

σ

.

When looking for dead states, we an simplify the reahability graph

by ring anyblak box transitions (t) before environmenttransitions

(

σ

).

Letthere be an enabled transition r in the blakbox and a path

π

leading to a dead marking.

Then

π

must ontain some transitiont from the blakbox.

π

(140)

Looser Priniples

⇒

Stubborn Sets

Againt is a transition fromthe blak box (stubborn set T

M

),and

σ

isa sequeneof transitions from the environment(T

\

TM).

Priniple 1* If M

σ

t

−→

,then M

−→

t

σ

.

Transitions of the stubborn setan be movedbefore

those of the environment.

Priniple 2* If M

− σ

→

, then M

−→ σ

t

′

for some xed transition t

′ ∈

TM.

In other words the stubborn setis never empty and the

environment annot disable its transitions.

(141)

P

∗

1 If M

σ

t

−→

, then M

−→

t

σ

.

P

∗

2 If M

− σ

→

, then M

−→ σ

t

′

.

LetM

− π

→

M

′

bea transitionsequene to a dead marking M

′

.

π

neessarily ontain a transitionfrom the stubborn set. (If it does

not,P

∗

2

implies that M

′ − →

t

′

and M

′

annot be dead.)

Therefore

π = σ

t

π ′

and by P

∗

1

we have M t

σπ ′

−−→

M

′

.

(142)

P

∗

1 If M

σ

t

−→

, then M

−→

t

σ

.

P

∗

2 If M

− σ

→

, then M

−→ σ

t

′

.

LetM

− π

→

M

′

bea transitionsequene to a dead marking M

′

.

π

neessarily ontain a transitionfrom the stubborn set. (If it does

not,P

∗

2

implies that M

′ − →

t

′

and M

′

annot be dead.)

Therefore

π = σ

t

π ′

and by P

∗

1

we have M t

σπ ′

−−→

M

′

.

(143)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(144)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(145)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(146)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(147)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(148)

a

0 b

0

t

a t

b

a

1

b

1

0

t

1

000 100

010 110

0-1 1-1

t

a

t

b t

b

t

a t

t

t

a

(149)

Anabbreviation:

∆(

t

,

s

) =

W

(

t

,

s

) −

W

(

s

,

t

)

.

Oneway to ompute a stubborn set for a non-dead marking M:

1

Pika transition t enabled in M (i.e. M

−

t

→

) and set TM

= {

t

}

.

2

For any transitiont in (and lateradded to) T

M :

If M

−

t

→

Addto TM anytransition that an disable t.

If

¬

M

− →

t Pika plae s

∈ •

t so that M

(

s

) < ∆(

t

,

s

)

.

∆( , ) >

(150)

A

C

D

F

G

t

a

t

b

t

t

d

(151)

Some words about about verifying innitebehaviors.

A

B

C

D t

a

t

b

t

t

d

T

AC

= {

ta

}

and TB

= {

tb

}

implies that

{

t

,

td

}

are neverred.

(152)

1 2 s

r

Client C

1

2 3

r

1

s

1

r

2

s

2

Server S

− ×

a

d

Canal B

Synhronizationrule forsystem

h

C

,

C

,

S

,

B

,

B

,

B

,

B

i

:

(

1

) h

s , . , . , . , . , a , .

i

(

2

) h

. , s , . , . , . , . , a

i

(

3

) h

r , . , . ,d , . , . , .

i

(

4

) h

. , r , . , . , d , . , .

i

(153)

111

− − −−

211

− − ×−

121

− − −×

212

− − −−

221

− − ××

123

− − −−

211

× − −−

222

− − −×

223

− − ×−

121

− × −−

221

× − −×

221

− × ×−

q0

q1 q2

q

3

q

4

q

5

q

6

q7 q8

q

9

q10 q11

(154)

¯

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

¯

d1

¯

d2

¯

r

1

¯

r2 d1d2

¯

r

1

¯

r2

¯

d1

¯

d2

r1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1r2

¯

d

1

¯

d

2

r

1

¯

r2

¯

d1d2

¯

r1 r

2

d1

¯

d2

(155)

¯

r

1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r

1

¯

r2 d1d2

¯

r

1

¯

r2

¯

d1

¯

d2

r1

¯

r2

¯

d

1

¯

d

2

¯

r1

¯

r2

¯

d

1 d

2

¯

r1

¯

r2

d

1

¯

d

2

¯

r1r2

¯

d

1

¯

d

2

r

1

¯

r2

¯

d1d2

¯

r1 r

2

d1

¯

d2

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