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12Time in Seconds108SATO TimeNDS TimeNDS Subsolution Generation TimeNDS Subsolution Processing TimeND TimeND Subsolution Generation TimeND Subsolution Processing Time6420100110120130Word Set Size140C RUvp S[cZ ²aØ HR qJZ ýi[cXYVSP150160[cZ T} S UNcT R L XYL9 TQX/NcR Tmx XYdSRU RUNM J SRUTcb\[cZ VSXYLSb\ Y] ü/øpøih T}PSnp S U [}Z T} S UN RUL9 § SRUTcb [}Z VuX/LSb\ Y ¡ ð § X/ NQPSZfNc[c SZfV7Z\[cb Z\L NDTQX/NcR T}xuXYdS Z lonp[@X xa *Z Xk#np[c TcZsN TcRU Z HPaZ Le g ÇwXY »*haX N{ RU ÃxuLS 3 ÀTcX/NQRUT}x X/dS Z PSZ L³ ST}R LSv ð øpøih nYlje g T T} SdSTcnY aNcR npLST\ KtlnpLSZ Snp SdS UZ T NQPSZ L- aq d7Z\[{n/l Tc SdaTcnp U aNQRUnpLST STcZ\ zNQn ü/øpiø h ] RUN R T Z SR YX/ Z L N NQnW[cZ V(Z X NQR LavNQPSZJVS[cn-b Z\ S S[}Z XYvpXYR LF RUNcP ð øpiø h nYl e g TiTc SdST}np *NQR nYLST nYLwXY U NQPaZWRULST}NQXYLSb Z\T NcPuX/N XY[}Z XYd(Z UZ SLSTQX NQR Tmx XYdS UZY ª LwNQPSRUT T}Z b\npLS ·X/N}NQZ qJVaN ;] § nYl)NQPSZWR LSTmN XYLSb\Z T RULSb nY[c[cZ\b\Nc U XYd(Z UZ SLSTQX NQR Tmx XYdS UZ R LkNcPSZ xu[cTmN NQ[}R XY RU d(ZW XYd(Z Z\ kTcX/NQRUT}x X/dS Z/ Dh*nfR xl R LaT}N X/LSb Z\TXY[cZ RULSb np[}[cZ\b\NQ F X/d7Z\ Z F aLSTQX/NcR Tmx XYdS UZ NQPSZ xu[}T}N NQR qJZY] XYLS nYl#NQPSnpT}ZJXY[cZ X/d7Z\ Z TQX/NcR T}xuXYdS Z NcPSZ TcZ\b npLa NQRUqWZ/] NQPSZ\L TcX/ NQRUT}x X/dSR URUNM lonp[ üYøpiø h TcPSnY S i Z uX/, ª « x 5 J§s]XYLS NQPSZiZ\[c[cnY[ )np a NQPSZ\L d( Z @ §s SR b » n-np» X NHNQPSZiTQX NQR Tmx XYdSRU R NM WvY[QXYVSPaT T}PSn TNQPuX/NÃNcPSR TÃT}Z Z\qWTÃNQn d(Z NQPSZ b XYTcZ/ UTcnS] XY NQPSnY SvpP RUN TcZ Z\qWT NQPSX/ N ûRUT xa *Z lonY[ÃXHvpRU YZ Li)np[}TcZ\N TcR \Z XYLa FL- aq d7Z\[ nYl T} SdST}np aNcR npLaT ]8R N qWX d7Z V7npT}TcRUdS Z NQn@ SZ\b [}Z XYT}Z RUN{d [cZ\qWn -R LavS SVS UR b X NQZ Tc SdaTcnp U aNQRUnpLSTilo[cnpq NcPSnpTcZWvpZ LaZ [QX NQZ wd ÇwXY »*haX/N XYLa [cZ\VS XYb\R LSvDNQPaZ q RUNcPLSZ\ynYLSZ T\ NQPSnp avpP NcPSR T RUT¡NcZ b PSLSRUb XY U U R LSb\[cZ X/TcR Lav NcPSZ)L Sq d(Z [¡nYlaTc SdST}np *NQR nYLST¡NcPuX/N%XY[}ZvpZ LSZ\[QX/NcZ ]1NQPaZ VS[cZs *RUnp ST )X nYl*R LSb\[cZ X/TcR Lav NQPSZ L Sq d(Z [ nYl-T} SdSTcnY aNcR npLSTÃRUL NQ[cn- S Sb\Z TÃqJnp[cZS SVS UR b X NQZ T\] PaZ [cZ XYT%NcPSR T qJZ\NcPSn* R T vp SXY[QXYL NcZ Z NQn{XY S XY S SR NQR nYLuXY Y aLS R SZjTc SdaTcnp U aNQRUnpLST HP- aT ] Z XYb P Tc adSTcnp U aNQRUnpL R U RU»YZ d(ZiqWnY[cZ ST}Z\lo S 9 : PDEf? EFA Ì BW A HPSZzRU SZ X nYl }b nYPSZ T}R npLS wR TJT}NQRU TcnYqWZs PuX/N@ /XYvp aZY] XYLS OR L np[} SZ [JNQn aZ\NQZ\[cqJR LSZPSZ\NcPSZ [Wnp[JLSnYN@R NWX/«8Z\b\NcT TcnY U -R LSv³NQRUqWZ/])R NW#np S U Od(Z ST}Z\lo S NQnkPuX pZFX³lonp[cqWXY SZ\xSLSRUNcR npLDXYLa fX qWXYLSLSZ\[HR L PSR b PDNQnJqJZ XYT} S[cZ R N e
10.9Percent Satisfiable0.80.70.60.50.4SATO SatisfiabilityND SatisfiabilityNDS Satisfiability0.30.2100110120130Word Set Size C R vY S[cZ %140150160)Ø h- 3ý [QXYVSP LFnpdaTcZ [m /X/NQRUnpLznYl xuvY S[cZ ²@T}PSn T NQPuX N e g XYLa ³e gih Z XYb P XY UnYN NQPSZ\R [ b\npqJVS aN}rR LSv NcR qJZ SRU«7Z [}Z L NQ p We g TcV(Z LS ST qWnY[cZ NQRUqWZ vYZ LSZ\[QX/NcR LSv Tc SdaTcnp U aNQRUnpLST XYLS kXYT X[cZ\Tc S N LaZ Z aT Ncn T}V7Z\LS w UZ TcT NcR qJZ VS[cn-b Z\TcTcRULSvDNQPSZ\q Je gihFTcV(Z La ST UZ TcT NcR qJZ vpZ L*rZ [cX/NQRULSvWT} SdST}np aNcR npLaT ](dS aN NcPSZ [cZ T} S UNcR LSvWvp[cnY SVST n/l Tc SdaTcnp U aNQRUnpLST N XY»/Z qJnp[cZiNQRUqWZNQnWVS[cn-b Z\TcT {K )nY S z UR »/Z NcnWNQ[m R LSb\np[cV(np[QX NQR Lav NcPSZ X/ a /XYL N XYvpZ\T nYl Z XYb PFZs -Nc[cZ qJZY HPSZ VS[cn-b Z\ S S[cZ )nY S vpZ LaZ [QX NQZ dS n-b » T{nYl Tc SdaTcnp U aNQRUnpLST ] XY[cvpZ\[ NQPuX/LfdS Un*b » T vpZ L*rZ [cX/NQZ\ wd e gih7] dS *N T}q XY U Z\[ NQPuXYL³NQPaZWT}Z\N n/lHXY U V7npT}TcRUdS Z T} SdSTcnY aNcR npLST\ @ HPSnpT}ZTc SdaTcnp U aNQRUnpLST b np a ·NQPSZ\LOd(Z T}np[}NcZ OXYTWRUL e g ] XYLS NQPSZ\LOT}q XY U Z\[ dS n-b » TWb\np S Ud7ZWVS[cn-b Z\TcT}Z ·np aN nYlHNQPSX/N XY[cvpZ\[ da n-b »(]¡N XY»/Z LynY« nYlHNcPSZWlo[}npL N f HPaZ@VS[cn-b Z\TcT}Z Tc SdaTcnp U aNQRUnpLST b\np S U wNQPSZ\L·d7ZW[cZ Va XYb\Z yR LkNQPSZ@ XY[cvYZ dS Un*b »³d yXfLaZ\ TcZ\N nYl Tc Sd-rTcnp U aNQRUnpLST{XYLS zNcPSZ X/[cvpZ dS n-b »D[cZ\Tcnp[mNQZ ³np[ NQPSZ XY[}vpZ da n-b » b\np S fd7Z b npqJVS ZsNQZ\ U Zs *PuXY aT}NQZ\ DXYLS JNQPSZ\L@[cZ\VS XYb\Z J RUNQP XYLSnYNcPSZ [\ HPSRUT)d(nYNQPWVS[cn -R SZ\T Xi XY[}vpZ [ SZ\vp[cZ\ZnYl(b npPaZ TcRUnpL NQPSXYL e gih XYLa WX/ Tcn Z RUqWRULuX/NcZ T¡T}NQZ\V ü n/l(e g Y { T}nS] R LSTmNQZ XY nYl(TcRUqWVS Tcnp[mNQRULSv NcPSZ@Tc adSTcnp U aNQRUnpLST\] lonp[ d7ZsNcNQZ\[ b nYPSZ T}R npLwNcPSZ XY Uvpnp[}RUNQPaq b np a kNc[} NQnfxSLSX· }qJnpT}N b\npPSZ\TcRU YZ Tc SdST}Z\Nc S]¡XY UNcPSnp SvpPzxSLS SR LavDXDb\npPSZ\TcRU YZ Tc SdST}Z\Ni#np S ³ UR »YZ\ U d(Zb npqJVS aNQX/NQRUnpLuXY U U S²R Jb S N h* ª b XYL d(Z b npLaxSvp S[cZ\ NQn STcZ aRU«7Z [cZ\L N qWZsNQPSn- ST) PSZ\L T}np U -RULSvS HPSZ\TcZiZs *V(Z [¶rR qJZ L NQT #Z [}Z b XY[}[cRUZ Jnp aN# STcRULSv g %5 ²a]*d(Z b XY ST}Z g %5 ü PuXYTjR LJNQPSZ VuXYTmN)LSn/NvpZ LaZ [QX NQZ Z\ YZ [} Tc adSTcnp U aNQRUnpL ³{n )Zs pZ\[ ]8RU9L SRUb » Zs *V(Z [cRUqWZ\L NQT ]7g %5 ü TcZ\Z qJTNQn d7ZJTcRUvpLSR xub XYL Nc U fl X/T}NQZ\[ X N T}np U -R Lav R LSTmN XYLab Z T\ KtN #np S U wd(Z R L NQZ\[cZ TmNQRULSv NcnTcZ\ZfPSn ST}R LSvyg ¡f ü #np S U Ob PuXYLavpZfNQPaZfV(Z [}lonY[cqWXYLSb Z SR «8Z\[cZ Lab Z TWd7ZsNM)Z\Z LNQPSZJ /XY[cRUnp ST VS[cn-b Z\ S S[cZ\T ] XYT Z\TcT NQR qJZ )np a w UR »/Z U ³d(Z T}V7Z\L N npL Va[cn-b Z T}TcRULSvfNQPaZTc SdaTcnp U aNQRUnpLST ]aV7Z\[cPuXYVaT{vpR -R LSv X/LfXY * YX/L NQXYvpZ NQn d(nYNQP h* ª XYLS e gih7
1400SSols processed by DecompSolve (ND)SSols processed by NDSSubsolutions/Iteration, DecompSolve (ND)Subsolutions/Iteration, NDS1200# of Subsolutions10008006004002000100110120130Word Set Size140150160C R vp a[cZ g*Ø#h* adSTcnp U aNQRUnpL ý [QXYVSPaT ) [cRUZ\l¶]pRULSRUNcR XY NQZ\T}NcTjnYLJÇwXY »*haX N XYLS We gih Yh* ª T}PSn )Z\ WNQPaZ q V(Z [mlonp[cqJR LSv q Sb P)nY[cTcZJNQPuX/L·h* ª np[ e g PSZ\LwNQPaZ TcZ\b npLa yTcZ\N nYlx "! # XYT}TcZ [mNQRUnpLST #Z [}Z LSn/NR LSb\ S aZ % HPSR T RULS SR b X/NQZ\T NQPuX N NQPSZHT}Z b nYLS WT}Z\NjnYlµ "! # XYT}TcZ [mNQRUnpLST XY[}ZHR qJV7np[mN XYL NNQn »/Z Z VaR LSvD[c aLSLSR Lav@NQR qJZ T n i Lalonp[mNQ SLuX NQZ p]8NcPSZ\ zXY[cZ\L ÆN XY U)X *T V(npT}TcR da Z lonp[XY VS[}npdS Z\q anpq X/R LST\ ½ {T Tc Sb P8] Zs *V(Z [}R qJZ L NQT npL VS[}npdS UZ q SnpqWXYR LSTWR L PSRUb PRUN R T}L N V7nYTcTcRUdS ZJNQnzvYZ LSZ\[QX/NcZ NQPSZ\TcZ@b\ XY aTcZ T )np a d(Z@R L NcZ [cZ\T}NcR LSvS KML·XY S SR NQRUnpL ]Zs *V(Z [cRUqWZ\L NQT RUNcPSnp aNiNQPaZ xS[cT}N TcZsN n/ l "! #·XYT}TcZ\[}NQRUnpLST\]% ST}R LSv npLS zNQPaZJT}Z b\npLSTcZsN ] q X kvpRU YZ@TcnYqWZ@RULSTcRUvpP N ³ HPSZ d(Z LSZsxSN NQn³e{gi h ÇwXY »*haX N )XYT vp[cZ X/NQZ\[ NQPSXYLNQPSZ d7Z\LSZ\xSN NQnDe giAh ph* ª ] XYLa FTcn@K Tc aTcV(Z b\NiNcPuX/N NcPSZ UZ\ pZ\ nYl b\npPSZ T}R npLFRULfNQPaZTc SdaTcnp U aNQRUnpLST qWX fd7Z RUqWV(np[mN XYL N RUL³ SZ\NcZ [cqJR LaR LSv@Pan1ÂPSZ UValo S NcPSZ TcZ\b npLa wTcZsN nYlXYTcT}Z [mNQR nYLSTWRUT UTcnS]#NQPSZ TcZ\b npLS ºT}Z\N nYliXYT}TcZ [mNQRUnpLST RUT SZ V(Z LS aZ L NDnpL NcPSZfT}Z\N nYlTc SdaTcnp U aNQRUnpLST [QX NQPSZ\[ NcPuXYL NQPSZ VS[}npdS UZ q SnYq XYRUL ] XYLa XYT T} Sb PWNcPSZ [}Z{qWX d7Z X )X NQn XY SvYqWZ\L ¶N "! #@Ncn XY *NQnpqWX/NQRUb XY U U SZsNQZ [}qWRULSZH PSZ\NcPSZ [jnp[ LSnYN Ncn XY a JNcPSZ q lonp[X vpRU YZ L TcZ\N nYl%Tc adSTcnp U aNQRUnpLST\ ª LSZ nYl8NQPSZ qW X }np[j S²R Wb\ S UNcR Z\TjRUL e g XYLS e{gihWR Tj SZ\NcZ [}qWRULSR LSv X lo SLab\NQRUnpL lonp[ NQPaZVuXY[cXYqWZsNQZ\[ ·KtN #np S ·d(ZD STcZslo S HRUl NQPaZ Va[cn-b Z a S[cZ\TWb\np S ·T}npqWZ\PSn UZ XY[}L·XYLnpVaNcR qWXY (lo aLSb\NcR npL lonp[H F RUNcPSnp aN aTcZ [ R L NcZ [QX/b\NQRUnpL · ¹ "º »5¼ ½ ¼5¾ ¿ À Á  ¿ à ½ Ä Å¶Æǽ Ä È Á5Ã@ÉÊà Ä"Á5ÅÌË Æ Í"Î5Ï Æǽ KtN )XYT qWZ\L NcR npLaZ wVa[cZ\ -RUnp STc FNQPuX N ÇwXY »*haX N qWX ³Z LS aV [cZsNQ S[}LSR LSvfXYLS kPSZ LSb\ZVS[cn-b Z\TcT}R LSv NQPSZ TQXYqJZ Tc SdaTcnp U aNQRUnpLWqJnp[}Z NcPuXYLWnpLSb Z/ KtNjR TjXY T}n P -V(nYNQPSZsNQRUb XY U U V(npTmrTcRUdS Z NcPuX/N ÇwXY »*haX Ni[cZ\Nc S[cLaT T} SdST}np aNcR npLaT RUNQP³LSnYL- aLSRUlonp[}q SRUT}Nc[cR da aNQRUnpL ]8 PSRUb Pb XY aTcZ T VS[cnpda Z qJTiRUl)Zs -NQZ\LS uXYdS UZ T} SdSTcnY aNcR npLST XY[cZ\L ÆN [}Z\NQ a[cLSZ\ RUNQP PSRUvpPwVS[}npduX rÐ
Time as Solutions Gathered Changes9SATO100%120%140%160%180%200%8Running Time765432100110120130Word Set Size C RUvp S[cZ e140150160jØ h*np U -RULSv HR qJZ lonp[ e g 1Ç XY U»*haX/NdSR URUNM [cZ X/NQR pZ NQn nYNQPSZ\[)T} SdSTcnY aNcR npLST\ HPSRUT)qWX d(Z Z\TcV(Z b\R XY U U VS[}npdS Z\q X NQR b PaZ LNQPSZ VS[}npdS Z\qWT XY[}Z RUL NcPSZ b [cR NQRUb XY U U rMb\npLST}Nc[QXYRULSZ [}Z vpRUnpL XYLS NQPSZ\[cZ X/[cZ loZ\ÖT} SdSTcn r aNcR npLaT NcPuX/N Z¼ *NcZ LS FNcn X@Tc SV(Z [}Tcnp U aNQRUnpL ]7Tcn@NQPSX/N RUT RUN{R qJV7nY[}N X/L N X R aZ /XY[cRUZ\NM nYl%Tc SdST}np *NQR nYLST XY[cZ lonp aLS KML·np[c SZ\[ NQnF SZsNQZ\[cqJR LSZDXFV(npR L N X/N PSR b P·NQnF[}Z\NQ a[cL SLSTQX NQR Tmx XYdS UZY]jXzVS[}n*b\Z S S[}Zb np a STcZ d7nYNcP ÇwXY U»ah*X/N{XYLS Fh- ª lonY[{ ÄX/LS f ST}Z NcPSZ T} SdST}np aNcR npLaT{vpZ\LSZ [cX/NQZ\d d(nYNQP n/l NcPSZ q ª LSb Z h- ª [} SLST np aN nYl Tc SdaTcnp U aNQRUnpLST ] R N RUT TQX/loZWNQnF[cZsNQ S[}L SLSTcX/NQRUT}x XYda ZY] dS aN S S[}R LSv NQPSZ#[c SL Ç X/ »*haX/N%q X id(Z)X/dS Z NQn{vpZ LSZ\[QX/NcZ)T} SdSTcnY aNcR npLSTqWnY[cfZ SR b » p HPSZ Tc SdaTcnp U aNQRUnpLWb\npLSb\Z VaN#b np a JXY T}nid7Z R L NcZ vp[cX/NQZ\ J R NQP n-b XY aTcZ X/[cb P@X/ vpnp[}RUNcPSqWT\ KMLST}NcZ XY nYl RULSTcZ\[}NcR LSv b XY ST}Z T NQPuX/N{XYT}TcZ\[}N NcPSZiTc SdaTcnp U aNQRUnpLST ] npLaZ b\np S D ST}Z T} SdSTcn r aNcR npLaT XYTjNcPSZ R LSR NQR X/ SXYT}TcR vYLSqWZ\L NQT lonp[ Ç XY U»*haX/N ]-XYLa [QX/LS SnpqJ U XYTcT}R vpL NQPSZ [cZ\T}N)nYlNQPSZ X/NQnpqJT HPaZ\ z#np S U FPuX pZ NQn d(Z ST}Z ³npLSZ X/NiX NQRUqWZ/]7da aNiTcRULSb Z NcPSZ\ zXY[cZ\L ÆNd7Z\R LSv T}Nc[cRUb\NQ ³XYT}TcZ [mNQZ\ ] ÇwXY »*haX/N qWX ³d(ZWXYdS Z Ncnº mxa * fXYL kZ\[c[}np[cT R L NQPSZJTc Sd-rTcnp U aNQRUnpL T X/NQnpqJT XYTi)Z\ #XYT RUL NcPSZJ SLuXYT}TcR vYLSZ yX NQnpqJT @KtN qWX ³d(Z T}A Jb RUZ L N NcnxuLS FXw }vpn-n* a WT} SdSTcnY aNcR npL Ktl NcPSR T X/T b nYq dSR LaZ f R NQPf aTcR Lav h* ª lonp[{ J] NQPaZVS[cn-b Z\ S S[}Z )nY S @PSX YZ X vpn-n- vp SZ\TcT n/là PSZ L@Ncn T}NcnpV ]aZs pZ\L RUlÃR N)TmNQR U u XYT}L ÆNHTc S[}ZPSZ\NcPSZ [{XYLD SLSTcX/NQRUT}x X/dS Z [}Z Tc a UN X/T b\np[c[}Z bsN np[HLanYN Ñ
Satisfiability as Solutions Gathered Changes10.9Percent 50.40.30.2100110C RUvp S[cZf *Ø h120130Word Set Size140150haX/NcR Tmx XYdS UZ lonp[ e g ÇwXY U»ah*X/N160 9 :tÒ Y ? ËzÌ AÓ BJA g S[cRULSv NcPSZ b\np S[cT}Z nYl NcPSZ Va[cnw }Z\b\N ]aK [cnYNcZ TcnYqWZ{b n- SZ{NcPuX/N aR SL ÆN#Z LS SV vpZ\N}NQR Lav ST}Z R L NQPSZ xuLSXY Zs *V(Z [}R qJZ L NQT\Ø NQPSnp avpPÖRUNF)XYTkLanYNFlonp[FNQPSRUTFVS[cn }Z b\NkTcV(Z b\RUxub XY p] X N NcPSZ·d7Z\vpR LaLSR LSvºn/l NQPaZTcZ\qWZ\T}NQZ\[@K [QX/L Zs *V(Z [cRUqWZ\L NQT@ ST}R LSvwe g Ncn Tcnp pZ [cnpT}T}#np[c S \ Z nYLST}Nc[c Sb¼rNQRUnpL R LSTmN XYLab Z T RUNcPSnp aN NcPSZ g SVS UR b X NQZ)Çknp[c npLSTmNQ[cXYR L N ] X/T#X/L XY a SZ LS
dS n-b » nYlTc adSTcnp U aNQRUnpLSTQ§ _Sn-ZT)e gihWLaZZ_@NcnvpZ\LSZ[cX/NQZ Z\ pZ\[} WTc SdaTcnp U aNQRUnpL Å pZ\LDRUl R N)_an*Z\T LSZZ\_·NcnzvpZ\LSZ[cX/NQZDXY U #T} SdST}np aNcR npLaT] NQPaZ[cZ R TNcPSZ XY_S_aZ_yd(ZLSZsxSN NQPuX N npLSb\Z XzdS n-b » nYl
