Modular construction of finite and complete prefixes of Petri net unfoldings

(1)

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.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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

0249-6399ISRNINRIA/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

(3)

(4)

Unité de recherche INRIA Rennes

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

!#"%$&'($)*+

#,-'.'/01#

23546879;:=<>:@?BAC47EDFAHGJIKLABMON:=PKL6

Q)RTSLUWV#X+Y[Z]\_^a`cbedfSUWVEb+gh8UiU#jlkTmngLopkqdHb

rtsfhpuvVdb+wxmybvdHsfmnz=Xh8U

{|op}T}=h8sed|~cVsfVEgHRlVsgHRTVk=JlO\qopkqj@opsH`p8\}loVLb

LT Q)RTmyb>}lo}=Vsghklbfmy~cVsb~TmnbedfsHmnzTjcdfVE~Wbf`cbvdHVUb8~cVlkTVL~oboghnVEgdHmh8kihpgh8Ui}=h8kTVkqdHb

mnkqdfVsogdfmnkTdfRTsHhjT8Rimnk8dHVsfo8gVEbXhUW}=hkTVLk8dbmnk8dHVsf ogVLbopk@~W~cmnbedfsHmnzTjcdfVE~be`cbedfVUbosfV+UihT~cVnVL~

o8b#rVdfsHm'klVdHbL¢¡£dimyb¤VL'¥akTh0¤¦dfRlopddHRTVjTkch8n~TmkT§h¨bfjlgR©o§~cmybvdHsfmnzTjcdHVL~©bf`abedfVLU( ogdHhsHmybeVEb

mnkªdfRTVbfVklbfVWdHRlo0d#mdgok¢z=V«V¬c}TsHVLbHbfVL~¢obdHRTVgh8UW}=h8bfmdHmh8k¢hp)jTkThy~cmkl8bhp)mdHb#ghUW}=hkTVLk8dHbL

Q)Rlmnb ogdHhsHmnbfVL~Wh8sfU®hpFdfRTV[jTkch8n~TmkTi}Tsfh0¯amy~cVLbo#UWhsHVxgh8UW}logd'sHV}TsHVLbfVkqdHopdfmnhkhdHRTVbe`cbedfVLU

dHsouvVEgdHhsHmVEbcokl~Uop¥VLb)md|}=h8bHbemnzTnVxdHhoklopn`cbeVxdHRTVbe`cbedfVU°zq`}losedb

Q)RTVx}@op}=VshcgjlbfVLbh8kdHRTV[~TVsHm¯po0dfmnhk«hpolkTmdfVokl~ghUW}TnVdHVx}TsHVT¬O±C²JXr³mkdfRTV[jTkch8n~a´

mnkTµhp+o&~TmnbedfsHmzljcdfVE~ªbe`cbedfVLU^a}=VEgm@gon`hkTV¤hjln~¢nm¥VWdfhO~cmnsHVLgdH`¶h8zcdHom[email protected]&²JXr·mnk

o8gdHhsHmnbfVL~µhsHU&¤+mdHRThjTdgh8UW}TjcdfmnkTlsbvdo²JXr;hptdfRTVWnhzlo~cmnbedfsHmnzTjcdfVE~§bf`cbvdHVU]opk@~µdHRTVk

o8gdHhsHmnbfmnkTmdLxQ)RTVgh8klbvdHsfj@gdfmnhkµhbfjlgHR¶o¸vUWhc~cjlnos|}TsHVT¬q¹myb|z@obfVL~&h8kO~cVLsfmn¯amkTbfjTUWUopsHmVEb

hgh8UW}=hkTVLk8d[z=VLRlo¯qmnhjlsHb¨¤sEdEdHRTVmnsmkqdfVsf ogVLbL=dfR@o0dopsHVighUWU#jTklmngLo0dfVE~µdfhdfRlVikTVmnRaz=hjTsf´

mnkTghUW}=h8kTVkqdHbLQ)RlVyo0dfdfVLs|mnk8dfVLso0dHV[dHRTVUºdfhdfRlVmns|nhcgLopFz=VRlo¯amh8jTsblopkl~&}TsfVL}lopsHVmnk8dfVLse o8gV

bfjTUWUopsHmVEbh8sWdfRTVµklV¬adghUW}=h8kTVk8db»Q)RTmybW8h8zlopn`.dHo¥VLbidfRlV&¼h8sfU½hpoªUiVEbfbHop8V}@obHbemnkT

onhsHmdfRTU&a¤+RTVLsfV[dHRTVnhzlobf`abedfVLU/myb)klV¯Vs|ghklbfmy~cVsHVL~

¾µ¿qÀ5ÁHÂ#Ã 0Ä8 rJVdHsfm|kTVdEjTkch8n~TmkT@tlkTmdfVOghUW}TnVdHV}lsfVT¬~cmybvdHsfmnzTjcdHVL~bf`abedfVLU&t}TjTnnzlog¥5

gLo0dHVh8sf`WdfRTVLhsH`

(5)

($)*«+//'#"%$&'/()$ 1+

XV&}@op}TmnVsghk@bemy~cSsHVµ~cVLb«bf`cbvdHSUWVLb«~cmnbedfsHmnzTjLbL)~lkTmyb«gh8UWUiVO~cVEb«gh8nVLgdfmnhklb

~cVghUW}=h8bHopkqdHb)sHVnmEb)}lops¨~cVLb|mkqdfVsf ogVLbL'XhUW}=h8bHopkqdHbLcmkqdfVLse o8gVLb+Vdxbe`cbedfSLUiVEb+~TmnbedfsHmzljLb|bfhk8d

UWhc~LmybEb}lops~cVLbFsEbeVEopjc¬[~cV>rVdHsfmB¡ VLbedzTmVLkghkTkajqjTV>nV~L}TnmnoV~jlk[dHV8bf`cbvdHSUWV~cmybvdHsfmnzTj

uvhjlmd)~jTkTVx}TsHh}TsHmd¨~cV| o8gdHhsHmnbHo0dHmh8kJmn=}=Vjcd'VLk«V=Vd)bV¬c}TsHmnUiVLsgh8UWUiV¨yo#gh8UW}=h8bfmdHmh8k«~cVLb

~L}Tmyop8VLb'~cVLb)~cmsHVkqdHb)gh8UW}=h8bHopk8db~cjbe`cbedfSLUiV8XVdedHV[sfVL}TsEbeVLk8dHopdfmnhk ogdfhsHmybLV¨~Tj~L}TnmnoV

VLbedVkWksopa}TnjlbghUW}logdfVqjTV+V|~L}Tmyop8V)nhzloCpVd}=VLsfUWVdVkWhjTdfsHV+~oklopn`cbeVLsV|bf`cbvdHSUWV

}los)UWh8sHgVEopjc¬

!h8zauvVEgdHm~cV¢gV¶}lo}TmnVsVLbedyo©gh8klbvdHsfj@gdfmnhk-~cV¶}Tsl¬aVEb«lkTmybghUW}TnVdHb¢±rt²JX)³~lopklbnVLb

~L}Tmyop8VLbt~cVxbe`cbedfSLUiVEbt~cmybvdHsfmnzTjEbJrtjlb}TsLgmybUWVk8dE8h8k«gHRlVsgHRTV#"hzcdHVkTmns~cmnsfVEgdHVUWVkqd'~cV|dfVyb

}TsT¬cVEbxbfhjlb¨h8sfUWV# ogdfh8sfmybV5Vk$L¯amdopk8d~cVWgh8klbvdHsHjTmsHV~opz=hs~µjTk§}TsT¬cVighUW}TnVd~Topklb[V

~L}Tmyop8V~cj be`cbedfSLUiV8h8zlop>}=h8jTsVLklbfjTmdHVnVW o8gdfh8sfmybfVsEXVdedHVghk@bvdHsfjlgdfmnhkª~cV«}TsT¬cVLbbeh8jlb

hsHUWV«UWhc~cjTyopmnsfVbeV«hkl~cVbfjTs#yoµkThpdHmh8k ~cV«sLbfjTU%~cj©gh8Ui}=h8sedHVUWVkqd~jlk©ghUW}=hqbfok8d}los

sop}T}=h8sed&"Obfhk»±CbeVEbH³mkqdfVLse o8gV8±CbH³§XVLb#sEbejlU'Eb#beh8k8dghUWU#jTkTm(qjLbopjc¬.gh8Ui}=hqbfok8db¯hmybfmklbL

qjTm+VEbimnk8dHSsHVkqd)"§nVjTsW}lsfh8}TsfV~}TnmnoV}=hjTsghklbedfsHjTmnsfV~cVEbisLbfjTU%Lbi¯VsbnVLbgh8Ui}=hqbfok8db

bfjTm¯popkqdHbL8Vd'omklbfm@~TV¨bfjTmdfV8 V+dHhjcd'}TsfVLkl~Wyoh8sfUWV¨~*jTko8hsHmdfRTUWV+"gmsgjlnopdfmnhk~TV|UWVEbfbHop8VLb

VkqdfsHVgh8UW}=h8bHopk8dbch,nVbf`abedfSLUWVnhzlo5k*VLbedueoUopmyb)UWokTmn}TjT

- Ã L Á /. © sEbeVLoj©~cVrVdfsHmBt~}TnmnoV8>}Tsl¬aVlkTm+ghUW}TnVdEtbf`cbvdHSUWV~cmybedfsHmzTj}lsfhc~TjTmd

lzTscdfRLhsHmV~cVEb+gLo0dLh8sfmnVLb

(6)

t2

t1

a 0

a 1

b 1

b 0

c 0

c 1

t3

t4

t7

t5 t₆

Component Component

Interface

A B

#%$&

N

t4

c 0

a 0

t1 t₂

t3

a 1

c 1

a 0 a₀ c₁

c 0

a 1

...

#%'&

UnfA

t5 t₆

b 1

a 0

c 1

b 0

t4

b 1

c 1 a₀

b 1

c 1 a₀ t4

t4

t7

c 0

t3

c 1

...

#(&

UnfB

²JmnjTsHVW>wxmybvdHsfmnzTjcdHVL~&bf`abedfVLU°okl~dfRlVjTkchy~cmnkT8b+hpJmdHb|gh8UW}=hkTVLk8dHb

) *

Wt#

rVdHsfmTkTVdHb|±r,+[³h8sfU ox¤+my~cVn`#jlbeVE~UWhc~cVlgyobHbhsopklon`abfmnkT[gh8klgjlsfsHVkqd>opkl~i~cmybvdHsfmnzTjcdHVL~Wbe`cbe´

dHVUbL.-|UWh8kT#dfRTV~cVE~cmygo0dHVL~dHhahybLabeh8UiVxsHVLgVk8d)o}T}TsHh8o8gHRTVEbRlo¯Vxz=VVkzlobfVL~«hk«dHRTVbeh´£gLopnVE~

rVdHsfm5kTVdbjTkchy~cmnkT8bL>Q)RTVxjTkchy~cmnkT#dfVEgHRTklmqjTV0/"T=1}Tsfh0¯amn~TVLbtok«V2«gmVLk8d'sHV}TsHVLbfVkqdHopdfmnhkhp

onsHjTklb¨hto«r3+mnkOdfRTV#dHsfjlVghklgjlsfsHVklg`¢±h8sx}losedHmnoh8sH~TVs³+bfVUopkqdfmygb 54¬aVEgjcdHmh8klb[opsHV#ghkc´

bfmy~cVsHVL~«o8bt}losedHmnon`ih8sH~cVLsfVE~bfVdHbhFVL¯VkqdHbso0dHRTVstdfR@opkbfV qjTVk@gVLbLq¤+RTmygHRsHVLbfjTdHbmkmUW}=hsfdHok8d

UWVLUih8sf`µbHoE¯amnkT8b|hso8hsHmdHRTUb¨oklopn`cbemnkTz@VLRlo¯qmnhjlsHb¨hpdHRTmnb[kTVdLZµh8bed[hpdfRlVLbfViopnh8sfmdfRlUWb

~ThklhpdsHVn`h8kdHRTV+jTljlkchy~cmnkThpor,+8RTh¤'V¯VsEpzTjcdtso0dHRTVs>h8kWo[lkTmdfV¨opk@~ighUW}TnVdfV+}lsfVT¬

±C²JXr³hmd6/87cl8p@9"1B>Q)RTVx~cVlkTmdfmnhk«hl¸eghUW}TnVdfVLkTVLbHb¹|~cVL}=Vkl~Tbth8kWdfRTV¨}TsHh}=VLsedv`#hmkqdfVLsfVEbvdE

zljcdxmdxmyb¨VklVsHon`z@obfVL~&hk&dHRTV o8gd¨dfRlopd[opnsfVEogHR@opzTnVUopsH¥amkTqb+hp>dfRTV#klVd[osfVsfVL}TsfVEbeVLk8dfVE~

mnkdfRTmyb+}TsHVT¬c¤+RTmygRmyb+bfj2«gmnVkqd+hs|bfV¯VLsHo5bedopkl~TosH~}Tjlsf}=hqbeVEb'mnkUWhc~cVL´ gHRlVLg¥qmnkT:/nLTFE1£

Q)RTmyb}lo}=Vs«hTgjlbfVLbhk»}lopsfdfmygjTyos«r,+(gopnnVL~ ¸f~cmybvdHsfmnzTjcdHVL~»bf`cbvdHVUb¹ª~TVlkTVE~ za` o8bfbfVUi´

zlmnkT§gh8UW}=hkTVLk8dHb± bfjTzc´£kTVdb³zq`ªmnk8dfVLse o8gVLb± bfRlosfVE~.bejTzc´£kTVdHb³;-°gVk8dHsHot}TsHh}=VLsedv`§hp+dHRTVLbfV

~TmnbedfsHmnzTjcdfVE~¶be`cbedfVUb¨myb¨dfRlopdxdfRlVmnsxjlkchy~cmnkTgLopkOz=V#jTsfdfRlVs+¸eghUW}TsHVLbHbeVE~q¹<9= ©o« ogdHhsHmnbHo0dHmh8k

}lsfh8}=Vsfd`>/FT?a>L@1£W^a}=VEgm@gon`85dHRTVWjTkch8n~cmnkT&hp'o~TmnbedfsHmnzTjcdfVE~§bf`cbedfVUgok§z=ViV¬c}TsfVEbfbfVL~

o8b|dfRlVigh8Ui}=hqbemdfmnhkOhpdfRTVjTkThy~cmkl8bxhptmdb[ghUW}=h8kTVk8dbLx¡k8dfVLsfVEbvdHmkln`@dHRTVigh8nVLgdfmnhkOhpdfRTV

nhcgLopajTkch8n~TmkTqbUWo`z=V)UWhsHV)ghUW}lo8gdJdHRlopkdHRTV)jTkT¼h8y~cmkThpldHRTV)8h8zlopcbf`abedfVLU&obJmnnjlbedHsHopdfVL~

mnkª²Jm8jTsHVWQ)RTVbf`cbvdHVU]mkª²JmnjTsHV8± o8³[ghklbfmnbedbxhtdv¤hµghUW}=h8kTVk8db

A ^okl~ BFmnk8dfVLsHo8gdHmkT

dHRTsHhjT8R¶ok¶mnk8dfVLse o8gV8Q)RTVLsfVV¬cmnbed

m = 2 }=hqbfbfmnzTmnmdfmnVLb¨dHh}TsfhT~cjlgV

t4

mk

Unf_A ±C²mn8jTsfV± z@³e³ okl~

n = 3 }=hqbfbfmzlmnmdHmnVLbdfh©}TsfhT~cjlgV

t4

mnk

Unf_B ±C²mn8jTsfV 8±CgL³e³|RTVk@gV

m·n = 6 ~cm=VLsfVLk8d ghU#zTmnklopdfmnhklb)mnkdHRTVjTkchy~cmnkThpdfRlVnhzlobe`cbedfVLU

Q)RTV+ ogdfh8sfmybfopdfmnhki}TsHh}=VLsedv`#hpjTkchy~cmnkT8b¤)oblsbedUWVkqdfmnhkTVE~Wzq`BA mklbf¥V@mk/nL1BqzTjcdhkTn`

h8jTkl~imdHb>lsHbedtop}T}Tnmygo0dHmh8klb>opsHhjTk@~iC7T0mk~cmyokTh8bfmybJ}TsHhzTnVUb5/D1BE-»gnh8bfVn`sfVLnopdfVE~}Tsfh8zTnVU

ghklbfmybvdbxmnkªghUW}TjcdHmkldfRlViUWmnkTmnUWo ogdfhsHmybeVE~µhsHU hp'ok§jTkchy~cmnkTlopkl~¶¤'o8bghk@bemy~cVsHVL~¶opd

dHRTVbHopUWV#}=Vsfmnhc~F/TG?a1£x^a}=VEgm@gon`@dfRTVLsfV#VLkTVsopnn`V¬amybedxbfV¯VLsHo ogdfh8sfmybfVL~h8sfUb|hs¨dfRTV

jlkchy~cmnkThp'o~TmnbedfsHmnzTjcdfVE~§bf`abedfVLU&#Q)RTV¸enosf8VLbvde¹h8kTVRlo8bxdfRTVijTghUW}=hklVk8djTkch8n~TmkTqbob

o8gdHhsbIH'jcdxhptgh8jTsHbfVklhpd[opnsfjTk@b|ho8m¯VLkOgh8Ui}=h8kTVkqd¨¤+mnJsfVLUWomnk&}=hqbfbfmnzTVmnkµdfRlV#nhzlo

bf`cbedfVU&+^a`aUWUWVdfsHmygopnn`TdfRlVx¸fbeUopnnVLbedv¹h8kTVmyb|hzTdHopmnkTVE~zq`&beVLVEgdHmkT«sfjlklb|hpVEogRµgh8UW}=hkTVLk8d

dHRlo0d~ch}@opsfdfmygmn}lo0dHVdHhosfjTk¶hdfRTVinhz@opJbf`cbvdHVU&J4!qjTmn¯0opnVkqdfn`5dfRlVLbfVUWmklmUopJ o8gdfh8sHb[gopk

z=VhzcdopmnkTVL~ zq`.sHVLbedfsHmngdfmnkT ± o8gdHjlopn`.}TsfhpuvVEgdfmnkTa³dHRTVnhzlojTkchy~cmnkT§h8k VEogR©ghUW}=hklVk8dE

ZOhc~cjTyopsopnh8sfmdfRlUWb5R@oE¯V>z=VLVk}Tsfh8}=h8bfVL~xmnk[dHRTVtoz@h0¯V>ghkqdfsHmzTjTdfmnhklb5dHh|ghUW}TjcdHV>dfRTVEbeVUWmkTmnUop

(7)

C

ogdfh8sHbL .PR R P9; P R @ WQ)RTmybopzTmnnmdv`µdfhµopklo`cbfV

nhz@op}TsHh}=VLsedHmVEbFhplo¨be`cbedfVLU;zq`nhcgo8ghUW}TjcdHopdfmnhk@bFh8kbfUopn8hzcuvVLgdHbmnbhplgh8jTsHbfVtopkV¬qdHsfVLUWVn`

~cVEbemnsHozTnVxVLopdfjTsHV[dfh«o~T~csHVLbHb'yopsHV[UWhc~cjTyops|be`cbedfVLUWbL

Q)RlV¶hzcuvVLgdfmn¯V¶hpdfRlmnb}lop}=VLsmyb«dHh jTsedHRTVsV¬c}TnhsHV&dHRTmybmy~cVLol¨opkl~»bfRTh¤dHRlo0ddfRlV§bHopUWV

}TsHmk@gmn}TVEbgopk z=V¶op}l}TmnVL~dHh©ghUW}TjTdfVµ o8gdfh8sfmybfVL~ hsHUWb«hpI,P h

jTkch8n~TmkTqb A mdfR.h8kTdHVsHU]h8zauvVEgdHm¯VidHRTV«~cVsHmn¯0o0dHmh8k§hpdfhah8nb[hsUWhc~cjlnosUWhc~cVLgHRlVLg¥qmnkT@

Q)RTV|}TsHhzlVLU oUih8jTk8dbdHhzTjTmnn~cmnkT#²JXr hpFgh8UW}=hkTVkqdHbLmnk«bfjlgHR«o¤'oE`#dHRlo0ddHRTVmnsgh8UW}=h8bfmdHmh8k

mybghUW}TnVdfVhs#dHRTV8h8zlopbf`cbvdHVU&;A mdHR dHRTVgh8klbvdHsHomnk8ddHRlo0ddHRTVLbfVhcgLop}TsHVT¬cVLb#U#jlbediz=V

gh8UW}TjcdfVE~mk#o|UihT~cjTyopsUopkTkTVLsL¤+mdfRTh8jcdsHVVsHVklgV>dfh|dfRTVnhzlo8be`cbedfVU&-xbo|bedfsopmnRqdeh8sf¤'opsH~

op}l}TmygopdfmnhkF@dHRTmybx¤h8jTy~µVk@opzTnV#h8kTVdHhsHV}lno8gVogh8UW}=hkTVkqdxmnk§obf`cbvdHVU opk@~OgRTVLg¥dfR@o0dxdHRTV

nhz@opbf`cbvdHVUºbvdHmnFz=VRlo¯VLb)¤VLBT¤+mdHRThjTd+R@oE¯amnkTdHhgRTVLg¥«dfRTV¤+RTh8Vbf`cbvdHVUºop8omk

Q)RlV«mdfVLsHopdfjTsHV«opz=h8jcdjlkchy~cmnkT8bRlo8bo8~T~csHVLbHbeVE~§}TsHhzTnVUbdHRlo0dUoE`§nhah¥¶sHVyo0dHVL~§dHh&dHRTmyb

qjTVLbedfmnhk²Th8sV¬TopUW}TnVodHVLgHRlkTmqjTV«dfh¶gh8klbvdHsfjlgd#o@kTmdHVghUW}TnVdfV«}TsHVT¬ ±C²JXr³hp|oµbf`akc´

gRTsfh8kThj@b}TsfhT~cjlgdhp+yopz=VnVL~¢dHsHoklbfmdHmh8k.bf`cbvdHVUb¤'ob}lsfVEbeVLk8dfVE~ªmk /1BOQ)RlmnbdfVEgHRlkTmqjTVjlbfVLb

dfRlV[}TsHhc~cjlgd)bedfsHjlgdHjTsHVxhpdfRTVbe`cbedfVLU¦dfhWbemnUW}Tnm`dfRlVgh8klbedfsHjlgdHmh8k«hFdHRTV²JXrczTjTd'dHRTmyb'}TsHVT¬

myb#kThpdigh8UW}TjcdfVE~.mk. o8gdfh8sfmybfVL~§h8sfU&H'`ªghkqdfsobedL/@1'}TsHh}=h8bfVL~ªoOUWhc~cjTyopsgh8klbedfsHjlgdHmh8kªh

gh8UW}TVdfV+}TsHVT¬cVEbpdHo¥qmnkTmkqdfhogLgh8jTk8d>dfRTV¨be`aUWUWVdHsfmnVLb>hpgh8Ui}=h8kTVkqdHbL ¨h¤VL¯VsE0dfRlV|bf`abedHVUb

gh8klbfmn~cVLsfVE~¢dfRTVLsfVopsHV«sHVLbvdHsHmngdfVL~ªdfh§kThkc´£sfVLVkqdfsopk8dbf`aklgRTsfh8kTmybfopdfmnhklbJmBV§hkTVdfsopklbfmdHmnhk.h¨o

gh8UW}=hkTVkqdgLopk«bf`aklgHRTsHhklmnbfV+hkl`i¤+mdHR«hkTV|dfsopklbfmdHmnhkWmnkopkThdfRTVLstghUW}=hklVk8dEQ)RTmybtsfVEbvdHsfmygdHmh8k

¥amnybdHRTV)ghUW}TsHVLbHbemnhk8omk#hT ogdfh8sfmybeVE~hsHUbFdfRTV+ghnnVLgdHmnhk#hpT o8gdHhsbhTdHRTV'8h8zlopqjTkchy~cmnkT

Rlo8b'dHRTVbHopUWVbem"!Vob)dfRTVnhz@opjTkch8n~cmnkTmdbeVvt¡ kOo~T~cmdfmnhkcdfRlV~cVLsfmn¯0opdfmnhkhphcgLop}lsfVT¬cVLb

jlbfVLbnhzlocmnkch8sfUo0dHmh8kF¡kWdHRTmyb}lo}=VsEpdfRTV¨hzcuvVLgdfmn¯V|mnb>dHhjlbfV|hkTn`hTgopTmnkch8sfUo0dHmh8kFmBV8dHRTV

gh8UW}=hkTVkqd'UWhc~cVL=}Tnjlb'mnkchsHUo0dHmh8kgh8UWU#jTkTmygo0dHVL~«za`kTVmnRaz=hjTsbqmnkh8sH~TVstdHhi~cVdfVsHUWmklV¨dHRTV

}TsHVT¬§dfhOsHVdHomk¢h8sVLo8gHR.ghUW}=hklVk8dEQ)RTV«mnkchsHUo0dfmnh8k§dHsHoklbeUWmdedHVL~ªzq`¢oµghUW}=hklVk8ddfhOmdb

kTVLm8Rqz=hjTs)¤+mn5dHo¥VxdfRTV[hsHU®hJo¸ebfjTUWUopsH`WkTVdE¹[mka¯hn¯amklhkTn`z=VLRloE¯amnhjTsbhpdfRlV[mnkqdfVsf ogV

dfR@o0d|sfVLnopdfVEbdfRlVU&

Q)RlVµbfjTzauvVEgdo8~T~csHVLbHbeVE~ RlVsfVµghU#zTmnkTVLbUWokq` dfVLgRTkTmygo|o8be}=VEgdHbL'beh8UWVh[¤+RTmngR osfV&kThpd

~cmnsfVEgdH`sfVLnopdfVL~dfh«dfRTV#}lsfh8zTVLUopk@~µgopkµz=Vgh8klbfmn~cVLsfVE~&ob¸emkl~TV}=Vkl~TVk8d~cm2gjTdHmnVLbL¹ªQ)RlVsHV´

hsHV@dHh}lsfVEbeVLsf¯VdHRTVigyopsHmd`h>dHRTmnbxlsHbedo0dedHVUW}cdE=dHRTVibeVdedHmklmybx¯hnjTk8dopsHmn`&bemnUW}TnmlVE~¶o0d[mdb

Uo0¬cmnU#jTU&Q)RTVopdedHVk8dHmh8kmyb)sHVLbedfsHmygdfVE~«dHhWjTkch8n~cmnkTqb)hbfopVr,+@okl~dfRTV~TmnbedfsHmzljcdfVE~bf`cbvdHVU

mybsHVL~cj@gVL~#dHh[dv¤hghUW}=hkTVLk8dbsfVLnopdfVE~#za`opkimnk8dfVLse o8gV±dfRTV|Vkl~ihp@dHRTV+}lo}@VLsbejlVEbvdbJRTh0¤ dHRTmyb

}losHo8~cm8U gLopkWdHRTVk«z=V¨V¬qdHVkl~TVL~l³Q)RTV¨mkqdfVLse o8gV|mybobHbejTUWVE~idfh#z=V¨o#bemnUW}TnV¨ojcdfh8Uo0dHhkWso0dHRTVs

dfR@opk o§dHsfjTVµr,+#opk@~©dHRTVµ}TsHVT¬ghklbedfsHjlgdHmh8ko8bfbfjTUWVLbWUosf¥amnkTªghUW}TnVdfVLkTVLbHb)okl~bfmnUi}TnV

o~TV qjlo0dHV[h8sH~cVLsHbL

Q)RlV}lop}=VLsimybihsH8okTmnbfVL~ obhnh0¤|bL©^cVLgdHmh8k OsHVLgLopnnb#dfRlVzlo8bemyg«dHRTVh8sfVdfmygop)zlog¥asHhjTkl~

gh8klgVLsfkTmnkTµr,+xbJdHRTVmns#jTkch8n~TmkTqb#opk@~§dHRTV~cVsHm¯po0dHmh8k¢h+oOgokThkTmygo>lkTmdfVgh8UW}TVdfV«}lsfVl¬5

^aVEgdfmnhk "¶~cVlklVLbi~cmybvdHsfmnzTjcdHVL~©be`cbedfVLUWbiopkl~.sHVLgLopnnbdfRTVo8bfbfhcgmyo0dHVL~§ o8gdHhsHmnbHo0dHmh8kª}TsHh}=VLsedv`§h8k

dfRlVmnsxjTkch8n~cmnkT@[^aVLgdfmnhkOghk8dopmnklb|dfRTVigh8k8dfsHmnzTjcdfmnhkOhdHRTmyb¨}lo}=Vs'md[}TsHVLbeVLk8db+dHRTVUWhc~cjTyops

~cVLsfmn¯0opdfmnhk hpxo¶lkTmdfV&okl~©gh8UW}TVdfV}TsfVT¬ mk© ogdfhsHmybeVE~ªhsHU>hTgjlbfmkl¶lsHbedih8k dfRTV&bfmnUi}TnV

go8beVhpdv¤hgh8UW}=hkTVkqdHbLopkl~µdHRTVk§}TsHh}=hqbemnkTdHRTViVLkTVsopnm#!Eo0dHmhk&dHhUWhsHVgh8UW}TV¬µkTVd ¤h8sf¥cb

hpghUW}=hklVk8db

$ %

'/ !01

¡k dHRTmnbbfVLgdfmnhk»zlobfmng¶~cV@kTmdHmh8klbghk@gVsHkTmnkT rJVdHsHmxklVdHbopk@~ klVdjTkchy~cmnkT8bopsHVO}TsHVLbeVLk8dHVL~

Q)RTVEbeVosfVUWomkl`o~To}cdfVE~sfh8U /87c8!1£

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

8µ¿ L:9 - kTVd«mnb«o qjlo~TsfjT}lV

N = (P, T,→, P⁰) ^bejlgR ^dfRlopd P ^opkl~ T opsHV&~cmnb uvhmnk8dbeVdHbWh

% R! opkl~; P != qsHVLbf}=VLgdfmn¯Vn`

→⊆(P×T)∪(T×P)^mnb'o< ^=lTopkl~ P⁰:P →

(8)

N={0,1,2, ...} mybxo«U#jTdfmybfVd[h8k

P sHV}TsHVLbfVk8dHmkldfRTV = &h>dfRTVklVdL²Th8s[o«kThc~cV

x∈P∪T^amdHb ^•xmyb'~TVlkTVE~zq`

•x={y|(y, x)∈→}okl~«mdHb

x^•myb)~cVlklVL~zq`

x^•={y|(x, y)∈→}^>¡kdfRTmyb)¤h8sf¥WdfRlVkTVdb+opsHVxnmUWmdfVE~dfh: UklVdHbLcmBVhs)V¯8Vsf`sHVLo8gHRlozTnV Uosf¥amnkT

M opkl~VL¯VsH`}Tyo8gV

p∈P M(p)⊆ {0,1}

- noz=VnVL~kTVd

N =¡

P, T,→, P⁰, λ,Λ¢mnbo[kTVdV¬adfVk@~cVL~i¤+mdHRo[noz=VTbfVd

Λokl~io[yopz=VnnmkT

jlklgdHmh8k

λ:T →Λ hkdHsHoklbemdHmh8klb

-Ã

9

- UWhsH}TRTmybfU

φ : A → B ^z=Vdv¤VVLk ^kTVdb x = ¡

Px, Tx,→_x, P_x⁰¢ /n71£'¤+RTVLsfV

x∈ {A, B}mnbo}lopmns

¡φ^P, φ^T¢ ^¤+mdfR

φ^P osHVyo0dfmnhk§hk¢}TyogVLbokl~

φ^T o}losedHmyopJjTklgdfmnhkªhk dHsopklbfmdHmh8klbEªQ)RTVmnkTmdfmyop'Uosf¥amnkTOmyb}TsHVLbfVsH¯VL~ªza`

φ ob#hnh0¤|b

P_B⁰ = φ¡ P_A⁰¢ ^opk@~

∀p_B ∈ P_B⁰,∃!p_A∈P_A⁰:pAφpB

>¡£

φ mnb|~cVlkTVE~h8k

pA∈PA

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

•pA

okl~

p^•_A φ }TsHVLbfVsH¯VLbdfRTVVkq¯amnsfh8kTUWVk8dhp|VLo8gHR.dfsopklbfmdfmnhk*

tB = φ(tA) mnUi}lmnVLb#dfRlopdsfVEbvdHsfmygdHmnhklb

φ^op:^•tB→^•tA

opkl~

φ^op:t^•_B→t^•_A opsHVxz=hdfRdfhdHop5jTklgdfmnhklbLc¤+RTVLsfV

φ^op ~cVkThdfVEbdfRlVsfVL¯VsbeVE~

sHVyo0dHmnhkhk}Tyo8gVLbL.+|hdfmygVdfRlopd+kTVd|UWhsH}TRTmybeUb)}TsHVLbfVsf¯8VxsHjTklbL

²Th8syopz=VnVL~§kTVdb

x=¡

Px, Tx,→x, P_x⁰, λx,Λx¢ dHRTV~cVlkTmdfmnhk¢h'oUWh8sf}TRlmnbfU

φ: A→ B

myb|sfVLmkThsgVL~zq`V¬adHsHoWsfVqjTmsHVUWVLk8dHb

ΛB⊆ΛA

±dfRTV#noz=VbfVd¨myb+sHVL~TjlgVL~&zq`

φ^³ Dom(φT) = λ⁻¹_A (ΛB)⊆TA

±φmyb~cVlkTVE~V¬co8gdH`xhkdfsopklbfmdHmh8klbRlo¯amkT|o|bfRlopsHVL~[noz=V¼³okl~m

φT(tA) =tB

dHRTVk

λA(tA) =λB(tB) ±yopz=VF}TsHVLbfVsH¯amkTa³

&'& 0 1 0 , 6 6 , 6

Q¤hklhc~cVLbxhokTVd

N y ^opkl~ y⁰FopsHV#mnk !2 < C~cVkThdfVL~¶zq`

y#y⁰mtdfRTVsHVV¬cmnbed

~Tmnbedfmnklgd|dHsHoklbfmdHmh8klb

t, t⁰ ∈T bfjlgRdfRlopd

•t∩^•t⁰6=∅^lokl~ (t, y)^opkl~ (t⁰, y⁰)osfVmkdHRTVsfVlV¬cmn¯V dHsopklbfmdHm¯Vgnh8bfjTsHV[hpdHRTVlh0¤-sHVyo0dHmh8k

→l~cVLkThpdHVL~zq`

¹

«/ ¿ ¿J¿ L:9 -|k ! R mybio¶kTVd

O = (C, E,→, C⁰)^¤+RTVLsfV C mybio§bfVdWhp

R =P ± }Tno8gVEbH³

E myb¨dfRTVibfVd[hp6 <@ 2 ±¼dHsHoklbemdfmnh8klbH³|opk@~

C⁰ =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@kTmdHVn`Uokq`

y⁰bej@gHRdfRlopd

y⁰≺y^¤+RTVsHV ≺~cVkThdfVEbdHRTV =l

dHRTVdHsHoklbemdHm¯VmnsfsHV lV¬cm¯VghqbejlsfVh

→)Q¤h«kThc~cVLb¨opsHVR ! 2B@~cVkThdfVL~

yky⁰@m>kTVLmdHRTVs

y#y⁰ ^klhs y¹y⁰ ^kThs y⁰¹y

! 0

"

"#

Ã ¿ L ¿ 9 - U R |hpo[kTVdtbf`cbvdHVU

N mybo[}@opmns

β = (O, f)^8¤+RTVsHV

UWh8sf}TRlmnbfU

f :O → N mybo¨dfhdHo8jlklgdHmh8kh8k

Oonbfh[gopnnVL~#o#hp

O^mnk8dfh NJQ)RlmnbJ¼h8y~cmkT

gLopk«z=V[beVVLkoboyopz=VnmnkT¼jlklgdHmh8khk«V¯VLk8dHb'opk@~«ghk@~cmdHmh8klbthp

Oazq`W¤+RTmygHRgh8kcljlsHopdfmnhklbhp

OsHV}TsHVLbfVkqdHb)sfjlklb+hp

N¡£d¨mnb)jTsfdfRTVLs|sHV qjTmnsfVE~«dHRlo0d

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

e1, e2∈E^=m ^•e1=^•e2

okl~

f(e1) =f(e2) ^dfRTVLk e1=e2

|Qh~cVlkTVlklmdHVgh8Ui}TnVdHV}TsHVl¬aVEb

md¨¤+mnz=VjlbfVjlFdfh«ghklbfmn~cVLs|oµ±¯amnsfdfjloy³'mnkTmdfmyopV¯VLk8d

⊥^mnk βl¤+RlmngR&Rlo8b|opk&VUW}cdv`}TsHVLbfVdL

C⁰

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

f^³

- zTsopklgRTmnkTª}TsHhcgVLbHbµ±=H'r³

β⁰ ^hp N ^mnbo »hoªzTsopklgRTmnkTª}TsHhcgVLbHb

β+~TVkThdfVL~ zq`

β⁰vβ^qm O⁰ mybtogojlbfon`ghqbeVE~WbfjTzc´£kTVdth

OghkqdHomkTmnkT#opnlmklmdHmno@ghkl~TmdHmh8klbtopkl~bfjlgHRdfRlopd

∀e∈E, e∈ O⁰ mnUi}TnmnVLb

e^•⊆ O⁰ ^okl~ f⁰ mnbtdfRTVxsHVLbedfsHmygdfmnhkh

f ^dHh C⁰∪E⁰>²lhsVEogRkTVd)be`cbedfVLU

N dHRTVsHVV¬cmnbedHb+ojTklmqjTV± jT}dHhimybeh8UWhsH}TRTmybeU«³tUWop¬cmUo>±¤sLd

v³'zTsHoklgHRlmkTi}TsHhcgVLbHb'~cVkThdfVE~

za` (Unf(N), fN)hsxh8sbeRlhsfd

(UnfN, fN)FgLopnVE~µdfRTV:Oh

N^¡^k¶dfRTVbeV qjTVLC¤+RTVLk

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

β ¤+mnFz=Vmn~TVk8dHmlVE~¤+mdfR

O

$

mn¯Vk¶oghkTljTso0dHmh8k

κhpo«zTsHoklgRTmkl«}TsHhcgVEbfb

β ^h N ² mybxgLopnVE~Oo ! % h

κ ^mk β ^m

dHRTVsHVV¬amybedHb+bfhUWVgh8kcljTso0dHmnhk

κ⁰AκbfjlgHRdfRlopd

²=κ⁰\κ

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

( .576 8

φ

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

L .L ,/5/<

$>=

)>?

φ^P

)+

φ^T^@

(9)

C

Ã # 0c

Ã . Ä°L:9 - R9 =»hpo¢zTsHoklgRTmkl¢}TsfhTgVLbHb

β mybo§lkTmdfVObfVdh

V¯VLk8db

κ ⊆ E bfjlgHRdfR@o0dhsopn

e, e⁰ ∈ κ ¬(e#e⁰) opkl~hs«V¯VLsf`

e ∈ κ e⁰ ≺ e mnUW}TnmVEb

e⁰ ∈ κ ^mnk o~l~cmdHmh8kmd«mybWsHV qjTmnsfVE~.dHRlo0d

⊥∈ κ ²lhsV¯VLsf` V¯VLk8d

e ∈ EdfRTVOgh8kcljlsHopdfmnhk

[e]

={e⁰|e⁰¹e}myb)gLopnVE~«dfRTV @ ! R9 U =[h

e^lopkl~ hei

= [e]\ {e} ~cVLkThpdHVLbdHRTVbfVd+hp

R 7 R R ! |Q)RTV#beVd¨hpopnlkTmdfV±sHVLbf}Ftz@obfmngE³)gh8kcljTso0dHmh8klb)hp>oWzTsopklgRTmnkTW}lsfhcgVLbHb

β mnb|~cVklhpdfVE~za`

κ^β

fin

± sfVEbe}F

κ^β

bas

³lokl~dfRlVbfjT}=VLsHbHgsHm}Td

β mnb|~csHh}T}=VE~¤+RlVk

β=Unf_N

-

myb)obfVd)hpJgh8kl~cmdfmnhk

C⁰ bejlgRdfRlopd'hs+opn~cmybvdHmk@gd

c, c⁰ ∈C⁰ ckc⁰Topkl~o-tmyb)o Uo0¬cmnUWo=ghp´beVdthsdfRTVbeVd'mk@gnjlbemnhk Vd

κz=Vo#ghkcl8jTsHopdfmnhkWdHRTVk

Cut(κ)

=¡

C⁰∪κ^•¢

\^•κ

myboOgjTd jTsfdfRTVLsfUWhsHVdHRTVbfVdh+}TyogVLb

f(Cut(κ)) mnboOsfVLo8gHR@opzTnVUopsH¥amklµhp

N>¤+RTmngR myb

~cVLkThpdHVL~¶za`

Mark(κ) ^- UWosf¥amnkT

M ^hp N ^myb ^! ² ^¶mnk β mdfRTVLsfVWmnbogh8kcl8jTsHopdfmnhk

κ

hp β bfjlgR&dfRlopd

M =M ark(κ)4t¯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 /nLTT8!1TdHhgh8}=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¥amkl8b³

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

¿

Ã ·gjcdfdfmnkT«ghkqdfV¬qd

Unf_N ^! Θ =¡

≈,C,{κ_e}_e∈E¢ ^P

≈ ^F ^<@ ⁰ ^=> κ_fin

!

C ^{R %} .o~cV8j@o0dfV 0 ! ,

R = .UC

κ_fin

U

⊂

κ⁰⊂κ⁰⁰ ^=! κ⁰Cκ⁰⁰

"

≈ C }TsfVEbeVLsf¯VE~ zq` @kTmdHV¢V¬adfVk@bemnhklb

!>< ; >U R =

κ⁰≈κ⁰⁰

I < R %F

²⁰ κ⁰

? 0 C P ! %

²⁰⁰ κ⁰⁰ ^R2-?

#

%$ κ⁰⁰⊕²⁰⁰≈κ⁰⊕²⁰

# &$

κ⁰⁰Cκ⁰ ^? κ⁰⁰⊕²⁰⁰Cκ⁰⊕²⁰

'

{κ_e}_e∈E ^B^: ^{J !2!} ⁼ κ

fin

Q)RTVio~cV qjlopdfV#hs~cVsxbf}=VLgmlVEb¨¤+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}=VLgm`dfRTVibeVdxhptgh8kcljTso0dHmnhklb+jlbfVL~µyo0dfVLs|dHh~TVLgmy~cV#¤+RTVdfRTVLs[opkOV¯VLk8d[gok

z=V~cVEbemnkTVE~ob|oWgjTde´£h ¶V¯8Vk8dEtQ)RTVgjcdedHmkTghkqdfV¬qd

ΘERV =©

≈mar,C_tot,{κ_e=κ_bas}_e∈Eª

gh8sfsHVLbf}=hkl~Tb[dfh&dHRTVsopUWV¤h8sf¥Omnk /7@1£¤+RlVsHV

≈mar

mybdHRTV«V qjTmn¯0oVLklgVsfVLnopdfmnhk¢h8k¢sHVLo8gHR@opzTnV

UopsH¥amkT#hp

N^amC^V κ⁰≈marκ⁰⁰ ^m Mark(κ⁰) =Mark(κ⁰⁰)^copkl~ C_tot mnb'odHhpdop5o~cVqjlo0dHV|h8sH~TVsE

VLo8bemnzTnVWV¯VLk8dHbopsHViV¯VLk8db¤+RThqbeVgojlbfo}TsHVL~cVEgVLbHbfhsb[opsHVikThpd#bvdo0dHmngWgjcde´£h ©V¯VLk8dbopk@~¶dfRqj@b

opsHV[mnklgnjl~cVE~mkdfRTV}lsfVT¬~cVdfVsHUWmklVL~zq`«dfRThqbeVgjcde´£h ¶VL¯Vk8db

¿

Ã )(+* VEobfmzTnVV¯VLk8db

fsble^Θ

? : U»bvdo0dfmyggjTde´£h

V¯VLk8db

C

cut^Θ

,0 = J < = J

Unf_N ^C=, ^RB ² ^<@

P..

, ,

6; <@ 2

e ^R ^<@ ²³ hei ∩cut^Θ =∅

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

L546/7894:736<;/=?><@A 8<BDCE6<;

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

$ 3 $ F 5 04)+;,

@IH

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

(U$

3L.1,

$ 0 ' .ON<FP

)<F4,84,.L (

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

$3(

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

$R(

<

=J' 6RQ

'@$

,.( @L

(10)

!

<

e ^< ^{R @ !}

4P #

-R % ¢ghsHsfV´

be}=hk@~cmkl $ 9 =

κ∈κ_e

R2 P

κ⊆fsble^Θ\cut^Θ

κ≈[e]

κC[e] ^#⁶

<

e⁰ ^U ^! : @ |gh8sfsHVLbf}=hkl~cmnkTVL¯Vk8dB

κ= [e⁰]^$

* P R! R

Pref_N^Θ ^2R ⁴ ^? ^I ^UI ^<@ ² fsble_N^Θ ^{R %} PxgLopkTh8kTmngLopq}lsfVT¬

U

UnfN

+¨hpdHVdfRlopd

Pref_N^Θ mybjTkTm(qjTVn`.~cVdfVsHUWmklVL~ zq`ªdfRTV&gjcdfdfmnkT¢ghkqdfV¬ad

Θ ^aV¯VLsHotjTkl~ToUiVLk8dHo

}lsfh8}=VsfdfmnVLbhp

Pref_N^Θ Rlo¯V|z=VLVk«}TsHh¯VLkWmnk /n1B¡ k«}losedHmngjTyopsE

Pref_N^Θ mybopn¤'o`cbghUW}TnVdfV¨¤sEdE

cut_N^Θlopk@~md|myb)lkTmdHVm

≈Rlo8b'lkTmdHVn`Uokq`«VqjTm¯popnVklgVgno8bfbfVLb)okl~

κ_e⊇κ_bas

&«+ J|"

wxmybvdHsfmnzTjcdHVL~«bf`cbvdHVUbtopsHV+UWhc~cVLVE~Wzq`WobfVdhpmnk8dHVsghkTklVLgdHVL~gh8UW}=hkTVLk8dHbtmkqdfVsogdfmnkTdfRTsHhjT8R

bfRlosfVE~bfjTzc´be`cbedfVUbLlgLopnVE~mkqdfVLse o8gVLbL'Q)RTmyb|UWhc~cjTyops¨bedfsHjlgdHjTsHVmnb¨onbfhgLopnVE~&o¸v ogdfh8sfmybfopdfmnhk

}lsfh8}=Vsfd`8¹+hdfRlV[be`cbedfVU&a~cjlV¨dfhdHRTVxjlbfV¨hF}TsHhc~cjlgdHbx± Uih8sfV¨}TsHVLgmnbfVn`ihp}TjTnzlo8gH¥cb³dfhigh8kTkTVLgd

ghUW}=hklVk8db Q)RTmyb o8gdfh8sfmybHo0dHmh8k }Tsfh8}=Vsfd `.mnbWmklRTVsHmdHVL~©za`.dfRTV&jTkch8n~cmnkTªhp¨dfRTVµ~cmybedfsHmzTjTdfVL~

bf`cbedfVU&T¤+RTmngRUo¥VLb)md|}=hqbfbfmnzTV[dfh}TsHhcgVLbHb)bfjlgHRµbe`cbedfVUb)mnk&oiUWhc~TjTnos)UokTkTVs /D1B

'*) 66:.,1

..TF:9 -·~cmybvdHsfmnzTjcdHVL~&bf`abedfVLU/myb'h8sfUon`«V¬c}TsHVLbHbeVE~o8b+oWbe}=VEgmyopFgo8beVhpoi}Tjlnzlog¥/1£

¿:

Ã

A,B C ^{B RR:} ⁼

P

A B ⁰ ^R ^P ^R

C ² φA:A →C φB:B →C ^{P (} ⁼ ^R ^=P ^UR<

@ R !

ΛC⊇ΛA∩ΛB

* i}ljTnzlo8gH¥ ? !

R P( #R =

A^φ→^AC^φ← B^B

C

N =A ×^C_N B ^C=, ^R ^:⁴

P = {(p_A, ?) :pA∈PA, pA∈/ Dom(φA)}

∪ {(?, p_B) :p_B∈P_B, p_B∈/Dom(φ_B)}

∪ {(p_A, p_B)∈P_A×P_B:φ_A(p_A) =φ_B(p_B)}

±v³

P !=P

T = {(t_A, ?) :t_A∈T_A, t_A∈/ Dom(φ_A), λ_A(t_A)∈Λ_A\Λ_B}

∪ {(?, t_B) :t_B∈T_B, t_B∈/ Dom(φ_B), λ_B(t_B)∈Λ_B\Λ_A}

∪ {(t_A, t_B)∈T_A×T_B:φ_A(t_A) =φ_B(t_B)}

±C8³

* P < = %E B R : , P0 ,P= U P R!

-xbHbehcgmnopdfVE~WdfhdfRTmyb'}TjTnnzlog¥«gh8klbvdHsfj@gdfmnhk8dfRTVLsfV[V¬cmybvd+gopklhkTmygo@UWhsH}TRTmybfUWb

ψ_A:N → A^okl~

ψB:N → BdHRlo0d>UWo}VnVUWVkqdHbhp

N dfhxdHRTV+ghsHsHVLbf}@h8kl~cmkl|VLVLUWVk8dbJmnk

A^opkl~ BpsHVLbf}=VLgdfmn¯Vn`

okl~dHRlo0d¨bHo0dfmybe`

φA◦ψA=φB◦ψB

L

¿ Ä- À ¿ :9 - kTVd

N mybxbHopmy~dfhz=Vo; ! @ mmd[gokµz=V#V¬a}lsfVEbfbfVL~

o8b+oi}ljTnzlo8gH¥hbeVL¯VsopFghUW}=hklVk8db)¤+RTVLsfVdfRTVmnk8dHVsHUWVL~cmyopsH`«kTVdb|opsHVmkqdfVsf ogVLbLQ)RTVnhzlo

=; !2 h>o~cmybedfsHmzTjTdfVL~µbf`cbvdHVUgok&z=VsfVL}TsHVLbfVk8dHVL~&ob|osop}TRFT¤+RlVsfV#opk&VE~cV

myb~csHo¤+k¶z=Vdv¤VVLkOdv¤h&ghUW}=h8kTVk8dbxmdHRTV`ORlo¯VoghUWUWhk¶mnk8dfVLse o8gV8²lhsbfmUW}Tnmygmd`Omk¶dfRlmnb

}@op}=Vso~TmnbedfsHmzljcdfVE~¢be`cbedfVLU

N mybmnUWmdfVL~§dfhdv¤hµgh8UW}=hkTVLk8dHb

A ^okl~ B opkl~¢opk¢mnk8dfVLse o8gV

C

¤+RlmngR gLopk.dHRTVk z=VV¬adfVk@~cVL~ dfh¢o¶sop}TRFª²JmnjTsHV¶~cVL}TmngdHb

N = A ×^C_N B ¤+mdfRobHbehTgmyo0dfVE~

UWh8sf}TRlmnbfUb