HAL Id: inria-00204548

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HAL Id: inria-00204548

https://hal.inria.fr/inria-00204548v2

Submitted on 15 Jan 2008 (v2), last revised 15 Jan 2008 (v3)

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Modular construction of finite and complete prefixes of Petri net unfoldings

Agnes Madalinski, Eric Fabre

To cite this version:

Agnes Madalinski, Eric Fabre. Modular construction of finite and complete prefixes of Petri net

unfoldings. [Research Report] 2007. �inria-00204548v2�

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a p p o r t

d e r e c h e r c h e

0249-6399 ISRN INRIA/RR--????--FR+ENG

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Modular construction

of finite and complete prefixes of Petri net unfoldings

Agnes Madalinski, Eric Fabre

N° ????

January 2008

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Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

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kTVLm8Rqz=hjTs)¡+mn5dHo¢VxdfRTV[hsHU«hJoµebfjTUWUopsH`WkTVdE¶[mka¬hn¬amklhkTn`z=VLRloE¬amnhjTsbhpdfRlV[mnkqdfVsfogV

dfR@o0d|sfVLnopdfVEbdfRlVU&

Q)RlV²bfjTzauvVEgdo8~T~csHVLbHbeVE~ RlVsfV²ghU#zTmnkTVLbUWokq` dfVLgRTkTmygo|o8be}=VEgdHbL'beh8UWVh[¡+RTmngR osfV&kThpd

~cmnsfVEgdH`sfVLnopdfVL~dfh¨dfRTV#}lsfh8zTVLUopk@~²gopk²z=Vgh8klbfmn~cVLsfVE~&obµemkl~TV}=Vkl~TVk8d~cm7gjTªdHmnVLbL¶§Q)RlVsHV±

hsHV@dHh}lsfVEbeVLsf¬VdHRTVigyopsHmd`h>dHRTmnbxlsHbedo0dedHVUW}cdE=dHRTVibeVdedHmklmybx¬hnjTk8dopsHmn`&bemnUW}TnmªlVE~³o0d[mªdb

Uo0©cmnU#jTU&Q)RTVopdedHVk8dHmh8kmyb)sHVLbedfsHmygdfVE~¨dHhWjTkch8n~cmnkTqb)hbfopVr.-@okl~dfRTV~TmnbedfsHmzljcdfVE~bf`cbvdHVU

mybsHVL~cj@gVL~#dHh[dv¡hghUW}=hkTVLk8dbsfVLnopdfVE~#za`opkimnk8dfVLseo8gV®dfRTV|Vkl~ihp@dHRTV+}lo}@VLsbejlVEbvdbJRTh0¡ dHRTmyb

}losHo8~cm8U gLopkWdHRTVk¨z=V¥V©qdHVkl~TVL~l°Q)RTV¥mkqdfVLseo8gV|mybobHbejTUWVE~idfh#z=V¥o#bemnUW}TnV¥ojcdfh8Uo0dHhkWso0dHRTVs

dfR@opk o¤dHsfjTV²r.-#opk@~¦dHRTV²}TsHVT©ghklbedfsHjlgdHmh8ko8bfbfjTUWVLbWUosf¢amnkT§ghUW}TnVdfVLkTVLbHb)okl~bfmnUi}TnV

o~TV qjlo0dHV[h8sH~cVLsHbL

Q)RlV}lop}=VLsimybihsH8okTmnbfVL~ obhnh0¡|bL¦^cVLgdHmh8k OsHVLgLopnnb#dfRlVzlo8bemyg¨dHRTVh8sfVdfmygop)zlog¢asHhjTkl~

gh8klgVLsfkTmnkT²r.-xbJdHRTVmns#jTkch8n~TmkTqb#opk@~¤dHRTV~cVsHm¬po0dHmh8kh+oOgokThkTmygo>lkTmdfVgh8UW}TVdfV¨}lsfVl©5

^aVEgdfmnhk "³~cVlklVLbi~cmybvdHsfmnzTjcdHVL~¦be`cbedfVLUWbiopkl~.sHVLgLopnnbdfRTVo8bfbfhcgmyo0dHVL~¤o8gdHhsHmnbHo0dHmh8k§}TsHh}=VLsedv`¤h8k

dfRlVmnsxjTkch8n~cmnkT@[^aVLgdfmnhk 4ghk8dopmnklb|dfRTVigh8k8dfsHmnzTjcdfmnhkOhdHRTmyb¥}lo}=Vs'md[}TsHVLbeVLk8db+dHRTVUWhc~cjTyops

~cVLsfmn¬0opdfmnhk hpxo³lkTmdfV&okl~¦gh8UW}TVdfV}TsfVT© mk¦ogdfhsHmybeVE~§hsHU>hTgjlbfmkl³lsHbedih8k dfRTV&bfmnUi}TnV

go8beVhpdv¡hgh8UW}=hkTVkqdHbLopkl~²dHRTVk¤}TsHh}=hqbemnkTdHRTViVLkTVsopnm#!Eo0dHmhk&dHhUWhsHVgh8UW}TV©²kTVd ¡h8sf¢cb

hpghUW}=hklVk8db

$ %

'/ !01

k dHRTmnbbfVLgdfmnhk¸zlobfmng³~cV@kTmªdHmh8klbghk@gVsHkTmnkT rJVdHsHmxklVdHbopk@~ klVdjTkchy~cmnkT8bopsHVO}TsHVLbeVLk8dHVL~

Q)RTVEbeVosfVUWomkl`o~To}cdfVE~sfh8U 1=<cR)?)!6

&('*) +-,/.021435,/.76

8²¼ L:9 / kTVd¨mnb¨o qjlo~TsfjT}lV

N = (P, T, →, P ⁰ )

^bejlgR ^dfRlopd

P

^opkl~

T

opsHV&~cmnbuvhmnk8dbeVdHbWh 3% `! opkl~; > !F qsHVLbf}=VLgdfmn¬Vn`

→⊆ (P × T ) ∪ (T × P)

^mnb'o< 5 FlTopkl~

P ⁰ : P →

(8)

N = {0, 1, 2, ...}

mybxo¨U#jTdfmybfVd[h8k

P

sHV}TsHVLbfVk8dHmkldfRTVK@F * &h>dfRTVklVdL¯Th8s[o¨kThc~cV

x ∈ P ∪ T

^amdHb ⁵

^• x

myb'~TVlkTVE~zq`

• x = {y | (y, x) ∈→}

okl~¨mdHb 5

x ^•

myb)~cVlklVL~zq`

x ^• = {y | (x, y) ∈→}

^>kdfRTmyb)¡h8sf¢WdfRlVkTVdb+opsHVxnmUWmdfVE~dfhB ]klVdHbLcmBVhs)V¬8Vsf`sHVLo8gHRlozTnV Uosf¢amnkT

M

opkl~VL¬VsH`}Tyo8gV

p ∈ P

M (p) ⊆ {0, 1}

/ noz=VnVL~kTVd

N = ¡

P, T, →, P ⁰ , λ, Λ ¢

mnbo[kTVdV©adfVk@~cVL~i¡+mªdHRo[noz=VTbfVd

Λ

okl~io[yopz=VnnmkT

jlklgdHmh8k

λ : T → Λ

hkdHsHoklbemdHmh8klb

-À

9

/ UWhsH}TRTmybfU

φ : A → B

^z=Vdv¡VVLk ^kTVdb

x = ¡

P x , T x , → _x , P _x ⁰ ¢

1%)<6 '¡+RTVLsfV

x ∈ {A, B}

mnbo}lopmns

¡ φ ^P , φ ^T ¢

^¡+mdfR

φ ^P

osHVyo0dfmnhk¤hk}TyogVLbokl~

φ ^T

o}losedHmyopJjTklgdfmnhk§hk dHsopklbfmªdHmh8klbE§Q)RTVmnkTmdfmyop'Uosf¢amnkTOmyb}TsHVLbfVsH¬VL~§za`

φ

ob#hnh0¡|b

P _B ⁰ = φ ¡ P _A ⁰ ¢

^opk@~

∀p _B ∈ P _B ⁰ , ∃!p _A ∈ P _A ⁰ : p A φp B

>

φ

mnb|~cVlkTVE~h8k

p A ∈ P A

cdfRlVk&mªd|myb+opybfhW~cVlklVL~hkz=hpdHR

• p A

okl~

p ^• _A φ

}TsHVLbfVsH¬VLbdfRTVVkq¬amnsfh8kTUWVk8dhp|VLo8gHR.dfsopklbfmdfmnhk*

t B = φ (t A )

mnUi}lmnVLb#dfRlopdsfVEbvdHsfmygdHmnhklb

φ ^op : ^• t B → ^• t A

opkl~

φ ^op : t ^• _B → t ^• _A

opsHVxz=hdfRdfhdHop5jTklgdfmnhklbLc¡+RTVLsfV

φ ^op

~cVkThdfVEbdfRlVsfVL¬VsbeVE~

sHVyo0dHmnhkhk}Tyo8gVLbL0-|hdfmygVdfRlopd+kTVd|UWhsH}TRTmybeUb)}TsHVLbfVsf¬8VxsHjTklbL

¯Th8syopz=VnVL~¤kTVdb

x = ¡

P x , T x , → x , P _x ⁰ , λ x , Λ x ¢

dHRTV~cVlkTmdfmnhkh'oUWh8sf}TRlmnbfU

φ : A → B

myb|sfVLmkThsgVL~zq`V©adHsHoWsfVqjTmsHVUWVLk8dHb

Λ B ⊆ Λ A

®dfRTV#noz=VbfVd¥myb+sHVL~TjlgVL~&zq`

φ

^°

Dom (φ T ) = λ ⁻¹ _A (Λ B ) ⊆ T A

®

φ

myb~cVlkTVE~V©co8gdH`xhkdfsopklbfmªdHmh8klbRlo¬amkT|o|bfRlopsHVL~[noz=V¹°okl~mª

φ T (t A ) = t B

dHRTVk

λ A (t A ) = λ B (t B )

®yopz=VF}TsHVLbfVsH¬amkTa°

&'& 0 1 0 , 6 6 , 6

Q¡hklhc~cVLbxhokTVd

N

y

^opkl~

y ⁰

FopsHV#mnk !2 < C~cVkThdfVL~³zq`

y#y ⁰

mtdfRTVsHVV©cmnbed

~Tmnbedfmnklgd|dHsHoklbfmªdHmh8klb

t, t ⁰ ∈ T

bfjlgRdfRlopd

• t ∩ ^• t ⁰ 6= ∅

^lokl~

(t, y)

^opkl~

(t ⁰ , y ⁰ )

osfVmkdHRTVsfVlV©cmn¬V dHsopklbfmªdHm¬Vgnh8bfjTsHV[hpdHRTVlh0¡-sHVyo0dHmh8k

→

l~cVLkThpdHVL~zq`

¹

¨/ ¼ ¼J¼ L:9 /|k !5 ` 3 mybio³kTVd

O = (C, E, →, C ⁰ )

^¡+RTVLsfV

C

mybio¤bfVdWhp

` F> ®}Tno8gVEbH°

E

myb¥dfRTVibfVd[hp;5EJ52 ®¹dHsHoklbemdfmnh8klbH°|opk@~

C ⁰ = c ∈ C : ^• c = ∅

mnb¥dfRlVbfVd hJmkTmdfmyoFgh8kl~cmdfmnhklb+bHo0dHmnbe`amkTidfRTVhnnh¡+mnkTJhs+VL¬VsH`

c ∈ C

| ^• c |≤ 1

hs+V¬VLsf`

y ∈ C ∪ E

¬ (y#y)

opkl~#dfRTVLsfV+opsHVt@kTmªdHVn`Uokq`

y ⁰

bej@gHRdfRlopd

y ⁰ ≺ y

^¡+RTVsHV

≺

~cVkThdfVEbdHRTV 5 5 Fl

dHRTVdHsHoklbemdHm¬VmnsfsHV lV©cm¬VghqbejlsfVh

→

)Q¡h¨kThc~cVLb¥opsHV` !52B@~cVkThdfVL~

y k y ⁰

@mª>kTVLmªdHRTVs

y#y ⁰

^klhs

y ¹ y ⁰

^kThs

y ⁰ ¹ y

! 0

"

"#

À ¼ L ¼ 9 / 5] ` |hpo[kTVdtbf`cbvdHVU

N

mybo[}@opmns

β = (O, f)

^8¡+RTVsHV

UWh8sf}TRlmnbfU

f : O → N

mybo¥dfhdHo8jlklgdHmh8kh8k

O

onbfh[gopnnVL~#o#hp

O

^mnk8dfh

N

JQ)RlmnbJ¹h8y~cmkT

gLopk¨z=V[beVVLkoboyopz=VnmnkT¹jlklgdHmh8khk¨V¬VLk8dHb'opk@~¨ghk@~cmªdHmh8klbthp

O

azq`W¡+RTmygHRgh8kcljlsHopdfmnhklbhp

O

sHV}TsHVLbfVkqdHb)sfjlklb+hp

N

d¥mnb)jTsfdfRTVLs|sHV qjTmnsfVE~¨dHRlo0d

β

bfopdfmybv`oW}lopsbemnUWhka`¨ghkl~cmdfmnhk>hs¥opn

e 1 , e 2 ∈ E

^=mª

^• e 1 = ^• e 2

okl~

f (e 1 ) = f (e 2 )

^dfRTVLk

e 1 = e 2

|Qh~cVlkTVlklmªdHVgh8Ui}TnVdHV}TsHVl©aVEb

md¥¡+mnz=VjlbfVjlFdfh¨ghklbfmn~cVLs|o²®¬amnsfdfjloy°'mnkTmdfmyopV¬VLk8d

⊥

^mnk

β

l¡+RlmngR&Rlo8b|opk&VUW}cdv`}TsHVLbfVdL

C ⁰

o8b)}=h8bede±beVd+okl~kThWyopz=VL®mBV>klhWmUoV[za`

f

^°

/ zTsopklgRTmnkT§}TsHhcgVLbHb²®FS'r°

β ⁰

^hp

N

^mnbo 3¸ho§zTsopklgRTmnkT§}TsHhcgVLbHb

β

+~TVkThdfVL~ zq`

β ⁰ v β

^qmª

O ⁰

mybtogojlbfon`ghqbeVE~WbfjTzc± kTVdth

O

ghkqdHomkTmnkT#opnlmklmªdHmno@ghkl~TmªdHmh8klbtopkl~bfjlgHRdfRlopd

∀e ∈ E, e ∈ O ⁰

mnUi}TnmnVLb

e ^• ⊆ O ⁰

^okl~

f ⁰

mnbtdfRTVxsHVLbedfsHmygdfmnhkh

f

^dHh

C ⁰ ∪ E ⁰

>¯lhsVEogRkTVd)be`cbedfVLU

N

dHRTVsHVV©cmnbedHb+ojTklmqjTV®jT}dHhimybeh8UWhsH}TRTmybeU¨°tUWop©cmUo>®¡sLd

v

°'zTsHoklgHRlmkTi}TsHhcgVLbHb'~cVkThdfVE~

za`

(Unf(N ), f N )

hsxh8sbeRlhsfd

(Unf N , f N )

FgLopnVE~²dfRTVBOh

N

^k³dfRTVbeV qjTVLC¡+RTVLk

h8n~TmkTqb+opsHV[jTk@opU#zTmnjTh8jlb'dfRTVL`¡+mnFz=VhUWmªdfdfVE~lbfh

β

¡+mnFz=Vmn~TVk8dHmªlVE~¡+mdfR

O

$

mn¬Vk³oghkTljTso0dHmh8k

κ

hpo¨zTsHoklgRTmkl¨}TsHhcgVEbfb

β

^h

N

²

mybxgLopnVE~Oo ! % h

κ

^mk

β

^m

dHRTVsHVV©amybedHb+bfhUWVgh8kcljTso0dHmnhk

κ ⁰ A κ

bfjlgHRdfRlopd

² = κ ⁰ \ κ

&('*)+-,/.102431.

( .576 8

φ

.1,:9-+;.5/5/<

W .W ,/5/<

$>=

)>?

φ ^P

)+

φ ^T

^@

(9)

H ? 5

À # 0c

À . ÁL:9 / `A F¸hpozTsHoklgRTmkl}TsfhTgVLbHb

β

mybo¤lkTmdfVObfVdh V¬VLk8db

κ ⊆ E

bfjlgHRdfR@o0dhsopn

e, e ⁰ ∈ κ

¬(e#e ⁰ )

opkl~hs¨V¬VLsf`

e ∈ κ

e ⁰ ≺ e

mnUW}TnmVEb

e ⁰ ∈ κ

^mnk o~l~cmªdHmh8kmªd¨mybWsHV qjTmnsfVE~.dHRlo0d

⊥∈ κ

¯lhsV¬VLsf` V¬VLk8d

e ∈ E

dfRTVOgh8kcljlsHopdfmnhk

[e]

= {e ⁰ |e ⁰ ¹ e}

myb)gLopnVE~¨dfRTV J ! `A ] F[h

e

^lopkl~

hei

= [e] \ {e}

~cVLkThpdHVLbdHRTVbfVd+hp

` 5 73` ` 5! |Q)RTV#beVd¥hpopnlkTmdfV®sHVLbf}Ftz@obfmngE°)gh8kcljTso0dHmh8klb)hp>oWzTsopklgRTmnkTW}lsfhcgVLbHb

β

mnb|~cVklhpdfVE~za`

κ ^β

fin

®sfVEbe}F

κ ^β

bas

°lokl~dfRlVbfjT}=VLsHbHgsHm}Td

β

mnb|~csHh}T}=VE~¡+RlVk

β = Unf _N

/

myb)obfVd)hpJgh8kl~cmdfmnhk

C ⁰

bejlgRdfRlopd'hs+opn~cmybvdHmk@gd

c, c ⁰ ∈ C ⁰

c k c ⁰

Topkl~o-tmyb)o Uo0©cmnUWo=ghp±beVdthsdfRTVbeVd'mk@gnjlbemnhk Vd

κ

z=Vo#ghkcl8jTsHopdfmnhkWdHRTVk

Cut(κ)

= ¡

C ⁰ ∪ κ ^• ¢

\ ^• κ

myboOgjTd jTsfdfRTVLsfUWhsHVdHRTVbfVdh+}TyogVLb

f (Cut(κ))

mnboOsfVLo8gHR@opzTnVUopsH¢amkl²hp

N

>¡+RTmngR myb

~cVLkThpdHVL~³za`

Mark (κ)

^*/ UWosf¢amnkT

M

^hp

N

^myb* ^{! 552} ^³mnk

β

mdfRTVLsfVWmnbogh8kcl8jTsHopdfmnhk

κ

hp

β

bfjlgR&dfRlopd

M = M ark(κ)

9t¬VsH`UWosf¢amnkTsfVL}TsfVEbeVLk8dHVL~mk

β

mnb¥sfVEogRTVL~mk

N

=opkl~&V¬VLsf`

sHVLogRlozTV[UopsH¢amklihp

N

mnb+sfVL}TsHVLbeVLk8dHVL~mnk

Unf _N

&(' 1 1. , ,/., 07,, 6

/|k¨opzlbedfsogd>}lopsopUWVdfsHmng'UihT~cVTRlo8b>z=VVkmkqdfsHhc~cjlgVE~mnk 1%)LT )?)!6TdHhgh8}=V)¡+mdfR¨~cm5VsHVkqdt¬0opsHmyopk8db

hp@dfsHjTklgopdfmnkT[jTkch8n~TmkTqbJ d>jlbfVLbJ}@opsopUWVdHVsHb¡+RTmygRi~TVdfVLsfUWmnkTV'dfRTV)mnkch8sfUo0dHmh8k#mnk8dHVkl~cVE~#dHh[z=V

}TsHVLbfVsH¬VL~imnkdfRTVxghUW}TnVdHV|}lsfVl©&®mk¨dfRTV[bvdopkl~TosH~gobfVdHRTmnbmnbdHRTVxbfVdhpFsfVEogRlopzTnV+UopsH¢amkl8b°

opk@~be}=VEgm`dHRTVgmnsgjTUbedHopk@gVLb)jTkl~cVLs+¡+RTmygR&okV¬Vkqd|gLopkz=V~cVLbfm8klo0dHVL~ob+oWgjcdf±BhOVL¬VkqdL

¼

À ´gjcdfdfmnkT¨ghkqdfV©qd

Unf _N

^!

Θ = ¡

≈, C , { κ _e } _e∈E ¢

^>

≈

^{Q 5EJ} ²⁵^FG

κ _fin

!

C

^{` %} .o~cV8j@o0dfV 5

2 ! .

` F 0]?*

κ _fin

] @

⊂

κ ⁰ ⊂ κ ⁰⁰

³^F!

κ ⁰ C κ ⁰⁰

"

≈

³

C

}TsfVEbeVLsf¬VE~ zq` @kTmªdHVV©adfVk@bemnhklb

!GE5; G] ` F

κ ⁰ ≈ κ ⁰⁰

T 5E ` %Q

² ⁰

κ ⁰

? 52? > ! %

² ⁰⁰

κ ⁰⁰

^`2-?

#

%$

κ ⁰⁰ ⊕ ² ⁰⁰ ≈ κ ⁰ ⊕ ² ⁰

3

# &$L

κ ⁰⁰ C κ ⁰

^? ⁵

κ ⁰⁰ ⊕ ² ⁰⁰ C κ ⁰ ⊕ ² ⁰

'

{ κ _e } _e∈E

^N^B ^{U !2! 5} ^F

κ

fin

Q)RTVio~cV qjlopdfV#hs~cVsxbf}=VLgmªlVEb¥¡+RTmygHR¤ghkTljTso0dHmh8klb|osfV}TsHVLbfVsH¬VL~&mkOdfRlVghUW}TnVdfV}TsfVT© o

C

± UWmklmUopghkTljTso0dHmh8klb¥mnk¤VLo8gHR¤V qjTmn¬0oVLk8dgyobHbxhp

≈

osfVi}TsfVEbeVLsf¬VE~Q)RTVWno8bvd}losHoUiVdfVLs

κ _e

myb¥kTVVE~cVL~&dHhbf}=VLgmª`dfRTVibeVdxhptgh8kcljTso0dHmnhklb+jlbfVL~²yo0dfVLs|dHh~TVLgmy~cV#¡+RTVdfRTVLs[opkOV¬VLk8d[gok

z=V~cVEbemnkTVE~ob|oWgjTde± h ³V¬8Vk8dEtQ)RTVgjcdedHmkTghkqdfV©qd

Θ ERV = ©

≈ mar , C _tot , { κ _e = κ _bas } _e∈E ª

gh8sfsHVLbf}=hkl~Tb[dfh&dHRTVsopUWV¡h8sf¢Omnk 1<J6 ¡+RlVsHV

≈ mar

mybdHRTV¨V qjTmn¬0oVLklgVsfVLnopdfmnhkh8ksHVLo8gHR@opzTnV

UopsH¢amkT#hp

N

^amC^V

κ ⁰ ≈ mar κ ⁰⁰

^m

Mark (κ ⁰ ) = Mark (κ ⁰⁰ )

^copkl~

C _tot

mnb'odHhpdop5o~cVqjlo0dHV|h8sH~TVsE

Q)RlVObedHopdfmyg²gjcdf±Bh-VL¬Vk8db¨osfVO~cVlkTVE~ mkl~TV}=Vkl~TVk8dH`hopk jTkch8n~cmnkT¦opnh8sfmdfRTU&'¡+RTVsHV

VLo8bemnzTnVWV¬VLk8dHbopsHViV¬VLk8db¡+RThqbeVgojlbfo}TsHVL~cVEgVLbHbfhsb[opsHVikThpd#bvdo0dHmngWgjcde± h ¦V¬VLk8dbopk@~³dfRqj@b

opsHV[mnklgnjl~cVE~mkdfRTV}lsfVT©~cVdfVsHUWmklVL~zq`¨dfRThqbeVgjcde± h ³VL¬Vk8db

¼

À )(+* VEobfmzTnVV¬VLk8db

53

fsble ^Θ

? B 5 ]¸bvdo0dfmyggjTde± h

V¬VLk8db

?

cut ^Θ

.2 5 F U E F U

Unf _N

^?F.3`N32 ^EJ5

>00

. ,

;D5EJ52

e

^`^55EJ528

hei ∩ cut ^Θ = ∅

-/.10 <<32 24+;<,/,.1)

W546/7894:736<;/=?><@A 8<BDCE6<;

.1,F4,< = ' 6 ,/<3G <(+

$ 3 $ F 5 04)+;,

@IH

) 9:<3G <(+(8J04<(+/<"5K0 <"5<(+;0ML31)

(]$

3L.1,

$ 0 ' .ON<FP

)<F4,84,.W (

<.5-+;<?<(+;,-5)

$8(

) 02 ) W <W 5 8*,/) .15 .1, +/<(243

$`(

<

=U' 6RQ

'J$

,.( @L

(10)

!

LE

e

^* ^5E ^{` J !} ⁵

K4>* *

#

5-` % ghsHsfV±

be}=hk@~cmkl $ A F

κ ∈ κ _e

`2 >

κ ⊆ fsble ^Θ \ cut ^Θ

κ ≈ [e]

κ C [e]

^#^;

E

e ⁰

^]5! B J |gh8sfsHVLbf}=hkl~cmnkTVL¬Vk8dN

κ = [e ⁰ ]

^$

* > 3 `! `

Pref _N ^Θ

^2` ⁴ ^? ^T ^]T5EJ52

fsble _N ^Θ

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HAL Id: inria-00204548