hal.inria.fr · hal id: inria-00070335 submitted on 19 may 2006 hal is a multi-disciplinary open...
TRANSCRIPT
HAL Id: inria-00070335https://hal.inria.fr/inria-00070335
Submitted on 19 May 2006
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Combining Data Structures with Nonstably InfiniteTheories using Many-Sorted Logic
Silvio Ranise, Christophe Ringeissen, Calogero Zarba
To cite this version:Silvio Ranise, Christophe Ringeissen, Calogero Zarba. Combining Data Structures with NonstablyInfinite Theories using Many-Sorted Logic. [Research Report] RR-5678, INRIA. 2005, pp.39. inria-00070335
ISS
N 0
249-
6399
ISR
N IN
RIA
/RR
--56
78--
FR
+E
NG
ap por t de r ech er ch e
Thème SYM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Combining Data Structures with Nonstably InfiniteTheories using Many-Sorted Logic
Silvio Ranise — Christophe Ringeissen — Calogero Zarba
N° 5678
Septembre 2005
Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois, Campus scientifique,
615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy (France)Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19
! #"#$%'&() *+-,/.01$2 %!3! # 45)6,789!:<;=) ) ∗
>?A@CB?CDFEG0H3?AIKJ† LNM6O3P ?AIQRDTS O JUEV?CHWTJX?CIKIYJXH ‡ LM G0@AD0WTJ P D[Z\G PK] G §
^`_*aRbdcegf9hi ekj0l-maRbdcnll-jkbo3prqtsvukw*cnlx!ypXz-cRm6`|rll-svl
|X~8~pry-m)8c9ycn_*cRy_*c\rFi5e0cR~8mcRbo8yci5~|Xcnl
n8 hplmNnprb~8w8mcRyN~8ypry|Xbdll-mpryccRqvcRbdcRkmlpX|`st cR|Xm-w8yc¡stkmpnprgm|Xst*cRy$¢o|rlcnU*|Xm|l-m-y-w*Rm-w8ycnl`l-w*_£|rlqtsvlml¤k|Xy-y|j0l¤0lcRml¤T|X*bVw8qtm-svlcRml¥!^pV cRy-s¦§jm-_*c6npry-ycnRm$¢*cnllpX!m-_*cnlc9~8ypry|Xbdl¤3pr*c¨*cncn8lmpdnprbo8st*c|dm-_*cnpry-j£bdpk8cRqtst8m-_*cV*|Xm|dlm-y-w*Rm-w8yc© stm-_ª|m-_*cnpry-j[bdpk8cRqtst8£m-_*ccRqvcRbdcRkml¥^`_8svl9nprbVo8st|Xm-svpr«|X¬o3c|r_8svcR cnw*l-st8=m-_*cRq®|rll-sv9¯cRqvlprT¢±°~8~3cRFbdcRm-_*p0Fpr8qtj²s¦oprm-_Fm-_*cnpry-svcnl6|Xyc9l-m|Xo8qtj=stT³*8stmc¥
^`_*cp|Xq´pXNm-_8svlµycR~pry-m¶svlµmpycRq®|·Um-_*c¸l-m|Xo8qvcstT³*8stmcR*cnll`ycnugw8stycRbdcRkm¥¡^Np|r_8svcR cm-_8svl¸p|Xq¹¤ © cstkm-ypkTw*ncm-_*c*prm-svprpXµ~3prqtstmcm-_*cnpry-svcnlº¤|X* © cl_*p © m-_|Xm9|Xm-w8y|XqcK·0¢|Xb~8qvcnlµpX~3prqtstmcm-_*cnpry-svcnlst*Rqtw*8cm-_*plc)bdp08cRqtst8*|Xm|Vl-m-y-w*Rm-w8ycnl¶lw*_|rlqtsvl-ml¤k|Xy-y|j0lº¤lcRml¤¶|X*»bw8qtm-svlcRml¥½¼8w8y-m-_*cRbdpryc¤ © c£~8yp 0sv8c¾|[bdcRm-_*p0»m-_|Xmdsvld|Xo8qvc£mp¬nprbVo8st*c[|~prqtstmc=m-_*cnpry-j © stm-_¿|Xgj«m-_*cnpry-j«pX¸m-_*c£cRqvcRbdcRgmlº¤ycRg|XyTqvcnllpX © _*cRm-_*cRydm-_*c=q®|Xm-mcRyUsvllm|Xo8qtjstT³*8stmc¨pry*prm¥
^`_*c)ycnl-w8qtml`pX´m-_8svlycR~pry-mµcR*cRy|XqtstÀcmpbU|Xgjk¢Álpry-mcnqvprsvm-_*plcycnncRgm-qtjUpro8m|Xst*cnokj9^`st*cRqtqtsT|X*Â|Xy-o|)ÃpryNnprbVo8st8st86m-_*clpX¢Á|Xqtqvcnl-_8stgj9m-_*cnpry-svcnl © stm-_V*pr*lm|Xo8qtj9stT³*8stmcm-_*cnpry-svcnlstFpr*cK¢Álpry-mcnFqvprsv¥ÄFÅÆÇÈÉ Ê µprbo8st|Xm-svpr[pX¡l|Xm-svl$³|Xo8stqtstmËj£~8ypkncnTw8ycnlº¤Ì¸cnRsvl-svpr¾~8ypro8qvcRbdlº¤hÍ|Xgjk¢lpry-mcn=qvprsv¤Îw8mprbU|XmcnF8cnTw*Rm-svpr
∗ ÏÐÑÓÒ`Ô3ÕYÖ¹×ÑØÒÙrÚÖÃÛ¹ÜÞÝdÒÃßXÙXÙÕÖÃÛ¹à-ádâÝãYÖÁÚYänÛ¹Òå\æçèAÏé¿ê´ê3éNëCìºíYí-îYïºíKíYðå!æºç¬ê´ê3éNëCìYìKñKòíKí±óðNÚYäXáÙÖ¹Õô§à$õ±Û¹ÒöNê3è´÷\øê´ê´ù!ù¶ðÚärá¸ú!æºûü8öûkýöNëAþÿ
† ‡ ! § " äÑ$#Kà±Ö¹ÒÃÑÞÛAÝ)Õ%*åà±Ô'&6à)(ºÑØõ$Õÿ3ø+*)ÚYÑØÜ-,+. / 013245672
) )<: 8N3! :(:)!! #)!:! %!3! )97g *+ V1! :) (+) )¸ +$3±7g !
|U~8qtw8~|Xy-m68cnl~8ypry|Xbbdcnl¸stT§pry-bU|Xm-svukw*cnl¸cRbbU|Xg|rl-st*cRgm98cnlRqRbdcRgml w8ncRy-m|XstmÁj0~3c¶*|X*l8cnll-m-y-w*Rm-w8ycnl\8c8pr8ncnlmcRqtqvcnl\ukw*c¡qtsvl-mcnl¤Xm|Xo8qvc|XwT·´¤cR*lcRbVo8qvcnlcRmVbVw8qtm-s¦¢ÁcR*lcRbo8qvcnl¥¾xprw8yV Ry-s¦³cRyVq®|¾npry-ycnRm-svpr«8cdncnl~8ypry|Xbbdcnl¤\pr«|£o3cnlprst 8cnprbo8st*cRyw8*c`m-_npry-svcbdp0Rqtsvl|Xkm¡q®|¸l-m-y-w*Rm-w8yc)8c8pr8ncnl| cnw8*cm-_npry-svc`bdp0Rqtsvl|XgmqvcnlRqRbdcRgml¥µµcRm-mcnprbVo8st|Xsvlpr²~cRw8mRm-ycccnRm-wnc6cRUw8m-stqtsvl|Xkm¶q®|9bRm-_*pk8c6Rq®|rll-svukw*c8c9¯6cRqvlprT¢±°~8~cRFw88svukw*cRbdcRgmlsncnl)m-_npry-svcnllprgmmprw8mcnl)qvcnl8cRwT·£lm|Xo8qvc¸stT³*8svcnl¥
ÞproTz$cnRm-s¦8cncy|X~8~3pry-m=cnl-m=8c¾ycRq®|r_*cRy=q _gj0~3prm-_*anlcÍl-m-st~8w8q®|Xkm£ugw*cqvcnlUm-_npry-svcnl8prst cRkmRm-yc¶l-m|Xo8qvc¡stT³*8svcnlº¥xprw8y\|Xm-mcRst*TycµncRmproTz-cnRm-s¦-¤*prw*lstgm-yp0Tw8svlpr*lq®|*prm-svpr8cm-_npry-svc~prqtsvc¤kcRmµ*prw*lbdprkm-ypr*lugw*c8cnl¶cK·8cRb~8qvcnlµ|Xm-w8ycRqvl¶8cm-_npry-svcnlµ~3prqtsvcnl¶st*Rqtw*cRgmncRqtqvcnl²bdp0Rqtsvl|XgmF8cnl=l-m-y-w*Rm-w8ycnl=8cÍ8pr8ncnl=mcRqtqvcnl=ugw*cqtsvl-mcnl¤)m|Xo8qvc|XwT·¤6cR*lcRbVo8qvcnlcRm9bw8qtm-s¦¢ÁcR*lcRbVo8qvcnlº¥Ì¸c~8qtw*l¤N*prw*l¨8pr8*pr*l¨w8*cdbRm-_*pk8cd|X~|Xo8qvcU8cdnprbVo8st*cRyw8*cm-_npry-svc~3prqtsvc|n cnw8*cm-_npry-svc~3prw8yqvcnlRqRbdcRgmlº¤kukw*c)ncRm-mc)8cRy-8svaRyclprstmprwd*prUlm|Xo8qvcstT³*8svc¥
cnlynlw8qtm|Xml8cnc¨y|X~8~pry-mRRy|XqtsvlcRkm9|Xw[|rl)bw8qtm-s¦¢Álpry-m8cnl)ynl-w8qtm|Xml¸pro8mcRgw*l~|Xy^`st*cRqtqtscRmÂ|Xy-o|¾~3cRy-bdcRm-m|Xkm8cnprbVo8st*cRy8cnlm-_npry-svcnlTstmcnlo8y-stqtq®|Xkmcnld|n cn²8cnlm-_npry-svcnl*prT¢Ál-m|Xo8qvc9stT³*8svcnl*|X*l)w8*c¸qvprsvukw*c9bdpr*pX¢Álpry-mnc¥ É Ç "! µprbVo8st|Xsvlpr8cµ~8ypk#nTw8ycnl8cl|Xm-svl$¹|Xsvl|Xo8stqtstm¤x!ypro8qvaRbdcnl¡8cnRsvlsvpr¤ prsvugw*c9bw8qtm-s¦¢Álpry-mnc¤Ì$nTw*Rm-svprÍ|Xw8mprbU|Xm-svugw*c
"!#$%&'!()+*,".-/01324$56'7!
8 .09:N)
9ËU~8ypry|Xb cRy-s¦³|Xm-svprFpr*c_|rl`pXAmcR²mp8cnRsv8c6m-_*cl|Xm-svl-³|Xo8stqtstmÁjpryµ |XqtsvTstmÁjpXqvprsv|XqÃpry-bVw8q®|rcdstg prqt 0st8*|Xm|£l-m-y-w*Rm-w8ycnlVl-w*_ |rl¨qtsvl-ml¤N|Xy-y|j0l¤lcRml¤N|X*Íbw8qtm-svlcRml¥d^`_*cnlc*|Xm|Ílm-y-w*Rm-w8ycnld|X½o3c£npr*lsv8cRycn |rldl-m-y-w*Rm-w8ycn nprkm|Xst*cRylUpX6cRqvcRbdcRkmlUpX9|[st cR|Xm-w8yc¥\¼*pry!st*lm|X*nc¤pr*cµbU|nj © |Xgm¡mp6yc|rlprd|Xoprw8m\qtsvl-ml!pXstgmcRcRylº¤glcRml\pXop0prqvc|X*l¤prybw8qtm-svlcRmlpX\yc|Xqvl¥
°*c © |j¬mp£yc|rlpr |Xoprw8mV*|Xm|Flm-y-w*Rm-w8ycnlVp cRycRqvcRbdcRgmlpX|=st cR|Xm-w8ycdsvlVmpw*lcVm-_*c¯6cRqvlprT¢±°~8~3cRÍbdcRm-_*p0[st[pry8cRy9mpbdpkTw8q®|Xy-qtj[nprbVo8st*cd|²8cnRsvl-svprÍ~8yp0ncnTw8ycÃpry|Vm-_*cnpry-j
Sbdp08cRqtst8m-_*c¸*|Xm|l-m-y-w*Rm-w8yc © stm-_F|8cnRsvlsvpr²~8yp0ncnTw8yc6Ãpry`|m-_*cnpry-j Tbdp08cRqtst8m-_*c¨cRqvcRbdcRkml¥/:6p © cR cRy¤m-_8svl)lprqtw8m-svpr£ycnukw8stycnlm-_|Xm)oprm-_ S |X*
To3c !()+*
.-/01 ¥<;TÃpry-m-w8|XmcRqtj¤km-_8svl!ycnugw8stycRbdcRkm¶svl¡*prml|Xm-svl$³cnokjbU|Xkj~8y|rRm-sv|XqtqtjycRqvcR |Xgmm-_*cnpry-svcnll-w*_[|rl¤TÃpryst*l-m|X*nc¤*m-_*c9m-_*cnpry-j£pX\o3p0prqvc|X*l¤8m-_*c¨m-_*cnpry-j=pX\stgmcRcRylbdp0Tw8qvpn¤*|X*=m-_*c9m-_*cnpry-j£pX³8·Tcn0¢ © svTm-_¾o8stm$¢¹ cnRmpryl>=Ø@?Á¥cnncRgm-qtj¤^`st*cRqtqts¸|X* Â|Xy-o|A=CB'?stgm-yp0Tw*ncn |¬cR*cRy|XqtstÀº|Xm-svpr pX9m-_*c¾pr*cK¢Álpry-mcn
cRylsvprUpX3m-_*c¯6cRqvlprT¢±°~8~cRdbdcRm-_*p0stUpry8cRy¡mp9nprbo8st*c)m-_*cnpry-svcnl¡m-_|Xmµ|Xyc*prml-m|Xo8qtjstT³*8stmc¥µh[pryc9~8ycnRsvlcRqtj¤*m-_*cRj=stkm-ypkTw*ncm-_*c9*prm-svprpX !"$(0* m-_*cnpry-svcnl¤3|X*£~8yp cVm-_|Xm|¾l-_8stgjªm-_*cnpry-j
S|X«ocUnprbVo8st*cn © stm-_½|Xgj prm-_*cRy|Xy-o8stm-y|Xy-j m-_*cnpry-j T ¤\cR cR s¦)m-_*cq®|Xm-mcRysvlµ*prm)l-m|Xo8qtjstT³*8stmc¥¡^`_*cRj²|Xqvlp~8ypº ksv8c|Vqtsvlm`pXl-_8stgj²m-_*cnpry-svcnl © _8sv_£st*Rqtw*8cnl
m-_*cm-_*cnpry-j¬pXcnugw|XqtstmËj¤m-_*cm-_*cnpry-j¬pXµ~|Xy-m-s®|Xq¶pry8cRyl¤Nm-_*cm-_*cnpry-jÍpXµmprm|Xqpry8cRyl¤|X*m-_*c¸m-_*cnpry-j£pX\o3prw8*8cnFq®|Xm-m-svncnl¥
Ì6cnl~8stmc9m-_*cnlc¸~8yprbsvl-st8dycnlw8qtml¤*^`st*cRqtqts|X*¾Â|Xy-o|ÞlbdcRm-_*pk£_|rl)m © pTy| © o|rED0l¥¼stylm¤ © _*cRnprbVo8st8st8V|l-_8stkj9m-_*cnpry-j S ¤rpr*c_|rlmp6nprb~8w8mc`|Aw8*Rm-svpr mincardS
¥^`_8svl§w8*Rm-svprm|FDcnl¡|rlst8~8w8m!|X
S¢Ál|Xm-svl-³|Xo8qvcµnprrz$w8*Rm-svpr
ΓpXqtstmcRy|Xqvlº¤|X*ycRm-w8y-*l\m-_*c
bst8stbU|Xq!|XyTst|XqtstmËjkÃpry © _8sv_m-_*cRycsvl¸| T ¢¹bdp08cRq\pX Γ
pX|XyTst|XqtstmËjk¥Îqtm-_*prw8_
mincardSsvl²nprb~8w8m|Xo8qvc§pry=| © sv8c[Rq®|rllUpX¨m-_*cnpry-svcnl¤)stmlnprb~8qvcK·TstmÁj¿svldst cR*cRy|Xq
NP¢¹_|Xy´¥Ì6w*c[mp«m-_8svl_8st_#nprb~8qvcK·TstmÁj¤stm=svl=|Xm-w8y|Xq¸mp«lm-w*Tj_*p © mp½|n prsv m-_*c
nprb~8w8m|Xm-svpr[pXmincardS
¥e0cnnpr*´¤8m-_*c¸*prm-svprFpXl-_8st8st*cnllsvl`mpkplm-ypr8*¤*|X*=stmbU|nj²o3c6 cRy-j=TsHGURw8qtmmp³**
§w8y-m-_*cRy¡cK·*|Xb~8qvcnl¡pX3~8y|rRm-sv|XqtqtjycRqvcR |Xgml_8stgjm-_*cnpry-svcnl¥JIc`o3cRqtsvcR c`m-_|Xm¡m-_8svl!TsHGURw8qtmËjsvl!Tw*c`mp¸m-_*cµ¹|rRm!m-_|Xm¡m-_*c`*prm-svprdpX´l-_8stgjm-_*cnpry-svcnl © |rl¡stgm-yp0Tw*ncnstdpr*cK¢Álpry-mcndqvprsv¥
9Ádm-_8svlycR~3pry-m © c6|Xyc)stgmcRycnlmcnUstdm-_*c~8ypro8qvcRb(pXnprbo8st8st8|¨m-_*cnpry-jSbdpk8cRqtst8
|*|Xm|dl-m-y-w*Rm-w8yc © stm-_¾|*pr*l-m|Xo8qtj²stT³*8stmc¨m-_*cnpry-j T bdp08cRqtst8dm-_*c9cRqvcRbdcRgml¥¶hpryc¨st8cRm|Xstq¹¤*m-_*c¨nprgm-y-sto8w8m-svpr*l6pXm-_8svl)ycR~3pry-m|XycK
B¥L9Á pry8cRy²mp l-sv8cnlmcR~¿m-_*cÍTsHGURw8qtm-svcnl£pX9³**Tst8½l-_8stgjm-_*cnpry-svcnl¤ © c[pr~cRy|Xmc[stbU|Xgjk¢Álpry-mcnFqvprsv¨y|Xm-_*cRy)m-_|XFstFpr*cK¢Álpry-mcn¾qvprsv¥
éé äMNYïOYò
'! ( !"!<'.6
T¥ Icstgm-yp0Tw*ncm-_*c*prm-svprpX ')+1 m-_*cnpry-svcnlº¤0|X* © c~8ypº cm-_|Xm¡|Xm-w8y|Xq3cK·8|Xb~8qvcnlpX~prqtstmc6m-_*cnpry-svcnl|Xyc6m-_*plc¸bdpk8cRqtst8d*|Xm|l-m-y-w*Rm-w8ycnll-w*_¾|rlµqtsvl-ml¤8|Xy-y|j0l¤8lcRml¤|X*£bVw8qtm-svlcRml¥
T¥ Ic6~8yp 0sv8c¸|V*c © nprbVo8st|Xm-svprFbdcRm-_*p0²m-_|Xm¶svl|Xo8qvc6mpnprbVo8st*c¨|~3prqtstmcm-_*cnpry-jS © stm-_|Xgj=m-_*cnpry-j T ¤TycRg|XyTqvcnllpX © _*cRm-_*cRy T svl)l-m|Xo8qtj²stT³*8stmc¨pry)*prm¥
*¥ Ic!cR*cRy|XqtstÀc¡m-_*c¡*prm-svprpX8l-_8st8st*cnll§yprb pr*cK¢Álpry-mcnVqvprsv!mpbU|Xgjk¢Álpry-mcnVqvprsv¤|X* © c)~8yp cm-_|Xm±iw8*8cRyµy|Xm-_*cRy © c|FDU|rll-w8b~8m-svpr*l±i¿l-_8st8st*cnllsvl¶cnugw8st |XqvcRgmµmp~3prqtstmcR*cnllst pr*cK¢Álpry-mcn qvprsv¥ ^`_*c[cnugw8st |XqvcR*nc[svlUqvcnllURqvc|Xy=stbU|Xkjg¢Álpry-mcnqvprsv¥
^`_*c²Ry-wT·½pXprw8ydnprbo8st|Xm-svpr bdcRm-_*pk»svlmp[bdp0Ts¦Aj m-_*c=¯cRqvlprT¢±°~8~cR½bdcRm-_*p0´¥^`_*cF*pr*8cRmcRy-bst8svl-m-svF cRyl-svpr¿pX¸m-_8svldbdcRm-_*pk npr*lsvl-mlst w*cnllst8 |X|Xy-y|X8cRbdcRgmpº cRym-_*clcRmpXl_|Xycnª |Xy-s®|Xo8qvcnl¥F^`_8svl|Xy-y|X8cRbdcRgmsvlVw*lcnªmp¾o8w8stqv cnukw|Xqtstm-svcnl|X*Tsvlcnugw|Xqtstm-svcnlocRm © cncR |Xy-s®|Xo8qvcnl¤¡mpÍnpr*l-m-y|XstlstbVw8qtm|X*cnprw*l-qtj«m-_*c=st8~8w8mldpX68cnRsvlsvpr~8yp0ncnTw8ycnl=Ãpry£nprb~3pr*cRkm¾m-_*cnpry-svcnl¥ °w8y£bdp0Ts¦³|Xm-svpr#svl=ycRq®|Xmcn mp«m-_*cÍ r|Xy-s®|Xo8qvcnlstk prqt cnFst¾|X|Xy-y|X8cRbdcRkm~8ycnRsvlcRqtj K É Ê \T É w*cnll|X|Xy-y|X8cRbdcRgm9pº cRy6|X¾cK·TmcR*8cn¾lcRmpX |Xy-s®|Xo8qvcnl¤|X*F*prm
z±w*lmdm-_*cFl-_|Xycn¿pr*cnl¥¿¼pryUnpry-ycnRm-*cnll¤`m-_*cFcK·TmcR*8cn¿|Xy-y|X8cRbdcRkm=bVw*l-m=|Xqvlpnprgm|Xst«pr~8~3pry-m-w8*cRqtj¬stkm-ypkTw*ncnªAycnl_¬ r|Xy-s®|Xo8qvcnl¤ © _*plcyprqvcdsvl9mp © stm-*cnllVm-_|XmncRy-m|Xst£¹|rRml_*prqv=Ãprym-_*c¨*|Xm|l-m-y-w*Rm-w8yc¥
°w8y)bdcRm-_*pk8pkcnl)*prmycnukw8styc9m-_*c¨nprb~8w8m|Xm-svprÍpX¡|mincardS
§w8*Rm-svpr¤|X*£stmsvlm-_*cRycKÃpryc¨c|rl-svcRympstb~8qvcRbdcRgm6m-_|X£m-_*c¨pr*c9~8ycnlcRkmcnFst =CBF?Á¥
²Å !AT Å Ê ÈÉ 9Ëb~8qtsvRstm) cRyl-svpr*lpX\h[pkTs¦³|Xm-svpr B © cRycV|Xqtyc|rTj=w*lcn=okj£Â|Xy-o|dstpry8cRymp¾nprbo8st*cm-_*cUm-_*cnpry-jpX)lcRml=CB?`|X*ªm-_*cUm-_*cnpry-jpX`bw8qtm-svlcRml =CB@? © stm-_»|Xgj|Xy-o8stm-y|Xy-j=m-_*cnpry-j
TpX\m-_*c9cRqvcRbdcRgml¤cR cRFs¦
Tsvl*prm)l-m|Xo8qtj=stT³*8stmc¥
^`_*c6³*yl-mcK·0~8qtsvRstm cRyl-svprFpXhp0Ts¦³|Xm-svpr B6svlTw*c6mp¼*prgm|Xst*cV|X* y-sto3prbdprkm&=ØF?© _*p¸nprbo8st*cm-_*cµm-_*cnpry-jpX3|Xy-y|j0l © stm-_U|Xkjprm-_*cRy\*pr*l-m|Xo8qtjstT³*8stmc¶m-_*cnpry-j T *prm\nprT¢m|Xst8st8m-_*c¨lpry-m
array¥¡^`_*cRstyycnl-w8qtm|X~8~8qtsvcnlmpdnprrz±w8*Rm-svpr*l6pX\qtstmcRy|Xqvl)*prmnprkm|Xst8st8
Tsvlcnugw|Xqtstm-svcnlo3cRm © cncR¾mcRy-bdlpX\lpry-m array¥
^`_*cq®|Xmcnlm6cK·T~8qtsvRstm¸ cRyl-svpr¬pX¶hp0Ts¦³|Xm-svprAB © |rl6w*lcn[ogj¾¼prgm|Xst*c¤6|X8svlc¤´|X*Â|Xy-o|3=Ó ?Á¤kstpry8cRyµmpVnprbVo8st*c9|V*pr*l-m|Xo8qtjstT³*8stmcm-_*cnpry-j
TpXm-_*ccRqvcRbdcRgml © stm-_Um-_*c
m-_*cnpry-jTlength
pX\qtsvl-ml)pX!cRqvcRbdcRgml © stm-_¾qvcR8m-_¾npr*l-m-y|Xstgmlº¥`||r8cRy|X* _8stq®|XyTsJ=CB¤0@?_| cycnncRkm-qtjdstgm-yp0Tw*ncn£|9*c © bdcRm-_*pkUÃpry¶nprbVo8st8st8
m-_*cnpry-svcnl9pº cRy9*pr*Tsvl§z-prstgmVl-st|Xm-w8ycnl¸w*lst8£bU|Xgjk¢Álpry-mcnÍqvprsv¥V^`_*cRsty9ycnl-w8qtm6Ãpry¸m-_*cnpX¢
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
y-svcnlµp cRy`*pr*Tsvl§z-prstgml-st|Xm-w8ycnl$imprcRm-_*cRy © stm-_²prw8yl¶Ãpry¶*pr*l-m|Xo8qtjUstT³*8stmcm-_*cnpry-svcnl$il_*p © lm-_|Xm)stmsvl cRy-j²nprk cR8svcRgm6mpnprbVo8st*cm-_*cnpry-svcnl)w*l-st8bU|Xkjg¢Álpry-mcnFqvprsv¥
gT É(É Å Å¡É º 9Ë e0cnRm-svpr¿ © c£stgm-yp0Tw*nclprbdc=~8ycRqtstbst|Xy-j½*pX¢m-svpr*lº¤*|rl © cRqtq|rl`m-_*c¸npr*ncR~8mpX~prqtstmc6m-_*cnpry-svcnl¥/9˾e0cnRm-svpr¾ © c¸~8yp c¨lprbdc¸|XwT·Tstqts®|Xy-j~8ypr~pl-stm-svpr*lVm-_|Xm|Xycw*lcK§w8q © _*cR ~8yp 0st8¾m-_|XmncRy-m|Xst m-_*cnpry-svcnl|Xycd~3prqtstmc¥ 9Á»e0cnK¢m-svpr © c~8ycnlcRkm¨prw8y¸nprbo8st|Xm-svprbdcRm-_*p0´¥ 9Áe0cnRm-svprª © cnprb~|Xycm-_*c*prm-svpr¬pX~prqtstmc=m-_*cnpry-svcnl © stm-_½m-_*c=*prm-svpr¿pX¸l-_8stkj m-_*cnpry-svcnlº¥A9Áe0cnRm-svpr*lB © c£~8yp cFm-_|Xm|Xm-w8y|Xq3cK·8|Xb~8qvcnl¶pX´~3prqtstmcm-_*cnpry-svcnl¶|Xyc)m-_*plcbdpk8cRqtst8*|Xm|¨l-m-y-w*Rm-w8ycnlº¤0|rl © cRqtq|rl¡m-_*cw8o8svukw8stmprw*l)m-_*cnpry-j£pX\cnugw|XqtstmËj¥ 9Ëe0cnRm-svpr BB © cVTy| © npr*Rqtw*lsvpr*l§yprb prw8y © pry6D¥
3!+ 3!
Î !"C '
ΣsvlV|=m-y-st~8qvc
(S, F, P ) © _*cRyc S svlV|FlcRm¨pX)lpry-ml¤Fsvl|£lcRmVpXµAw8*Rm-svpr
ljkbo3prqvl¤Psvl|FlcRmVpX)~8ycnTsv|Xmc=l-jkbo3prqvlº¤|X*«|Xqtq¶m-_*cl-jkbo3prqvlst
F, P_|n c=|Xy-stm-svcnl
npr*lm-y-w*Rmcndw*l-st8¨m-_*clpry-ml!stS¥ st cR|¨l-st|Xm-w8yc
Σ = (S, F, P )¤ © c © y-stmc ΣS §pry S ¤
ΣF Ãpry F ¤´|X* ΣP ÃpryP¥ 9Á
Σ1 = (S1, F1, P1)|X*
Σ2 = (S2, F2, P2)|Xycl-st|Xm-w8ycnl¤
m-_*cRsty 7' svlm-_*c¨l-st|Xm-w8ycΣ1 ∪Σ2 = (S1 ∪ S2, F1 ∪ F2, P1 ∪ P2)
¥ st cRª|=l-st|Xm-w8yc
Σ¤ © cU|rll-w8bdcm-_*cl-m|X**|Xy*prm-svpr*l¨pX Σ 1 ¤ Σ )+1 ') ¤N|X*
Σ " )H ¥ÎΣ !.01" svl¡| Σ
¢ÃÃpry-bVw8q®| © stm-_U*p6Aycncµ |Xy-s®|Xo8qvcnl¥\ÎqtstmcRy|Xq8svl"! s¦stm!svlpXm-_*cÃpry-b
x ≈ y¤x 6≈ y
¤x ≈ f(y1, . . . , yn)
¤p(y1, . . . , yn)
¤8|X*¬p(y1, . . . , yn)
¤ © _*cRycx, y, y1, . . . , yn
|Xyc9 |Xy-s®|Xo8qvcnl¤fsvl|§w8*Rm-svpr¾l-j0bVoprq¹¤|X*
psvl|~8ycnTsv|XmcVl-jkbo3prq¹¥
9¹tsvl|¾mcRy-bF¤ © c²8cR*prmc © stm-_ varsσ(t)
m-_*c²lcRmpX |Xy-s®|Xo8qvcnlpXlpry-mσpknRw8y-y-st8
stt¥ekstbstq®|Xy-qtj¤Ns¦
ϕsvl¨|Ãpry-bVw8q®|T¤ © c8cR*prmc © stm-_ varsσ(ϕ)
m-_*clcRm¸pXµ§ycnc |Xy-s®|Xo8qvcnlpXlpry-m
σp0nRw8y-y-st8²st
t¥L9Á
ϕsvlcRstm-_*cRy9|dmcRy-b pry6|dÃpry-bVw8q®|T¤ © c8cR*prmc © stm-_ vars(ϕ)m-_*clcRm ⋃
σ varsσ(ϕ)¥ ¼st|Xqtqtj¤)s¦
Φsvl=|lcRm=pX9mcRy-bdl²pry=|ªlcRmpX9Ãpry-bVw8q®|rc¤ © c¾qvcRm
varsσ(Φ) =⋃
ϕ∈Φ varsσ(ϕ)|X*
vars(Φ) =⋃
ϕ∈Φ vars(ϕ)¥
9Á=m-_*c¸ycnl-mpXm-_8svl)~|X~3cRy¤ © c¸sv8cRgm-s¦§j²nprrz$w8*Rm-svpr*l)pXNÃpry-bVw8q®|rc ϕ1 ∧ · · · ∧ ϕn© stm-_
m-_*c9lcRmϕ1, . . . , ϕn
¥
$#%&"('*),+-£Å C É cRm
Σo3cF|[l-st|Xm-w8yc¤`|X*»qvcRm
XocF|[lcRmdpX |Xy-s®|Xo8qvcnl © _*plc£lpry-ml
|Xyc=stΣS ¥½Î Σ .0/21"3"465"765"463893"/2:;1 A p cRy
XsvlU|[bU|X~ © _8sv_stgmcRy-~8ycRml²c|r_lpry-m
σ ∈ ΣS |rl|£*prT¢ÁcRb~8mËj 8prbU|Xst Aσ¤\c|r_« r|Xy-s®|Xo8qvc
x ∈ XpXlpry-m
σ|rl|X«cRqvcRbdcRgm
xA ∈ Aσ¤\c|r_ Aw8*Rm-svpr½l-j0bVoprq
f ∈ ΣF pX|Xy-stmÁjσ1 × · · · × σn → τ
|rl|FAw8*Rm-svpr
éé äMNYïOYò
'! ( !"!<'.6
fA : Aσ1× · · · ×Aσn
→ Aτ¤T|X*=c|r_£~8ycnTsv|Xmc9l-j0bVoprq
p ∈ ΣP pX|Xy-stmËj σ1× · · · × σn|rl|l-w8o*lcRmpA
pXAσ1× · · · ×Aσn
¥Î
Σ . 3"535"4 svl| Σ¢¹stgmcRy-~8ycRm|Xm-svprÍp cRy|XcRb~8mÁj£lcRm)pX\ r|Xy-s®|Xo8qvcnl¥
ÎΣ¢ÃÃpry-bVw8q®|
ϕp cRy²|lcRm
XpX |Xy-s®|Xo8qvcnlsvl ! !- () s¦stmsvlm-y-w*c=st lprbdc
Σ¢
stkmcRy-~8ycRm|Xm-svprpº cRyX¥!^ © p Σ
¢ÃÃpry-bVw8q®|rcϕ|X*
ψpº cRy|lcRm
XpX\ r|Xy-s®|Xo8qvcnl|Xyc )C s¦
ϕA = ψA ¤0§pry|Xqtq Σ¢¹stgmcRy-~8ycRm|Xm-svpr*l¸p cRy
X¥
cRmAoc`|X
Ω¢¹stgmcRy-~8ycRm|Xm-svprpº cRylprbdc`lcRm
VpX3 r|Xy-s®|Xo8qvcnlº¥¼pry|¸l-st|Xm-w8yc
Σ ⊆ Ω|X* |lcRm²pX9 r|Xy-s®|Xo8qvcnlU ⊆ V
¤ © c¾8cR*prmc © stm-_ AΣ,U m-_*c¾stkmcRy-~8ycRm|Xm-svpr pro8m|Xst*cn§yprb
Aogj¾ycnlm-y-svRm-st8²stm¸mpstgmcRy-~8ycRm¨pr8qtjm-_*cljkbo3prqvl6st
Σ|X*[m-_*c |Xy-s®|Xo8qvcnlst
U¥
¼*w8y-m-_*cRy-bdpryc¤ © cVqvcRm AΣ = AΣ,∅ ¥
#' # +¼prqtqvp © st8 |X8Ànst8cRy =Ø@?Á¤ © c8cK³**cm-_*cnpry-svcnl¨|rllcRml¸pXl-m-y-w*Rm-w8ycnl6y|Xm-_*cRy¸m-_|X|rllcRmlpX!lcRkmcR*ncnl¥¡h[pryc¸§pry-bU|Xqtqtj¤ © c9st c9m-_*c6Ãprqtqvp © st8²8cK³*8stm-svpr¥-£Å C É Î
Σ . 3 4:;5 svl²|ª~|Xsty(Σ,A) © _*cRyc Σ
svl²| l-st|Xm-w8ycÍ|X*Asvl=|
Rq®|rllpXΣ¢Ál-m-y-w*Rm-w8ycnlº¥ st cR|[m-_*cnpry-j
T = (Σ,A)¤`|
T . /1 3 45"765"4,38 3"/:1 svld|Σ¢
stkmcRy-~8ycRm|Xm-svprAl-w*_£m-_|Xm
AΣ ∈ A¥
st cR |Σ¢¹m-_*cnpry-j
T¤|
Σ¢ÃÃpry-bVw8q®|
ϕpº cRy=|ªlcRm
XpX9 r|Xy-s®|Xo8qvcnl²svl
T ! !- () s¦stmsvlm-y-w*c²st½lprbdc
T¢¹stkmcRy-~8ycRm|Xm-svprp cRy
X¥ Ic © y-stmc A |=T ϕ © _*cR A svl|
T¢
stkmcRy-~8ycRm|Xm-svprl|Xm-svl-Aj0st8ϕ¥ st cR¾|
Σ¢¹m-_*cnpry-j
T¤Tm © p Σ
¢ÃÃpry-bVw8q®|rcϕ|X*
ψp cRy|lcRm
XpX |Xy-s®|Xo8qvcnl|Xyc
T @') 0 s¦ ϕA = ψA ¤0§pry|Xqtq T ¢¹stgmcRy-~8ycRm|Xm-svpr*l¸p cRy X ¥ st cRF|Σ¢¹m-_*cnpry-j
T¤km-_*c '0 - 6 &! !- ()+* 6() pX
Tsvlµm-_*c6~8ypro8qvcRb
pX!8cnRsvTst8*¤TÃpryc|r_ugw|Xkm-s¦³cRy$¢ÃAycncΣ¢Ã§pry-bw8q®|
ϕ¤ © _*cRm-_*cRypry*prm ϕ svl T ¢ÁlY|Xm-svl$³|Xo8qvc¥
-£Å C ÉVÉ [ T É $ cRmTi = (Σi,Ai)
o3c£|m-_*cnpry-j¤¡Ãpryi = 1, 2
¥»^`_*c :"!$# /21 8 3 /2:1 pX
T1|X*
T2svl)m-_*cVm-_*cnpry-j
T1 ⊕ T2 = (Σ,A) © _*cRyc Σ = Σ1 ∪ Σ2|X*
A =
A | AΣ1 ∈ A1|X*AΣ2 ∈ A2
¥
9¹Φsvl|lcRmµpX
Σ¢ÁlcRgmcR*ncnl¤ © c¸qvcRm TheoryΣ(Φ) = (Σ,A)
ocm-_*cm-_*cnpry-jl-w*_=m-_|XmAsvlm-_*c¨Rq®|rll)pX!|Xqtq
Σ¢Ál-m-y-w*Rm-w8ycnll|Xm-svl-Aj0st8
Φ¥
% É ¡É C É'& )( Φi
! Σi !01E"! i = 1, 2 * 24$5
TheoryΣ1(Φ1)⊕ TheoryΣ2(Φ2) = TheoryΣ1∪Σ2(Φ1 ∪ Φ2) .
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
5:: cRm
(Σ,A1) = TheoryΣ1(Φ1) ,
(Σ,A2) = TheoryΣ2(Φ2) ,
(Σ,A) = TheoryΣ1∪Σ2(Φ1 ∪ Φ2) ,
(Σ,B) = TheoryΣ1(Φ1)⊕ TheoryΣ2(Φ2)
^`_*cR K
A ∈ B ⇐⇒ AΣ1 ∈ A1|X*AΣ2 ∈ A2
⇐⇒ AΣ1lY|Xm-svl$³cnl
Φ1|X*AΣ2
l|Xm-svl$³cnlΦ2
⇐⇒ Al|Xm-svl-³cnl
Φ1 ∪ Φ2
⇐⇒ A ∈ A .
Icdstgm-yp0Tw*nc²o3cRqvp © lcR cRy|Xq¶Rq®|rllcnlpX)m-_*cnpry-svcnl¥ Ic © stqtq¶lcncd_*p © m-_*cRjycRq®|XmcstcRbU|Xy6D BT¥-£Å C É' C Å É Ê Å ! É¡Å º Æ $ cRm
Σo3c`|l-st|Xm-w8yc¤qvcRm
S ⊆ ΣS o3c`|lcRmpX\lpry-ml¤*|X*=qvcRm
Toc¨|
Σ¢¹m-_*cnpry-j¥Ic¨l|jUm-_|Xm
T_|rlm-_*c
/21 /3 4 ! :4 765:764653© stm-_£ycnl~3cnRmmp S s¦§pry)cR cRy-j T ¢Ál|Xm-svl$³|Xo8qvc9ukw|Xgm-s¦³cRy$¢Ã§ycnc Σ¢ÃÃpry-bVw8q®|
ϕm-_*cRyc9cK·Tsvl-ml|
T¢¹stkmcRy-~8ycRm|Xm-svpr
Al|Xm-svl$§jkst8
ϕl-w*_Fm-_|Xm
Aσsvl`³*8stmc¤TÃpry)c|r_lpry-m
σ ∈ S¥
-£Å C É 8 ! Å C Å Å cRmΣocV|dlst|Xm-w8yc¤qvcRm
S ⊆ ΣS oc|dlcRmpXlpry-ml¤0|X*²qvcRm
Toc¸|
Σ¢¹m-_*cnpry-j¥ Ic6lY|njdm-_|Xm
Tsvl 38 # /21 /21 / 3"4 © stm-_=ycnl-~cnRm¶mp Ss¦¶§pry9cR cRy-j
T¢Ál|Xm-svl$³|Xo8qvcukw|Xgm-s¦³cRy$¢Ã§ycnc
Σ¢ÃÃpry-bVw8q®|
ϕm-_*cRyccK·0svl-ml¨|
T¢¹stkmcRy-~8ycRm|Xm-svpr
Al|Xm-svl$§jkst8
ϕl-w*_Fm-_|Xm
Aσsvl)stT³*8stmc¤T§pry)c|r_lpry-m
σ ∈ S¥
-£Å C É É´É NÅ cRmΣo3c|dl-st|Xm-w8yc¤qvcRm
S = σ1, . . . , σn ⊆ ΣS oc|lcRm)pX\lpry-ml¤*|X*=qvcRm
Toc¸|
Σ¢¹m-_*cnpry-j¥Ic9l|nj²m-_|Xm
Tsvl ! ::3 © stm-_£ycnl~3cnRmmp Ss¦6K
§pry)cR cRy-j T ¢Ál|Xm-svl-³|Xo8qvc¨ugw|Xkm-s¦³cRy$¢ÃAycnc Σ¢ÃÃpry-bVw8q®|
ϕ¤
§pry)cR cRy-j T ¢¹stkmcRy-~8ycRm|Xm-svpr A l|Xm-svl$§jkst8ϕ¤
§pry)cR cRy-j£|XyTst|Xqkw8bVocRy κ1, . . . , κnl-w*_£m-_|Xm
κi ≥ |Aσi|¤TÃpry
i = 1, . . . , n¤
m-_*cRycVcK·0svl-ml)|T¢¹stgmcRy-~8ycRm|Xm-svpr
Bl|Xm-svl$§j0st8
ϕlw*_Fm-_|Xm
|Bσi| = κi ,
§pryi = 1, . . . , n .
éé äMNYïOYò
'! ( !"!<'.6
Finite model propertyPoliteSmoothStably infinite Finite witnessable
2 Å cRq®|Xm-svpr*l-_8st~*lo3cRm © cncRRq®|rllcnlpXm-_*cnpry-svcnlº¥
-£Å C É C Å¾È C Å º8 ! C Æ cRmΣoc|l-st|Xm-w8yc¤qvcRm
S ⊆ ΣS oc|UlcRmpXlpry-ml¤\|X*¬qvcRm
To3cU|
Σ¢¹m-_*cnpry-j¥Icl|nj[m-_|Xm
Tsvl /1 /3"4 / 3"1 4 8 #4 © stm-_
ycnl~3cnRm)mpSs¦!m-_*cRycVcK·Tsvl-ml|Unprb~8w8m|Xo8qvc¨§w8*Rm-svpr
witnessm-_|Xm§prycR cRy-jFugw|Xkm-s¦³cRy$¢
§ycncΣ¢ÃÃpry-bVw8q®|
ϕycRm-w8y-*l6|ugw|Xkm-s¦³cRy$¢ÃAycnc
Σ¢Ã§pry-bw8q®|
ψ = witness(ϕ)l-w*_Fm-_|Xm.K
sϕ|X*
(∃v)ψ|Xyc
T¢Ácnukw8st r|XqvcRkm¤ © _*cRyc v = vars(ψ) \ vars(ϕ)
sts²s¦ψsvlT¢Ál|Xm-svl$³|Xo8qvc=m-_*cR½m-_*cRyc£cK·Tsvl-ml|
T¢¹stgmcRy-~8ycRm|Xm-svpr
Al|Xm-svl$§j0st8
ψlw*_
m-_|XmAσ = [varsσ(ψ)]A
¤0Ãpry)c|r_σ ∈ S
¥
-£Å C É %É ! C Å NÅ cRmΣo3c¸|¨l-st|Xm-w8yc¤TqvcRm
S ⊆ ΣS oc6|lcRm¶pXlpry-ml¤T|X*qvcRm
Toc6|
Σ¢¹m-_*cnpry-j¥Ic9lY|njdm-_|Xm
Tsvl 76:/3 4 © stm-_£ycnl-~3cnRm`mp S s¦stm`svlµoprm-_Fl-bdpkprm-_|X*£³*8stmcRqtj © stm-*cnllY|Xo8qvc © stm-_£ycnl-~cnRm)mp S ¥
²Å ª8 cRmΣocV|lst|Xm-w8yc¤*qvcRm
S ⊆ ΣS ¤|X*£qvcRm T ocV|Σ¢¹m-_*cnpry-j¥`^`_*cRFm-_*c
Ãprqtqvp © st8U_*prqv8l K-¥!¼stw8ycBK
9¹Tsvl)l-bdp0prm-_ © stm-_Fycnl-~cnRmmp S m-_*cR
Tsvll-m|Xo8qtj=stT³*8stmc © stm-_Fycnl-~cnRmmp S ¥
9¹Tsvl¶³*8stmcRqtj © stm-*cnll|Xo8qvc © stm-_=ycnl-~3cnRm`mp S m-_*cR T _|rl`m-_*c³*8stmc6bdpk8cRq~8ypr~T¢
cRy-mÁj © stm-_Fycnl-~cnRmmp S ¥
+` +$\)+
9Ë£m-_*c¸ycnl-mpX\m-_8svlycR~pry-m © c © stqtq~8yp c9m-_|XmlcR cRy|Xqm-_*cnpry-svcnl|Xyc¨~3prqtstmc¥¼*pry¶nprk cR8svcR*nc¤ © _*cR~8ypº kst8m-_|Xm`| Σ
¢¹m-_*cnpry-jTsvlµl-bdp0prm-_ © stm-_ycnl~3cnRm¶mp|¨lcRm
SpXlpry-ml¤ © c © stqtqycnl-m-y-svRm6prw8ylcRqt cnl6mpUnprrz$w8*Rm-svpr*l9pX |Xm Σ
¢¹qtstmcRy|Xqvlº¥)¼8w8y-m-_*cRy-bdpryc¤© _*cR~8yp 0st8ªm-_|Xm T svld³*8stmcRqtj © stm-*cnll|Xo8qvc © stm-_ycnl~3cnRmmp S ¤ © c © stqtq8cK³**c£m-_*c
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
§w8*Rm-svprwitnessT
ogjycnlm-y-svRm-st8Íprw8ylcRqt cnlmp[nprrz$w8*Rm-svpr*lΓpX |Xm
Σ¢¹qtstmcRy|Xqvll-w*_
m-_|Xmvarsσ(Γ) 6= ∅
¤TÃpry)c|r_[lpry-mσ ∈ S
¥^`_*cVÃprqtqvp © st8=m © p=~8ypr~pl-stm-svpr*l9l-_*p © m-_|Xm¸m-_8svl¸|XÍo3c8pr*c © stm-_*prw8m6qvpll6pX¶cRT¢
cRy|XqtstmËj¥% É ¡É C É )(
Σ !"C ' 6 )
S = σ1, . . . , σn ⊆ ΣS E !. !'! )C
T
Σ $5 '"* * !E!$ ' F* E'.'Γ
! Σ )+1 ') ! ' F*
T 01 7' A ! ! *' Γ ' F* !
τ ∈ S ' F* ')
0 κ > |Aτ |
$5 ' !!>T 01 6 B ! ! *' Γ
!5$ $
|Bτ | = κ ,
|Bσ| = |Aσ | , σ ∈ S \ τ .24$5
T! !.$ #$ 6! 6.
S *
5:: 9Ëml-w5GUncnlmpd*prmcVm-_|Xm6cR cRy-jFukw|Xgm-s¦³cRy$¢Ã§ycncΣ¢Ã§pry-bw8q®|dsvl
T¢Ácnukw8st r|XqvcRkmmp²|
̸¯6¼Γ1 ∨ · · · ∨ Γn
l-w*_Fm-_|Xm|Xqtq´m-_*cΓiocRqvpr8dmp
L¥
% É ¡É C É )( Σ
!"C ' 6 ) S = σ1, . . . , σn ⊆ ΣS E !. !'!
) T
E>Σ $6'* * ) ! J)
L >$5 !. )) '" 7'!
Γ !
Σ ) 16') !!5$ $5varsσ(Γ) 6= ∅
'>6 $ !'σ ∈ S *
!E! $5L$5 ' !! ' 5() w $ ' F* '" 7'Γ
L/6 !> 50 - Σ " )H
ψ = w(Γ)!"$ $
Γ
' (∃v)ψ
' T @)C # $5
v = vars(ψ) \ vars(Γ) ψ !
T ! !- () $5 $5 ' !! T 01 7' A ! ! *' ψ !"$$
Aσ = [varsσ(ψ)]A '>6 $
σ ∈ S *24$5
T! - 1) *# "!"!()C#$3 "!
S *
5:: Ic © |Xkmmp 8cK³**c | © stm-*cnll¾§w8*Rm-svpr witnesspr |Xqtqukw|Xgm-s¦³cRy$¢Ã§ycnc
Σ¢
Ãpry-bVw8q®|rc¨ogj=w*l-st8|rl|o8q®|rED=o3p·²m-_*c¸§w8*Rm-svprw8cK³**cnFpr
L¥
^Npd8pdlp8¤TqvcRmϕoc¨|ukw|Xgm-s¦³cRy$¢Ã§ycnc
Σ¢ÃÃpry-bVw8q®|T¤3|X*£~3cRy$Ãpry-b m-_*c¸Ãprqtqvp © st8K
nprg cRy-m ϕ stkmpF|T¢Ácnugw8st |XqvcRgm9̸¯¼
Γ1 ∨ · · · ∨ Γnl-w*_Ím-_|Xm
varsσ(Γ) 6= ∅¤3§pry
c|r_σ ∈ S
qvcRm witness(ϕ) = w(Γ1) ∨ · · · ∨ w(Γn)¥
éé äMNYïOYò
B '! ( !"!<'.6
2 %(`' %9:
cRmTioc¨|
Σi¢¹m-_*cnpry-j¤*§pry
i = 1, 2¤|X*£qvcRm
S = ΣS1 ∩ ΣS
2
¥!Îll-w8bdc¸m-_|Xm.K
m-_*c¨ukw|Xgm-s¦³cRy$¢Ã§ycncVl|Xm-svl$³|Xo8stqtstmËj=~8ypro8qvcRb pXTisvl)8cnRsv*|Xo8qvc¤8Ãpry
i = 1, 2
ΣF1 ∩ ΣF
2 = ∅|X*
ΣP1 ∩ ΣP
2 = ∅
T2svl)~3prqtstmc © stm-_Fycnl~3cnRm)mp S ¥
9Ë[m-_8svl6lcnRm-svpr © c8cnlRy-sto3c|UbdcRm-_*pkÃpry9nprbVo8st8st8£m-_*c8cnRsvl-svprÍ~8ypkncnTw8ycnl6Ãpry6m-_*cukw|Xgm-s¦³cRy$¢Ã§ycncl|Xm-svl$³|Xo8stqtstmËj~8ypro8qvcRbdl!pX
T1|X*pX
T2stpry8cRy!mp68cnRsv8c`m-_*cµukw|Xgm-s¦³cRy$¢
§ycnc)l|Xm-svl-³|Xo8stqtstmÁj~8ypro8qvcRb pXT1⊕T2
¥JI¿stm-_*prw8m¶qvpll¡pX´cR*cRy|XqtstmÁj¤ © cycnl-m-y-svRmprw8ylcRqt cnlmpnprrz$w8*Rm-svpr*l6pXqtstmcRy|Xqvl¥
^`_*c²nprbo8st|Xm-svpr bdcRm-_*p0«npr*lsvl-mlpX)§prw8y~8_|rlcnl K @'7() ("! ( ¤ # "!E!06 7' ¤ ' F!7' ¤|X* $56 ¥
AXn N8 Å 88A8 ! Å 8nX8 É cRmΓoc|nprrz±w8*Rm-svpr£pX
(Σ1 ∪Σ2)¢¹qtstmcRy|Xqvl¥
^`_*c9prw8m-~8w8mpXm-_*c¸ |Xy-s®|Xo8qvcV|Xo*l-m-y|rRm-svprF~8_|rlc9svl)|nprrz±w8*Rm-svprΓ1 ∪ Γ2
l|Xm-svl-Aj0st8m-_*cÃprqtqvp © st8U~8ypr~3cRy-m-svcnl.K | £c|r_FqtstmcRy|Xqst
Γisvl|
Σi¢¹qtstmcRy|Xq¹¤8Ãpry
i = 1, 2
o Γ1 ∪ Γ2
svl(T1 ⊕ T2)
¢Ál|Xm-svl$³|Xo8qvc9s¦¡|X*Fpr8qtj²s¦Γsvl
(T1 ⊕ T2)¢Ál|Xm-svl-³|Xo8qvc¥
¯6prmcm-_|Xm¨~8ypr~cRy-m-svcnl | V|X* o ¸|Xocdc´cnRm-st cRqtj¬cRT§pryncn © stm-_m-_*c_*cRqt~ pXµ§ycnl-_ |Xy-s®|Xo8qvcnl¥Ic9|Xqtq
Γ1 ∪ Γ2|nprrz$w8*Rm-svprpX\qtstmcRy|Xqvl)st !. '61 Ãpry-bF¥
´Å É Ê 8 Å È C Å É Ê É cRmΓ1∪Γ2
oc)|¸nprrz±w8*Rm-svpr²pX3qtstmcRy|XqvlstlcR~|Xy|Xmcµ§pry-b ycRm-w8y-*cnstm-_*c` |Xy-s®|Xo8qvc)|Xo*l-m-y|rRm-svpr~8_|rlc¥ 9Ám-_*c © stm-*cnll!stkm-ypkTw*Rm-svpr~8_|rlc © cdnprb~8w8mc ψ2 = witnessT2
(Γ2)¤|X* © cdprw8m-~8w8m Γ1 ∪ ψ2
¥ 9Ákm-w8stm-st cRqtj¤m-_8svl~8_|rlcstkm-ypkTw*ncnlVm-_*cAycnl_[ |Xy-s®|Xo8qvcnl9st
vars(ψ2) \ vars(Γ)¤ © _*plcyprqvcsvl¸mp © stm-*cnll
m-_|XmncRy-m|Xst£Ã|rRml)_*prqv=§prym-_*c¨~3prqtstmc¸m-_*cnpry-jT2¥
çXÕÖÑØäXÒ§ÛÁÚäXõ$àKðÑØäÛ¹ÐXà´Û¹Ðà$ÕYÖÃݵÕ%0ÚÖÃÖÁÚ$ݺÒÚ!ÜØÑØÛ¹àËÖÁÚYÜ a 6≈array b Ñ$*ÙÜØÑÓà±ÒÛ¹ÐrÚÛÛ¹Ðà±Ö¹àNÑÓÒ3ÚYä)ÑØäráºà)( i ÒÃßXõÁÐ`Û¹ÐrÚÛread(a, i) 6≈ read(b, i) ÿÏÐà$äkð i õ-ÚYä6âàÛ¹ÐXÕYßXãKÐnÛÕ%ÚYÒNÚ¶ÔÑÞÛ¹äXà$ÒÃÒNÕ% a 6≈array b ÿ
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! BB
AXÊ 8 Å Ê Å É ¡É C É cRmΓ1 ∪ ψ2
o3c£m-_*cnprrz±w8*Rm-svpr pro8m|Xst*cn¿stm-_*c © stm-*cnllVstkm-ypkTw*Rm-svpr«~8_|rlc¥ cRm Vσ = varsσ(ψ2)
ÃpryVc|r_σ ∈ S
¤|X*qvcRmV =
⋃
σ∈S Vσ¥ 9Ë m-_*c[8cnnprb~pl-stm-svpr#~8_|rlc © cÍ*pr*8cRmcRy-bst8svl-m-sv|Xqtqtj w*cnllF|ª¹|Xbstqtj
EpX9cnukw8st r|XqvcR*nc£ycRq®|Xm-svpr*lE = Eσ ⊆ Vσ × Vσ | σ ∈ S
¥ ^`_*cR¤ © c¾npr*l-m-y-w*Rmdm-_*c"6' (0 pX
Vst*Tw*ncnFokj
E¤*8cK³**cn£ogj
arr (V,E) = x ≈ y | (x, y) ∈ Eσ|X*
σ ∈ S ∪
x 6≈ y | (x, y) ∈ (Vσ × Vσ) \ Eσ|X*
σ ∈ S ,
|X* © c¨prw8m-~8w8m)m-_*cVnprrz±w8*Rm-svpr Γ1 ∪ ψ2 ∪ arr (V,E)¥
É º 8 Å ¸ Å cRmΓ1∪ψ2∪arr (V,E)
oc|9nprrz$w8*Rm-svprpro8m|Xst*cndstUm-_*c8cnnprb~pl-stm-svprF~8_|rlc¥¡^`_*c¨_*cnED£~8_|rlc9npr*l-svl-ml)stF~3cRy$Ãpry-bst8dm-_*c6Ãprqtqvp © st8²l-mcR~*l.K Å 9Á
Γ1∪arr (V,E)svlT1¢Ál|Xm-svl$³|Xo8qvc`pmp6m-_*c¶*cK·Tm!l-mcR~ prm-_*cRy © svlc`prw8m-~8w8m*¥
Å 9Áψ2 ∪ arr (V,E)
svlT2¢Ál|Xm-svl$³|Xo8qvcp=mp=m-_*c*cK·Tm9l-mcR~ Nprm-_*cRy © svlcUprw8m-~8w8m
*¥ Å prw8m-~8w8m ¥
#,"%# cRm
Σ1o3cdm-_*cl-st|Xm-w8ycUnprkm|Xst8st8[|£lpry-m
elem¤|rl © cRqtqµ|rl¨m © pnpr*l-m|Xgml-j0bVoprqvl a|X*
bpX!lpry-m
elem¥µpr*l-sv8cRy)m-_*c
Σ1¢¹m-_*cnpry-j
T1 = TheoryΣ1(Φ1)¤ © _*cRyc
Φ1 = (∀elem x)(x ≈ a ∨ x ≈ b) .
¶qvc|Xy-qtj¤ÃprydcR cRy-jT1¢¹stgmcRy-~8ycRm|Xm-svpr
A¤ © c£_|n c |Aelem| ≤ 2
¥«^`_*cRycKÃpryc¤T1
svl*prmlm|Xo8qtjstT³*8stmc © stm-_¾ycnl-~3cnRmmp elem ¥¯cK·Tm¤´npr*l-sv8cRy9m-_*c
Σset
¢¹m-_*cnpry-jTset
pX`lcRml9pXµcRqvcRbdcRkml¥^`_*cl-st|Xm-w8ycΣset
nprT¢m|Xst*lº¤|Xbdpr8£prm-_*cRy9lcRm$¢¹m-_*cnprycRm-sv|Xq¡l-j0bVoprqvl¤´|lpry-m
elem§pry¸cRqvcRbdcRgml¤|X*[|²lpry-m
setÃprylcRmlpXcRqvcRbdcRgmlº¥^`_*c¨m-_*cnpry-jTset
© stqtqo3c8cK³**cn¾bdprycV§pry-bU|Xqtqtj£st¬ekw8o*lcnRm-svprT¥¼pry)m-_8svl)cK·*|Xb~8qvc¤8stml-w5GUncnlmpDk*p © m-_|Xm
Tset
svl~prqtstmc © stm-_Fycnl-~cnRm)mp elem ¥¯cK·Tm¤8npr*l-sv8cRy)m-_*c¸Ãprqtqvp © st8nprrz±w8*Rm-svpr ΓpX
(Σ1 ∪Σset)¢¹qtstmcRy|Xqvl K
Γ =
a ≈ b ,x 6≈ ∅ ,y 6≈ ∅ ,x ∩ y ≈ ∅
,
éé äMNYïOYò
B '! ( !"!<'.6
© _*cRyc x |X* y |Xyc set¢¹ |Xy-s®|Xo8qvcnl¥
¯prmcm-_|XmΓsvl
(T1⊕Tset)¢¹w8*l|Xm-svl$³|Xo8qvc¥^NpVlcncm-_8svl¤k|rll-w8bdcogjnprkm-y|rTsvRm-svpr²m-_|Xm
Asvl¨|
(T1 ⊕ Tset)¢¹stkmcRy-~8ycRm|Xm-svpr«l-w*_Ím-_|Xm
Γsvl6m-y-w*cst
A¥ µj¾m-_*c³*yl-m¸qtstmcRy|Xq!st
Γ¤
© c¨_| c |Aelem| = 1¥ :p © cR cRy¤ogj£m-_*c9m-_8ycncVq®|rl-mqtstmcRy|Xqvlst Γ
¤ © cV_|n c |Aelem| ≥ 2¤|
nprkm-y|rTsvRm-svpr¥Ic © |Xkm)mpÃpry-bU|Xqtqtj=8cRmcnRmm-_|Xm Γ
svl(T1 ⊕ Tset)
¢¹w8*l|Xm-svl$³|Xo8qvcVogj=w*l-st8dprw8ynprb¢o8st|Xm-svpr¾bdcRm-_*pk´¥
ekst*nc|XqtqqtstmcRy|XqvlstΓ|XyccRstm-_*cRy
Σ1¢¹qtstmcRy|Xqvl¶pry
Σset
¢¹qtstmcRy|Xqvlº¤stdm-_*c |Xy-s®|Xo8qvc|Xo*lm-y|rK¢m-svprU~8_|rlc © c8p9*prm¡*cncnUmp9stgm-yp0Tw*ncAycnl-_U r|Xy-s®|Xo8qvcnl¤0|X* © cl-stb~8qtjycRm-w8y-Um-_*c)m © pnprrz$w8*Rm-svpr*l.K
Γ1 =
a ≈ b
, Γset =
x 6≈ ∅ ,y 6≈ ∅ ,x ∩ y ≈ ∅
.
9Á m-_*c © stm-*cnll£stkm-ypkTw*Rm-svpr ~8_|rlc © cÍ*cncn mp½nprb~8w8mcwitness set(Γset)
¥ ^`_*cstkm-w8stm-svpr¾ocR_8st*£m-_*c9nprb~8w8m|Xm-svprÍpX
witness set(Γset)svl|rl`§prqtqvp © l¥
^`_*c`qtstmcRy|Xqx 6≈ ∅
stb~8qtsvcnl!m-_*ccK·Tsvl-mcR*ncpX|XdcRqvcRbdcRkmwx
stx¥ sCDc © svlc¤m-_*c`qtstmcRy|Xq
y 6≈ ∅stb~8qtsvcnlm-_*cVcK·Tsvl-mcR*ncVpX|XcRqvcRbdcRkm
wysty¥`^`_*cVprw8m-~8w8m¸pX
witness set(Γset)svl
|nprrz$w8*Rm-svpr∆set
m-_|XmbU|FDcnlcK·T~8qtsvRstmm-_*c¸cK·0svl-mcR*nc¸pXm-_*c6cRqvcRbdcRkmlwx
|X*wy¥JIc
|X8pm-_8svlogj=qvcRm-m-st8
∆set =
wx ∈ x ,wy ∈ y ,x ∩ y ≈ ∅
.
¯6prmc¸m-_|XmΓset
|X*(∃elem wx)(∃elem wy)∆set
|XycTset
¢Ácnukw8st r|XqvcRkm¥9Ám-_*c8cnnprb~pl-stm-svpr[~8_|rlc © cV*cncnmpdw*cnll6|XÍcnugw8st |XqvcR*ncycRq®|Xm-svpr Eelem
pº cRym-_*c |Xy-s®|Xo8qvcnl¸st
varselem(∆set)¥¨ekst*nc
vars elem(∆set) = wx, wy¤m-_*cRyc|Xycm © p²~3pl-¢
lsto8qvc9_*prsvncnl.K¶cRstm-_*cRy © c9w*cnll (wx, wy) ∈ Eelem
pry © c9w*cnll (wx, wy) /∈ Eelem
¥9¹ © c w*cnll (wx, wy) ∈ Eelem
m-_*cR © c _|n c m-_|Xm ∆set ∪ wx ≈ wysvlTset
¢w8*lY|Xm-svl$³|Xo8qvc¤9|X* © c © stqtqVprw8m-~8w8m «st#l-mcR~ »pXm-_*c_*cnED ~8_|rlc¥ 9Ást*lmc|r© cw*cnll (wx, wy) /∈ Eelem
m-_*cR © c_| cm-_|Xm Γ1 ∪wx 6≈ wysvlT1¢¹w8*lY|Xm-svl$³|Xo8qvc¤8|X*
© c © stqtqNprw8m-~8w8m stFl-mcR~ B9pX\m-_*c¨_*cn D=~8_|rlc¥ekst*nc!m-_*c¡_*cnED¸~8_|rlc¡prw8m-~8w8ml µÃpryN|Xgj¸cnugw8st |XqvcR*nc¡ycRq®|Xm-svpr
Eelem
pXvarselem(∆set)
¤prw8y)nprbo8st|Xm-svprbdcRm-_*p0Fnpry-ycnRm-qtj£npr*Rqtw*8cnlm-_|Xm
Γsvl
(T1 ⊕ Tset)¢¹w8*l|Xm-svl-³|Xo8qvc¥
ö %¦ÕYÖ *)ÚÜgáàXäXÑÞÛ¹ÑØÕKä6Õ%8Ú %¦ßXäõ±Û¹ÑØÕKäwitness set
õ-ÚYäâà % ÕKßXäXáÑØä9æßXâÒÃà$õ±Û¹ÑØÕKäñºÿçXÕÖ´Û¹ÐXÑØÒ´à)(Ú*ÙÜØàKðnÔàÙÖ¹à)% à±ÖÛ¹Õ`Ò§Û¹ÑØõË×Û¹Õ`ÑØänÛ¹ßXÑÞÛ¹Ñ #Rà!ÚÖ¹ãKß*à±äÛ¹Ò$ÿ
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! B
(#)9( #+ + " ) %#,' ^`_*cnpry-ycnRm-*cnll9pXµprw8y¸nprbo8st|Xm-svprbdcRm-_*p0svl¸o|rlcnprÍm-_*c§prqtqvp © st8µprbVo8st|Xm-svpr^`_*cnprycRbF¤ © _8sv_£svl)|~|Xy-m-svRw8q®|Xy)|rlc¸pX\|nprbVo8st|Xm-svprycnl-w8qtm`_*prqvTst8§prypry8cRy$¢Álpry-mcnqvprsv=CBB?Á¥ ÅkÉ Å VÉ T É" )(
Σ1'
Σ2E !"C ' "! !5$ $
ΣF1 ∩ΣF
2 = ∅ ΣP
1 ∩ ΣP2 = ∅ * ) ! )C
Φi
E3!.L Σi " )H( ' i = 1, 2 * 24$5Φ1 ∪ Φ2! ! !- () ' '0) * $5 '!! ' 01 7'
A!! *F Φ1
' 01 7'
B!! *F Φ2
!5$ $5 |Aσ| = |Bσ|
'> F*σ ∈ ΣS
1 ∩ ΣS2
xA = yA
' ')+* xB = yB > F*
x, y ∈ vars(Φ1) ∩ vars(Φ2) *
% É ¡É C É & )( Ti
Σi $56'* !"$ $
ΣF1 ∩ ΣF
2 = ∅'
ΣP1 ∩ ΣP
2 = ∅
' i = 1, 2 * !E!>$ T2
! ')+1#$ "! 6LS = ΣS
1 ∩ ΣS2 *
) ! /)CΓ1 ∪ Γ2E " 7' )+1 ') ! !. '61 ' ' )C
ψ2 = witnessT2(Γ2) *
')) *)
Vσ = varsσ(ψ2) '6($
σ ∈ SL )
V =⋃
σ∈S Vσ *24$5 $5 '))H'# '6
@') 0 * Γ1 ∪ Γ2
!(T1 ⊕ T2) !!- ()
* 24$5 '!!> ' ) *E
@') " )H '!
E = Eσ ⊆ Vσ × Vσ | σ ∈ S ,
!"$3$ Γ1∪ arr(V,E)
!T1 !!- () '
ψ2∪ arr(V,E)!T2 ! !- () *
5:: 1 ⇒ 2K¥Î6ll-w8bdcm-_|Xm
Γ1 ∪ Γ2svl
(T1 ⊕ T2)¢Ál|Xm-svl$³|Xo8qvc¥ cRm
v = vars(ψ2) \vars(Γ2)
¥Fekst*ncΓ2
|X*(∃v)ψ2
|XycT2¢Ácnugw8st |XqvcRgm¤\stmV§prqtqvp © lVm-_|Xm Γ1 ∪ ψ2
svl|Xqvlp(T1⊕T2)
¢Ál|Xm-svl$³|Xo8qvc¥^`_gw*lº¤ © c¡|X¨³8·V| (T1⊕T2)¢¹stkmcRy-~8ycRm|Xm-svpr
Al|Xm-svl$§j0st8
Γ1∪ψ2¥
¯6cK·Tm¤TqvcRmE = Eσ | σ ∈ S
© _*cRyc
Eσ = (x, y) | x, y ∈ Vσ|X*
xA = yA ,Ãpry
σ ∈ S .
¶j¨npr*lm-y-w*Rm-svpr¤ © c¶_|n cµm-_|Xm Γ1∪arr(V,E)svlT1¢ÁlY|Xm-svl$³|Xo8qvcµ|X*
ψ2∪arr(V,E)svlT2¢Ál|Xm-svl$³|Xo8qvc¥
2 ⇒ 1
K¥ cRmAocF|
T1¢¹stgmcRy-~8ycRm|Xm-svpr l|Xm-svl$§jkst8
Γ1 ∪ arr (V,E)¤`|X*½qvcRm
BocF|
T2¢¹stgmcRy-~8ycRm|Xm-svpr£l|Xm-svl$§j0st8
ψ2∪arr (V,E)¥¶ekst*nc
T2svl\³*8stmcRqtj © stm-*cnll|Xo8qvc¤ © c|X
|rll-w8bdc © stm-_*prw8mqvpllpX\cR*cRy|XqtstmÁj£m-_|Xm Bσ = V Bσ
¤TÃpry)c|r_σ ∈ S
¥
éé äMNYïOYò
B '! ( !"!<'.6
^`_gw*l¤TÃpry)c|r_σ ∈ S
¤ © c9_| c
|Bσ| = |VBσ |
l-st*ncBσ = V B
σ
= |V Aσ |
l-st*nc¸o3prm-_A|X*Bl|Xm-svl$§j
arr (V,E)
≤ |Aσ|l-st*nc
V Aσ ⊆ Aσ .
¶w8m6m-_*cR¤3ogjFm-_*cl-bdp0prm-_8*cnll6pXT2¤m-_*cRyccK·Tsvl-ml6|
T2¢¹stkmcRy-~8ycRm|Xm-svpr
Cl|Xm-svl$§j0st8
ψ2 ∪ arr(V,E)l-w*_ m-_|Xm
|Cσ| = |Aσ|¤§pryc|r_
σ ∈ S¥ Ic²|X«m-_*cRycKÃpryc=|X~8~8qtj
^`_*cnprycRb BÍmpA|X*
C¤µpro8m|Xst8st8ªm-_*c¾cK·0svl-mcR*nc¾pX9|
(T1 ⊕ T2)¢¹stgmcRy-~8ycRm|Xm-svpr
FlY|Xm-svl$Aj0st8Γ1 ∪ ψ2 ∪ arr (V,E)
¥dekst*ncΓ2
|X*(∃v)ψ2
|XycT2¢Ácnugw8st |XqvcRgm¤NstmÃprqtqvp © l
m-_|XmF|XqvlplY|Xm-svl$³cnl
Γ1 ∪ Γ2¥
;l-st8²x!ypr~pl-stm-svpr B£|X*Fm-_*c¨Ã|rRmm-_|Xmprw8y¸nprbVo8st|Xm-svprÍbdcRm-_*pk¾svlmcRy-bst|Xm-st8*¤© c¨pro8m|XstFm-_*c¨npry-ycnRm-*cnllpX!prw8ynprbVo8st|Xm-svprbdcRm-_*pk´¥ ÅkÉ Å 'VÉ Å Å ºd Ê¿ É ! Å ® Æ (
Ti
E Σi $6'* i = 1, 2 *
!E!$
$5 '0 - 6 !!- )+* )C Ti
! 6 () ' i = 1, 2 ΣF
1 ∩ ΣF2 = ∅
' ΣP
1 ∩ ΣP2 = ∅
T2 ! ) 1#$ 6! 6.
ΣS1 ∩ ΣS
2 *24$ $ 5' - ! !- () * 6() ! () *
' 6 F $5 '0 - &!!- )+* ()C!> T1' T2
' &NP
witnessT2
! ' 5() ')+*'' 7') 4$ $5 5' - ! !- ()+* ()C T1 ⊕ T2
!NP ' ) 1 *
5:: ¶qvc|Xy-qtj¤Xm-_*c¡8cnRsv*|Xo8stqtstmËj¨pXTm-_*cugw|Xkm-s¦³cRy$¢ÃAycnc¶l|Xm-svl$³|Xo8stqtstmÁj9~8ypro8qvcRb pXT1⊕T2Ãprqtqvp © l)ogjx¡ypr~3pl-stm-svpr BU|X*=m-_*c¹|rRmm-_|Xm)prw8y)nprbVo8st|Xm-svprbdcRm-_*pk=svl`mcRy-bst|Xm-st8*¥
µpr*ncRy-8st8NP
¢¹_|XyT*cnll¤X*prmc!m-_|XmNs¦ © c¶|XVlprqt cm-_*c¡ugw|Xkm-s¦³cRy$¢ÃAycncµl|Xm-svl$³|Xo8stqtstmËj~8ypro8qvcRb pX
T1 ⊕ T2¤Tm-_*cR © c¨|X¾|Xqvlpdlprqt c¸~8ypr~pl-stm-svpr|Xql|Xm-svl-³|Xo8stqtstmÁj¥
µpr*ncRy-8st8bdcRbo3cRyl-_8st~stNP
¤X|rllw8bdc!m-_|XmNm-_*c¡ukw|Xgm-s¦³cRy$¢Ã§ycncµl|Xm-svl$³|Xo8stqtstmÁj9~8yproT¢qvcRbdlpX
T1|X*pX
T2|Xycst
NP¤|X*m-_|Xm
witnessT2
svlnprb~8w8m|Xo8qvcµst~3prqtj0*prbs®|Xq0m-stbdc¥I stm-_*prw8m`qvpll`pXNcR*cRy|XqtstmÁj¤stm¶svl`cR*prw8_£mpl-_*p © m-_|Xm`st=*pr*8cRmcRy-bst8svl-m-sv9~prqtj0*prbs®|Xqm-stbdc © cd|X«_*cn DÍm-_*c (T1 ⊕ T2)
¢Ál|Xm-svl$³|Xo8stqtstmËjpX)nprrz±w8*Rm-svpr*lpX(Σ1 ∪ Σ2)
¢¹qtstmcRy|Xqvl¥^plcnc6m-_8svl¤0*prmc¸m-_|Xmµm-_*c¸cK·8cnRw8m-svprFpXprw8ynprbVo8st|Xm-svpr¾bdcRm-_*pk=ycnukw8stycnl`mpw*cnll|X|Xy-y|X8cRbdcRkm9p cRy¨|dlcRm6pX¡ r|Xy-s®|Xo8qvcnl © _*plc|XyTst|XqtstmÁjsvl~prqtjk*prbs®|Xq © stm-_ycnl~3cnRm6mpm-_*cl-stÀcpXm-_*cVst8~8w8m¥6^`_8svl6w*cnll6|Xoc8pr*cst[*pr*8cRmcRy-bst8svl-m-sv~3prqtj0*prbs®|Xqm-stbdc¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! B
^`_*cnprycRb B²|X[o3cycR~c|XmcnTqtj[|X~8~8qtsvcn[mp=npr*l-sv8cRy¸m-_*cVw88svpr¬pXnm-_*cnpry-svcnl
T1 ⊕· · · ⊕ Tn
¤ © _*cRyc T2, . . . , Tn|Xycd~prqtstmc © stm-_ ycnl-~3cnRmVmpFm-_*cUlcRmpX)l-_|Xycnªlpry-ml¥=^`_8svl
qvc|r8l)mpm-_*c6§prqtqvp © st8UcR*cRy|XqtstÀº|Xm-svprÍpX\^`_*cnprycRb B§prynm-_*cnpry-svcnl¥
ÅkÉ Å )( n ≥ 2
' ) Ti
Σi $56'* ' 1 ≤ i ≤ n * ) !)
S =⋃
i6=j(ΣSi ∩ ΣS
j ) * !E!>$
$5 '0 - 6 !!- )+* )C Ti
! 6 () ' 1 ≤ i ≤ n ⋃
i6=j(ΣSi ∩ ΣS
j ) =⋂
i ΣSi
ΣFi ∩ ΣF
j = ∅'
ΣPi ∩ ΣP
j = ∅ ' 1 ≤ i < j ≤ n
Ti
! ')+1>#$3 "! 6 S ' 2 ≤ i ≤ n *
24$ $ 5' - ! !- () * 6() T1 ⊕ · · · ⊕ Tn
! 6 ()C * ' 6 F 4 $ '0 - 6E ! !- ()+* 6() Ti
! NP
' 1 ≤ i ≤ n
witnessTi
!E' )C ')+*F ' ')J 2 ≤ i ≤ n $5 $ '0 -
!!- () * 6() T1 ⊕ · · · ⊕ Tn
!NP E' )C1 *
5:: Ic=~8yp0ncncn«ogjªst*Tw*Rm-svpr¿prn¥ 9Á
n = 2 © c£|X|X~8~8qtj«prw8ynprbo8st|Xm-svprbdcRm-_*p0 mp
T1|X*
T2¤!|X* m-_*cRq®|Xstb§prqtqvp © logjª^`_*cnprycRb BT¥ 9Á)st*l-mc|r
n > 2¤\stm
lw5GUncnlmpÍ|X~8~8qtjªprw8ynprbVo8st|Xm-svpr bdcRm-_*p0ª³*yl-mmpT1
|X*T2¤¡|X*«l-w8o*lcnukw*cRgm-qtj mp
T1 ⊕ T2¤T3¤´¥n¥n¥n¤
Tn¥
¸%6,/%!3!
ek_8stkjUm-_*cnpry-svcnl © cRycstkm-ypkTw*ncn=okj^`st*cRqtqts|X*FÂ|Xy-o| =CB@?st=pry8cRy`mpcK·0mcR*m-_*c¸pr*cK¢lpry-mcnU cRyl-svpr£pXNm-_*c¸¯6cRqvlprT¢±°~8~cR²bdcRm-_*p0²mpVm-_*c¸nprbo8st|Xm-svpr¾pXN*pr*l-m|Xo8qtjstT³*8stmcm-_*cnpry-svcnlº¥\ek_8stkj¨m-_*cnpry-svcnl|XycstgmcRycnl-m-st8¸ocn|Xw*lccR cRy-j¨l_8stgj9m-_*cnpry-j
S|Xo3c¶nprbVo8st*cn
© stm-_[|Xgj=prm-_*cRy)m-_*cnpry-j T ¤8cR cRFs¦m-_*c¨q®|Xm-mcRy)svl*prml-m|Xo8qtj²stT³*8stmc¥^`_*c¸*prm-svprFpXl-_8st8st*cnll © |rl)pry-stst|Xqtqtjstkm-ypkTw*ncnFst=pr*cK¢Álpry-mcn£qvprsv¤*|X*²st£m-_8svllcnRm-svpr © ccR*cRy|XqtstÀc6stmµmp¨bU|Xgjk¢Álpry-mcn²qvprsv¥Ic6|XqvlpV~8ypº c6m-_|Xm¤gw8*8cRy`y|Xm-_*cRy © c|FD|rll-w8b~8m-svpr*l¤rl-_8st8st*cnllNsvlcnukw8st r|XqvcRkmmp~3prqtstmcR*cnllNstVpr*cK¢Álpry-mcnVqvprsv¥\^`_*ccnukw8st |XqvcR*ncsvl)qvcnllRqvc|Xy)stFbU|Xkjg¢Álpry-mcn¾qvprsv¥-£Å C É cRm
Toc|
Σ¢¹m-_*cnpry-j¤¸qvcRm
S ⊆ ΣS ¤¸|X* qvcRmϕoc|
T¢Ál|Xm-svl$³|Xo8qvc
ukw|Xgm-s¦³cRy$¢Ã§ycncΣ¢ÃÃpry-bVw8q®|T¥Ic8cR*prmc © stm-_ mincardT,S(ϕ)
m-_*cbst8stbw8bpX`m-_*c§prq¦¢qvp © st8lcRm)pX!|XyTst|Xqkw8bVocRyl.K
(
maxσ∈S|Aσ|
)
| A |=T ϕ
.
éé äMNYïOYò
B '! ( !"!<'.6
²Å ª8 cRmToc |
Σ¢¹m-_*cnpry-j#m-_|Xm[_|rl¾m-_*cª³*8stmcªbdp08cRq~8ypr~3cRy-mËj © stm-_ ycK¢
l~3cnRm[mpS¥ ^`_*cR¤§pry¬cR cRy-j
T¢Ál|Xm-svl$³|Xo8qvc»ukw|Xgm-s¦³cRy$¢Ã§ycnc
Σ¢Ã§pry-bw8q®|
ϕ¤ © c»_|n c
mincardT,S(ϕ) ∈ N+ ¥
-£Å C É ' Å º cRmΣoc|Vl-st|Xm-w8yc¤gqvcRm
S ⊆ ΣS oc|VlcRmpXlpry-ml¤0|X*qvcRm
Toc¨|
Σ¢¹m-_*cnpry-j¥/Ic9l|nj=m-_|Xm
Tsvl /1 © stm-_¾ycnl-~3cnRmmp S s¦ K
Tsvl)l-bdp0prm-_ © stm-_Fycnl-~cnRm)mp S
T_|rlm-_*c¸³*8stmc9bdp08cRq~8ypr~cRy-mÁj © stm-_Fycnl-~cnRmmp S
mincard T,Ssvl)nprb~8w8m|Xo8qvc¥
^`_*c¸§prqtqvp © st8d~8ypr~pl-stm-svprl-_*p © lm-_|Xml-_8st8st*cR*cnll|Xq © |j0lstb~8qtsvcnl)~prqtstmcR*cnll¥% É ¡É C É )(
T 3!"$(* $6'* #$ "! 6 !
S !'! * 24$5
T! ')+1#$ "! 6 S *
5:: µj |rllw8b~8m-svpr¤ T svllbdpkprm-_ © stm-_»ycnl-~cnRmmp S ¥¬^p~8ypº c=m-_|Xm T svl|Xqvlp³*8stmcRqtj © stm-*cnll|Xo8qvc © stm-_ ycnl~3cnRm£mp
S¤qvcRm
ϕo3c¬|
T¢Ál|Xm-svl$³|Xo8qvc¬ukw|Xgm-s¦³cRy$¢Ã§ycnc
Σ¢
Ãpry-bVw8q®|T¤|X*£qvcRmn = mincardT,S(ϕ)
¥^`_*cRFm-_*cRyc¨cK·Tsvl-ml|
T¢¹stgmcRy-~8ycRm|Xm-svpr
Al|Xm-svl$§jkst8
ϕl-w*_¾m-_|Xm
|Aσ| ≤ n¤kÃpryc|r_
σ ∈ S¥!^`_*cRycKÃpryc¤8Ãpry)c|r_
σ ∈ S¤ © cV|XFqvcRm Aσ = aσ
1 , . . . , aσkσ¤ © stm-_ kσ < n
¥¼*pryc|r_[lpry-m
σ ∈ S¤*qvcRm
wσ1 , . . . , w
σn
oc¸§ycnl-_F |Xy-s®|Xo8qvcnlpXlpry-mσ*prmp0nRw8y-y-st8²st
ϕ¥¶µpr*l-sv8cRym-_*c6§pry-bw8q®|5K
ψ : ϕ ∧∧
σ∈S
n∧
i=1
(wσi ≈ w
σi ) .
¶qvc|Xy-qtj¤ϕ|X*
(∃w)ψ|Xyc
T¢Ácnugw8st |XqvcRgm¤ © _*cRyc w = vars(ψ) \ vars(ϕ)
¥6hprycnpº cRy¤ψsvl)m-y-w*c9st£m-_*c
T¢¹stgmcRy-~8ycRm|Xm-svpr
Bpro8m|Xst*cn¾ogj£cK·TmcR*Tst8
A|rl`Ãprqtqvp © l K
(wσi )B =
aσi ,
s¦i ≤ kσ ,
aσ1 ,
s¦i > kσ .
¶w8mm-_*cR¤ © cV|XF8cK³**c| © stm-*cnll`§w8*Rm-svpr£Ãpry T ogj=qvcRm-m-st8witness(ϕ) = ψ
¥
^`_*c`§prqtqvp © st8V~8ypr~pl-stm-svprcnlm|Xo8qtsvl-_*cnl¡l-w5GURsvcRkm¶npr*Tstm-svpr*l¶w8*8cRy © _8sv_~3prqtstmcR*cnllstb~8qtsvcnll-_8st8st*cnll¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! Bº
% É ¡É C É )( Σ
!H ' )CS ⊆ ΣS !. !'! ' )
T
Σ $56'* * !"!" $
ΣS = S Σ
!-/1
' 6($Σ 01 6 A !"$ $ ⋃
σ∈S Aσ
! - 1 ! 6 () $6# $5$5
A !
T 01 7' '& T
! ')+1#$ "! 6 S *
24$T
! !"$(0* #$ 6! 6. S *
5:: µj|rll-w8b~8m-svpr¤ T svl!lbdpkprm-_ © stm-_ycnl~3cnRm!mp S ¥Îqvlp8¤ T _|rl!m-_*cµ³*8stmc)bdpk8cRq~8ypr~cRy-mÁj © stm-_Fycnl~3cnRmmp S ¥¯cK·Tm¤ © c9Rq®|Xstb m-_|Xmm-_*c6§w8*Rm-svprmincardT,S
|X=oc6c´cnRm-st cRqtj=nprb~8w8mcn£ogjm-_*c~8yp0ncnTw8yc /1 85 st¬¼stw8ycdT¥V^Np=lcncm-_8svl¤ © c*cncnÍmp=l-_*p © m-_|Xm¸m-_*c~8yp0ncnTw8yc /21 8;5 svlmcRy-bst|Xm-st8²|X*£~|Xy-m-s®|Xqtqtj£npry-ycnRm¥µpr*ncRy-8st8[mcRy-bst|Xm-svpr¤´z±w*l-m*prmcUm-_|Xml-st*nc
ΣS = S¤stm¨Ãprqtqvp © lm-_|Xm¤§pryc|r_
k > 0¤*m-_*c9kw8bVocRy6pX!stkmcRy-~8ycRm|Xm-svpr*l6cRkw8bdcRy|Xmcn bdpkTw8qvpUsvlprbdpry-~8_8svl-b `st¾qtst*c dsvl
³*8stmc¥µpr*ncRy-8st8 ~|Xy-m-s®|Xq¸npry-ycnRm-*cnll¤)|rllw8bdcFm-_|XmUm-_*c~8yp0ncnTw8yc /1 85 cK·0stml=|Xm
qtst*c)T¥\^`_*cRdm-_*cRyccK·0svl-ml¡|XstkmcRcRykl-w*_dm-_|Xm
ϕA = true¤rÃpry¡lprbdc
T¢¹stkmcRy-~8ycRm|Xm-svpr
Al-w*_¾m-_|Xm
|Aσ | ≤ k¤0Ãpry|Xqtq
σ ∈ S¥¡h[prycnp cRy¤*§pry|Xqtq
T¢¹stkmcRy-~8ycRm|Xm-svpr*l
Bl-w*_Fm-_|Xm
|Bσ| ≤ k − 1¤0§pry|Xqtq
σ ∈ S¤ © c9_| c ϕB = false
¥^`_*cRycK§pryc¤mincard T,S(ϕ) = k
¥¯cK·Tm¤8|rll-w8bdc6m-_|Xm`m-_*c~8yp0ncnTw8yc /1 85 cK·Tstml`|Xm`qtst*cV0¥9ËmÃprqtqvp © l`m-_|Xm`Ãpry|Xqtq
T¢¹stkmcRy-~8ycRm|Xm-svpr*l
Bl-w*_Fm-_|Xm
|Bσ| ≤ n− 1¤TÃpry|Xqtq
σ ∈ S¤ © c9_| c ϕB = false
¥^`_8svlstb~8qtsvcnlm-_|Xm
mincardT,S(ϕ) ≥ n¥!h[prycnp cRy¤l-st*nc
n = max j | j = |varsσ(witness(ϕ))||X*
σ ∈ S ,
stm)§prqtqvp © lm-_|Xm)m-_*cRyccK·0svlml| T ¢¹stkmcRy-~8ycRm|Xm-svpr A l-w*_¾m-_|XmϕA = true
|X*|Aσ| ≤ n
¤Ãpry|Xqtq
σ ∈ S¥!^`_8svlstb~8qtsvcnlm-_|Xm
mincard T,S(ϕ) ≤ n¥
ekst*ncmincard T,S(ϕ) ≥ n
|X*mincard T,S(ϕ) ≤ n
¤ © cpro8m|Xst mincardT,S(ϕ) = n¥
I¿_*cR ∣
∣ΣS∣
∣ = 1¤¡x!ypr~pl-stm-svpr 5B²mcRqtqvlw*lm-_|Xmst»m-_*c=pr*cK¢Álpry-mcn»|rlc²~3prqtstmcR*cnll
|X* l-_8st8st*cnllF|Xycm-_*cÍl|Xbdc[npr*ncR~8mÃpry£|Xqtq6~8y|rRm-sv|Xq¸~8w8y-~plcnl¥ I _*cR ∣
∣ΣS∣
∣ > 1¤
m-_*c_kjk~prm-_*cnl-svlΣS = S
bU|nj[ocmp0p£l-m-ypr8*¥µpr*lcnukw*cRgm-qtj¤m-_*ccnugw8st |XqvcR*nco3cRm © cncR~prqtstmcR*cnll|X*¾l-_8st8st*cnllsvlqvcnll)Rqvc|Xy)st£m-_*c9bU|Xkjg¢Álpry-mcn|rlc¥
éé äMNYïOYò
B '! ( !"!<'.6
; N ÎT¢Ál|Xm-svl-³|Xo8qvc¨ugw|Xkm-s¦³cRy$¢ÃAycnc
Σ¢ÃÃpry-bVw8q®|
ϕN NmincardT,S(ϕ)
í, É Å Ê Å /1 85 ϕ þ, n← max j | j = |varsσ(witness(ϕ))|
|X*σ ∈ S
î, ËÉ k ← 1
Én− 1
Ê Éó, ËÉ d! !
Σ¢¹stkmcRy-~8ycRm|Xm-svpr*l
Ap cRy
vars(ϕ)l¥ m¥
k = maxσ∈S |Aσ|Ê É
N, A |=T ϕ
Å ï, Å N
kO, Å
n
N Å Î ~8ypkncnTw8yc¸Ãpry)nprb~8w8m-st8mincard T,σ
¥
¸+$k,
-£Å C É ^`_*c 3 4:;5 : 4 8 / 3 © stm-_»l-st|Xm-w8yc Σsvl¨m-_*cm-_*cnpry-j
TΣ≈ =
〈Σ,A〉¤ © _*cRyc
Asvlm-_*c¨Rq®|rll)pX!|Xqtq
Σ¢Álm-y-w*Rm-w8ycnl¥
% #+ +% É ¡É C É )(
Σ 3 !"C ' ' )
τ ∈ ΣS * ) ! )CΓ
!!- ()CE'.'A !
Σ ) 16') ! ) A
Σ 01 7' ! ! *' Γ
' )Cκ >
|Aτ | *24$5 $5 '!!Σ 01 6 B ! ! *' Γ
!"$ $
|Bσ| =
κ , σ = τ ,
|Aσ| ,$"#! *
5:: cRm Vσ = varsσ(Γ)¤§pry
σ ∈ ΣS ¤¡|X* qvcRm V =⋃
σ∈ΣS Vσ¥ Ic²npr*l-m-y-w*Rmd|
Σ¢¹stkmcRy-~8ycRm|Xm-svpr
Bp cRy
V|rl`Ãprqtqvp © l¥¼s¦·F|lcRm A′ l-w*_Fm-_|Xm |Aτ ∪A
′| = κ¤|X*£qvcRm
Bσ =
Aτ ∪A′ ,
s¦σ = τ ,
Aσ ,prm-_*cRy © svlc ,
|X*
xB = xA ,§pryc|r_F |Xy-s®|Xo8qvc
x ∈ V .9Á«pry8cRymp[8cK³**c
Bp cRym-_*cljkbo3prqvlst
Σ¤³8·«|X«cRqvcRbdcRkm
aσ0
stAσ¤Ãpryc|r_
σ ∈ ΣS ¥!^`_*cR¤ © c9qvcRm.K
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! B
§pry`§w8*Rm-svpr¾l-j0bVoprqvl f pX¡|Xy-stmÁj σ1 × · · · × σn → σK
fB(a1, . . . , an) =
fA(a1, . . . , an) ,s¦a1 ∈ Aσ1
, . . . , an ∈ Aσn,
aσ0 ,
prm-_*cRy © svlc , §pry~8ycnTsv|Xmcl-j0bVoprqvl p pX!|Xy-stmËj σ1 × · · · × σn
K
(a1, . . . , an) ∈ pB ⇐⇒ a1 ∈ Aσ1, . . . , an ∈ Aσn
|X*(a1, . . . , an) ∈ pA .
¶j[npr*l-m-y-w*Rm-svpr¤Bsvl¨|
TΣ≈
¢¹stkmcRy-~8ycRm|Xm-svpr«l-w*_¬m-_|Xm|Bτ | = κ
|X*|Bσ| = |Aσ|
¤Ãpry
σ 6= τ¥¡¯6cK·0m¤ © c¨l-_*p © m-_|Xm
Bl|Xm-svl-³cnl|XqtqqtstmcRy|Xqvl)st
Γ¥
C Å X!C É Å ËÉ x = y
Êx 6= y
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ÁÉ x = f(y1, . . . , yn)
È Å ÅfA É Æ [ É ! É 8C Æ
σ1 × · · · × σn → σ Ic9_| cK
xB = xA
= fA(
yA1 , . . . , yAn
)
= fB(
yB1 , . . . , yBn
) l-st*nc (
yA1 , . . . , yAn
)
∈ Aσ1× · · · ×Aσn
.
C Å X!C É Å ÁÉ p(y1, . . . , yn)
ʬp(y1, . . . , yn)
È Å ÅpA= Å ÊAT Å
Æ É ! É 8 C Æσ1×· · ·×σn
w*l-m¶pro*lcRy- cm-_|Xm¤gl-st*nc (
yA1 , . . . , yAn
)
∈ Aσ1×· · ·×
Aσn
¤ © c9_|n cVm-_|Xm (yA1 , . . . , yAn ) ∈ pA
s (yB1 , . . . , y
Bn ) ∈ pB
¥
% É ¡É C É & É´É Å '&6 $ !"C ' Σ/' ' '* ' *!
!'!S ⊆ ΣS $5>$56'*
TΣ≈
! !.$ #$ "! S *
5:: µj=x!ypr~3plstm-svpr*lBBV|X*FT¥
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnllU§w8*Rm-svpr witness≈
ÃpryTΣ≈
|X o3c[8cK³**cn |rlÃprq¦¢qvp © l¥>I¿stm-_*prw8m¨qvpll9pX¶cR*cRy|XqtstmÁj¤NqvcRm
Γo3cd|²nprrz$w8*Rm-svprªpX |Xm
Σ¢¹qtstmcRy|Xqvl¨l-w*_¬m-_|Xm
varsσ(Γ) 6= ∅¤0Ãpry)c|r_[lpry-m
σ ∈ S¥!^`_*cR © c¨l-stb~8qtj²qvcRm witness≈(Γ) = Γ
¥% É ¡É C É )(
Γ 3 " 7' !
Σ )+1 ) ! ' ) S ⊆ ΣS * !"!"
$5varsσ(Γ) 6= ∅
'6 $ !'σ ∈ S * 24$5 $5 '))H'# ' @') 0
éé äMNYïOYò
'! ( !"!<'.6
* Γ! !!- () *
* 24$5 !! Σ 01 7' A ! ! *' Γ
!"$ $5Aσ = [varsσ(Γ)]A
'6 $ !'σ ∈ S *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤§pryσ ∈ ΣS ¤r|X*VqvcRm V =
⋃
σ∈ΣS Vσ¥\ekst*nc
ΓsvlNl|Xm-svl$³|Xo8qvc¤
m-_*cRyccK·Tsvl-mlV|Σ¢¹stkmcRy-~8ycRm|Xm-svpr
Bl|Xm-svl$§jkst8
Γ¥Ic © stqtq!w*lc Γ
stpry8cRy¨mp£npr*l-m-y-w*Rm|XFpr~8~pry-m-w8*c
TΣ≈
¢¹stgmcRy-~8ycRm|Xm-svprA¥
¼*pryσ ∈ S
¤8cR*prmc © stm-_ ∼σm-_*ccnukw8st r|XqvcR*ncycRq®|Xm-svpr¬p cRy¸m-_*c
Σ¢¹mcRy-bdl¸p cRy
VpX
lpry-mσ¤8cK³**cnokj
s ∼σ ts sB = tB
¥39Á¬m-_*cÃprqtqvp © st8*¤ © c © y-stmc ∼ st~8q®|rncdpX∼σ© _*cR¾m-_*c9lpry-m σ svl)Rqvc|Xy§yprb m-_*c¨nprgmcK·Tm¥
¯cK·Tm¤0qvcRmTσ
ocm-_*c¸lcRmpXNmcRy-bdlpXlpry-mσm-_|XmpknRw8y`st
Γ¥ 9ÁÃpry`|lpry-m
σ © c¸_|n cTσ = ∅
¤8³8·£|§ycnl-_£ |Xy-s®|Xo8qvcxσ
0
pX\lpry-mσ¥
cRmAocµm-_*c
Σ¢¹stgmcRy-~8ycRm|Xm-svpr²p cRy
V|X*m-_*c` r|Xy-s®|Xo8qvcnl
xσ0
npr*l-m-y-w*RmcnU|rl\Ãprqtqvp © lº¥¼styl-m¤ © c9qvcRm
Aσ =
Tσ/ ∼ ,s¦Tσ 6= ∅ ,
[xσ0 ]∼ ,
prm-_*cRy © svlc .|X*
xA = [x]∼ ,Ãpryc|r_F |Xy-s®|Xo8qvc
x .
9Á«pry8cRymp8cK³**cAp cRym-_*cl-jkbo3prqvlst
Σ¤³8· |X»cRqvcRbdcRgm
aσ0
stAσ¤Ãpryc|r_
σ ∈ ΣS ¥!^`_*cR¤ © c9qvcRm.K
§pry`§w8*Rm-svpr¾l-j0bVoprqvl f pX¡|Xy-stmÁj σ1 × · · · × σn → σK
fA([t1]∼, . . . , [tn]∼) =
[f(t1, . . . , tn)]∼ ,s¦f(t1, . . . , tn) ∈ Tσ ,
aσ0 ,
prm-_*cRy © svlc ,
§pry~8ycnTsv|Xmcl-j0bVoprqvl p pX!|Xy-stmËj σ1 × · · · × σnK
([t1]∼, . . . , [tn]∼) ∈ pA ⇐⇒(
tB1 , . . . , tBn
)
∈ pB .
¯prmcm-_|XmAsvl¨| © cRqtq¦¢Á8cK³**cn Σ
¢¹stgmcRy-~8ycRm|Xm-svpr¤\|X*¬m-_|XmAσ = [varsσ(Γ)]A
¤Ãpryc|r_¾lpry-m
σ ∈ S¥!¯6cK·Tm¤ © c¨l-_*p © m-_|Xm
Al|Xm-svl$³cnl|Xqtq´qtstmcRy|Xqvlst
Γ¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! 5B
C Å X!C É Å ËÉ x = y
Êx 6= y
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ÁÉ x = f(y1, . . . , yn)
È Å ÅfA É Æ [ É ! É 8C Æ
σ1 × · · · × σn → σ Ic9_| cK
xA = [x]∼
= [f(y1, . . . , yn)]∼
= fA([y1]∼, . . . , [yn]∼)l-st*nc
f(y1, . . . , yn) ∈ Tσ
= fA(
yA1 , . . . , yAn
)
.
C Å X!C É Å ÁÉ p(y1, . . . , yn)
ʬp(y1, . . . , yn)
È Å ÅpA= Å ÊAT Å
Æ É ! É 8 ® Æσ1 × · · · × σn
w*l-mpro*lcRy- c9m-_|Xm.K(
yB1 , . . . , yBn
)
∈ pB ⇐⇒ ([y1]∼, . . . , [yn]∼) ∈ pA
⇐⇒(
yA1 , . . . , yAn
)
∈ pA .
% É ¡É C É ' ® ÅÈ C NÅ 8 ! C Æ ' $&!H ' Σ ' '* ' *
!. !'!S ⊆ ΣS $5>$56"*
TΣ≈
!-/1 # "!"!() &#$ 6! 6. S *
5:: µj=x!ypr~3plstm-svpr*lBd|X*FT¥
*' # #+ + ÅkÉ Å %6É ! C Å Å ' 6 $ !"C '
Σ3' ' '* ' * !. !'!
S ⊆ ΣS $5>$56'*TΣ≈
! ) 1#$ 6! 6. S *
5:: µj=x!ypr~3plstm-svpr*l|X*FT¥
;= 8
cRmAo3c|²*pr*cRb~8mËjÍlcRm¥VÎ )+!
xp cRy
Asvl¨|lcnugw*cR*nc
〈a1, . . . , an〉¤ © _*cRyc n ≥ 0|X*
a1, . . . , an ⊆ A¥Ic98cR*prmc © stm-_ A∗ m-_*c¨lcRm)pX\qtsvl-ml)p cRy A ¥^`_*cm-_*cnpry-jpXqtsvl-mlTlist
_|rl9|Ulst|Xm-w8ycΣlist
nprkm|Xst8st8¾|lpry-melem
Ãpry6cRqvcRbdcRkml|X*|lpry-m
list§pry)qtsvlmlpX\cRqvcRbdcRkml¤*~8qtw*lm-_*c6§prqtqvp © st8l-j0bVoprqvl.K
m-_*c¨npr*l-m|Xkml-j0bVoprq nil¤8pX!lpry-m
list
m-_*c¸§w8*Rm-svpr¾l-j0bVoprqvl
éé äMNYïOYò
'! ( !"!<'.6
car¤8pX¡|Xy-stmÁj
list→ elem
cdr¤8pX¡|Xy-stmÁj
list→ list
cons¤8pX¡|Xy-stmÁj
elem× list→ list¥
-£Å C É Î 3(8;1 8;5 list . /1 3 45"765"4,38 3"/:1 A svl`|Σlist
¢¹stgmcRy-~8ycRm|Xm-svprFlY|Xm-svl$¢§j0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Alist = (Aelem)∗
nilA = 〈〉
carA(〈e1, . . . , en〉) = e1¤0Ãpry)c|r_
n > 0|X*
e1, . . . , en ∈ Aelem
cdrA(〈e1, . . . , en〉) = 〈e2, . . . , en〉¤8§pry)c|r_
n > 0|X*
e1, . . . , en ∈ Aelem
consA(e, 〈e1, . . . , en〉) = 〈e, e1, . . . , en〉¤g§pry¶c|r_
n ≥ 0|X*
e, e1, . . . , en ∈ Aelem
¥^`_*c 3 46:5 : /3 svlm-_*c¡~|Xsty Tlist = 〈Σlist,A〉
¤ © _*cRycAsvlm-_*c¶Rq®|rllpX|Xqtqkl-m|X**|Xy
list¢Ál-m-y-w*Rm-w8ycnlº¥
% #+ +% É ¡É C É )(
A
Tlist 01 7'A!! *F E'.'Γ
!
Σlist )+16') ! L' !5$ $ Aelem
! - 1 * 24$5 $5 ' !! Tlist 01 7' B!! *F Γ
!5$ $ |Belem| = κ
'>6($ ' ')40 κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤§pry
σ ∈ elem, list¥ Icnpr*l-m-y-w*Rm9|
Tlist
¢¹stgmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vlist
|rl`Ãprqtqvp © l¥¼s¦·F|lcRm A′ l-w*_Fm-_|Xm |Aelem ∪A′| = κ
¤|X*=qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xB = xA ,§pry)c|r_
list¢¹ r|Xy-s®|Xo8qvc
x ∈ Vlist .
¶j=npr*l-m-y-w*Rm-svpr¤Bsvl|
Tlist
¢¹stkmcRy-~8ycRm|Xm-svpr[l-w*_Fm-_|Xm|Belem| = κ
¥hprycnpº cRy¤*stm)svlm-y-st 0s®|Xqmp_*cn D£m-_|Xm
BlY|Xm-svl$³cnl|XqtqqtstmcRy|Xqvlst
Γ¥
% É ¡É C É ' É´É Å 24$5 $5 '"*Tlist
! !.$3#$ "! 6 elem *
5:: µj=x!ypr~3plstm-svpr*lBBV|X*FT¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
' ' )9(' ; +9ËFpry8cRymp~8ypº c¨m-_|Xmm-_*c9m-_*cnpry-j
Tlist
svl³*8stmcRqtj © stm-*cnll|Xo8qvc © stm-_Fycnl~3cnRm)mp elem ¤© c © stqtqw*lc¸m-_*c|XwT·0stqts®|Xy-jAw8*Rm-svpr*l compress|X*
δ¥
st cRx ∈ A∗ |X* X ⊆ A
¤m-_*c¨Aw8*Rm-svprcompress
ycRm-w8y-*lm-_*cVqtsvl-mpro8m|Xst*cn¾§yprbxogj²ycRbdp 0st8²|XqtqNcRqvcRbdcRkml)m-_|Xm|Xyc9*prmst
X¥¡¼*pry-bU|Xqtqtj K
compress(〈α1, . . . , αn〉, X) =
〈〉 ,s¦n = 0 ,
〈α1〉 compress(〈α2, . . . , αn〉, X) ,s¦n > 0
|X*α1 ∈ X ,
compress(〈α2, . . . , αn〉, X) ,prm-_*cRy © svlc¤
© _*cRyc svl`m-_*c¨npr*|XmcR|Xm-svpr[pr~cRy|Xmprypº cRy)qtsvl-ml¥ st cRx, y ∈ A∗ ¤Tm-_*c¸§w8*Rm-svpr δ mcnlml © _*cRm-_*cRy x = y
pryx 6= y
¥!¼*pry-bU|Xqtqtj K
δ([α1, . . . , αn], [β1, . . . , βm]) =
∅ ,s¦n = 0
|X*m = 0 ,
α1 ,s¦n > 0
|X*m = 0 ,
β1 ,s¦n = 0
|X*m > 0 ,
α1, β1 ,s¦n,m > 0
|X*α1 6= β1 ,
δ([α2, . . . , αn], [β2, . . . , βm]) ,prm-_*cRy © svlc¥
% É ¡É C É '>')) ) !!x, y
$5 '))C'# $ ) !
x 6= y ' ) * δ(x, y) 6= ∅
' '*!X x 6= y
' δ(x, y) ⊆ X
$5compress(x,X) 6= compress(y,X) *
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witness list
§prym-_*c¶m-_*cnpry-jTlist
|Xocµ8cK³**cn|rlÃprqtqvp © l¥ I stm-_*prw8m`qvpll¶pXcR*cRy|XqtstmËj¤TqvcRm
Γoc6|Vnprrz±w8*Rm-svpr£pX |Xm
Σlist
¢¹qtstmcRy|Xqvl`l-w*_m-_|Xm
varselem(Γ) 6= ∅¥<Ic¶qvcRm
witness list(Γ)oc¡m-_*c¶ycnl-w8qtm\pX|X~8~8qtjkst8¸mp
Γm-_*cÃprqtqvp © st8
m-y|X*l-§pry-bU|Xm-svpr*l.K
cR~8q®|rncdc|r_ªqtstmcRy|XqµpXµm-_*cÃpry-b e ≈ car(x)st
Γ © stm-_m-_*cÃpry-bVw8q®| x 6≈ nil →x ≈ cons(e, y′)
¤ © _*cRyc y′ svl|Aycnl-_ list¢¹ |Xy-s®|Xo8qvc¥
cR~8q®|rncdc|r_qtstmcRy|XqpX`m-_*c§pry-b x ≈ cdr(y)st
Γ © stm-_ªm-_*c§pry-bw8q®| x 6≈ nil →y ≈ cons(e′, x)
¤ © _*cRyc e′ svl|Aycnl-_ elem¢¹ |Xy-s®|Xo8qvc¥
éé äMNYïOYò
'! ( !"!<'.6
¼*pryc|r_»qtstmcRy|XqpXm-_*cU§pry-b x 6≈list yst
Γ¤\cR*cRy|Xmc²m © p[§ycnl-_ elem
¢¹ |Xy-s®|Xo8qvcnlw′
x,y
|X*w′′
x,y
¤*|X*|r8=m-_*c9qtstmcRy|Xqvlw′
x,y ≈ w′x,y
|X*w′′
x,y ≈ w′′x,y
mpΓ¥
²Å ª8 cRmΓo3c9|nprrz$w8*Rm-svprFpX |Xm
Σlist
¢¹qtstmcRy|Xqvl¤TqvcRm∆ = witness list(Γ)
¤8|X*qvcRm
v = vars(∆) \ vars(Γ)¥¡^`_*cR
Γ|X*
(∃v)∆|Xyc
Tlist
¢Ácnukw8st r|XqvcRkm¥
% É ¡É C É )( Γ
& ' ! Σlist ) 16') ! !5$ $
vars elem(Γ) 6=∅ ' '0 '0*)+16') $5 ' e ≈ car(x)
' x ≈ cdr(y) * ) ! )
kE&$5& > )+1 ') ! $5 ' x 6≈list y.E
Γ * 24$ $5 '))C'# 6 @') 0 * Γ
!Tlist ! !- () *
* 24$5 '!!>Tlist 1 6 ' A ! ! *' Γ
!"$ $
Aelem = [vars elem(Γ)]A ∪A′ ,
# $5 |A′| ≤ 2k *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1 ⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤Ãpryσ ∈ elem, list
¥£ekst*ncΓsvlTlist
¢Ál|Xm-svl$³|Xo8qvc¤m-_*cRyccK·Tsvl-mlV|
Tlist
¢¹stkmcRy-~8ycRm|Xm-svprBl|Xm-svl-Aj0st8
Γ¥3Ic © stqtqw*lc B st pry8cRyVmp¾npr*l-m-y-w*Rm|Xpr~8~pry-m-w8*c
Tlist
¢¹stgmcRy-~8ycRm|Xm-svprA¥
hpryc¡l~3cnRs¦³|Xqtqtj¤ © cqvcRm A o3c\m-_*cw88svugw*c Tlist
¢¹stkmcRy-~8ycRm|Xm-svprnpr*l-m-y-w*Rmcn¨okj9qvcRm-m-st8
Aelem = V Belem ∪
α ∈ δ(
xB, yB)
|m-_*c9qtstmcRy|Xq
x 6≈ ysvlst
Γ
,
|X*
eA = eB ,Ãpry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xA = compress(
xB, Aelem
)
,Ãpry)c|r_
list¢¹ |Xy-s®|Xo8qvc
x ∈ Vlist .
¯prmcm-_|XmAsvl| © cRqtq¦¢Á8cK³**cn Tlist
¢¹stgmcRy-~8ycRm|Xm-svpr¤|X*m-_|XmAelem = V A
elem ∪A′ ¤Ãpry
|lcRmA′ l-w*_Fm-_|Xm |A′| ≤ 2k
¥¡¯6cK·Tm¤ © c¨l-_*p © m-_|XmAl|Xm-svl$³cnl|Xqtq´qtstmcRy|Xqvlst
Γ¥
C Å X!C É Å ËÉ e1 ≈elem e2
Êe1 6≈elem e2
9ÁbbdcnTs®|Xmc¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
C Å X!C É Å ËÉ x ≈list y
Ic¸_|n c
xA = compress(
xB, Aelem
)
= compress(
yB, Aelem
)
= yA .
C Å X!C É Å ËÉ x 6≈list y
µj²x!ypr~pl-stm-svpr5B © stm-_ X = Aelem
K¥
C Å X!C É Å ËÉ x ≈ nil
Ic9_|n c
xA = compress(
xB, Aelem
)
= compress(〈〉, Aelem)
= 〈〉 .
C Å X!C É Å ËÉ x ≈ cons(e, y)
Ic¸_|n c
xA = compress(
xB, Aelem
)
= compress(⟨
eB⟩
yB, Aelem
)
=⟨
eB⟩
compress(
yB, Aelem
) l-st*nceB ∈ Aelem
=⟨
eA⟩
yA
% É ¡É C É & ® ÅÈ C NÅ 8 ! C Æ 24$5 $5 '"*Tlist
!-/1)+* # !E!)C#$ 6"! 6
elem *
5:: µj=x!ypr~3plstm-svpr*lBT¤*cRbU|Xy6D=T¤|X*£x¡ypr~3pl-stm-svprÍT¥
*' # #+ + ÅkÉ Å ' %6É ! C Å Å 24$5$56'*
Tlist
! ')+1#$ "! 6 elem *
5:: µj=x!ypr~3plstm-svpr*ld|X*F*¥
) #)9 (# Ic`nprrz-cnRm-w8yc`m-_|Xm|bdpryccGURsvcRgm © stm-*cnll§w8*Rm-svpr witness ′list
ÃpryTlist
|Xoc`8cK³**cn|rlÃprqtqvp © l¥ I stm-_*prw8m`qvpll¶pXcR*cRy|XqtstmËj¤TqvcRm
Γoc6|Vnprrz±w8*Rm-svpr£pX |Xm
Σlist
¢¹qtstmcRy|Xqvl`l-w*_m-_|Xm
vars elem(Γ) 6= ∅¥ Ic¨qvcRm
witness ′listoc¨m-_*cVycnl-w8qtm¸pX¶|X~8~8qtjkst8²mp
Γm-_*c9Ãprqtqvp © st8
m-y|X*l-§pry-bU|Xm-svpr K
éé äMNYïOYò
'! ( !"!<'.6
cR~8q®|rncdc|r_qtstmcRy|XqpX`m-_*c§pry-b x ≈ cdr(y)st
Γ © stm-_ªm-_*c§pry-bw8q®| x 6≈ nil →y ≈ cons(e′, x)
¤ © _*cRyc e′ svl|Aycnl-_ elem¢¹ |Xy-s®|Xo8qvc¥
Ic98pd*prm)_|n c9jcRm6|§pry-bU|Xq~8yp0pX!pXm-_8svl)Rq®|XstbF¥
¡,9
^`_*cm-_*cnpry-jdpX|Xy-y|njTlTarray
_|rl`|9l-st|Xm-w8ycΣarray
nprkm|Xst8st8d|¨lpry-melem
§prycRqvcRbdcRkml¤|²lpry-m
index§pry¸st*Tsvncnl¤|X*¬|²lpry-m
array§pry9|Xy-y|njTl¤´~8qtw*l6m-_*cÃprqtqvp © st8£m © p²Aw8*Rm-svpr
ljkbo3prqvl.K
read¤8pX\lpry-m
array × index→ elem
write¤8pX\lpry-m
array × index× elem→ array¥
FÉ T É st cRa : I → E
¤i ∈ I
|X*e ∈ E
¤ © c8cK³**c ai7→e : I → E|rlÃprqtqvp © l.K
ai7→e(i) = e|X*
ai7→e(j) = a(j)¤8Ãpry
j 6= i¥
-£Å C É Î 38;1 8;5 array . /21"3"465"765"4,3(893"/2:;1 A svl=|Σarray
¢¹stgmcRy-~8ycRm|Xm-svprlY|Xm-svl$Aj0st8m-_*c6Ãprqtqvp © st8npr*Tstm-svpr*l K
Aarray = (Aelem)Aindex
readA(a, i) = a(i)¤*§pry)c|r_
a ∈ Aarray
|X*i ∈ Aindex
writeA(a, i, e) = ai7→e¤0Ãpry)c|r_
a ∈ Aarray
¤i ∈ Aindex
¤*|X*e ∈ Aelem
¥^`_*c 3 46:5 : 8;5"58 svlm-_*cV~|Xsty Tarray = 〈Σarray,A〉
¤ © _*cRycAsvlm-_*cRq®|rllpX|Xqtq
lm|X**|Xyarray
¢Ál-m-y-w*Rm-w8ycnl¥
% #+ +% É ¡É C É )(
A
Tarray 01 7' !! *F '" 7'Γ
!
Σarray ) 16') ! ' !"$ $ Aindex
!J-/1 * 24$5 $5 > !!&Tarray 1 6 '
B! ! *' Γ
!"$ $ |Belem| = |Aelem|
' |Bindex| = κ
>6 $ ')4
κ > |Aindex| *
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
5:: cRm Vσ = varsσ(Γ)¤¶Ãpry
σ ∈ elem, index, array¥ IcFnpr*l-m-y-w*Rml|
Tarray
¢stkmcRy-~8ycRm|Xm-svpr
Bpº cRy
Velem ∪Vindex∪Varray
|rl¡Ãprqtqvp © l¥!¼s¦·|VlcRm A′ l-w*_Um-_|Xm |Aindex ∪A′| = κ
¤*|X*£qvcRm
Bindex = Aindex ∪A′ ,
Belem = Aelem ,
|X*
iB = iA ,Ãpry)c|r_
index¢¹ r|Xy-s®|Xo8qvc
i ∈ Vindex ,
eB = eA ,Ãpry)c|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem .
9Ëpry8cRyVmp¾8cK³**cBpº cRyVm-_*c
array¢¹ |Xy-s®|Xo8qvcnl¤N³8·ª|XªcRqvcRbdcRgm
e0 ∈ Aelem
¥^`_*cR¤Ãpryc|r_
array¢¹ |Xy-s®|Xo8qvc
a ∈ Varray
¤ © c9qvcRm
aB(i) =
aA(i) ,s¦i ∈ Aindex ,
e0 ,prm-_*cRy © svlc .
¶j npr*l-m-y-w*Rm-svpr¤Bsvl[|
Tarray
¢¹stgmcRy-~8ycRm|Xm-svpr l-w*_ m-_|Xm|Belem| = |Aelem|
|X*|Bindex| = κ
¥¡¯6cK·0m¤ © c¨l-_*p © m-_|XmBlY|Xm-svl$³cnl|Xqtq´qtstmcRy|Xqvlst
Γ¥
C Å X!C É Å ÁÉ e1 ≈elem e2 e1 6≈elem e2 i ≈index j
Êi 6≈index j
9ÁbbdcnTs¦¢|Xmc¥
C Å X!C É Å ËÉ a ≈array b
cRmi ∈ Bindex
¥9¹i ∈ Aindex
m-_*cRaB(i) = aA(i) =
bA(i) = bB(i)¥/9Á
i /∈ Aindex
m-_*cRaB(i) = e0 = aB(i)
¥^`_gw*lº¤aB = bB
¥
C Å X!C É Å ÁÉ a 6≈array b
ekst*ncaA 6= bA
¤m-_*cRyccK·Tsvl-ml9|XÍst*8cK·i ∈ Aindexlw*_Fm-_|Xm
aA(i) 6= bA(i)¥ 9Ám`Ãprqtqvp © lm-_|Xm aB(i) 6= bB(i)
¤ © _8sv_Fstb~8qtsvcnl aB 6= bB¥
C Å X!C É Å ËÉ e ≈ read(a, i)
Ic¸_|n cK
eB = eA
= [read(a, i)]A
= aA(
iA)
= aB(
iB) lst*nc
iA ∈ Aindex .
éé äMNYïOYò
'! ( !"!<'.6
C Å X!C É Å ËÉ a = write(b, i, e)
Ic²_|n caB(iB) = aA(iA) = eA = eB
¥¯6cK·Tm¤qvcRm
j ∈ Bindex
l-w*_¬m-_|Xmj 6= iB
¥>9¹j ∈ Aindex
m-_*cR © c_|n c aB(j) = aA(j) =bA(j) = bB(j)
¥ 9¹st*l-mc|rj /∈ Aindex
m-_*cRaB(j) = e0 = bB(j)
¥¶^`_gw*l¤aB = bB
iB 7→eB
¥
% É ¡É C É )( A
Tarray 01 7' !! *F '" 7'
Γ !
Σarray ) 16') ! !5$ $ Aelem
!-/01 * 24$ $5 '!!>Tarray 1 6 '
B! ! *' Γ
!5$ $ |Bindex| = |Aindex|
|Belem| = κ
>6 $κ > |Aelem| *
5:: ekstbstq®|Xympm-_*c9~8ypkpX!pX!x!ypr~3plstm-svpr0¥
% É ¡É C É ' É´É Å ' * * ! !'!S ⊆ elem, index
$$6'*
Tarray
! !.$ #$ 6! 6. S *
5:: µj=x!ypr~3plstm-svpr*lBB¤0¤*|X*¾T¥
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witnessarray
Ãprym-_*c²m-_*cnpry-jTarray
|X»oc8cK³**cn|rl6Ãprqtqvp © lº¥I stm-_*prw8m9qvpll9pX¶cR*cRy|XqtstmËj¤qvcRm
Γoc|nprrz$w8*Rm-svprªpX |Xm
Σarray
¢qtstmcRy|Xqvl`l-w*_m-_|Xm
vars index(Γ) 6= ∅|X*
vars elem(Γ) 6= ∅¥ IcqvcRm
witnessarray(Γ)oc)m-_*c
ycnlw8qtm)pX!|X~8~8qtj0st8UmpΓm-_*c9§prqtqvp © st8dm-y|X*l$Ãpry-bU|Xm-svpr K
cR~8q®|rnc¬c|r_ qtstmcRy|Xq¨pXm-_*c§pry-ba 6≈array b
stΓ © stm-_ | qtstmcRy|Xq¨pXm-_*c[Ãpry-b
read(a, i′) 6≈ read(b, i′)¤ © _*cRyc i′ svl|§ycnl-_ index
¢¹ r|Xy-s®|Xo8qvc¥²Å ª8 & cRm
Γo3c|²nprrz±w8*Rm-svprpX |Xm
Σarray
¢¹qtstmcRy|Xqvl¤´qvcRm∆ = witnessarray(Γ)
¤|X*FqvcRm
v = vars(∆) \ vars(Γ)¥!^`_*cR
Γ|X*
(∃v)∆|Xyc
Tarray
¢Ácnukw8st r|XqvcRkm¥
% É ¡É C É'& )( Γ
E E'.' ! Σarray ) 16') !/!"$$
vars index(Γ) 6=∅
varselem(Γ) 6= ∅
' L '0 '* ) 16')/ $5 ' x 6≈array y *24$5
$ '))H# 3' @)C * Γ
!Tarray ! !- () *
* 24$5 '!! Tarray 01 7' A !! *' Γ
!"$$ Aindex = [vars index(Γ)]A'
Aelem = [vars elem(Γ)]A *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤0Ãpryσ ∈ elem, index, array
¥ekst*ncΓsvlTarray
¢Ál|Xm-svl-³|Xo8qvc¤m-_*cRyccK·0svl-ml¡|
Tarray
¢¹stgmcRy-~8ycRm|Xm-svprBlY|Xm-svl$Aj0st8
Γ¥JIc © stqtq*w*lc B stUpry8cRy!mp¨npr*l-m-y-w*Rm|XFpr~8~pry-m-w8*c
Tarray
¢¹stgmcRy-~8ycRm|Xm-svprA¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
hpryc¨l~3cnRs¦³|Xqtqtj¤ © c9qvcRm A o3c¸m-_*cTarray
¢¹stkmcRy-~8ycRm|Xm-svpr[pº cRyVelem ∪ Vindex ∪ Varraynpr*lm-y-w*Rmcn¾|rl`Ãprqtqvp © l¥¡¼styl-m¤ © c9qvcRm
Aindex = V Bindex ,
Aelem = V Belem ,
|X*
iA = iB ,Ãpry)c|r_
index¢¹ r|Xy-s®|Xo8qvc
i ∈ Vindex ,
eA = eB ,Ãpry)c|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem .
9Ë¿pry8cRy²mpª8cK³**cAp cRy²m-_*c
array¢¹ |Xy-s®|Xo8qvcnl¤`³8·¿|X
e0 ∈ Aelem
¥ ^`_*cR¤`Ãpryc|r_a ∈ Varray
|X*i ∈ Vindex
¤ © c9qvcRm
aA(
iA)
=
aB(iB) ,s¦aB(iB) ∈ Aelem ,
e0 ,prm-_*cRy © svlc .
¯prmcm-_|XmAsvl| © cRqtq¦¢Á8cK³**cn Tarray
¢¹stgmcRy-~8ycRm|Xm-svpr¤¡|X*Ím-_|XmAσ = [varsσ(Γ)]A
¤Ãpry
σ ∈ index, elem¥¯cK·Tm¤ © c¨l_*p © m-_|Xm
Al|Xm-svl$³cnl|XqtqNqtstmcRy|Xqvl)st
Γ¥
C Å X!C É Å ÁÉ e1 ≈elem e2 e1 6≈elem e2 i ≈index j
Êi 6≈index j
9ÁbbdcnTs¦¢|Xmc¥
C Å X!C É Å ÁÉ a ≈array b
cRmi ∈ Aindex
¥9ÁaB
(
iB)
∈ Aelem
m-_*cRaA
(
iA)
=aB
(
iB)
= bB(
iB)
= bA(
iA) ¥9¹\st*l-mc|r
aB(
iB)
/∈ Aelem
m-_*cRaA(iA) = e0 = bA(iA)
¥
C Å X!C É Å ËÉ e ≈ read(a, i)
Ic¸_|n cK
eA = eB
= [read(a, i)]B
= aB(
iB)
= aA(
iA) l-st*nc
aB(
iB)
∈ Aelem .
C Å X!C É Å ÁÉ a = write(b, i, e)
ekst*ncaB
(
iB)
= eB ∈ Aelem
¤ © cd_|n c © c_| c
aA(
iA)
= bB(
iB)
= eB = eA¥ ¯6cK·0m¤µqvcRm
j ∈ Aindex
lw*_m-_|Xmj 6= iA
¥ 9¹aB(j) ∈ Aelem
m-_*cRaA(j) = aB(j) = bB(j) = bA(j)
¥ 9Á¶st*l-mc|raB(j) /∈ Aelem
m-_*cRaA(j) = e0 = bA(j)
¥
éé äMNYïOYò
'! ( !"!<'.6
% É ¡É C É'& ® ÅÈ C NÅ 8 ! C Æ ' * *L!. !!S ⊆ elem, index
$$6'*
Tarray
! - 1&# "!E!() #$ 6"! 6 S *
5:: µj=x!ypr~3plstm-svpr*lBT¤*cRbU|Xy6D gT¤|X*£x¡ypr~3pl-stm-svpr B¥
*' # #+ + ÅkÉ Å & %6É ! C Å Å ' '* * ! !'!
S ⊆ elem, indexL$5
$6'*Tarray
! ') 1#$ "! S *
5:: µj=x!ypr~3plstm-svpr*ld|X* gT¥
¸3
^`_*cVm-_*cnpry-jFpX¡lcRmlTset
_|rl|Ul-st|Xm-w8ycΣset
nprkm|Xst8st8=|Ulpry-melem
ÃprycRqvcRbdcRgml9|X*|lpry-m
setÃpry)lcRml)pX!cRqvcRbdcRgml¤8~8qtw*lm-_*c6§prqtqvp © st8l-j0bVoprqvl.K
m-_*c¨npr*l-m|Xkml-j0bVoprq ∅ ¤8pX\lpry-m set
m-_*c¸§w8*Rm-svpr¾l-j0bVoprqvl.K
·¤pX\lpry-m
elem→ set
∪¤∩¤*|X*
\¤8pX!lpry-m
set× set→ set
m-_*c9~8ycnTsv|Xmc¨l-j0bVoprq ∈ ¤8pX!lpry-m elem× set¥
-£Å C É'&& Î 3(8;1 8;5 set . /1 3 45"765"4,38 3"/:1 A svlV|Σset
¢¹stgmcRy-~8ycRm|Xm-svpr»l|Xm$¢svl-Aj0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Aset = P(Aelem)
m-_*cVl-j0bVoprqvl ∅ ¤ · ¤ ∪ ¤ ∩ ¤ \ ¤|X* ∈ |Xyc¨stkmcRy-~8ycRmcn[|rnnpryTst8²mpm-_*cRsty6l-m|X**|XystgmcRy-~8ycRm|Xm-svprÍp cRylcRml¥^`_*c 3 46:5 : 4,3 svlm-_*cµ~|Xsty Tset = 〈Σset,A〉
¤ © _*cRycAsvl\m-_*c`Rq®|rll\pX|Xqtq8l-m|X**|Xy
set¢Ál-m-y-w*Rm-w8ycnl¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7! 5B
% #+ +% É ¡É C É'& )(
AE
Tset 01 7' ! ! *' E'.'Γ
!
Σset )+1 ) ! L !"$ $ Aelem
! - 1 * 24$5 $56 '!! Tset 01 7' B!! *F Γ
!5$ $ |Belem| = κ
'>6($κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤Ãpry
σ ∈ elem, set¥ Icnpr*l-m-y-w*Rm¸|
Tset
¢¹stgmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vset
|rl`§prqtqvp © l¥¡¼s¦·F|lcRm A′ l-w*_Fm-_|Xm |Aelem ∪A′| = κ
¤*|X*£qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
xB = xA ,§pry)c|r_
set¢¹ |Xy-s®|Xo8qvc
x ∈ Vset .
¶jFnpr*lm-y-w*Rm-svprBsvl|
Tset
¢¹stkmcRy-~8ycRm|Xm-svprl-w*_m-_|Xm|Belem| = κ
¥)h[prycnp cRy¤3stmsvlm-y-st 0s®|Xqmp_*cn D£m-_|Xm
BlY|Xm-svl$³cnl|XqtqqtstmcRy|Xqvlst
Γ¥
% É ¡É C É'& ' É´É Å 24$5 $56"*Tset
! !".$ #$ "! 6 $5 !'
elem *
5:: µj=x!ypr~3plstm-svpr*lBBV|X* gT¥
'* ' # ' ( #+ + ' ' C NÅ É Î © stm-*cnll§w8*Rm-svpr witness set
§pry´m-_*c!m-_*cnpry-jTset
|X¨o3c8cK³**cn|rlÃprqtqvp © l¥ I stm-_*prw8m`qvpll¶pXcR*cRy|XqtstmÁj¤TqvcRm
Γoc|nprrz$w8*Rm-svpr=pX |Xm
Σset
¢¹qtstmcRy|Xqvl`l-w*_m-_|Xm
varselem(Γ) 6= ∅¥<Ic¶qvcRm
witness set(Γ)o3c¶m-_*c¶ycnl-w8qtmpX|X~8~8qtjkst86mp
Γm-_*cÃprqtqvp © st8
m-y|X*l-§pry-bU|Xm-svpr K
cR~8q®|rncdc|r_ªqtstmcRy|XqµpXµm-_*cÃpry-b x 6≈set yst
Γ © stm-_ |£qtstmcRy|XqpXµm-_*cd§pry-b e′ ∈(x \ y) ∪ (y \ x)
¤ © _*cRyc e′ svl|§ycnl-_ elem¢¹ r|Xy-s®|Xo8qvc¥
²Å ª8 & cRmΓoc¸|nprrz$w8*Rm-svprFpX |Xm
Σset
¢¹qtstmcRy|Xqvlº¤TqvcRm∆ = witness set(Γ)
¤*|X*qvcRm
v = vars(∆) \ vars(Γ)¥¡^`_*cR
Γ|X*
(∃v)∆|Xyc
Tset
¢Ácnugw8st |XqvcRgm¥
% É ¡É C É'& )( Γ
' ! Σset ) 16') ! !"$ $
vars elem(Γ) 6=∅ ' ''0 '*A)+16')& $5 " x 6≈set y * 24$5 $ '))H'# 6
@') 0
éé äMNYïOYò
'! ( !"!<'.6
* Γ!Tset !!- )C *
* 24$5 '!! Tset 1 6 ' A ! ! *' Γ
!"$$ Aelem = [vars elem(Γ)]A *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1 ⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤§pryσ ∈ elem, set
¥Fekst*ncΓsvlTset
¢ÁlY|Xm-svl$³|Xo8qvc¤m-_*cRyccK·Tsvl-ml|
Tset
¢¹stgmcRy-~8ycRm|Xm-svprBl|Xm-svl$§jkst8
Γ¥ Ic © stqtq¶w*lc B stªpry8cRyVmpnpr*l-m-y-w*Rm|X
pr~8~pry-m-w8*cTset
¢¹stkmcRy-~8ycRm|Xm-svprA¥
hprycl-~3cnRs¦³|Xqtqtj¤ © cqvcRm A ocm-_*cTset
¢¹stgmcRy-~8ycRm|Xm-svprFp cRyVelem∪Vset
npr*l-m-y-w*RmcnokjqvcRm-m-st8
Aelem = V Belem ,
|X*
eA = eB ,Ãpryc|r_
elem¢¹ |Xy-s®|Xo8qvc
e ∈ Velem ,
xA = xB ∩Aelem ,Ãpryc|r_
set¢¹ |Xy-s®|Xo8qvc
x ∈ Vset .
¶j¸npr*l-m-y-w*Rm-svpr¤AsvlN| © cRqtq¦¢Á8cK³**cn Tset
¢¹stgmcR~8ycRm|Xm-svprl-w*_Vm-_|XmAelem = [vars elem(Γ)]A
¥¯6cK·Tm¤ © c¨l-_*p © m-_|Xm
AlY|Xm-svl$³cnl|Xqtq´qtstmcRy|Xqvlst
Γ¥
C Å X!C É Å ËÉ e1 ≈elem e2
Êe1 6≈elem e2
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈set y
Ic¸_| cxA = xB ∩Aelem = yB ∩Aelem = yA
¥
C Å X!C É Å ËÉ x ≈ ∅
Ic9_| cxA = xB ∩Aelem = ∅ ∩Aelem = ∅
¥
C Å X!C É Å ÁÉ x ≈ e
Ic_| cxA = xB ∩Aelem =
eB
∩Aelem =
eB ¥
C Å X!C É Å ÁÉ x ≈ y ∪ z
Ic¸_|n cxA = xB ∩Aelem =
(
yB ∪ zB)
∩Aelem =(
yB ∩Aelem
)
∪(
zB ∩Aelem
)
= yA ∪ zA¥
C Å X!C É Å ÁÉ x ≈ y ∩ z
Ic¸_|n cxA = xB ∩Aelem =
(
yB ∩ zB)
∩Aelem =(
yB ∩Aelem
)
∩(
zB ∩Aelem
)
= yA ∩ zA¥
C Å X!C É Å ÁÉ x ≈ y \ z
Ic¨_|n cxA = xB ∩Aelem =
(
yB \ zB)
∩Aelem =(
yB ∩Aelem
)
\(
zB ∩Aelem
)
= yA \ zA¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
C Å X!C É Å ËÉ e ∈ x
Êe 6∈ x
w*lmpro*lcRy- c9m-_|XmeA ∈ Aelem
¥
% É ¡É C É'& ' ® ÅÈ C NÅ 8 ! C Æ 24$5 $5 '"*Tset
!>-/1)+* # !E!)C#$ 6"! 6
elem *
5:: µj=x!ypr~3plstm-svpr BT¤*cRbU|Xy6D k0¤|X*Fx¡ypr~3pl-stm-svpr gT¥
*' # #+ + ÅkÉ Å %6É ! C Å Å 24$5$56'*
Tset
! ) 1#$ 6! 6. elem *
5:: µj=x!ypr~3plstm-svpr*l gd|X* gT¥
8 45+$
hw8qtm-svlcRml±i |Xqvlp Dk*p © |rl!o|Xl±i |XycnprqtqvcnRm-svpr*l¶m-_|Xm!bU|njnprgm|Xst²Tw8~8qtsv|Xmc)cRqvcRbdcRgml¥¼pry-bU|Xqtqtj¤|bVw8qtm-svlcRm
xsvl|§w8*Rm-svpr
x : A→ N¤0Ãpry)lprbdc¨lcRm
A¥
Icw*lc`m-_*c)l-j0bVoprq[[ ]]mp¨8cR*prmcm-_*c)cRb~8mËjbVw8qtm-svlcRm¥ I _*cR
n ≥ 0¤ © c © y-stmc [[e]](n)
mp8cR*prmcm-_*cbVw8qtm-svlcRm`nprgm|Xst8st8cK·*|rRm-qtjnpknRw8y-ycR*ncnlµpX
e|X*d*prm-_8st8cRqvlc¥JI¿_*cR
n < 0¤ © c9qvcRm [[e]]n = [[ ]]
¥ cRm
x, yo3c¸m © pdbw8qtm-svlcRml¥¡^`_*cR K
m-_*cRsty 7'x ∪ y
svlm-_*cbVw8qtm-svlcRmzl-w*_[m-_|Xm¤§pry6c|r_ÍcRqvcRbdcRgm
e¤m-_*ccnukw|XqtstmÁj
z(e) = max(x(e), y(e))_*prqv8l
m-_*cRsty !x ] y
svlVm-_*cbVw8qtm-svlcRmzl-w*_ m-_|Xm¤ÃpryVc|r_«cRqvcRbdcRkm
e¤m-_*cUcnukw|XqtstmÁj
z(e) = x(e) + y(e)_*prqv8l
m-_*cRsty 01"!67'x ∩ y
svlm-_*c=bw8qtm-svlcRmzl-w*_ m-_|Xm¤§pryc|r_¿cRqvcRbdcRgm
e¤¡m-_*c
cnugw|XqtstmËjz(e) = min(x(e), y(e))
_*prqv8lº¥^`_*c¡m-_*cnpry-jVpX8bVw8qtm-svlcRml
Tbag
_|rl|l-st|Xm-w8ycΣbag
nprgm|Xst8st8¨|lpry-mint
§prystkmcRcRyl¤|lpry-m
elemÃpry)cRqvcRbdcRgml¤3|X*¾|lpry-m
bagÃprybVw8qtm-svlcRml¤*~8qtw*lm-_*c9§prqtqvp © st8Ul-j0bVoprqvl.K
m-_*c¨npr*l-m|Xkml-j0bVoprqvl.K
0|X*
1¤*pX!lpry-m
int
[[ ]]¤pX\lpry-m
bag
m-_*c¸§w8*Rm-svpr¾l-j0bVoprqvl.K
éé äMNYïOYò
'! ( !"!<'.6
+¤−¤max
¤*|X*min
¤8pX!lpry-mint× int→ int
[[·]](·)¤8pX\lpry-m
elem× int→ bag
∪¤]¤*|X*
∩¤TpX!lpry-m
bag× bag→ bag
count¤8pX\lpry-m
elem× bag→ int
m-_*c9~8ycnTsv|Xmc¨l-j0bVoprq < ¤8pX!lpry-m int× int¥
-£Å C É Î 38;1 8;5 bag . /1 3 45"765"4,38 3"/:1 A svl)|Σbag
¢¹stgmcRy-~8ycRm|Xm-svprÍl|Xm$¢svl-Aj0st8m-_*c6§prqtqvp © st8npr*Tstm-svpr*l.K
Aint = Z
Abag = NAelem
m-_*c=l-jkbo3prqvl 0¤1¤+¤−¤max
¤min
¤¡|X*<|XycstkmcRy-~8ycRmcn |rnnpryTst8¬mp¾m-_*cRsty
l-m|X**|Xy£stkmcRy-~8ycRm|Xm-svpr[pº cRym-_*c¸stgmcRcRyl
m-_*c¨l-j0bVoprq [[ ]] ¤ ∪ ¤ ∩ ¤ \ ¤ [[·]](·) |Xyc¸stkmcRy-~8ycRmcn[|rnnpryTst8dmpdm-_*cRsty)l-m|X**|Xy£stgmcRy$¢~8ycRm|Xm-svprpº cRy)bVw8qtm-svlcRml
countA(e, x) = x(e)¤T§pry)c|r_
e ∈ Aelem
|X*x ∈ Abag
¥^`_*c 3 4:;5&: !$ 3"/ 4,3 svl`m-_*c¸~|Xsty Tbag = 〈Σbag,A〉
¤ © _*cRycAsvl`m-_*c9Rq®|rllpX\|Xqtq
lm|X**|Xybag
¢Ál-m-y-w*Rm-w8ycnl¥
$% " # ++% É ¡É C É )(
AE
Tbag 01 6 !! *' ' 7'Γ
!
Σbag )+16') ! ' !5$ $ Aelem
! -/1 * 24$5 $6 !!&Tbag 01 6 B!! *F Γ
!5$ $ |Belem| = κ
'>6($κ > |Aelem| *
5:: cRm Vσ = varsσ(Γ)¤ºÃpry
σ ∈ elem, int, bag¥<Ic!npr*l-m-y-w*Rm|
Tbag
¢¹stkmcRy-~8ycRm|Xm-svprBp cRy
Velem ∪ Vint ∪ Vbag
|rl`Ãprqtqvp © lº¥¡¼s¦·£|lcRm A′ l-w*_¾m-_|Xm |Aelem ∪A′| = κ
¤|X*=qvcRm
Belem = Aelem ∪A′ ,
|X*
eB = eA ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
uB = uA ,§pry)c|r_
int¢¹ |Xy-s®|Xo8qvc
u ∈ Vint .
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
¼*w8y-m-_*cRy-bdpryc¤8ÃprycR cRy-jbag
¢¹ |Xy-s®|Xo8qvcx ∈ Vbag
|X*e ∈ Belem
¤ © c9qvcRm
xB(e) =
xA(e) ,s¦e ∈ Aelem ,
0 ,prm-_*cRy © svlc .
¶jnpr*l-m-y-w*Rm-svpr¤Bsvl|
Tbag
¢¹stkmcRy-~8ycRm|Xm-svpr¾l-w*_²m-_|Xm|Belem| = κ
¥!¯6cK·Tm¤ © c¸l-_*p ©m-_|Xm
Bl|Xm-svl$³cnl|XqtqNqtstmcRy|Xqvl)st
Γ¥
C Å X!C É Å ËÉ e1 ≈elem e2 e1 6≈elem e2 u ≈int v
Êu 6≈int v
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈bag y
cRme ∈ Belem
¥9Áe ∈ Aelem
m-_*cRxB(e) = xA(e) =
yA(e) = yB(e)¥ 9Á
e /∈ Aindex
m-_*cRxB(e) = 0 = yB(e)
¥^`_gw*lº¤xB = yB
¥
C Å X!C É Å ÁÉ x 6≈bag y
ekst*ncxA 6= yA
¤Nm-_*cRyccK·0svlml|Xe ∈ Aelem
l-w*_m-_|Xm
xA(e) 6= yA(e)¥/9Ám`Ãprqtqvp © lm-_|Xm xB(e) 6= yB(e)
¤ © _8sv_¾stb~8qtsvcnl xB 6= yB¥
C Å X!C É Å ÁÉ u ≈ 0 u ≈ 1 u ≈ v + w u ≈ v − w u ≈ max(v, w)
u ≈ min(v, w) 9ËbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈ [[e]](u) ekst*nc
eB ∈ Aelem
¤ © c_|n c xB(eB) = xA(eA) =uA = uB
¥¯cK·Tm¤rqvcRme′ ∈ Belem
l-w*_m-_|Xme′ 6= eB
¥J9Áe′ ∈ Aelem
m-_*cRxB(e′) = xA(e′) =
0¥/9Ást*lmc|r
e′ /∈ Aelem
m-_*cRxB(e′) = 0
¥¡^`_kw*lxB = [[eB]](uB) ¥
C Å X!C É Å ËÉ x ≈ y ∪ z x ≈ y ] z
Êx ≈ y ∩ z
^`_8svl|rlc6svl`l-stbstq®|Xympm-_*c9|rlc¨pXqtstmcRy|XqvlpXm-_*c6Ãpry-b
x ≈ [[e]](u) ¥
C Å X!C É Å&ÁÉ u ≈ count(x, e)
ekst*nceB ∈ Aelem
¤ © c£_|n c uB = uA =xA
(
eA)
= xB(
eB) ¥
% É ¡É C É ' É´É Å 24$5 $56'*Tbag
! !.$#$ "! 6/elem *
5:: µj=x!ypr~3plstm-svpr*lBBV|X*FT¥
éé äMNYïOYò
'! ( !"!<'.6
' ' # ' #+ + ' *' C NÅ É Î © stm-*cnll)Aw8*Rm-svpr witnessbag
§prym-_*cVm-_*cnpry-jTbag
|X¾oc98cK¢³**cnd|rl!§prqtqvp © l¥JI¿stm-_*prw8m!qvpllpXcR*cRy|XqtstmÁj¤kqvcRm
Γoc|¸nprrz±w8*Rm-svprUpX |Xm
Σbag
¢¹qtstmcRy|Xqvllw*_Ím-_|Xm
vars elem(Γ) 6= ∅¥&IcqvcRm
witnessbag(Γ)ocm-_*cycnl-w8qtm9pXµ|X~8~8qtjkst8£mp
Γm-_*c
Ãprqtqvp © st8Um-y|X*l$§pry-bU|Xm-svpr K cR~8q®|rncc|r_ qtstmcRy|XqVpXm-_*cÍÃpry-b
x 6≈bag yst
Γ © stm-_ |»qtstmcRy|XqVpXm-_*cÍÃpry-bcount(e′, x) 6≈ count(e′, y)
¤ © _*cRyc e′ svl|§ycnl-_ elem¢¹ r|Xy-s®|Xo8qvc¥
²Å ª8 & cRmΓoc|¸nprrz±w8*Rm-svprUpX |Xm
Σbag
¢¹qtstmcRy|Xqvl¤gqvcRm∆ = witnessbag(Γ)
¤g|X*qvcRm
v = vars(∆) \ vars(Γ)¥¡^`_*cR
Γ|X*
(∃v)∆|Xyc
Tbag
¢Ácnukw8st r|XqvcRkm¥
% É ¡É C É )( Γ
' 7' ! Σbag )+16') !L!"$$
vars elem(Γ) 6=∅ 0' A'* ) 16') $5 " x 6≈bag y * 24$ $5 '))H# A6
@') 0 * Γ
!Tbag ! !- () *
* 24$5 '!! Tbag 01 7' A !! *' Γ
!5$$5Aelem = [vars elem(Γ)]A *
5:: 2⇒ 1K¥/9ËbbdcnTs®|Xmc¥
1 ⇒ 2
K¥ cRmVσ = varsσ(Γ)
¤!§pryσ ∈ elem, int, bag
¥ªekst*ncΓsvlTbag
¢Ál|Xm-svl-³|Xo8qvc¤m-_*cRyc¸cK·Tsvl-ml`|
Tbag
¢¹stgmcRy-~8ycRm|Xm-svprBlY|Xm-svl$Aj0st8
Γ¥JIc © stqtqw*lc B st£pry8cRy¶mpnpr*l-m-y-w*Rm|XFpr~8~pry-m-w8*c
Tbag
¢¹stkmcRy-~8ycRm|Xm-svprA¥
hpryc[l-~cnRs¦³|Xqtqtj¤ © cqvcRm A oc¾m-_*cTbag
¢¹stgmcRy-~8ycRm|Xm-svpr p cRyVelem ∪ Vint ∪ Vbagnpr*lm-y-w*Rmcn£ogj=qvcRm-m-st8
Aelem = V Belem ,
|X*
eA = eB ,§pry)c|r_
elem¢¹ r|Xy-s®|Xo8qvc
e ∈ Velem ,
uA = uB ,§pry)c|r_
int¢¹ |Xy-s®|Xo8qvc
u ∈ Vint .
¼*w8y-m-_*cRy-bdpryc¤8ÃprycR cRy-jbag
¢¹ |Xy-s®|Xo8qvcx ∈ Vbag
¤ © c9qvcRm
xA(e) = xB(e) ,Ãpryc|r_
e ∈ Aelem .
¶jnprgm-y-w*Rm-svprAsvl!|
Tbag
¢¹stgmcRy-~8ycRm|Xm-svpr²l-w*_m-_|XmAelem = [vars elem(Γ)]A
¥¯cK·Tm¤© c¨l-_*p © m-_|Xm
Al|Xm-svl-³cnl|XqtqqtstmcRy|Xqvl)st
Γ¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
C Å X!C É Å ËÉ e1 ≈elem e2 e1 6≈elem e2 u ≈int v
Êu 6≈int v
9ÁbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈bag y
ekst*ncµÃpry!c|r_e ∈ Aelem
¤ © c_| c xA(e) = xB(e) =yB(e) = yA(e)
¤*stm`Ãprqtqvp © lm-_|Xm xA = yA¥
C Å X!C É Å ÁÉ u ≈ 0 u ≈ 1 u ≈ v + w u ≈ v − w u ≈ max(v, w)
u ≈ min(v, w) 9ËbbdcnTs®|Xmc¥
C Å X!C É Å ËÉ x ≈ [[e]](u) ekst*nc
eB ∈ Aelem
¤ © c_|n c xA(eA) = xB(eB) =uB = uA
¥!¯6cK·0m¤TqvcRme′ ∈ Aelem
|X*e′ 6= eB
¥!^`_*cRxA(e′) = xB(e′) = 0
¥
C Å X!C É Å ËÉ x ≈ y ∪ z x ≈ y ] z
Êx ≈ y ∩ z
^`_8svl|rlc6svl`l-stbstq®|Xympm-_*c9|rlc¨pXqtstmcRy|XqvlpXm-_*c6Ãpry-b
x ≈ [[e]](u) ¥
C Å X!C É Å&ÁÉ u ≈ count(x, e)
ekst*nceB ∈ Aelem
¤ © c£_|n c uA = uB =xB
(
eB)
= xA(
eA) ¥
% É ¡É C É ' ® ÅÈ C NÅ 8 ! C Æ 24$5 $6'*Tbag
! -/1)+* # !E!)C#$ 6"! 6 3$5&!
elem *
5:: µj=x!ypr~3plstm-svpr BT¤*cRbU|Xy6D£*¤|X*Fx¡ypr~3pl-stm-svpr[T¥
' # #+ + ÅkÉ Å %6É ! C Å Å 24$5$56'*
Tbag
! ')+1#$ "! 6 elem *
5:: µj=x!ypr~3plstm-svprd|X*¾T¥
8 8 )+ )
Ic|r8Tycnllcn¾m-_*cV~8ypro8qvcRb pX¡nprbVo8st8st8F|m-_*cnpry-jSbdpk8cRqtst8£|d*|Xm|Ul-m-y-w*Rm-w8ycnprT¢
m|Xst8st8dcRqvcRbdcRkml)pX\|st cR=|Xm-w8yc © stm-_¾|m-_*cnpry-j T pXm-_*c¸cRqvcRbdcRkml¥Ic © cRyc¸~|Xy-m-svK¢w8q®|Xy-qtj²stgmcRycnlmcnFstFm-_*c9|rlc¨st © _8sv_ T svl)*prm)l-m|Xo8qtj=stT³*8stmc¥
^NpFlprqt cdm-_8svl9~8ypro8qvcRbF¤ © cd8cK³**cn¬m-_*c*prm-svpr pXµ~prqtstmcm-_*cnpry-svcnl¤|X* © cdl_*p © cnm-_|Xm6|~3prqtstmc9m-_*cnpry-j
S|XFoc9nprbVo8st*cn © stm-_|Xgj=m-_*cnpry-j T ¤TycRg|XyTqvcnllpX © _*cRm-_*cRy Tsvlµl-m|Xo8qtjstT³*8stmc¸pryµ*prm¥ Ic)m-_*cR=~8yp cnm-_|Xm`|Xm-w8y|Xq3cK·*|Xb~8qvcnl`pX~3prqtstmcm-_*cnpry-svcnl|Xyc
st cR¾ogj²m-_*c9m-_*cnpry-j£pX\cnukw|XqtstmÁj¤8qtsvlml¤*|Xy-y|njTl¤*lcRmlº¤*|X*£bVw8qtm-svlcRmlº¥
éé äMNYïOYò
'! ( !"!<'.6
°w8y¶ycnl-w8qtml © cRyc68cR cRqvpr~cnw*l-st8bU|Xkjg¢Álpry-mcn=qvprsvy|Xm-_*cRy`m-_|X=pr*cK¢Álpry-mcn²qvprsv¥9ËVprw8yNcK·T~cRy-svcR*nc¤Xnprbo8st8st8¸*pr*l-m|Xo8qtj¸stT³*8stmcm-_*cnpry-svcnlstVpr*cK¢Álpry-mcnqvprsv!svlNTsHGURw8qtm¥µj bdpº kst8»mp bU|Xgjk¢Álpry-mcn qvprsv¤ © c © cRyc|Xo8qvc[mp³** bU|Xkj~8y|rRm-sv|XqtqtjycRqvcR |Xgmm-_*cnpry-svcnl c¥ *¥t¤kqtsvl-ml¤k|Xy-y|j0lº¤klcRmlº¤k|X*dbw8qtm-svlcRml !m-_|Xm¶|XUocnprbo8st*cn © stm-_*pr*l-m|Xo8qtjstT³*8stmc¨m-_*cnpry-svcnl¥
µpr*ncRy-8st8§w8m-w8ycdycnlc|Xy_¤ © c © svl-_ªmpl-m-w*Tj¬_*p © ~prqtstmcdm-_*cnpry-svcnlycRq®|XmcdmpproT¢lcRy- r|Xo8qvc²m-_*cnpry-svcnl =Ø@?|X*«qvpk|Xq`m-_*cnpry-j cK·TmcR*l-svpr*l =CBF?Á¥ Ic£|Xqvlp © svl-_»mp[stb~8qvcRbdcRgmprw8ynprbo8st|Xm-svpr¾bdcRm-_*p0st =Ø@?Á¤*|X*£|X~8~8qtjUstm`mpm-_*c6 cRy-s¦³|Xm-svpr¾pXlcRm$¢¹o|rlcnl~3cnRs¦³|Xm-svpr*lpX\l-bU|Xy-m$¢Á|Xy8l>= '?Á¥
9¡"#+!: !
Ic|Xycy|XmcKAw8qmp=x|rl|Xq\¼prgm|Xst*c¤Ì¸cncR~|FD ¨|X~8w8y¤|X*µcnl|Xyc^`st*cRqtqts§pry6~8qvc|rl|XgmTsvlRw*ll-svpr*ldpr½m-_*c=~8ypro8qvcRb'pX6nprbVo8st8st8«*pr*l-m|Xo8qtj«stT³*8stmc£m-_*cnpry-svcnl¥AIcF|Xyc|Xqvlpy|XmcK§w8qmpUm-_*c|X*prgj0bdprw*l¸ycR ksvc © cRyl§pry6m-_*cRsty6npr*lm-y-w*Rm-st cVÃcncnTo|r D¥6¼8w8qtqRycnTstm6Ãprym-_*c¸ycnlw8qtmlpXe0cnRm-svprpkcnl)mpm-_*c9ycR 0svc © cRyl¥
n\!!
=CB?¼8y|X8À ||r8cRy|X*¾ekstqt ksvp _8stq®|XyTs¹¥µpr8*cnRm-st8dbU|Xgjk¢Álpry-mcnFl-m-y-w*Rm-w8ycnl|X*=m-_*cK¢pry-svcnl=m-_8yprw8_ |rXz$prstkm=Aw8*Rm-svpr*l¥ 9Á `cRy-8_|Xy y|Xbqtsv_¤¸cnTstmpry¤ 6'0"! '0 0*F!1! ¤ cnRm-w8ycV¯6prmcnlstµprb~8w8mcRy¸e0RsvcR*nc¥ek~8y-st8cRy¤T¥
=Ø@?¼8y|X8À ||r8cRy|X*£ekstqt 0svp _8stq®|XyTs¹¥¡µpr8*cnRm-st8dbU|Xkjg¢Álpry-mcn=m-_*cnpry-svcnlº¥J9Ë£procRy-m¯svcRw © cR8_gw8svlº¤*cnTstmpry¤ '1 > 5.' ¤k prqtw8bdc¸pX ( %&1"!& 1& E ¥ek~8y-st8cRy¤T¥
=Ø@?h¾sv_*cRq µsv8prstm|X* prq¦ :cR88svEDcRy¥ µcR_| ksvprw8y|Xq`m-_*cnpry-svcnld|X* m-_*cU~8yp0pXpXocK¢_|n 0svprw8y|XqN~8ypr~cRy-m-svcnl¥ 24$5667 ') ' 51 " ¤ B BKØ0T¤JBT¥
= '? c|XT¢Á¼8y|Xnprsvl¡µprw*_*prm¤rÌ9|n 0svVÌ$R_|Xy-oc¤Îq®|Xst svpry-cRm-m-s¹¤|X*ekstqt 0svp)|X8svlc¥ge0|Xq¦¢|Xo8qvcU|Xw8mprbU|Xmcn~8ypº kst8Í|X*8cRo8w8st8pXlcRm$¢¹o|rlcnl~3cnRs¦³|Xm-svpr*l¥ ' " ') $5 6)+'A' 51 7* ¤ KCBº 0T¤*¥
=Ø@?̸| ksv£Ì$R_|Xy-ocV|X*¾ekstqt 0svpd|X8svlc¥ st_km$¢ © cRst_km6m-_*cnprycRb ~8yp 0st8§pry)8cRo8w8st8|X* cRy-s¦Aj0st89w88stml¡pX´npk8c¥9Ë # ' '
'') $ ! ¤~|Xcnl
0T¥09 !! µprb~8w8mcRy9e0pkRsvcRmËj¤T¥
èAåéNè§ö
"!#$%&'!()+*,".-/01324$56'7!
=Ø@?x|rl|Xq6¼*prgm|Xst*c[|X* x\|rl|Xq y-stoprbdprgm¥ µprbVo8st8st8«*prT¢Ál-m|Xo8qtj»stT³*8stmc¤µ*prT¢³*yl-mpry8cRy6m-_*cnpry-svcnl¥ 9ËÍekstqt 0svp²|X8svlc|X*͵cnl|Xyc^`st*cRqtqts¹¤3cnTstmpryl¤ '! >6 !' 6.E "! 5'1 6F!' ¤*¥
=Ó ?x|rl|Xq`¼*prkm|Xst*c¤¶ekstqt ksvp|X8svlc¤\|X*«`|XqvprcRyp ¥\Â|Xy-o|T¥µprbVo8st8st8Íqtsvl-ml © stm-_*prT¢Ál-m|Xo8qtj½stT³*8stmc¾m-_*cnpry-svcnlº¥ 9Á ¼8y|X8À ||r8cRy=|X*¿Î*TycRs¡pryprDpº 3¤)cnTstmprylº¤( '7 ' - 7')4,"01)) C (" ' 6'!' ¤N prqtw8bdcUg£pX( 6 &%&1! A' 1 E ¤8~|Xcnl5B 0T¥ek~8y-st8cRy¤3T¥
=Ø@?s z|nj |X*cnl_¤e0cRy-cRj `cRycRÀnst¤|X* ̸| ksv ¥µÌ6stqtq¹¥ Î8cnRsvl-svpr ~8ypkncnTw8ycÃpry³8·Tcn0¢ © svTm-_Fo8stm$¢¹ cnRmprylº¥ ;8~8w8o8qtsvl-_*cn´¤T¥
=Ø@?:6|Xy|Xqv |X8Ànst8cRy¥ek_*pl-m|FD[qtst_gm¥9ˬÎ*TycRs¡pryprDpº 3¤NcnTstmpry¤ 1 5.' ¤T prqtw8bdcpX ( %& 1"! ' 1 7E ¤8~|Xcnl0gT¥3ek~8y-st8cRy¤T¥
=CB@?svpry-sv|=e0pXAypr8svcK¢±ekmp'D DcRy-bU|X*l¥ :svcRy|Xy_8svyc|rlpr8st8=stqvp0|XqNm-_*cnpry-jcK·0mcR*l-svpr*lº¥9Á²procRy-m¶¯svcRw © cR8_kw8svl¤8cnTstmpry¤ 1 7' ¤k prqtw8bdc¸pX ( 6 6&%1"! ' 51 7E ¥ek~8y-st8cRy¤3T¥
=CBB?µcnl|Xyc`^`st*cRqtqts|X*`|XqvprcRyp ¥Â|Xy-o|T¥µprbVo8st8st8V8cnRsvl-svpr~8yp0ncnTw8ycnl\Ãprylpry-mcnm-_*cnpry-svcnl¥ 9Ë pl qtsvp»Îq¦ÃcRycnl²|X* prp«ÎqvcK·8|X*Tyc cRstmc¤cnTstmpryl¤ ( '7!
- 7')(,"01)) C ( E ¤k prqtw8bdc)9pX ( 6 6L%1"! ' 51 7 E ¤~|XcnlB 0T¥3ek~8y-st8cRy¤*¥
=CB@?µcnl|Xycd^`st*cRqtqts¶|X* `|XqvprcRyp ¥NÂ|Xy-o|T¥µprbo8st8st8*pr*l-m|Xo8qtj[stT³*8stmcdm-_*cnpry-svcnlº¥ ' " ')< 5'1 6'!' ¤T¥¡^NpU|X~8~c|Xy¥
=CB@?`|XqvprcRyp ¥Â|Xy-o|T¥¬µprbVo8st8st8ÍbVw8qtm-svlcRml © stm-_«stgmcRcRylº¥ 9Ë Î*TycRs¡pryprDpº 3¤cnTstmpry¤ 5'1 > 57' ¤ prqtw8bdc`)pX ( 6 6 %&1! 1 E ¤*~|Xcnl0rT¥ek~8y-st8cRy¤T¥
=CB'?`|XqvprcRyp ¥Â|Xy-o|T¥µprbVo8st8st8²lcRml © stm-_FcRqvcRbdcRgmlº¥/9Á£¯¸|r_kw8b ̸cRyl-_*p © stm-À¤cn0¢stmpry¤ - 7' 24$56'* ' 6 7E ¤ prqtw8bdcrUpX ( 6 6>%&1!>A 1 E ¤*~|XcnlrTrT¥ek~8y-st8cRy¤*¥
éé äMNYïOYò
Unité de recherche INRIA LorraineLORIA, Technopôle de Nancy-Brabois - Campus scientifique
615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399