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HAL Id: inria-00204548

https://hal.inria.fr/inria-00204548v3

Submitted on 15 Jan 2008

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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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] RR-6412, INRIA. 2007. �inria-00204548v3�

(2)

a p p o r t

d e r e c h e r c h e

0249-6399 ISRN INRIA/RR--6412--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° 6412

January 2008

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Unité de recherche INRIA Rennes

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

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dHRlo0d¨VklbfjTsHVLb

φ A ◦ f A = f C ◦ φ O A

—amBŸVƒŸJdHRlo0d)Uop¥ƒVEbdHRTV™~cmyopŽ8sHoƒU¦gh8UiU#jTdHo0dHm›¯ƒV8Ÿ.H`«bf`aUiUWVdHsH`ƒ—qh8kTV[oƒšnbfh#Rlo8b'o UWhƒsH}TRTmybeU

φ O B : O B → Unf C

Ÿ $ mn¯ƒVk&dHRTVLbfVUWhƒsH}TRTmybfUWb+dfh

Unf C

—ldfRTV #R hp

O A

oƒkl~

O B

mnkdHRTVgopdfVŽ8hƒsH`WhƒJhcggjTsfsHVklgVxklVdHb+myb+~TVsHm›¯ƒVE~«žsHhƒU/dfRTVh8sH~cmnkloƒsf`}TjTšnšnzloƒg¥«hƒJkTVdb+zq`

O A × C O O B = Unf ³

O A × Unf N C O B

´

± ‚q³

Zµh8sfVLh†¯ƒVLsL—am›d|myb+VEoƒbfm›šn`‰gRTVLg¥ƒVL~dHRlo0d|dfRTVsHVV¬cmybvdb|ožh8šn~cmnkTŽ

f

žsHhƒU

O A × C O O B

dfR@o0d|Uoƒ¥ƒVLb)m›dxo zTsopklgRTmnkTŽi}lsfhcgVLbHb'hp

A × C N B

Ÿ,+|hƒdfmygV[dHRlo0d#± ‚q³'VkqdHoƒm›šyb'dHRTVV¬cmybedfVk@gV™hp>oisfVEgjTsbfm›¯ƒV[}TsHhcgVE~cjTsHV dfhgh8Ui}ljcdfV¶dfRTV§}ljTš›šnzlo8gH¥ hpzTsopk@gHRTmnkTŽ©}TsfhTgVLbHbfVLbL—)zlo8beVL~ h8k dfRTV§sHVLgjTsHbfmn¯ƒVOghƒklbedfsHjlgœdHmnhƒk hƒ

jTkcžh8šn~Tm›kTŽqb+oƒkl~hƒkžh8sfU#jTšyopV±v„ƒ—…ƒ³œŸ

“ ÃE¿ •ƒ’

à ‘:9 $ mn¯ƒVk

Unf N = Unf A × C O Unf B

—beh8UWVkThc~cVEbxhƒ

Unf N

opsHVšnoƒz=Všnš›VE~Ozq`O}TšyoƒgVEb opk@~dHsHoƒklbem›dHm›h8klb>hp

C

—ƒdHRTsHhƒjTŽ8Ropkq`hƒ=dHRTV± gh8UWU#jcdfmnkTŽq³>}lopdfRlbsHVšyo0dHm›klŽ

Unf N

dfh

C

mnk«²Jm›Ž8jTsHVI"TŸ

Q)RTV0 ! =&hƒ

Unf N

dfh#dfRlVLbfVxkThc~cVEbtmyb'~cVLkThpdHVL~«za`

(Unf N ) |C

Ÿ.H'`gh8k8dfsoƒbedL—8dfRlV U hp

Unf N

hƒkz=VLRloE¯amnhƒjTsb'hp

C

mnb|~cV˜@kTVL~&oƒb'dHRTVm›UoƒŽƒV™hp

Unf N

m›k

Unf C

Π C (Unf N ) = φ O A ◦ ψ A O (Unf N ) = φ O B ◦ ψ B O (Unf N ) .

±7ƒ³

(12)

f

f C f A

f B

ψ A O ψ B O ψ A ψ B

φ B

φ A

φ O A φ O B

C

A B

U nf C

O A O B

O N = O A × U nf O C O B A × C N B

²JmnŽƒjTsHV6"XhƒUWU#jcdo0dHm›¯ƒVx~TmnoƒŽƒsopU hpdHRTVx}TjTšnš›zlo8gH¥cbthƒFzTsopklgRTm›klŽ}TsHhcgVEbfbfVLbtopkl~klVdHbL—aoƒkl~WdfRlVmns

sHVšyo0dHmnhƒkzq`o8bfbfhcgmyo0dHVL~«¼h8šy~cm›kTŽqbŸ

-¨š›dfVsHklopdfmn¯ƒVšn`ƒ—8dfRTmybt}TsHhpuvVLgdfmnhƒk«gLopkz=VxhƒzcdopmnkTVL~zq`idHoƒ¥qmnkTŽ˜@sHbed

(Unf N ) |C

opkl~WdfRlVk«}=VLsežh8sfUWmnkTŽ

o ! 0

mCŸVƒŸzq`µUWVsHŽƒmnkTŽ‰mybfhƒUWhƒsH}TRTmyg#gh8kc˜lŽƒjlsHopdfmnhƒklb¨m›k

(Unf N ) |C

mnk¶h8sH~TVs¨dHhŽ8Vd™o‰¯popšnmn~

zlsHoƒklgHRlm›kTŽ#}lsfhcgVLbHbhƒ

C

ŸrsHh0uvVEgœdHm›h8klbhƒk

A

opkl~

B

hƒ

Unf N

opsHVx~cV˜lklVL~bemnUWm›šyopsHš›`oƒbm­dbmnUopŽƒVEb za`

ψ A O

opkl~

ψ O B

sHVLbf}=VLgdfmn¯ƒVšn`ƒŸrsHhpuvVLgœdHm›h8klb)klopdfjTsopšnš›`«V¬qdHVkl~dfh«opkq` H'rhƒ

N

Ÿ

+|hƒdfmygVdfRlopd¨dfRTV#sHVLbedfsHmygœdfmnhƒk&hƒ

Unf N

dfh‰o‰bejTz@beVdxhƒ>kThc~cVEb|UoE`VsoƒbfVgopj@bfoƒš›m›dv`‰h8sxghƒklmngd sHVšyo0dHmnhƒklb|z=Vd¤'VVkOdHRTVLbfV#kThT~cVLbL—=¤+RTmygHR¶UoE`dfRlVk§op}T}=VEops[oƒbxgh8klgjTsHsHVk8dxm›k

(Unf N ) |C

¤+RTVLsfVEoƒb

dHRTV`W¤'VsHV|klhpdmnk

Unf N

Ÿ>Q)RTVxbHopUWV¨}TRTVklhƒUWVkTh8k«hcggjTsHbto8bt¤Všnš=¤+m­dHR«}lsfhpuvVLgdfmnhƒklbLŸQ)RTVsHVžh8sfV¨o }lsfhpuvVLgdfmnhƒk

Π C (Unf N )

mnbxbfoƒmn~dfh«z=V 0 !R ©m¢VL¯ƒVsH`ghƒkc˜lŽ8jTso0dfmnhƒk

κ 0

mnk

Π C (Unf N )

myb'dfRlV™mnUWoƒŽƒV[hƒJoWghƒkc˜@ŽƒjTso0dHm›h8k«hƒ

κ

mnk

Unf N

—lopkl~goƒjlbfoƒš›m›dv`sHVšyo0dfmnhƒk@bh8kV¯ƒVLk8db)hp

κ 0

oƒsfV[kThƒd

šnhqbvdEŸYxzlbfVsH¯ƒVdHRlo0d}lsfhpuvVLgdfmnhƒklbhƒk¢mnk8dHVsf oƒgVEb

C

dHRlo0d#oƒsfVopjTdfhƒUopdHoT—mBŸV8Ÿ«rJVdfsHmkTVdHb¤+m›dfRlhƒjcd ghƒklgjTsHsfVLklg`8—aoƒsfV™zq`~cV˜lkTm›dfmnhƒk&kThƒkUWmybešnVLo8~cmnkTŽlŸ

Q)RTV« oƒgœd#dfRlopd#dfRTVbfm›UW}TšnV«}TsHhpuvVLgdfmnhƒklb#~cV˜@kTVL~ opz=h0¯ƒVUoE`¢z=V«UWmnbfšnVLoƒ~Tm›kTŽOmybdfRlV«VEbfbfVkqdfmyopš

sHVLo8beh8k§¤+Rq`¶dfRTmyb}lop}=Vsšnm›UWm›dHbm­dbbHgh8}=VidfhObfjlgR.bemnUW}Tš›V«m›kqdfVsf oƒgVLboƒbopjcdHhƒUo0doTŸ‰²Thƒs#bemnUWm›šyoƒs

sHVLo8beh8klbL— /n„1mnk8dHsfhT~cjlgVE~> ! <9 !2 @—5¤+RTmnš›V:/8?1}lsfh8}@hqbeVE~dfh¤h8sf¥¤+m›dHR 2 R

; R! R —|m›k hƒs~cVLsidHh.}=VsfžhƒsHU½UWhc~cjTšyops‰ghƒUW}TjTdHo0dHm›h8klbLŸ»¡kqdfsHhc~cjlgm›kTŽ.RTVLsfV&dHRTVLbfV

dHVLgRTkTmygoƒšasHV˜@kTVUWVkqdHb¤h8jTšn~igVsfdopmnkTš›`z=V)}=hqbfbfmnzTš›V8—0zTjcd¤h8jTšn~igh8klbemy~cVsopzlš›`š›hqoƒ~~cV¯ƒVLš›h8}TUWVk8db

¤+m›dHR«dfVEgHRlkTmngLopš5~TVdHoƒm›šybtdfR@o0d)oƒsfV¨kThƒd'sHVLopšnšn`Wo0ddHRTV[gVkqdfVs'hƒdHRTV™bvdHjl~c`8Ÿ¡£d'mybtdfRlVsHVžhƒsHVxgRTh8bfVkdHh

žhTgjlbh8k«dfRTV™bf}=VLgm­˜@g[~cm2«gjTš­dHm›VEb'sfVLšnopdfVE~WdfhihƒzcdopmnkTmnkTŽ˜@kTm­dHV™gh8Ui}lš›VdfVx}lsfV˜l¬aVEb¤+m­dHRo#UWhc~cjlšnoƒs

oƒ}T}TsHh8o8gHRF—@opd|dHRTV#V¬a}=VLklbfVhptošnm›UWm›dfVE~ObfVdedHm›klŽ«¤+RlVsHVm›kqdfVLse o8gVkTVdHb[opsHVkThƒdxŽƒVklVsopšFklVdHbxzTjcd

bfmnUW}Tš›VopjcdHhƒUopdHoTŸ

-

.”.

.. J”l•µ• Ã=¿ “

"# 9 rtsfhpuvVEgœdfmnhƒk@bJopšnšnh†¤ jlbdfh[zTjTmnšn~dHRTVBP #R R<!

hƒ

Unf N

—=¤+RTmygHROsfVEbvdHsfmygœdb)dHRTV oƒgdfh8sHb

Unf A

opkl~

Unf B

dfhdHRTV#z=VRlo†¯amnhƒjTsb+hpghƒUW}=hƒkTVLk8dHb

A, B

dHRlo0d|sHVUopmnk¼VEoƒbfm›zlš›V™mnkdfRTVVLk8dfmnsHVbf`abedfVLU

N

ŸYxkTVgoƒk¤+sHm›dfV

Unf N = Π A (Unf N ) × C O Π B (Unf N )

± q³

oƒkl~OVEoƒgR 0

Π x (Unf N )

—¤+RTVsHV

x ∈ {A, B}

—myb[o}TsHV˜T¬Ohp

Unf x

Ÿ#Q)RTVEbeV oƒgœdHhƒsb

oƒsfV[UWmnkTm›UoƒšmnkdfRTVbeVLklbeV[dfRlopd)dop¥am›klŽWopkq`‰bedfsHmygœd+}TsHV˜T¬‰hpdfRTVLU°¤hƒjlšn~}TsHV¯ƒVkqd)sHVLgh†¯ƒVLsfmnkTŽ#dfRTV

žjlš›š

Unf N

za`‰}TjTšnš›zlo8gH¥5Ÿ

ZµmnkTm›Uoƒš5 oƒgdfhƒsb)goƒkz=Vgh8Ui}ljcdfVE~m›k&oiUihT~cjTšyops)UopkTkTVLs)¤+m›dfRTh8jcd|ghƒUW}TjcdHm›klŽ

Unf N

m›dHbfVš›vŸ

Y[kTV™Rloƒb

Π A (Unf N ) = Unf A × C O Π C (Unf B )

±?ƒ³

±Copkl~ bf`aUiUWVdfsHmngLopšnš›`.žh8s

Π B (Unf N )

³ /T—?1£Ÿ-Q)RTmnbsHVšyo0dHm›h8k V¬a}lsfVLbHbfVLbdHRlo0d«¥akTh0¤+mnkTŽ¢dfRTVµz=V´

R@oE¯amnhƒjTsbhƒ

B

hƒk«dfRlVxmnk8dHVsfžo8gVxkTVd

C

myb'bfj2«gmnVkqddHh

A

dHhi~cVdHVsHUimnkTVx¤+RlmngR‰hpm›dHb'sfjlklbsfVLUWoƒm›k }=hqbfbfmnzTš›V[mnk‰dHRTV™Žƒšnhƒz@opš5bf`cbvdHVU

N

Ÿ|±?p³RTh8šn~lb—Topkl~«dHRqjlb|opšnš›h0¤|bh8kTVxdfhW}=VsfžhƒsHU/oUWhc~cjTšyops+ghƒUi´

}ljcdHopdfmnhƒkhƒJUWm›kTmnUopš o8gœdfh8sHbL—co8b+behchƒkoƒb)}TsHhpuvVLgdfmnhƒklb)hƒk

C

opsHV[kTh8kUimybfš›VEoƒ~cmnkTŽlŸ>¡£d+myb+}TsfVEgmybeVLš›`

(13)

„LŠ C

dfh™h8zcdHoƒm›kWo[bfm›UWmnšnoƒs}Tsfh8}=Vsfd `™žh8s>oƒkq`mnk8dfVLse o8gV)kTVd

C

dfRlopd 2R R J¤VsHV

~cVL¯ƒVšnhƒ}=VL~mnk /?1B—T¤+RTmnšnV /n„1šnopdfVs+}TsHhƒ}=hqbeVE~dHRTVopš›dfVLsfk@o0dfV[dfVEgHRTklmqjTVhp !R < !2 œŸ

²lhƒsUih8sfV‰ghƒUW}TšnV¬.~cmybvdHsfmnzTjcdHVL~ bf`cbvdHVUb—JdHoƒ¥am›kTŽµdHRTVbfRloƒ}@V‰hp¨oµkTVd ¤h8sf¥¢hp¨ghƒUW}=h8kTVkqdHbL—

UWm›klm›Uopšl oƒgdfhƒsbgoƒkWz=V|h8zcdHoƒm›kTVE~iza`V¬adHVkl~cmnkTŽdHRTV¨oƒz=h†¯ƒV+}TsHm›k@gmn}Tš›V|hƒ5UWhc~cjlšnoƒstghƒUW}TjTdHo0dHm›h8klbLŸ

¡£ddfRTVLk™dop¥ƒVEb5dHRTVžhƒsHU hpTo+UWVLbHbfoƒŽƒV}@oƒbHbemnkTŽ|opšnŽƒh8sfm›dfRlU—E¤+RTmngRsHjTklbFh8k[dHRTVtm›kqdfVsoƒgdfmnhƒkbvdHsfjlgdfjTsHV

hp

N

opkl~}Tsfh8ŽƒsHVLbfbfm›¯ƒVšn`|jl}5~To0dHVLbFmnkcžh8sfUo0dHm›h8k™hƒkmnk8dHVsf oƒgVEbŸQ)RTVopšnŽƒh8sfm›dfRlU goƒk™z=Vt}TsHh†¯ƒVE~[V¬Toƒgd

žhƒs#bf`cbvdHVUbš›mn¯amnkTŽµhƒk.odfsHVVƒ—Joƒkl~¢mybhƒklš›`§oƒ}T}TsHhE¬cmnUo0dfVWmnkªhƒdfRTVLsgLoƒbfVLbLŸ A¢VsHVžVLs™dHRTV«sfVEoƒ~cVLs

dfh/T— ?1Fžhƒs|~cVdHoƒm›šybŸ

W º' ' $)‰*«+

Q)RTV+hƒzcuvVLgœdHm›¯8V)hp@dHRTmyb>}lop}=Vs>mybJdHhgh8UW}TjcdfV)˜lkTm›dfV|oƒkl~igh8UW}Tš›VdfV+}TsfV˜T¬cVLb+± ²JXr³hƒ5o™~cmybvdHsfmnzTjcdHVL~

bf`abedfVLU

N

mnk  oƒgdfhƒsHmybeVE~©žh8sfU&Ÿ Qhªžhcgjlb«hƒk dfRTV¶VLbHbeVLk8dHmnoƒš¨~cm2«gjTš›dfmnVLb‰hƒ[dHRTmnb‰}TsHhƒzTšnVU&—'dHRTV

~cmybfgjlbHbemnhƒk mnbi˜lsbedWšnm›UWm›dfVE~©dfh¢dfRTVOgoƒbfVhp¨dv¤h.gh8Ui}=h8kTVkqdHb

A

opkl~

B

mnk8dHVsoƒgdfmnkTŽ§dfRTsHhƒjTŽ8Ro gh8UWUih8kOm›kqdfVsf oƒgV

C

—5¤+RTmygHR¶Vk@bejTsHVLb

N = A × C N B

o8b¨m­d[¤)oƒbxbfVVk§opz=h0¯ƒVƒŸ6-d[dfRlVVkl~Ohp>dHRTmyb bfVLgœdHm›h8kF—ERTh0¤V¯ƒVLsL—dHRTVŽ8VkTVLsHoƒš›mybfo0dHmnhƒk¨dHh|Uih8sfVghƒUW}TšnV¬W±¼dHsfVLV´ beRloƒ}=VL~l³be`cbedfVLUWbF¤+mnš›š8z=VV¬TopUWmnkTVL~Ÿ

¡£d[myb¨žjTsedHRTVsHUWhƒsHV#o8bfbfjTUWVL~&dHRlo0dxdHRTVmnk8dfVLse o8gV

C

myb[opk¶oƒjcdfh8UWopdfh8kF—=kThpd™o«Ž8VkTVLsHoƒšbfopžV#kTVdL—5mnk hƒs~cVLs'dfho†¯ƒhƒmy~«dHRTVV¬adfso#dHVLgRTkTmygopš~cm2«gjTš›d`hƒ~cVLopšnmnkTŽW¤+m­dHRUWmybfš›VEoƒ~cmnkTŽW}TsfhpuvVEgœdfmnhƒk@bŸ

-·dfsHmn¯qmyopšbeh8š›jcdHmnhƒkdfh«h8zcdHoƒm›kOopkµ²JXr»hƒ

N

mnk& oƒgdfhƒsHmybeVE~‰žhƒsHUº¤h8jTšn~&z=VdHh«bedHoƒsed|žsfh8Ubfo†`

oiŽƒšnhƒz@opš²JXr—

Pref N

—loƒkl~‰dfh}TsfhpuvVEgœd+m›d|hƒkdHRTVghƒUW}=h8kTVkqdHb+oƒb'žh8š›šnh0¤|b

Pref N v Π A (Pref N ) × C O Π B (Pref N )

±C‹8³

|h0¤VL¯ƒVsE—LdfRTmybmnUi}=hqbeVEbFdfh¨¤'hƒsH¥¨˜lsbvdJhƒkdfRTVŽ8š›h8zlopšqbe`cbedfVU&—†¤+RlmngR#ghƒjTšy~z=VtV¬adfsHVUWVšn`[V¬c}=Vk@bemn¯ƒV

m›dHRTV™bf`cbedfVU/mnb)šnoƒsfŽ8VƒŸJ¡k}loƒsedHmngjTšnoƒsL—qdfRTmyb)}TsHV¯ƒVkqdHbdHoƒ¥am›kTŽWoƒ~c¯popkqdHopŽ8V¨hpdHRTVgh8Ui}lsfVLbHbfm›h8k«Ž8oƒm›k

}TsHh†¯amy~cVL~«zq`W oƒgdfhƒsHmybeVE~ižh8sfUbL—aoƒkl~dfRTmyb'}TsfVL¯ƒVkqdHb'oƒb¤VLš›š5}TsHhcgVEbfbfm›klŽdfRTV™bf`cbvdHVU¦za`}lopsfdHbLŸQ)RTV

hƒzcuvVLgdfmn¯ƒV™hƒdfRTmyb|bfVLgœdHm›h8kmnb)dfhWzljTm›šy~>§dfRTV™ o8gœdHhƒsb'hpoW²JXr hp

Unf N

Ÿ

Qhµh8zcdHoƒm›kªdfRTVLU—m›d#mnbkThƒd#VkTh8jTŽƒR¢dfh¶bfm›UW}Tšn`§zTjlm›šy~.²JXr hp'dHRTV‰ghƒUW}=hƒkTVLk8dHb mnk.ŽƒVklVsopšB—

dfRlVmns+ghƒU#zTmnklo0dHm›h8kzq`«}TjTšnš›z@oƒg¥¤'hƒjTšy~‰kThƒd)z=VghƒUW}TšnVdHVxžh8s'dfRTV™Ž8šnhƒzlopšbe`cbedfVLUŸQ)RqjlbL—adfRTV™my~cVEo

mybidfhªzTjlm›šy~²JXr¦hƒ[ghƒUW}=hƒkTVLk8dbdHRlo0d«¤+m›šnš¨opšybeh¢}TsHh0¯qmy~cV&bfj2«gmnVkqdfšn` sHmygHRz=VRlo†¯am›h8jTsbh8k©dHRTV

mnk8dfVLse o8gVim›k§h8sH~cVLs¨dHhVLklbejTsHVŽ8š›h8zlopšgh8Ui}lš›VdfVklVLbfbLŸQ)RTmyb™~cVLUWoƒkl~Tb[hp'gh8jTsHbfVoƒk§opŽ8sfVLVUWVkqdxhƒ

dfRlVdv¤h™ghƒUW}=h8kTVk8dbopz=hƒjcd¤+RlopdJz=VLRlo†¯qmnhƒjlsHbdfRTV)m›kqdfVsf oƒgV'bfRThƒjlšn~#}TsHh†¯amy~cVƒ—popkl~bfh[opkV¬TgHR@opkTŽ8V

hpJmnkcžh8sfUo0dHm›h8kz=Vd ¤VLVkµgh8Ui}=h8kTVkqdHbLŸ

- bedfsopmnŽƒRqdežhƒsH¤'opsH~‰¤)oE`«dHhm›UW}TšnVUWVkqd|dHRTmnb|my~cVLoWmyb)dHh«bvdopsfd+¤+m›dHR&šnhcgoƒš›šn`‰ghƒUW}TšnVdHV}TsfV˜T¬cVLbL—

opk@~dfRTVLk&dHhdfsopk@beUWm›d± opk@~&sHVLgVLm›¯ƒV†³dHRTV#z=VRlo†¯am›h8jTsb)sHV qjTmnsHVL~hƒk&dHRTV#m›kqdfVsf oƒgVƒ—@z=Vžh8sfVsfVEgh8U´

}TjcdHm›klŽ&š›hTgopš>²JXr ¤+m­dHR§dfRlVLbfViV¬adfso‰sHV qjTmnsHVUWVkqdHbL—VdgpŸ¨h†¤'V¯ƒVsE—5dHRTmnbUWm›Ž8R8dsHVLbfjTš›d™mnk¢UWoƒkq`

m›dfVso0dHmnhƒklb'z=Vdv¤VVLk‰dHRTVgh8UW}@h8kTVkqdHbLŸtXhƒklbfmy~cVs'dfRTV™V¬TopUW}TšnV[m›k&²Jm›Ž8jTsHVx‚l—c¤+RlmngR~TV}Tmygœdb

Pref A

opk@~

Pref B

—pdv¤h™šnhcgoƒšT²JXr—p¤+m›dfRo™bfmnUi}lš›V+mnk8dfVLse o8gV)h8kTš›`gh8klbemybedfmnkTŽ[hƒ@dfRlV)dfsopklbfm­dHmnhƒk

t 1

oƒkl~#dHRTV

}TšyoƒgV

c 0

Ÿ

Pref A

kTVVE~TbhƒkTV|hcgLgjTsHsfVLklgV)hƒ=dHRTV|mnk8dHVsfžo8gV+dfsopklbfm›dfmnhƒk

t 1

ŸEH'`igh8k8dfsoƒbedL—

Pref B

kTVLVL~Tb

dv¤hihcgLgjTsHsfVLklgVEbthp

t 1

dfhWz=V™ghƒUW}TšnVdfV8Ÿ>Q)Rqj@b—

Pref A

Rlo8bdfhiz=V™V¬adfVLkl~cVL~zq`«h8kTV[hcggjlsfsHVklgV¨hƒ

t 1

dfRTmybkTVL¤©z=VRlo†¯am›h8jTsJh8kdHRTV)mnk8dfVLse o8gV'Rlo8bdfh™z@V)}TsHhƒ}loƒŽ8o0dHVL~dfh

Pref B

ŸJ¡kio8~T~cm›dfmnhƒkF—ph8kTV'kTVLVL~Tb dfhWgRTVLg¥Wžhƒs'Ž8š›h8zlopš5gh8Ui}TšnVdHVkTVEbfb'zq`«ghƒklbfmn~TVsHm›kTŽŽƒšnhƒzloƒš@UopsH¥am›kTŽqb—aopk@~dfRTmyb'UWmnŽƒRqd)sfV8jlm›sHV|dHRTV

V¬adHVklbfm›h8khpdHRTV™}TsfV˜T¬cVLb'hpdfRTVEbeVgh8Ui}=h8kTVkqdHb'jTkqdfmnšFŽƒšnhƒzloƒš@dfsHjTklgLo0dfmnhƒk}=h8m›kqdHb+opsHVxsHVLoƒgRTVL~FŸJ¡k

dfRlVWVkl~—FdfRTmyb™UWm›Ž8R8dsHVLbfjTš›d™m›k¢Uopka`µV¬cgRloƒkTŽƒVEbŸQh&o†¯ƒhƒmy~µdHRTmnbbem›dfjlopdfmnhƒkF—dHRTVWz=VRlo†¯am›h8jTs[hƒ'o

gh8UW}=hƒkTVkqdhƒk.m›dHbm›kqdfVLse o8gV‰mnbgLop}cdHjTsfVE~ªjTkl~TVs#dfRlV«žh8sfU(hƒ|o> R B—JdfsopklbfUWm›dedfVE~ªmnk

¸ehƒkTV#beRThƒdLŸ¹¢Q)RTVbejTUWUopsH`kTVdxmyb|h8zcdHoƒm›kTVE~žsfh8UopkµV¬adfVkl~TVL~µgLopkTh8kTmygopš}TsfV˜T¬za`W¸esHVžhƒšy~cmnkTŽ0¹

m›dHb+sfVEbvdHsfmygœdHmnhƒk‰dfhWdfRTVmnk8dHVsfžo8gV8Ÿ

>

)>5/< 5 0

$

5.

L

# & ) L

<0

$

,< F $ 31.576.?

Pref N

.1,

$

31+;<

$>=

6 N.G <

L .L 2F4313

'$R(

?)+/08

$

,.

L #O &

@ F 5.

L

N<

L

<(+

$ 3

5 04< 0.

L

.10

$ 3 2JF4313 '@$!( (

)/G <(+;.

L

N)>?

Pref N

.1,-3

$ +N<(+

@

(14)

a 0

a 1

a 0 t 2

t 1 t 2 e 2

e 3

e 4 c 0 a 0

a 1 t 1

e 1 c 0 c 0

#%$&

Pref A

c 0

c 0 t 1

c 0 t 1 t 1

c 0 c 0

c 0 t 1

#'&

.L 5/<(+

$!(

5/.1)

L

t 3 b 1

b 0

b 1 d 1 b 0

b 1

b 0

d 2

b 1 b 1 b 1

b 0 b 1

d 1 d 0 e 2

t 4

e 4 t 1

t 5 t 3

e 1

e 3 t 1

t 3 e 5

t 4 e 8

d 0 e 6

t 6 t 1 e 9 e 7 t 3

e 11 e 10 c 0

c 0 c 0

c 0

#(&

Pref B

²Jm›Ž8jTsfV™‚>¡š›šnjlbedfso0dHm›klŽisHV qjTmnsfVE~‰m›kqdfVsoƒgdfmnhƒk¯amnodfRlV™mnk8dHVsf oƒgV™z=Vdv¤VVLk&ghƒUW}=hƒkTVLk8db

'*) .,-, 1 07,(, 6

Q)RlV#V¬adHVkl~cVE~OgLopkTh8kTmygopš}lsfV˜T¬&hpto‰ghƒUW}=hƒkTVLk8d

A

hƒs

B

mnbxzTjTmnš­d[¤+m›dHROsHVŽqops~dfh‰m›dHbxmnk8dHVsfžo8gV8Ÿ

^cjlgHR©oµ}TsHV˜T¬ªgLop}cdHjTsHVLbdfRTV‰z=VRlo†¯amnhƒjTshƒ'dHRlo0dmnk8dHVsfžo8gV«mnk.sHVšyo0dHm›h8k¢dfhOm›dHbgh8Ui}=h8kTVkqdLŸ&¡£dmyb

h8zcdHoƒmnkTVL~«zq`sfVEbvdHsfmygœdHm›kTŽdHRTV[gjcdedHm›klŽighƒk8dHV¬adL—amnk‰}loƒsedHmngjTšnoƒsdHRTV™beVd'hpgh8kc˜lŽƒjlsHopdfmnhƒklb¤+RTmygRopsHV

j@beVE~‰žhƒs)dHRTVgjcdf´Bh¶gsfm›dHVsfmnh8kFŸ

¿:

’ Ã

A

R R 2.? F2 :

C

&*

% " B R

C

?

2:

Θ C = ¡

≈ mar , C tot , { κ e } e∈E A ¢

C=, RE?

∀e ∈ E A

κ e =

½ { [e 0 ] : e 0 ∈ E A , Π C (e 0 ) 6= ∅ }

Π C (e) 6= ∅

# %$

{ [e 0 ] : e 0 ∈ E A , Π C ([e] 4 [e 0 ]) = ∅ }

. # &$

±=Dq³

P

4

P ! U

Q)RlV™sHVLbedfsHmngdfmnhƒkhƒdfRTVgjcdedHm›kTŽ«ghƒk8dHV¬ad

Θ C

mnk

A

UWVLoƒklb'dfRlopd

±Co8³:-|k mnk8dHVsf oƒgVV¯ƒVLk8dµ±Copk V¯ƒVLk8dghƒsHsfVEbe}=hƒkl~Tm›kTŽOdHh¢o¶dfsopklbfm›dfmnhƒk©hp¨dfRlVmnk8dfVLse o8gVE³gLopk z=V

~cVLbfm›Ž8klo0dHVL~oƒb+oWgjcdf´BhOV¯ƒVLk8d+h8kTš›`«m›Jm›dHb|ghƒsHsHVLbf}=hƒkl~cmnkTŽV¯ƒVkqd+mnb|opšybehWopkmnk8dfVLse o8gV™V¯ƒVkqdLŸ

±žz=³ A RlVsHVLoƒbL—@dfRTVWghƒsHsfVEbe}=hƒk@~cm›klŽV¯ƒVLk8d

e 0

hpto‰}Tsfmn¯0opdfVgjcdf´BhªV¯ƒVLk8d

e

±žmBŸV8Ÿ™opkOV¯ƒVLk8d[¤+RTmygR

~chaVLbWkThpd«ghƒsHsfVLbf}=hƒkl~.dfh.oƒkq`.dfsopkqdfm›dfmnhƒkhp¨dHRTV&mnk8dHVsf oƒgVE³Rloƒbidfhªz=VµgRTh8bfVkbfjlgR©dHRlo0d

dfRTVsHVopsHV[kThWm›kqdfVsf oƒgV[VL¯ƒVkqdHb¸ez=Vdv¤VVLka¹

e

opk@~

e 0

—cmBŸV8Ÿmnk

Π C ([e] 4 [e 0 ])

ŸQ)RTmyb+gh8kl~cm›dfmnhƒk mnb+kTVLgVLbHbfoƒsf`Wdfh«gjcd|mnkc˜lkTm›dfV#gHR@opmnklb)hpJ}TsHmn¯0o0dHV[VL¯ƒVkqdHb+m›kdfRTVjTkTžhƒšy~cm›klŽlŸ

¿:

’ à +* P

Pref A Θ C

2 R; -? 6U < =

fsble A Θ C

R

? V¬adfVk@~cVL~&goƒkThƒkTmygoƒš5}TsHV˜T¬

A

% "6= 2

C

²JmnŽƒjTsHVLb 7T± oq³[oƒkl~± gE³™beRlh†¤dfRTVV¬adHVkl~cVE~¢gLopkTh8kTmngLopš>}TsfV˜T¬cVLb™hpdHRTV«ghƒUW}=h8kTVk8db

A

opkl~

B

m›k

²JmnŽƒjTsHV…cŸ¢¡k dHRTVbfV qjTVLšdfRTV&gjcdf´Bh V¯ƒVkqdHbioƒsfV~csoE¤+k©o8b~ch8jTzTšnVz=hE¬cVLbm›k ˜lŽ8jTsHVLbL—¤+RTVLsfVEoƒb

mnk§dHRTVgLoƒbfVihp'V¬adHVkl~cVE~¢gjcdf´Bh V¯ƒVLk8db[dfRTVhƒjTdfVsz=h†¬Omyb~cso†¤+k§¤+m­dHRªo&~ToƒbfRTVE~¶šnm›kTV8Ÿ

Pref A Θ C

(15)

„E… C

a 0 c 0

a 0 c 1

t 3

t 2

e 1

e 3

c 0 e 2 t 1

a 1 a 1

t 1

e 0

#$&

Pref A Θ C

t 2 y 0

t 3 y 1

s A 0 (a 0 ,c 0 )

s A 1 (a 0 ,c 1 )

#%'&

Π ˆ C ( A )

c 0

t 2

e 0

b 0

t 3

e 2 c 1

c 0

t 2

e 3

b 2

c 1

t 3

e 6

b 2 c 0

e 1 t 4

b 1

e 4 t 5

b 0 e 5 t 4

b 1

#(&

Pref B Θ C

t 2 y 0

y 1

t 3

y 2

t 2

t 3 y 3

s B 1 (c 1 , b 0 )

s B 2 (c 0 , b 2 ) s B 0 (c 0 , b 0 )

s B 3 (c 1 , b 2 )

#=&

Π ˆ C ( B )

²mnŽ8jTsfV 7.4>¬adfVLkl~cVL~&}TsfV˜T¬cVLb+opkl~dfRTVmns¨gh8sfsHVLbf}=hƒkl~cmnkTŽibfjTUiUoƒsf`«kTVdHb

gh8m›k@gmy~cVLb|¤+m­dHR&dHRTV#gLopkThƒklmngLopšF}TsHV˜T¬&bfm›klgVdfRTV#hƒkTšn`gjcdf´Bh§V¯ƒVLk8dxoƒkl~&m­db¨gh8sfsHVLbf}=hƒkl~cmnkTŽWV¯ƒVLk8d

opsHV[mnk8dfVLse o8gV™V¯ƒVLk8dbŸ |h0¤V¯ƒVLsL—adfRTmyb+mnb+kThpd)dHRTVgoƒbfVxžh8s

Pref B Θ C

m›k&²JmnŽƒjTsHV 7T±CgL³ŸQ)RTV™V¬adfVk@~cVL~

}TsHV˜T¬‰mnb)šnoƒsfŽ8VsdfRloƒk‰dfRlV™bedHoƒkl~ToƒsH~«}TsHV˜l¬5—c¤+RTmygHR¤hƒjlšn~‰z=V™hƒzcdopmnkTVL~‰za`‰beVdfdfmnkTŽ

e 4

oƒb)oigjTde´£h

V¯ƒVLk8d+bfm›klgV

[⊥] ≈ mar [e 4 ]

opkl~

[⊥] C tot [e 4 ]

Ÿ ¨h†¤'V¯ƒVsE—

e 4

~ThaVLbkThƒd)ghƒsHsfVEbe}=hƒkl~WdfhWopk‰V¬adfVk@~cVL~

gjTde´£h ¢VL¯ƒVk8d™bemnklgV#m›dxmybxo«}TsHm›¯po0dfVVL¯ƒVkqd[opkl~

Π C ([e 4 ] 4 [⊥]) = {e 2 , e 3 }

Ÿ¨Q)Rlmnbxoƒ}T}Tšnm›VEbxopšybehdfh dfRlV™V¯ƒVLk8d

e 5

Ÿ>Q)RTVV¯ƒVkqd

e 6

mnb+oƒkV¬qdHVkl~TVL~gjcde´£h ¶bfmnklgV™m­dbeVLš­>opkl~m­db+ghƒsHsfVEbe}=hƒkl~Tm›kTŽV¯ƒVLk8d

e 2

opsHV[mnk8dHVsf oƒgV™VL¯ƒVkqdHbLŸ

-¨k¶V¬adfVk@~cVL~¶gLopkTh8kTmygopš}TsHV˜l¬µmnb™ghƒUW}TšnVdHVibfm›klgVm­d™myb[ogopklhƒkTmygoƒš}TsfV˜T¬ ± bfVVrtsfh8}=h8bfm­dHm›h8k

…cŸDmnkF/›„EŠ@1ž³œŸ>¡£d|Rlo8b'dfhz@VbfRTh†¤+kdHRlo0d|m›d|mnb)˜lkTm›dfV8Ÿ

“ à à ‘ ’ Ã

Pref A Θ C

,P

“ ÃFà 9 H'`«rsfh8}=h8bfm­dHm›h8k…TŸ›„EŠ#mnk /›„EŠ@1Fm›d+myb)VklhƒjTŽ8R‰dfhbeRlh†¤»dHRlo0d+VEoƒgRmnkc˜lkTm›dfV

´£gRlopmnkmnk

Unf A

goƒk z=VµgjcdLŸ Q¤hªgLoƒbfVLbWopsHV&gh8klbemy~cVLsfVE~Ÿ ¡£xdHRTVsHV&oƒsfVmnkc˜lkTm›dfVLš›` UWoƒkq` mnk8dHVsf oƒgV&V¯ƒVLk8db—mBŸV8Ÿ

V¯ƒVLk8db¨¤+RTmygR¶ghƒsHsfVEbe}=hƒkl~dfh

C

—=mnkOdfRTVigRloƒm›kµdHRTVkOdHRTVigHR@opmnk¶goƒkµz=VigjTd™bemnklgV#dfRTVkajTU#z=Vs[hƒ

UopsH¥am›kTŽqb'mnb)˜lkTm›dfVoƒkl~‰dfRajlb|bfhƒUWV™UWoƒsf¥amnkTŽimyb+o˜lk@opšFUWoƒsf¥amnkTŽihpbfV¯ƒVLsHoƒš5m›kqdfVLse oƒgV™V¯ƒVLk8dHbLŸ

Y¨dHRTVsH¤+mybeV8—8dfRlVsHVxmybh8kTšn`o#˜lkTm›dfV™kqjTU#z=Vs+hpmnk8dHVsf oƒgV[V¯ƒVLk8dHb'mnk‰dfRTVgHRloƒm›kŸ^am›k@gVxdfRlV™gHR@opmnk

mybm›kT˜lkTm›dfV¨m›d'ghƒkqdHoƒm›klbtopkmnkc˜lkTm›dfV|dopmnšlhph8kTš›`i}Tsfmn¯0opdfV± kThƒkT´Bmnk8dHVsfžo8gV†³JVL¯ƒVkqdHbLŸ>^amnklgV|dfRTV¨kqjlU#z=Vs

hpUoƒsf¥amnkTŽ8b>myb>˜@kTm­dHVxbfhƒUWV|UWoƒsf¥amnkTŽmybto™˜lk@opš@UWoƒsf¥amnkTŽhpFbfV¯ƒVLsHoƒšT}TsHm›¯po0dfV|V¯ƒVLk8dbmnkWdHRTV|dHoƒm›š=opk@~

dfRajlb)dHRTVgHRloƒm›kµgopkz=VgjcdEŸ

¤

'& 0 ,/.76

Q)RTV)bfjTUWUWoƒsf`kTVd>hp@o[gh8Ui}=h8kTVkqdgLop}cdHjTsfVEbdfRTV)z=VRlo†¯am›h8jTshƒ@o[ghƒUW}=h8kTVkqdJ¤ŸsLŸdLŸJm­dbJm›kqdfVLse o8gVƒ—

opk@~m­d|myb+~TVsHm›¯ƒVE~«žsHhƒU°m­db+V¬adfVLkl~cVL~&goƒkThƒkTmygoƒš5}TsHV˜T¬Ÿ

Referências

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