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�
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
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=Jl O\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`¶h8zcdHomk.bej@gHR.o&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#à 0Ä8 rJVdHsfm|kTVdEjTkch8n~TmkT@tlkTmdfVOghUW}TnVdHV}lsfVT¬~cmybvdHsfmnzTjcdHVL~bf`abedfVLU&t}TjTnnzlog¥5
gLo0dHVh8sf`WdfRTVLhsH`
($)*«+//'#"%$&'/()$ 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$L¯amdopk8d~cVWgh8klbvdHsHjTmsHV~opz=hs~µjTk§}TsT¬cVighUW}TnVd~Topklb[V
~L}Tmyop8V~cj be`cbedfSLUiV8h8zlop>}=h8jTsVLklbfjTmdHVnVW o8gdfh8sfmybfVsEXVdedHVghk@bvdHsfjlgdfmnhkª~cV«}TsT¬cVLbbeh8jlb
hsHUWV«UWhc~cjTyopmnsfVbeV«hkl~cVbfjTs#yoµkThpdHmh8k ~cV«sLbfjTU%~cj©gh8Ui}=h8sedHVUWVkqd~jlk©ghUW}=hqbfok8d}los
sop}T}=h8sed&"Obfhk»±CbeVEbH³mkqdfVLse o8gV8±CbH³§XVLb#sEbejlU'Eb#beh8k8dghUWU#jTkTm(qjLbopjc¬.gh8Ui}=hqbfok8db¯hmybfmklbL
qjTm+VEbimnk8dHSsHVkqd)"§nVjTsW}lsfh8}TsfV~}TnmnoV}=hjTsghklbedfsHjTmnsfV~cVEbisLbfjTU%Lbi¯VsbnVLbgh8Ui}=hqbfok8db
bfjTm¯popkqdHbL8Vd'omklbfm@~TV¨bfjTmdfV8 V+dHhjcd'}TsfVLkl~Wyoh8sfUWV¨~*jTko8hsHmdfRTUWV+"gmsgjlnopdfmnhk~TV|UWVEbfbHop8VLb
VkqdfsHVgh8UW}=h8bHopk8dbch,nVbf`abedfSLUWVnhzlo5k*VLbedueoUopmyb)UWokTmn}TjT
- Ã L Á /. © sEbeVLoj©~cVrVdfsHmBt~}TnmnoV8>}Tsl¬aVlkTm+ghUW}TnVdEtbf`cbvdHSUWV~cmybedfsHmzTj}lsfhc~TjTmd
lzTscdfRLhsHmV~cVEb+gLo0dLh8sfmnVLb
t 2
t 1
a 0
a 1
b 1
b 0
c 0
c 1
t 3
t 4
t 7
t 5 t 6
Component Component
Interface
A B
#%$&
N
t 4
t 4
c 0
c 0
a 0
t 1 t 2
t 3
t 3
a 1
c 1
a 0 a 0 c 1
c 0
a 1
...
#%'&
Unf A
t 5 t 6
b 1
a 0
c 1
b 0
t 4
b 1
c 1 a 0
b 1
c 1 a 0 t 4
t 4
t 7
c 0
t 3
c 1
...
#(&
Unf B
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:/nLTFE 1£
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~B
Fmnk8dfVLsHo8gdHmkTdHRTsHhjT8R¶ok¶mnk8dfVLse o8gV8Q)RTVLsfVV¬cmnbed
m = 2
}=hqbfbfmnzTmnmdfmnVLb¨dHh}TsfhT~cjlgVt 4
mk
Unf A
±C²mn8jTsfV± z@³e³ okl~n = 3
}=hqbfbfmzlmnmdHmnVLbdfh©}TsfhT~cjlgVt 4
mnk
Unf B
±C²mn8jTsfV 8±CgL³e³|RTVk@gVm · n = 6
~cm=VLsfVLk8d ghU#zTmnklopdfmnhklb)mnkdHRTVjTkchy~cmnkThpdfRlVnhzlobe`cbedfVLUQ)RTV+ ogdfh8sfmybfopdfmnhki}TsHh}=VLsedv`#hpjTkchy~cmnkT8b¤)oblsbedUWVkqdfmnhkTVE~Wzq`BA mklbf¥V@mk/nL1BqzTjcdhkTn`
h8jTkl~imdHb>lsHbedtop}T}Tnmygo0dHmh8klb>opsHhjTk@~i C7T0mk~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
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, →, P 0 )
bejlgR dfRlopdP
opkl~T
opsHV&~cmnb uvhmnk8dbeVdHbWh% R! opkl~; P != qsHVLbf}=VLgdfmn¯Vn`
→⊆ (P × T ) ∪ (T × P)
mnb'o< =lTopkl~P 0 : P →
N = {0, 1, 2, ...}
mybxo«U#jTdfmybfVd[h8kP
sHV}TsHVLbfVk8dHmkldfRTV = &h>dfRTVklVdL²Th8s[o«kThc~cVx ∈ P ∪ T
amdHb• x
myb'~TVlkTVE~zq`• x = {y | (y, x) ∈→}
okl~«mdHbx •
myb)~cVlklVL~zq`x • = {y | (x, y) ∈→}
>¡kdfRTmyb)¤h8sf¥WdfRlVkTVdb+opsHVxnmUWmdfVE~dfh: UklVdHbLcmBVhs)V¯8Vsf`sHVLo8gHRlozTnV Uosf¥amnkTM
opkl~VL¯VsH`}Tyo8gVp ∈ P
M (p) ⊆ {0, 1}
- noz=VnVL~kTVd
N = ¡
P, T, →, P 0 , λ, Λ ¢
mnbo[kTVdV¬adfVk@~cVL~i¤+mdHRo[noz=VTbfVdΛ
okl~io[yopz=VnnmkTjlklgdHmh8k
λ : T → Λ
hkdHsHoklbemdHmh8klb-Ã
9
- UWhsH}TRTmybfU
φ : A → B
z=Vdv¤VVLk kTVdbx = ¡
P x , T x , → x , P x 0 ¢
/n71£'¤+RTVLsfVx ∈ {A, B}
mnbo}lopmns¡ φ P , φ T ¢
¤+mdfRφ P
osHVyo0dfmnhk§hk¢}TyogVLbokl~φ T
o}losedHmyopJjTklgdfmnhkªhk dHsopklbfmdHmh8klbEªQ)RTVmnkTmdfmyop'Uosf¥amnkTOmyb}TsHVLbfVsH¯VL~ªza`φ
ob#hnh0¤|bP B 0 = φ ¡ P A 0 ¢
opk@~∀p B ∈ P B 0 , ∃!p A ∈ P A 0 : p A φp B
>¡£
φ
mnb|~cVlkTVE~h8kp A ∈ P A
cdfRlVk&md|myb+opybfhW~cVlklVL~hkz=hpdHR
• p A
okl~
p • A φ
}TsHVLbfVsH¯VLbdfRTVVkq¯amnsfh8kTUWVk8dhp|VLo8gHR.dfsopklbfmdfmnhk*t B = φ (t A )
mnUi}lmnVLb#dfRlopdsfVEbvdHsfmygdHmnhklbφ op : • t B → • t A
opkl~
φ op : t • B → t • A
opsHVxz=hdfRdfhdHop5jTklgdfmnhklbLc¤+RTVLsfVφ op
~cVkThdfVEbdfRlVsfVL¯VsbeVE~sHVyo0dHmnhkhk}Tyo8gVLbL.+|hdfmygVdfRlopd+kTVd|UWhsH}TRTmybeUb)}TsHVLbfVsf¯8VxsHjTklbL
²Th8syopz=VnVL~§kTVdb
x = ¡
P x , T x , → x , P x 0 , λ x , Λ x ¢
dHRTV~cVlkTmdfmnhk¢h'oUWh8sf}TRlmnbfUφ : A → B
myb|sfVLmkThsgVL~zq`V¬adHsHoWsfVqjTmsHVUWVLk8dHb
Λ B ⊆ Λ A
±dfRTV#noz=VbfVd¨myb+sHVL~TjlgVL~&zq`
φ
³Dom (φ T ) = λ −1 A (Λ B ) ⊆ T A
±
φ
myb~cVlkTVE~V¬co8gdH`xhkdfsopklbfmdHmh8klbRlo¯amkT|o|bfRlopsHVL~[noz=V¼³okl~mφ T (t A ) = t B
dHRTVk
λ A (t A ) = λ B (t B )
±yopz=VF}TsHVLbfVsH¯amkTa³&'& 0 1 0 , 6 6 , 6
Q¤hklhc~cVLbxhokTVd
N
y
opkl~y 0
FopsHV#mnk !2 < C~cVkThdfVL~¶zq`y#y 0
mtdfRTVsHVV¬cmnbed~Tmnbedfmnklgd|dHsHoklbfmdHmh8klb
t, t 0 ∈ T
bfjlgRdfRlopd• t ∩ • t 0 6= ∅
lokl~(t, y)
opkl~(t 0 , y 0 )
osfVmkdHRTVsfVlV¬cmn¯V dHsopklbfmdHm¯Vgnh8bfjTsHV[hpdHRTVlh0¤-sHVyo0dHmh8k→
l~cVLkThpdHVL~zq`¹
«/ ¿ ¿J¿ L:9 -|k ! R mybio¶kTVd
O = (C, E, →, C 0 )
¤+RTVLsfVC
mybio§bfVdWhpR =P ± }Tno8gVEbH³
E
myb¨dfRTVibfVd[hp6 <@ 2 ±¼dHsHoklbemdfmnh8klbH³|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@kTmdHVn`Uokq`y 0
bej@gHRdfRlopdy 0 ≺ y
¤+RTVsHV≺
~cVkThdfVEbdHRTV =ldHRTVdHsHoklbemdHm¯VmnsfsHV lV¬cm¯VghqbejlsfVh
→
)Q¤h«kThc~cVLb¨opsHVR ! 2B@~cVkThdfVL~y k y 0
@m>kTVLmdHRTVsy#y 0
klhsy ¹ y 0
kThsy 0 ¹ y
! 0
"
"#
à ¿ L ¿ 9 - U R |hpo[kTVdtbf`cbvdHVU
N
mybo[}@opmnsβ = (O, f)
8¤+RTVsHVUWh8sf}TRlmnbfU
f : O → N
mybo¨dfhdHo8jlklgdHmh8kh8kO
onbfh[gopnnVL~#o#hpO
mnk8dfhN
JQ)RlmnbJ¼h8y~cmkTgLopk«z=V[beVVLkoboyopz=VnmnkT¼jlklgdHmh8khk«V¯VLk8dHb'opk@~«ghk@~cmdHmh8klbthp
O
azq`W¤+RTmygHRgh8kcljlsHopdfmnhklbhpO
sHV}TsHVLbfVkqdHb)sfjlklb+hpN
¡£d¨mnb)jTsfdfRTVLs|sHV qjTmnsfVE~«dHRlo0dβ
bfopdfmybv`oW}lopsbemnUWhka`«ghkl~cmdfmnhk>hs¨opne 1 , e 2 ∈ E
=m• e 1 = • e 2
okl~
f (e 1 ) = f (e 2 )
dfRTVLke 1 = e 2
|Qh~cVlkTVlklmdHVgh8Ui}TnVdHV}TsHVl¬aVEb
md¨¤+mnz=VjlbfVjlFdfh«ghklbfmn~cVLs|oµ±¯amnsfdfjloy³'mnkTmdfmyopV¯VLk8d
⊥
mnkβ
l¤+RlmngR&Rlo8b|opk&VUW}cdv`}TsHVLbfVdLC 0
o8b)}=h8bede´beVd+okl~kThWyopz=VL±mBV>klhWmUoV[za`
f
³- zTsopklgRTmnkTª}TsHhcgVLbHbµ±=H'r³
β 0
hpN
mnbo »hoªzTsopklgRTmnkTª}TsHhcgVLbHbβ
+~TVkThdfVL~ zq`β 0 v β
qmO 0
mybtogojlbfon`ghqbeVE~WbfjTzc´£kTVdthO
ghkqdHomkTmnkT#opnlmklmdHmno@ghkl~TmdHmh8klbtopkl~bfjlgHRdfRlopd∀e ∈ E, e ∈ O 0
mnUi}TnmnVLbe • ⊆ O 0
okl~f 0
mnbtdfRTVxsHVLbedfsHmygdfmnhkhf
dHhC 0 ∪ E 0
>²lhsVEogRkTVd)be`cbedfVLUN
dHRTVsHVV¬cmnbedHb+ojTklmqjTV± jT}dHhimybeh8UWhsH}TRTmybeU«³tUWop¬cmUo>±¤sLdv
³'zTsHoklgHRlmkTi}TsHhcgVLbHb'~cVkThdfVE~za`
(Unf(N ), f N )
hsxh8sbeRlhsfd(Unf N , f N )
FgLopnVE~µdfRTV:OhN
¡k¶dfRTVbeV qjTVLC¤+RTVLkh8n~TmkTqb+opsHV[jTk@opU#zTmnjTh8jlb'dfRTVL`¤+mnFz=VhUWmdfdfVE~lbfh
β
¤+mnFz=Vmn~TVk8dHmlVE~¤+mdfRO
$
mn¯Vk¶oghkTljTso0dHmh8k
κ
hpo«zTsHoklgRTmkl«}TsHhcgVEbfbβ
hN
²
mybxgLopnVE~Oo ! % hκ
mkβ
mdHRTVsHVV¬amybedHb+bfhUWVgh8kcljTso0dHmnhk
κ 0 A κ
bfjlgHRdfRlopd² = κ 0 \ κ
&('*)+-,/.102431.
( .576 8
φ
.1,:9-+;.5/5/<
L .L ,/5/<
$>=
)>?
φ P
)+
φ T
@ C
à # 0c
à . Ä°L:9 - R9 =»hpo¢zTsHoklgRTmkl¢}TsfhTgVLbHb
β
mybo§lkTmdfVObfVdhV¯VLk8db
κ ⊆ E
bfjlgHRdfR@o0dhsopne, e 0 ∈ κ
¬(e#e 0 )
opkl~hs«V¯VLsf`e ∈ κ
e 0 ≺ e
mnUW}TnmVEbe 0 ∈ κ
mnk o~l~cmdHmh8kmd«mybWsHV qjTmnsfVE~.dHRlo0d⊥∈ κ
²lhsV¯VLsf` V¯VLk8de ∈ E
dfRTVOgh8kcljlsHopdfmnhk[e]
= {e 0 |e 0 ¹ e}
myb)gLopnVE~«dfRTV @ ! R9 U =[he
lopkl~hei
= [e] \ {e}
~cVLkThpdHVLbdHRTVbfVd+hpR 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~¤+RlVkβ = Unf N
-
myb)obfVd)hpJgh8kl~cmdfmnhk
C 0
bejlgRdfRlopd'hs+opn~cmybvdHmk@gdc, c 0 ∈ C 0
c k c 0
Topkl~o-tmyb)o Uo0¬cmnUWo=ghp´beVdthsdfRTVbeVd'mk@gnjlbemnhk Vdκ
z=Vo#ghkcl8jTsHopdfmnhkWdHRTVkCut(κ)
= ¡
C 0 ∪ κ • ¢
\ • κ
myboOgjTd jTsfdfRTVLsfUWhsHVdHRTVbfVdh+}TyogVLb
f (Cut(κ))
mnboOsfVLo8gHR@opzTnVUopsH¥amklµhpN
>¤+RTmngR myb~cVLkThpdHVL~¶za`
Mark (κ)
- UWosf¥amnkTM
hpN
myb ! 2 ¶mnkβ
mdfRTVLsfVWmnbogh8kcl8jTsHopdfmnhkκ
hp
β
bfjlgR&dfRlopdM = M ark(κ)
4t¯VsH`UWosf¥amnkTsfVL}TsfVEbeVLk8dHVL~mkβ
mnb¨sfVEogRTVL~mkN
=opkl~&V¯VLsf`sHVLogRlozTV[UopsH¥amklihp
N
mnb+sfVL}TsHVLbeVLk8dHVL~mnkUnf 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
¿
à ·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
⊂
κ 0 ⊂ κ 00
=!κ 0 C κ 00
"
≈
C
}TsfVEbeVLsf¯VE~ zq` @kTmdHV¢V¬adfVk@bemnhklb!>< ; >U R =
κ 0 ≈ κ 00
I < R %F
² 0
κ 0
? 0 C P ! %
² 00
κ 00
R2-?#
%$
κ 00 ⊕ ² 00 ≈ κ 0 ⊕ ² 0
# &$
κ 00 C κ 0
?κ 00 ⊕ ² 00 C κ 0 ⊕ ² 0
'
{ 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}losHoUiVdfVLsκ 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
Θ ERV = ©
≈ mar , C tot , { κ e = κ bas } e∈E ª
gh8sfsHVLbf}=hkl~Tb[dfh&dHRTVsopUWV¤h8sf¥Omnk /7@1£¤+RlVsHV
≈ mar
mybdHRTV«V qjTmn¯0oVLklgVsfVLnopdfmnhk¢h8k¢sHVLo8gHR@opzTnV
UopsH¥amkT#hp
N
amCVκ 0 ≈ mar κ 00
mMark (κ 0 ) = Mark (κ 00 )
copkl~C tot
mnb'odHhpdop5o~cVqjlo0dHV|h8sH~TVsEQ)RlVObedHopdfmygµgjcdf´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@~¶dfRqj@b
opsHV[mnklgnjl~cVE~mkdfRTV}lsfVT¬~cVdfVsHUWmklVL~zq`«dfRThqbeVgjcde´£h ¶VL¯Vk8db
¿
à )(+* 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 <@ 23hei ∩ 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
!
<
e
< R @ !4P #
-R % ¢ghsHsfV´
be}=hk@~cmkl $ 9 =
κ ∈ κ e
R2 P
κ ⊆ fsble Θ \ cut Θ
κ ≈ [e]
κ C [e]
#6<
e 0
U ! : @ |gh8sfsHVLbf}=hkl~cmnkTVL¯Vk8dBκ = [e 0 ]
$* P R! R
Pref N Θ
2R 4 ? I UI <@ 2fsble N Θ
R % PxgLopkTh8kTmngLopq}lsfVT¬U
Unf N
+¨hpdHVdfRlopd
Pref N Θ
mybjTkTm(qjTVn`.~cVdfVsHUWmklVL~ zq`ªdfRTV&gjcdfdfmnkT¢ghkqdfV¬adΘ
^aV¯VLsHotjTkl~ToUiVLk8dHo}lsfh8}=VsfdfmnVLbhp
Pref N Θ
Rlo¯V|z=VLVk«}TsHh¯VLkWmnk /n1B¡ k«}losedHmngjTyopsEPref N Θ
mybopn¤'o`cbghUW}TnVdfV¨¤sEdEcut N Θ
lopk@~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 RC
2φ A : A → C
φ B : B → C
P ( = R =P UR<
@ R !
Λ C ⊇ Λ A ∩ Λ B
* i}ljTnzlo8gH¥ ? !
R P( #R =
A φ → A C φ ← B B
C
N = A × C N B
C=, R : 4P = {(p A , ?) : p A ∈ P A , p A ∈ / Dom (φ A )}
∪ {(?, 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 )}
±C 8³
* P < = %E B R : , P0 ,P= U P R!
-xbHbehcgmnopdfVE~WdfhdfRTmyb'}TjTnnzlog¥«gh8klbvdHsfj@gdfmnhk8dfRTVLsfV[V¬cmybvd+gopklhkTmygo@UWhsH}TRTmybfUWb
ψ A : N → A
okl~ψ B : N → B
dHRlo0d>UWo}VnVUWVkqdHbhpN
dfhxdHRTV+ghsHsHVLbf}@h8kl~cmkl|VLVLUWVk8dbJmnkA
opkl~B
psHVLbf}=VLgdfmn¯Vn`okl~dHRlo0d¨bHo0dfmybe`
φ A ◦ ψ A = φ B ◦ ψ B
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}=hkTVLk8dHbA
okl~B
opkl~¢opk¢mnk8dfVLse o8gVC
¤+RlmngR gLopk.dHRTVk z=VV¬adfVk@~cVL~ dfh¢o¶sop}TRFª²JmnjTsHV ¶~cVL}TmngdHb
N = A × C N B
¤+mdfRobHbehTgmyo0dfVE~UWh8sf}TRlmnbfUb
C
1
1 2
a t
a
t 2
0 3
1
1 5
2 0 4
t
c t c
b t b t
b
t 3 0
t 2
c 1
t 3
c 0
t 1
a 0
a 1
t 2
c 1
t 3
c 0
t 4
b 0
t 5
b 2
b 1
t 2
c 1
c 0
t 3 N
A B
ψ A
φ A φ B
C
ψ B
²Jm8jTsHV >Q)RTV}TjTnz@og¥
N = A × C N B
('& . 021*6 .71( > /1 6
¡£d+¤'o8b)bfRTh¤+kmnk /nL@1dHRlo0d+dfRTV[ o8gdHhsHmnbfVL~«hsHU®hJoklVd|bf`cbvdHVU/`amVLn~Tb+o# o8gdHhsHmnbfVL~«hsHU®hdHRTV
jTkch8n~TmkThpbfjlgR&oWbf`abedfVLU& $ mn¯Vk
N = A × C N B
hkTVh8zcdHomk@bUnf N = Unf A × C O Unf B
±"8³
¤+RTVLsfV
Unf N
Unf A
okl~
Unf B
opsHVtdfRTV)jTkThy~cmkl8bJh
N
A
opk@~B
okl~× C O
~cVkThdfVEbdfRTV)}Tjlnzlog¥hkzTsHoklgRTmkTx}TsHhcgVLbHbeVEb¤+mdHR
Unf C
o8bmkqdfVLse o8gVJQ)Rajlb0dfRlV'jTkch8n~cmnkT[hTdfRlV'nhzloqbf`cbvdHVU gok
z=V¨V¬c}TsHVLbHbeVE~WobdfRTV¨}ljTnzlo8gH¥ih5dfRTVxjTkchy~cmnkT8bthmdHbghUW}=hkTVLk8dHbx± o8bmnnjlbedfso0dHVL~Wmnk²mn8jTsfV "8³
..T F9 Q)RTVghUW}=h8bfmdfmnhkza`}TjTnzlo8gH¥hd¤hzTsopklgRTmnkTi}TsHhcgVLbHbeVLb'myb+~cVLsfmn¯VL~o8b'hnh0¤|b
±sHVVLs'dfhW²JmnjTsHVJ"³ FVd
β A = (O A , f A )
okl~β B = (O B , f B )
z=V[zTsopklgRTmnkT}TsHhcgVLbHbeVEbthpA
opkl~B
5sfVEbe}=VEgdfmn¯VL`8¨Q)RTVUih8sf}lRTmnbfUφ A ◦ f A
sHVyo0dHVLb+dHRTVhcggjTsfsHVklgVkTVd
O A
dfhkTVdC
behzq`dHRTV jTkTmn¯VsbHop}TsHh}=VLsedv`«hUnf C
TdHRTVsHVV¬cmybvdb+oWjTklmqjTVUWhsH}TRTmnbfU
φ O A : O A → Unf C
dHRlo0d¨VklbfjTsHVLb
φ A ◦ f A = f C ◦ φ O A
amBVJdHRlo0d)Uop¥VEbdHRTV~cmyop8sHoU¦gh8UiU#jTdHo0dHm¯V8.H`«bf`aUiUWVdHsH`qh8kTV[onbfh#Rlo8b'o UWhsH}TRTmybeUφ O B : O B → Unf C
$ mn¯Vk&dHRTVLbfVUWhsH}TRTmybfUWb+dfh
Unf C
ldfRTV #R hp
O A
okl~O B
mnkdHRTVgopdfV8hsH`WhJhcggjTsfsHVklgVxklVdHb+myb+~TVsHm¯VE~«sHhU/dfRTVh8sH~cmnklosf`}TjTnnzlog¥«hJkTVdb+zq`
O A × C O O B = Unf ³
O A × Unf N C O B
´
± q³Zµh8sfVLh¯VLsLamd|myb+VEobfmn`gRTVLg¥VL~dHRlo0d|dfRTVsHVV¬cmybvdb|oh8n~cmnkT
f
sHhUO A × C O O B
dfR@o0d|Uo¥VLb)mdxo zTsopklgRTmnkTi}lsfhcgVLbHb'hpA × C N B
,+|hdfmygV[dHRlo0d#± q³'VkqdHomyb'dHRTVV¬cmybedfVk@gVhp>oisfVEgjTsbfm¯V[}TsHhcgVE~cjTsHV dfhgh8Ui}ljcdfV¶dfRTV§}ljTnzlo8gH¥ hpzTsopk@gHRTmnkT©}TsfhTgVLbHbfVLbL)zlo8beVL~ h8k dfRTV§sHVLgjTsHbfmn¯VOghklbedfsHjlgdHmnhk hjTkch8n~TmkTqb+okl~hkh8sfU#jTyopV±v ³
ÃE¿
à :9 $ mn¯Vk
Unf N = Unf A × C O Unf B
beh8UWVkThc~cVEbxhUnf N
opsHVnoz=VnVE~Ozq`O}TyogVEb opk@~dHsHoklbemdHmh8klb>hpC
dHRTsHhjT8Ropkq`h=dHRTV± gh8UWU#jcdfmnkTq³>}lopdfRlbsHVyo0dHmklUnf N
dfh
C
mnk«²Jm8jTsHVI"TQ)RTV0 ! =&h
Unf N
dfh#dfRlVLbfVxkThc~cVEbtmyb'~cVLkThpdHVL~«za`
(Unf N ) |C
.H'`gh8k8dfsobedL8dfRlV U hpUnf N
hkz=VLRloE¯amnhjTsb'hp
C
mnb|~cV@kTVL~&ob'dHRTVmUoVhpUnf N
mk
Unf C
Π C (Unf N ) = φ O A ◦ ψ A O (Unf N ) = φ O B ◦ ψ B O (Unf N ) .
±7³f
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
²JmnjTsHV6"XhUWU#jcdo0dHm¯Vx~TmnosopU hpdHRTVx}TjTnzlo8gH¥cbthFzTsopklgRTmkl}TsHhcgVEbfbfVLbtopkl~klVdHbLaokl~WdfRlVmns
sHVyo0dHmnhkzq`o8bfbfhcgmyo0dHVL~«¼h8y~cmkTqb
-¨dfVsHklopdfmn¯Vn`8dfRTmybt}TsHhpuvVLgdfmnhk«gLopkz=VxhzcdopmnkTVL~zq`idHo¥qmnkT@sHbed
(Unf N ) |C
opkl~WdfRlVk«}=VLseh8sfUWmnkTo ! 0
mCVzq`µUWVsHmnkTmybfhUWhsH}TRTmyg#gh8kcljlsHopdfmnhklb¨mk
(Unf N ) |C
mnk¶h8sH~TVs¨dHh8Vdo¯popnmn~zlsHoklgHRlmkT#}lsfhcgVLbHbh
C
rsHh0uvVEgdHmh8klbhkA
opkl~B
hUnf N
opsHVx~cVlklVL~bemnUWmyopsH`obmdbmnUopVEb za`ψ A O
opkl~ψ O B
sHVLbf}=VLgdfmn¯Vn`rsHhpuvVLgdHmh8klb)klopdfjTsopn`«V¬qdHVkl~dfh«opkq` H'rhN
+|hdfmygVdfRlopd¨dfRTV#sHVLbedfsHmygdfmnhk&h
Unf N
dfhobejTz@beVdxh>kThc~cVEb|UoE`VsobfVgopj@bfomdv`h8sxghklmngd sHVyo0dHmnhklb|z=Vd¤'VVkOdHRTVLbfV#kThT~cVLbL=¤+RTmygHR¶UoE`dfRlVk§op}T}=VEops[obxgh8klgjTsHsHVk8dxmk(Unf N ) |C
¤+RTVLsfVEob
dHRTV`W¤'VsHV|klhpdmnk
Unf N
>Q)RTVxbHopUWV¨}TRTVklhUWVkTh8k«hcggjTsHbto8bt¤Vn=¤+mdHR«}lsfhpuvVLgdfmnhklbLQ)RTVsHVh8sfV¨o }lsfhpuvVLgdfmnhkΠ C (Unf N )
mnbxbfomn~dfh«z=V 0 !R ©m¢VL¯VsH`ghkcl8jTso0dfmnhkκ 0
mnkΠ C (Unf N )
myb'dfRlVmnUWoV[hJoWghkc@jTso0dHmh8k«h
κ
mnkUnf N
lopkl~gojlbfomdv`sHVyo0dfmnhk@bh8kV¯VLk8db)hpκ 0
osfV[kThdnhqbvdEYxzlbfVsH¯VdHRlo0d}lsfhpuvVLgdfmnhklbhk¢mnk8dHVsf ogVEb
C
dHRlo0d#osfVopjTdfhUopdHoTmBV8«rJVdfsHmkTVdHb¤+mdfRlhjcd ghklgjTsHsfVLklg`8aosfVzq`~cVlkTmdfmnhk&kThkUWmybenVLo8~cmnkTlQ)RTV« ogd#dfRlopd#dfRTVbfmUW}TnV«}TsHhpuvVLgdfmnhklb#~cV@kTVL~ opz=h0¯VUoE`¢z=V«UWmnbfnVLo~TmkTOmybdfRlV«VEbfbfVkqdfmyop
sHVLo8beh8k§¤+Rq`¶dfRTmyb}lop}=VsnmUWmdHbmdbbHgh8}=VidfhObfjlgR.bemnUW}TV«mkqdfVsf ogVLbobopjcdHhUo0doT²Ths#bemnUWmyos
sHVLo8beh8klbL /n1mnk8dHsfhT~cjlgVE~> ! <9 !2 @5¤+RTmnV:/8?1}lsfh8}@hqbeVE~dfh¤h8sf¥¤+mdHR 2 R
; R! R |mk hs~cVLsidHh.}=VsfhsHU½UWhc~cjTyopsghUW}TjTdHo0dHmh8klbL»¡kqdfsHhc~cjlgmkT.RTVLsfV&dHRTVLbfV
dHVLgRTkTmygoasHV@kTVUWVkqdHb¤h8jTn~igVsfdopmnkT`z=V)}=hqbfbfmnzTV80zTjcd¤h8jTn~igh8klbemy~cVsopzl`hqo~~cV¯VLh8}TUWVk8db
¤+mdHR«dfVEgHRlkTmngLop5~TVdHomybtdfR@o0d)osfV¨kThd'sHVLopnn`Wo0ddHRTV[gVkqdfVs'hdHRTVbvdHjl~c`8¡£d'mybtdfRlVsHVhsHVxgRTh8bfVkdHh
hTgjlbh8k«dfRTVbf}=VLgm@g[~cm2«gjTdHmVEb'sfVLnopdfVE~WdfhihzcdopmnkTmnkT@kTmdHVgh8Ui}lVdfVx}lsfVl¬aVEb¤+mdHRo#UWhc~cjlnos
o}T}TsHh8o8gHRF@opd|dHRTV#V¬a}=VLklbfVhptonmUWmdfVE~ObfVdedHmkl«¤+RlVsHVmkqdfVLse o8gVkTVdHb[opsHVkThdxVklVsopFklVdHbxzTjcd
bfmnUW}TVopjcdHhUopdHoT
-
..
.. Jlµ Ã=¿
"# 9 rtsfhpuvVEgdfmnhk@bJopnnh¤ jlbdfh[zTjTmnn~dHRTVBP #R R<!
h
Unf N
=¤+RTmygHROsfVEbvdHsfmygdb)dHRTV ogdfh8sHb
Unf A
opkl~
Unf B
dfhdHRTV#z=VRlo¯amnhjTsb+hpghUW}=hkTVLk8dHb
A, B
dHRlo0d|sHVUopmnk¼VEobfmzlVmnkdfRTVVLk8dfmnsHVbf`abedfVLU
N
YxkTVgok¤+sHmdfVUnf N = Π A (Unf N ) × C O Π B (Unf N )
± q³okl~OVEogR 0
Π x (Unf N )
¤+RTVsHVx ∈ {A, B}
myb[o}TsHVT¬OhpUnf x
#Q)RTVEbeV ogdHhsb
osfV[UWmnkTmUomnkdfRTVbeVLklbeV[dfRlopd)dop¥amklWopkq`bedfsHmygd+}TsHVT¬hpdfRTVLU°¤hjln~}TsHV¯Vkqd)sHVLgh¯VLsfmnkT#dfRTV
jl
Unf N
za`}TjTnzlo8gH¥5ZµmnkTmUo5 ogdfhsb)gokz=Vgh8Ui}ljcdfVE~mk&oiUihT~cjTyops)UopkTkTVLs)¤+mdfRTh8jcd|ghUW}TjcdHmkl
Unf N
mdHbfVvY[kTVRlob
Π A (Unf N ) = Unf A × C O Π C (Unf B )
±?³±Copkl~ bf`aUiUWVdfsHmngLopn`.h8s
Π B (Unf N )
³ /T?1£-Q)RTmnbsHVyo0dHmh8k V¬a}lsfVLbHbfVLbdHRlo0d«¥akTh0¤+mnkT¢dfRTVµz=V´R@oE¯amnhjTsbh
B
hk«dfRlVxmnk8dHVsfo8gVxkTVdC
myb'bfj2«gmnVkqddHhA
dHhi~cVdHVsHUimnkTVx¤+RlmngRhpmdHb'sfjlklbsfVLUWomk }=hqbfbfmnzTV[mnkdHRTVnhz@op5bf`cbvdHVUN
|±?p³RTh8n~lbTopkl~«dHRqjlb|opnh0¤|bh8kTVxdfhW}=VsfhsHU/oUWhc~cjTyops+ghUi´}ljcdHopdfmnhkhJUWmkTmnUop o8gdfh8sHbLco8b+behchkob)}TsHhpuvVLgdfmnhklb)hk
C
opsHV[kTh8kUimybfVEo~cmnkTl>¡£d+myb+}TsfVEgmybeVL`L C
dfhh8zcdHomkWo[bfmUWmnnos}Tsfh8}=Vsfd `h8s>okq`mnk8dfVLse o8gV)kTVd
C
dfRlopd 2R R J¤VsHV~cVL¯Vnh}=VL~mnk /?1BT¤+RTmnnV /n1nopdfVs+}TsHh}=hqbeVE~dHRTVopdfVLsfk@o0dfV[dfVEgHRTklmqjTVhp !R < !2
²lhsUih8sfVghUW}TnV¬.~cmybvdHsfmnzTjcdHVL~ bf`cbvdHVUbJdHo¥amkTµdHRTVbfRlo}@Vhp¨oµkTVd ¤h8sf¥¢hp¨ghUW}=h8kTVkqdHbL
UWmklmUopl ogdfhsbgokWz=V|h8zcdHomkTVE~iza`V¬adHVkl~cmnkTdHRTV¨oz=h¯V+}TsHmk@gmn}TV|h5UWhc~cjlnostghUW}TjTdHo0dHmh8klbL
¡£ddfRTVLkdop¥VEb5dHRTVhsHU hpTo+UWVLbHbfoV}@obHbemnkT|opnh8sfmdfRlUE¤+RTmngRsHjTklbFh8k[dHRTVtmkqdfVsogdfmnhkbvdHsfjlgdfjTsHV
hp
N
opkl~}Tsfh8sHVLbfbfm¯Vn`|jl}5~To0dHVLbFmnkch8sfUo0dHmh8khkmnk8dHVsf ogVEbQ)RTVopnh8sfmdfRlU gokz=Vt}TsHh¯VE~[V¬Togdhs#bf`cbvdHVUbmn¯amnkTµhk.odfsHVVJokl~¢mybhkl`§o}T}TsHhE¬cmnUo0dfVWmnkªhdfRTVLsgLobfVLbL A¢VsHVVLsdHRTV«sfVEo~cVLs
dfh/T ?1Fhs|~cVdHomyb
W º' ' $)*«+
Q)RTV+hzcuvVLgdHm¯8V)hp@dHRTmyb>}lop}=Vs>mybJdHhgh8UW}TjcdfV)lkTmdfV|okl~igh8UW}TVdfV+}TsfVT¬cVLb+± ²JXr³h5o~cmybvdHsfmnzTjcdHVL~
bf`abedfVLU
N
mnk ogdfhsHmybeVE~©h8sfU& Qhªhcgjlb«hk dfRTV¶VLbHbeVLk8dHmno¨~cm2«gjTdfmnVLbh[dHRTmnb}TsHhzTnVU&'dHRTV~cmybfgjlbHbemnhk mnbilsbedWnmUWmdfVE~©dfh¢dfRTVOgobfVhp¨dv¤h.gh8Ui}=h8kTVkqdHb
A
opkl~B
mnk8dHVsogdfmnkT§dfRTsHhjT8Ro gh8UWUih8kOmkqdfVsf ogVC
5¤+RTmygHR¶Vk@bejTsHVLbN = A × C N B
o8b¨md[¤)obxbfVVk§opz=h0¯V6-d[dfRlVVkl~Ohp>dHRTmyb bfVLgdHmh8kFERTh0¤V¯VLsLdHRTV8VkTVLsHomybfo0dHmnhk¨dHh|Uih8sfVghUW}TnV¬W±¼dHsfVLV´ beRlo}=VL~l³be`cbedfVLUWbF¤+mn8z=VV¬TopUWmnkTVL~¡£d[myb¨jTsedHRTVsHUWhsHV#o8bfbfjTUWVL~&dHRlo0dxdHRTVmnk8dfVLse o8gV
C
myb[opk¶ojcdfh8UWopdfh8kF=kThpdo«8VkTVLsHobfopV#kTVdL5mnk hs~cVLs'dfho¯hmy~«dHRTVV¬adfso#dHVLgRTkTmygop~cm2«gjTd`h~cVLopnmnkTW¤+mdHRUWmybfVEo~cmnkTW}TsfhpuvVEgdfmnhk@b-·dfsHmn¯qmyopbeh8jcdHmnhkdfh«h8zcdHomkOopkµ²JXr»h
N
mnk& ogdfhsHmybeVE~hsHUº¤h8jTn~&z=VdHh«bedHosed|sfh8Ubfo`oinhz@op²JXr
Pref N
lokl~dfh}TsfhpuvVEgd+md|hkdHRTVghUW}=h8kTVkqdHb+ob'h8nh0¤|bPref N v Π A (Pref N ) × C O Π B (Pref N )
±C8³|h0¤VL¯VsELdfRTmybmnUi}=hqbeVEbFdfh¨¤'hsH¥¨lsbvdJhkdfRTV8h8zlopqbe`cbedfVU&¤+RlmngR#ghjTy~z=VtV¬adfsHVUWVn`[V¬c}=Vk@bemn¯V
mdHRTVbf`cbedfVU/mnb)nosf8VJ¡k}losedHmngjTnosLqdfRTmyb)}TsHV¯VkqdHbdHo¥amkTWo~c¯popkqdHop8V¨hpdHRTVgh8Ui}lsfVLbHbfmh8k«8omk
}TsHh¯amy~cVL~«zq`W ogdfhsHmybeVE~ih8sfUbLaokl~dfRTmyb'}TsfVL¯VkqdHb'ob¤VL5}TsHhcgVEbfbfmkldfRTVbf`cbvdHVU¦za`}lopsfdHbLQ)RTV
hzcuvVLgdfmn¯VhdfRTmyb|bfVLgdHmh8kmnb)dfhWzljTmy~>§dfRTV o8gdHhsb'hpoW²JXr hp
Unf N
Qhµh8zcdHomkªdfRTVLUmd#mnbkThd#VkTh8jTR¢dfh¶bfmUW}Tn`§zTjlmy~.²JXr hp'dHRTVghUW}=hkTVLk8dHb mnk.VklVsopB
dfRlVmns+ghU#zTmnklo0dHmh8kzq`«}TjTnz@og¥¤'hjTy~kThd)z=VghUW}TnVdHVxh8s'dfRTV8nhzlopbe`cbedfVLUQ)RqjlbLadfRTVmy~cVEo
mybidfhªzTjlmy~²JXr¦h[ghUW}=hkTVLk8dbdHRlo0d«¤+mn¨opybeh¢}TsHh0¯qmy~cV&bfj2«gmnVkqdfn` sHmygHRz=VRlo¯amh8jTsbh8k©dHRTV
mnk8dfVLse o8gVimk§h8sH~cVLs¨dHhVLklbejTsHV8h8zlopgh8Ui}lVdfVklVLbfbLQ)RTmyb~cVLUWokl~Tb[hp'gh8jTsHbfVok§op8sfVLVUWVkqdxh
dfRlVdv¤hghUW}=h8kTVk8dbopz=hjcd¤+RlopdJz=VLRlo¯qmnhjlsHbdfRTV)mkqdfVsf ogV'bfRThjln~#}TsHh¯amy~cVpopkl~bfh[opkV¬TgHR@opkT8V
hpJmnkch8sfUo0dHmh8kz=Vd ¤VLVkµgh8Ui}=h8kTVkqdHbL
- bedfsopmnRqdehsH¤'opsH~¤)oE`«dHhmUW}TnVUWVkqd|dHRTmnb|my~cVLoWmyb)dHh«bvdopsfd+¤+mdHR&nhcgon`ghUW}TnVdHV}TsfVT¬cVLbL
opk@~dfRTVLk&dHhdfsopk@beUWmd± opk@~&sHVLgVLm¯V³dHRTV#z=VRlo¯amh8jTsb)sHV qjTmnsHVL~hk&dHRTV#mkqdfVsf ogV@z=Vh8sfVsfVEgh8U´
}TjcdHmkl&hTgop>²JXr ¤+mdHR§dfRlVLbfViV¬adfsosHV qjTmnsHVUWVkqdHbLVdgp¨h¤'V¯VsE5dHRTmnbUWm8R8dsHVLbfjTdmnk¢UWokq`
mdfVso0dHmnhklb'z=Vdv¤VVLkdHRTVgh8UW}@h8kTVkqdHbLtXhklbfmy~cVs'dfRTVV¬TopUW}TnV[mk&²Jm8jTsHVxlc¤+RlmngR~TV}Tmygdb
Pref A
opk@~
Pref B
pdv¤hnhcgoT²JXrp¤+mdfRobfmnUi}lV+mnk8dfVLse o8gV)h8kT`gh8klbemybedfmnkT[h@dfRlV)dfsopklbfmdHmnhkt 1
okl~#dHRTV
}TyogV
c 0
Pref A
kTVVE~TbhkTV|hcgLgjTsHsfVLklgV)h=dHRTV|mnk8dHVsfo8gV+dfsopklbfmdfmnhkt 1
EH'`igh8k8dfsobedL
Pref B
kTVLVL~Tbdv¤hihcgLgjTsHsfVLklgVEbthp
t 1
dfhWz=VghUW}TnVdfV8>Q)Rqj@b
Pref A
Rlo8bdfhiz=VV¬adfVLkl~cVL~zq`«h8kTV[hcggjlsfsHVklgV¨ht 1
dfRTmybkTVL¤©z=VRlo¯amh8jTsJh8kdHRTV)mnk8dfVLse o8gV'Rlo8bdfhz@V)}TsHh}lo8o0dHVL~dfh
Pref B
J¡kio8~T~cmdfmnhkFph8kTV'kTVLVL~Tb dfhWgRTVLg¥Whs'8h8zlop5gh8Ui}TnVdHVkTVEbfb'zq`«ghklbfmn~TVsHmkTnhzlo@UopsH¥amkTqbaopk@~dfRTmyb'UWmnRqd)sfV8jlmsHV|dHRTVV¬adHVklbfmh8khpdHRTV}TsfVT¬cVLb'hpdfRTVEbeVgh8Ui}=h8kTVkqdHb'jTkqdfmnFnhzlo@dfsHjTklgLo0dfmnhk}=h8mkqdHb+opsHVxsHVLogRTVL~FJ¡k
dfRlVWVkl~FdfRTmybUWm8R8dsHVLbfjTdmk¢Uopka`µV¬cgRlokTVEbQh&o¯hmy~µdHRTmnbbemdfjlopdfmnhkFdHRTVWz=VRlo¯amh8jTs[h'o
gh8UW}=hkTVkqdhk.mdHbmkqdfVLse o8gVmnbgLop}cdHjTsfVE~ªjTkl~TVs#dfRlV«h8sfU(h|o> R BJdfsopklbfUWmdedfVE~ªmnk
¸ehkTV#beRThdL¹¢Q)RTVbejTUWUopsH`kTVdxmyb|h8zcdHomkTVE~sfh8UopkµV¬adfVkl~TVL~µgLopkTh8kTmygop}TsfVT¬za`W¸esHVhy~cmnkT0¹
mdHb+sfVEbvdHsfmygdHmnhkdfhWdfRTVmnk8dHVsfo8gV8
>
)>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<(+
@
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/<(+
$!(
5/.1)
L
t 3 b 1
b 0
b 1 d 1 b 0
b 1
b 0
d 2
b 1 b 1 b 1
b 0 b 1
d 1 d 0 e 2
t 4
e 4 t 1
t 5 t 3
e 1
e 3 t 1
t 3 e 5
t 4 e 8
d 0 e 6
t 6 t 1 e 9 e 7 t 3
e 11 e 10 c 0
c 0 c 0
c 0
#(&
Pref B
²Jm8jTsfV>¡njlbedfso0dHmklisHV qjTmnsfVE~mkqdfVsogdfmnhk¯amnodfRlVmnk8dHVsf ogVz=Vdv¤VVLk&ghUW}=hkTVLk8db
'*) .,-, 1 07,(, 6
Q)RlV#V¬adHVkl~cVE~OgLopkTh8kTmygop}lsfVT¬&hptoghUW}=hkTVLk8d
A
hsB
mnbxzTjTmnd[¤+mdHROsHVqops~dfhmdHbxmnk8dHVsfo8gV8^cjlgHR©oµ}TsHVT¬ªgLop}cdHjTsHVLbdfRTVz=VRlo¯amnhjTsh'dHRlo0dmnk8dHVsfo8gV«mnk.sHVyo0dHmh8k¢dfhOmdHbgh8Ui}=h8kTVkqdL&¡£dmyb
h8zcdHomnkTVL~«zq`sfVEbvdHsfmygdHmkTdHRTV[gjcdedHmklighk8dHV¬adLamnk}losedHmngjTnosdHRTVbeVd'hpgh8kcljlsHopdfmnhklb¤+RTmygRopsHV
j@beVE~hs)dHRTVgjcdf´Bh¶gsfmdHVsfmnh8kF
¿:
Ã
A
R R 2.? F2 :C
&*% " B R
C
?
2:
Θ C = ¡
≈ mar , C tot , { κ e } e∈E A ¢
C=, RE?∀e ∈ E A
κ e =
½ { [e 0 ] : e 0 ∈ E A , Π C (e 0 ) 6= ∅ }
Π C (e) 6= ∅
# %${ [e 0 ] : e 0 ∈ E A , Π C ([e] 4 [e 0 ]) = ∅ }
. # &$±=Dq³
P
4
P ! UQ)RlVsHVLbedfsHmngdfmnhkhdfRTVgjcdedHmkT«ghk8dHV¬ad
Θ C
mnk
A
UWVLoklb'dfRlopd±Co8³:-|k mnk8dHVsf ogVV¯VLk8dµ±Copk V¯VLk8dghsHsfVEbe}=hkl~TmkTOdHh¢o¶dfsopklbfmdfmnhk©hp¨dfRlVmnk8dfVLse o8gVE³gLopk z=V
~cVLbfm8klo0dHVL~ob+oWgjcdf´BhOV¯VLk8d+h8kT`«mJmdHb|ghsHsHVLbf}=hkl~cmnkTV¯Vkqd+mnb|opybehWopkmnk8dfVLse o8gVV¯VkqdL
±z=³ A RlVsHVLobL@dfRTVWghsHsfVEbe}=hk@~cmklV¯VLk8d
e 0
hpto}Tsfmn¯0opdfVgjcdf´BhªV¯VLk8de
±mBV8opkOV¯VLk8d[¤+RTmygR~chaVLbWkThpd«ghsHsfVLbf}=hkl~.dfh.okq`.dfsopkqdfmdfmnhkhp¨dHRTV&mnk8dHVsf ogVE³Rlobidfhªz=VµgRTh8bfVkbfjlgR©dHRlo0d
dfRTVsHVopsHV[kThWmkqdfVsf ogV[VL¯VkqdHb¸ez=Vdv¤VVLka¹
e
opk@~e 0
cmBV8mnkΠ C ([e] 4 [e 0 ])
Q)RTmyb+gh8kl~cmdfmnhk mnb+kTVLgVLbHbfosf`Wdfh«gjcd|mnkclkTmdfV#gHR@opmnklb)hpJ}TsHmn¯0o0dHV[VL¯VkqdHb+mkdfRTVjTkThy~cmkll¿:
à +* P
Pref A Θ C
2 R; -? 6U < =fsble A Θ C
R? V¬adfVk@~cVL~&gokThkTmygo5}TsHVT¬
A
% "6= 2C
²JmnjTsHVLb 7T± oq³[okl~± gE³beRlh¤dfRTVV¬adHVkl~cVE~¢gLopkTh8kTmngLop>}TsfVT¬cVLbhpdHRTV«ghUW}=h8kTVk8db
A
opkl~B
mk²JmnjTsHV c¢¡k dHRTVbfV qjTVLdfRTV&gjcdf´Bh V¯VkqdHbiosfV~csoE¤+k©o8b~ch8jTzTnVz=hE¬cVLbmk l8jTsHVLbL¤+RTVLsfVEob
mnk§dHRTVgLobfVihp'V¬adHVkl~cVE~¢gjcdf´Bh V¯VLk8db[dfRTVhjTdfVsz=h¬Omyb~cso¤+k§¤+mdHRªo&~TobfRTVE~¶nmkTV8
Pref A Θ C
E C
a 0 c 0
a 0 c 1
t 3
t 2
e 1
e 3
c 0 e 2 t 1
a 1 a 1
t 1
e 0
#$&
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 )
²mn8jTsfV 7.4>¬adfVLkl~cVL~&}TsfVT¬cVLb+opkl~dfRTVmns¨gh8sfsHVLbf}=hkl~cmnkTibfjTUiUosf`«kTVdHb
gh8mk@gmy~cVLb|¤+mdHR&dHRTV#gLopkThklmngLopF}TsHVT¬&bfmklgVdfRTV#hkTn`gjcdf´Bh§V¯VLk8dxokl~&mdb¨gh8sfsHVLbf}=hkl~cmnkTWV¯VLk8d
opsHV[mnk8dfVLse o8gVV¯VLk8db |h0¤V¯VLsLadfRTmyb+mnb+kThpd)dHRTVgobfVxh8s
Pref B Θ C
mk&²JmnjTsHV 7T±CgL³Q)RTVV¬adfVk@~cVL~}TsHVT¬mnb)nosf8VsdfRlokdfRlVbedHokl~TosH~«}TsHVl¬5c¤+RTmygHR¤hjln~z=VhzcdopmnkTVL~za`beVdfdfmnkT
e 4
ob)oigjTde´£h
V¯VLk8d+bfmklgV
[⊥] ≈ mar [e 4 ]
opkl~[⊥] C tot [e 4 ]
¨h¤'V¯VsEe 4
~ThaVLbkThd)ghsHsfVEbe}=hkl~WdfhWopkV¬adfVk@~cVL~
gjTde´£h ¢VL¯Vk8dbemnklgV#mdxmybxo«}TsHm¯po0dfVVL¯Vkqd[opkl~
Π C ([e 4 ] 4 [⊥]) = {e 2 , e 3 }
¨Q)Rlmnbxo}T}TnmVEbxopybehdfh dfRlVV¯VLk8de 5
>Q)RTVV¯Vkqd
e 6
mnb+okV¬qdHVkl~TVL~gjcde´£h ¶bfmnklgVmdbeVL>opkl~mdb+ghsHsfVEbe}=hkl~TmkTV¯VLk8d
e 2
opsHV[mnk8dHVsf ogVVL¯VkqdHbL
-¨k¶V¬adfVk@~cVL~¶gLopkTh8kTmygop}TsHVl¬µmnbghUW}TnVdHVibfmklgVmdmyb[ogopklhkTmygo}TsfVT¬ ± bfVVrtsfh8}=h8bfmdHmh8k
cDmnkF/E@1³>¡£d|Rlo8b'dfhz@VbfRTh¤+kdHRlo0d|md|mnb)lkTmdfV8
à à Ã
Pref A Θ C
,P ÃFà 9 H'`«rsfh8}=h8bfmdHmh8k TE#mnk /E@1Fmd+myb)VklhjT8RdfhbeRlh¤»dHRlo0d+VEogRmnkclkTmdfV
≺
´£gRlopmnkmnkUnf A
gok z=VµgjcdL Q¤hªgLobfVLbWopsHV&gh8klbemy~cVLsfVE~ ¡£xdHRTVsHV&osfVmnkclkTmdfVL` UWokq` mnk8dHVsf ogV&V¯VLk8dbmBV8
V¯VLk8db¨¤+RTmygR¶ghsHsfVEbe}=hkl~dfh
C
=mnkOdfRTVigRlomkµdHRTVkOdHRTVigHR@opmnk¶gokµz=VigjTdbemnklgV#dfRTVkajTU#z=Vs[hUopsH¥amkTqb'mnb)lkTmdfVokl~dfRajlb|bfhUWVUWosf¥amnkTimyb+olk@opFUWosf¥amnkTihpbfV¯VLsHo5mkqdfVLse ogVV¯VLk8dHbL
Y¨dHRTVsH¤+mybeV88dfRlVsHVxmybh8kTn`o#lkTmdfVkqjTU#z=Vs+hpmnk8dHVsf ogV[V¯VLk8dHb'mnkdfRTVgHRlomk^amk@gVxdfRlVgHR@opmnk
mybmkTlkTmdfV¨md'ghkqdHomklbtopkmnkclkTmdfV|dopmnlhph8kT`i}Tsfmn¯0opdfV± kThkT´Bmnk8dHVsfo8gV³JVL¯VkqdHbL>^amnklgV|dfRTV¨kqjlU#z=Vs
hpUosf¥amnkT8b>myb>@kTmdHVxbfhUWV|UWosf¥amnkTmybtolk@op@UWosf¥amnkThpFbfV¯VLsHoT}TsHm¯po0dfV|V¯VLk8dbmnkWdHRTV|dHom=opk@~
dfRajlb)dHRTVgHRlomkµgopkz=VgjcdE
¤
'& 0 ,/.76
Q)RTV)bfjTUWUWosf`kTVd>hp@o[gh8Ui}=h8kTVkqdgLop}cdHjTsfVEbdfRTV)z=VRlo¯amh8jTsh@o[ghUW}=h8kTVkqdJ¤sLdLJmdbJmkqdfVLse o8gV
opk@~md|myb+~TVsHm¯VE~«sHhU°mdb+V¬adfVLkl~cVL~&gokThkTmygo5}TsHVT¬