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