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Loop 2 Actives: {23, 40} Passives: { 25 , 27, 39, 44} Subsumed: {41, 42, 43} Loop 3 Actives: {23, 40, 45} Passives: { 27 , 39, 44, 51} Subsumed: {46, 47, 48, 49, 50} Loop 4 Actives: {23, 27, 40, 45} Passives: { 39 , 44, 51, 59, 60} Subsumed: {52, 53, 54, 55, 56, 57, 58} Loop 5 Actives: {23, 27, 40, 45} Passives: { 44 , 51, 59, 60} Subsumed: ; Loop 6 Actives: {23, 27, 40, 44, 45} Passives: { 51 , 59, 60, 65} Subsumed: {61, 62, 63, 64} Loop 7 Actives: {23, 27, 40, 44, 45, 51} Passives: { 59 , 60, 65, 70} Subsumed: {66, 67, 68, 69} Loop 8 Actives: {23, 27, 40, 44, 45, 51, 59} Passives: { 60 , 65, 70, 82, 84, 85} Subsumed: {71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81} Loop 9 Actives: {23, 27, 40, 44, 45, 51, 59, 60} Passives: {65, 70, 82, 84, 85, 97, 99, 100} Subsumed: {86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96} 'JHVSF  " UZQJDBM FYFDVUJPO PG -Əƙ** JO UFSNT PG UIF DPOUFOUT PG JUT NBJO EBUBT USVDUVSFT UIF BDUJWF BOE QBTTJWF DMBVTF TFUT ĦF FYFDVUJPO JT TIPXO BT B TFRVFODF PG TUBUFT &BDI TUBUF TIPXT UIF DPOUFOUT PG UIF EBUBTUSVDUVSFT BU UIF FOE PG -Əƙ**ÙT NBJO MPPQ $MBVTFT BSF SFQSFTFOUFE CZ UIFJS OVNFSJDBM JOEJDFT 'PMMPXJOH UIF HJWFO DMBVTF BMHPSJUIN UIF DPOUFOUT PG UIF BDUJWF BOE QBTTJWF TFUT QFSTJTU BDSPTT MPPQT XIJMF UIF TFU PG TVCTVNFE DMBVTFT JT DPNQVUFE BU FBDI MPPQ /FX DMBVTFT UIBU BSF OPU TVCTVNFE BSF BEEFE UP UIF QBTTJWF TFU "U FBDI MPPQ B QBTTJWF DMBVTF JT TFMFDUFE ĦF JOEFY PG B DMBVTF TPNFUJNFT DIBOHFT XIFO UIF DMBVTF JT NPWFE GSPN UIF QBT TJWF UP UIF BDUJWF TFU TP IFSF XF JOEJDBUF UIF QBTTJWF DMBVTFT UIBU BSF BDUJWBUFE JO UIF OFYU MPPQ CZ DJSDMJOH UIFN 'PS FYBNQMF DMBVTF  JT DJSDMFE JO UIF ǪSTU TUBUF CFDBVTF JU JT BDUJWBUFE EVSJOH UIF TFDPOE MPPQ CVU -Əƙ** SFOBNFT JU UP  XIFO JU JT BEEFE UP UIF BDUJWF TFU JO UIF TFDPOE MPPQ /PUF UIBU -Əƙ** JT BMXBZT QJDLJOH UIF PMEFTU QBTTJWF DMBVTF UIBU JT UIF POF XJUI UIF TNBMMFTU JOEFY ĦF FYFDVUJPO TUBSUT XJUI B TFU PG DMBVTFT DPOTJTUJOH PG UIF DPOKFDUVSF BOE B DPMMFDUJPO PG TFUUIFPSFUJDBM EFǪOJUJPOT TVDI BT UIBU GPS TVCTFU BOE VOJPO  $MBVTFT BSF OVNCFSFE TFRVFOUJBMMZ -Əƙ** TUBSUT CZ QSFQSPDFTTJOH UIF QSPCMFN /PUF UIBU EFǪOJUJPOT BSF VOGPMEFE EVS JOH QSFQSPDFTTJOH %VSJOH UIF ǪSTU FYFDVUJPO PG UIF NBJO MPPQ -Əƙ** BDUJWBUFT UIF SE DMBVTF BT XF DBO TFF JU JT UIF TPMF NFNCFS PG UIF BDUJWF TFU  8F DBO TFF UIBU OPOF PG UIF DMBVTFT QSPEVDFE CZ DMBVTF  JO UIF ǪSTU MPPQ BSF TVCTVNFE %VSJOH UIF TFDPOE SVO PG UIF MPPQ -Əƙ** QJDLT DMBVTF  GPS BDUJWBUJPO $MBVTF  IBT JUT WBSJBCMFT SFOBNFE BOE UIF SFTVMUJOH DMBVTF JT DBMMFE  XIFO JU JT BEEFE UP UIF BDUJWF TFU ĦF QSPDFTTJOH PG DMBVTF  BEET B DMBVTF  UP UIF QBTTJWF TFU ĦSFF PUIFS DMBVTFT JU QSPEVDFT    BSF TVCTVNFE BXBZ ĦJT DPOUJOVFT PO VOUJM B SFGVUBUJPO JT GPVOE EVSJOH UIF UFOUI JUFSBUJPO XIJDI JT OPU TIPXO IFSF  ĦF USBOTMBUJPOT BSF VTFE CZ -Əƙ** UP DPMMBCPSBUF XJUI '0- "51 TZTUFNT -Əƙ**ÙT QSPPG DBMDVMVT JT EFUBJMFE JO "QQFOEJY " "MNPTU BMM PG JUT SVMFT BSF WBMJEJUZQSFTFSWJOH ĦBU JT UIF DPODMVTJPO JT WBMJE TBUJTǪFE CZ BMM NPEFMT JG UIF QSFNJTFT BSF WBMJE 8F DBO FNCFE UIF SVMFT JO *TBCFMMF)0- VTJOH TDIFNBUJD NFUB SVMFT PS UBDUJDT XIFO NPSF FYQSFTTJWFOFTT JT OFFEFE ĦF POMZ UXP FYDFQUJPOT BSF 4LPMFNJ[BUJPO BOE TQMJUUJOH ĦFTF SVMFT BSF DPOTJTUFODZQSFTFSWJOH OPU WBMJEJUZ QSFTFSWJOH BOE UIFZ DBOOPU CF VTFE EJSFDUMZ JO B WBMJEJUZQSFTFSWJOH DBMDVMVT TVDI BT UIBU PG *TBCFMMF)0- 8F IBOEMF UIFTF SVMFT CZ BQQMZJOH QSPPG USBOTGPSNBUJPOT XF XJMM GPDVT PO UIJT JO $IBQUFS  'PSNVMBT JO -Əƙ** BSF SFQSFTFOUFE VTJOH HFOFSBM DMBVTFT $MBVTFT DPOTJTU PG DPMMFD UJPOT PG MJUFSBMT " MJUFSBM DPOTJTUT PG BO BUPNJD GPSNVMB BOE B QPMBSJUZ ĦF QPMBSJUZ TJNQMZ JOEJDBUFT XIFUIFS UIF BUPN JT CFJOH BTTFSUFE QPTJUJWFMZ PS OFHBUJWFMZ (FOFSBM DMBVTFT DBO IBWF BSCJUSBSZ GPSNVMBT JO MJUFSBMT SBUIFS UIBO TJNQMZ BUPNJD GPSNVMBT *O XIBU GPMMPXT XF SFGFS UP HFOFSBM DMBVTFT TJNQMZ BT ÛDMBVTFTÜ 8F XJMM VTF UIF GPMMPXJOH OPUBUJPO GPS MJUFSBMT + T − T XIFSF T JT B QSPQPTJUJPOBMMZUZQFE UFSN 4FNBOUJDBMMZ + T JT FRVJWBMFOU UP UIF GPSNVMB T = True BOE − T JT FRVJWBMFOU UP T = False  'PS FYBNQMF UIF MJUFSBM − a ∨ x IBT UIF TBNF NFBOJOH BT UIF GPSNVMB ¬ (a ∨ x)  4FNBOUJDBMMZ B DMBVTF DPSSFTQPOET UP UIF EJTKVODUJPO PG JUT MJUFSBMT "O FNQUZ DMBVTF JT VTVBMMZ EFOPUFE CZ ! 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'PSNVMBT BOE JOGFSFODF OBNFT BSF OPU TIPXO IFSF TJODF UIFJS QSFDJTF WBMVFT BSF OPU JNQPSUBOU ĦF HSBQI JT JN QMJDJUMZ EJSFDUFE GSPN MFĤ UP SJHIU ĦF EPVCMFDJSDMFE OPEF PO UIF SJHIU JT UIF Ǫ OBM DMBVTF×UIF EFSJWBUJPO PG GBMTJUZ ĦF OPEF DPOOFDUFE UP JU JT UIF DPOUSBEJDUJPO EFSJWFE CZ & GSPN UIF  DMBVTFT UP XIJDI JU JT DPOOFDUFE ĦF EJBHSBN VTFT UIF QSFTFOUBUJPO DVFT VTFE CZ 5SBD FU BM  JO UIFJS *%7 TZTUFN ĦF MFĤQPJOUJOH QFOUBHPOBM OPEF JT UIF DPOKFDUVSF UIF SJHIUQPJOUJOH QFOUBHPO JT UIF OFHBUFE DPO KFDUVSF USJBOHMFT BSF BYJPNT BOE DJSDVMBS OPEFT BSF EFSJWFE DMBVTFT  9U : a! b. 8V : a. U V = cA (9U : a! b. 8V : a. U V = cA) = False (9U : a! b. 8V : a. U V = cA) = False (¬9U : a! b. 8V : a. U V = cA) = True (8U : a! b. ¬(U (sK1V U) = cA)) = True (8U : a! b. ¬(U (sK1V U) = cA)) = True 8SV1 : a! b. ¬(SV1 (sK1V SV1) = cA) = True 8SV1 : a! b. (SV1 (sK1V SV1) = cA) = False False = True False negate conjecture unfold def polarity switch extcnf combined clause copy extcnf forall pos extcnf not pos ext uni solved all splits 'JHVSF  (SBQI GPS -Əƙ**ÙT QSPPG PG 4&5? ĦJT QSPPG JT OPUBCMF GPS JUT MBDL PG SFMJBODF PO BO FYUFSOBM "51 TZTUFN  XBSET RVJDL USBOTGPSNBUJPO PG FTTFOUJBMMZ IJHIFSPSEFS DMBVTFT JOUP FT TFOUJBMMZ ǪSTUPSEFS PS QSPQPTJUJPOBM POFT <á> ĦFO <á> XF XBOU UP DPPQFSBUF XJUI TQFDJBMJTU SFBTPOFST GPS FǬDJFOU SFGVUBUJPO PG UIFTF TVC TFUT PG DMBVTFT *O TVNNBSZ XF BSF SBUIFS JOUFSFTUFE JO UIF TUSFOHUIT PG UIF DPPQFSBUJWF BQQSPBDI UIBO JO JTPMBUFE DPNQMFUFOFTT PG -Əƙ** BOE CPUI BTQFDUT NBZ FWFO UVSO PVU UP CF JO DPOǫJDU XJUI FBDI PUIFSÜ #FO[N»MMFS FU BM  g /BUVSBMMZ UIF OVNCFS PG DMBVTFT HJWFO UP UIF ǪSTUPSEFS QSPWFS WBSJFT GSPN QSPC MFN UP QSPCMFN 'PS JOTUBODF JO UIF QSPPG PG /6.? -Əƙ** HJWFT & B TJOHMF GBDU UP QSPWF JODPOTJTUFOU ∀SV1 : nat. (pl x y = pl x SV1) = False. ĦJT JT FBTZ UP SFGVUF TJNQMZ JOTUBOUJBUF SV1 UP y BOE XF PCUBJO BO BCTVSE TUBUF NFOU *O UIJT DBTF -Əƙ** TIPVME USZ BOE SFGVUF UIJT DMBVTF JUTFMG SBUIFS UIBO QBZ UIF QFSGPSNBODF QFOBMUZ PG DBMMJOH & GPS TVDI BO FBTZ SFGVUBUJPO .BOZ QSPPGT JOWPMWF TFOEJOH & NBOZ NPSF GBDUT×GPS JOTUBODF JO UIF QSPPG PG 4&7? & JT HJWFO  DMBVTFT #Z NPEFSO TUBOEBSET UIJT JT OPU B MBSHF OVNCFS PG DMBVTFT UP HJWF UP BO "51 TZTUFN .PSF NBUVSF "51 TZTUFNT BSF GBJSMZ SPCVTU UP MBSHFS OVNCFS PG DMBVTFT DPNQBSFE UP OFXFS QSPUPUZQFT UIJT DBO CF TFFO JO 'JHVSF  &ÙT JOEJTQFOTJCJMJUZ UP -Əƙ** VOEFSTDPSFT UIF JNQPSUBODF PG UIF FODPEJOH VTFE CZ -Əƙ** UP DPNNVOJDBUF QSPCMFNT UP & %VSJOH FBSMJFS XPSL UIJT FODPEJOH XBT GPVOE UP IBWF UIF GPMMPXJOH TIPSUDPNJOHT 4VMUBOB FU BM  #FO[N»MMFS BOE 4VMUBOB   à "T QSFWJPVTMZ PCTFSWFE 4VMUBOB FU BM  UIF PME USBOTMBUJPO XBT WFSCPTF BOE JUT VTF SFTVMUFE JO ǪSTUPSEFS GPSNVMBT UIBU XFSF GBJSMZ MBSHF CFDBVTF PG IPX UZQF JOGPSNBUJPO XBT FODPEFE ĦJT WFSCPTJUZ DBVTFE BEEJUJPOBM PWFS IFBE UP "51 TZTUFNT *U DPOUSJCVUFE UP "51 TZTUFNT NJTTJOH UIFJS UJNFPVU UP ǪOE B SFGVUBUJPO * XPSLFE PO B SFJNQMFNFOUBUJPO PG UIF USBOTMBUJPO UP '0- ĦF OFX USBOTMBUJPO JT DPNQMFUF JO UIF TFOTF UIBU λUFSNT BSF GVMMZ λMJĤFE BOE BMM SFRVJSFE UZQF JOGPSNBUJPO JT FODPEFE 4JODF UIF OFX USBOTMBUJPO UFOET UP JODMVEF NPSF JOGPSNBUJPO UIBO UIF PME USBOTMBUJPO VTFE JO -Əƙ** JU NBZ XFMM CF MBSHFS 5P BEESFTT UIJT QSPCMFN B NPSF DPNQBDU FODPEJOH PG UZQF JO GPSNBUJPO XBT JNQMFNFOUFE DMPTFMZ GPMMPXJOH $MBFTTFO FU BM  UP JNQMF  NFOU UIFJS NPOPUPOJDJUZ BOBMZTJT PG UZQFT " UZQF JT NPOPUPOJD JG JUT EPNBJO DBO CF FYUFOEFE XJUI OFX FMFNFOUT $MBFTTFO FU BM TIPX UIBU JU JT TBGF UP FSBTF UIF UZQF JOGPSNBUJPO PG NPOPUPOJD UZQFT MFBEJOH UP TNBMMFS FODPEJOHT PG λUFSNT BOE EFTDSJCF B NFUIPE UP DIFDL XIFUIFS B UZQF JT NPOPUPOJD 3F DFOUMZ #MBODIFUUF BOE 1PQFTDV  GPSNBMJTFE BOE WFSJǪFE UIF DPSSFDUOFTT QSPPGT PG $MBFTTFO FU BM  VTJOH *TBCFMMF)0- à ĦF PME USBOTMBUJPO XBT JODPNQMFUF TJODF JU EJE OPU QSPQFSMZ FODPEF λUFSNT OPS EJE JU FODPEF UFSNMFWFM QSPYJFT PG MPHJDBM DPOTUBOUT ĦJT JT EFTDSJCFE JO NPSF EFUBJM OFYU 6TJOH UIF OFX USBOTMBUJPO XIFO -Əƙ** JT BTLFE UP QSPWF i?7U;QH- +QMD2+im`2- UU4V 4 U4VVVX -Əƙ** QSPEVDFT UIF DMBVTF ȨU0i`m2V QSPCBCMZ CZ VTJOH UIF VOJ@USJW SVMF PO UIF OFHBUFE DPOKFDUVSF BOE HJWFT JU UP & UP SFGVUF 6OGPSUVOBUFMZ -Əƙ** EPFT OPU SFDPH OJTF UIBU JUT XPSL JT DPNQMFUF 6TJOH UIF PME USBOTMBUJPO JU XPVME TFOE & UIF TJOHMF DMBVTF 7Q7Ud-tBQK-UU H2QGBiUH2QhBUi`m2-QVVVVVX )FSF UIF DPOTUBOU TZNCPM i`m2 JT B QSPYZ B UFSNMFWFM DPOTUBOU XIJDI TIPVME IBWF UIF TBNF TFNBOUJDT BT B GPSNVMBMFWFM MPHJDBM DPOTUBOU 1SPYJFT BSF VTFE TJODF JO '0- XF NVTU NBJOUBJO B EJTUJODUJPO CFUXFFO GPSNVMBT BOE UFSNT 1SPYJFT IFMQ VT HFU BSPVOE UIJT EJTUJODUJPO GPS UFSNT XJUI QSPQPTJUJPOBM TFNBOUJDT 6OGPSUVOBUFMZ JO UIJT DBTF -Əƙ** EJE OPU JODMVEF BO BYJPN UP HJWF TFNBOUJDT UP i`m2 *U TIPVME IBWF JODMVEFE BO BYJPN TVDI BT UIF GPMMPXJOH 7Q7Ui`m2- tBQK- H2QGBiUH2QhBUi`m2-QVVVX )BE TVDI BO BYJPN CFFO JODMVEFE UIF '0- "51 TZTUFN XPVME IBWF CFFO BCMF UP ǪOE B SFGVUBUJPO ĦF PUIFS TQFDJBM DPOTUBOUT TFFO BCPWF XJMM CF EFTDSJCFE OFYU *O CPUI UIF PME BOE OFX USBOTMBUJPOT VTFE CZ -Əƙ** H2QhB JT VTFE UP BTTJHO UZQFT UP UFSNT×IFSF JU JT TBZJOH UIBU UIF UFSN i`m2 JT PG UZQF Q XIFSF ØQÙ JUTFMG JT B DPOTUBOU JO UIF MBOHVBHF ĦJT DPOTUBOU JT TQFDJBM UP UIF USBOTMBUJPO JU JT VTFE UP FODPEF UIF #PPMFBO UZQF ĦVT UIF USBOTMBUJPO FODPEFT UZQFT BT ǪSTUPSEFS UFSNT ĦJT LJOE PG UZQF FODPEJOH VTJOH B DPOTUBOU TVDI BT H2QhB JT DBMMFE UBHHJOH JO 4MFEHFIBNNFS #MBODIFUUF   BOE JT DSFEJUFE CZ .FOH BOE 1BVMTPO  UP )VSE   ĦF DPOTUBOU H2QGBi JT VTFE UP MJĤ QSPQPTJUJPOBM UFSNT UIBU JT UFSNT UIBU BSF UZQFE ØQÙ JOUP GPSNVMBT (JWFO UIF TBNF 5)' QSPCMFN UIF OFX USBOTMBUJPO TFOET UIF GPMMPXJOH PVUQVU UP UIF "51 TZTUFN 7Q7Ud- tBQK- U0i`m2VVX ĦBU JT UIF USBOTMBUJPO OPUJDFT UIBU JOTUFBE PG VTJOH H2QGBi UP FODPEF True JU DBO EJSFDUMZ VTF UIF '0' DPOTUBOU XJUI UIBU EFOPUBUJPO 8IFO JU CFDPNFT OFDFTTBSZ UP VTF QSPYZ UFSNT TVDI BT i`m2 UIFO UIF TFNBOUJDT PG FBDI QSPYZ BSF TQFDJǪFE VTJOH BEEJUJPOBM BYJPNT TVDI BT 7Q7UT`Qtni`m2R- THBM- U0i`m2 I4= H2QGBiUH2QhBUi`m2- QVVVVX ĦF PME USBOTMBUJPO XBT BMTP JODPNQMFUF JO B EJǩFSFOU TFOTF TJODF JU EJE OPU GVMMZ USBOTMBUF λUFSNT JOUP ǪSTUPSEFS GPSN ĦF QSFWJPVT USBOTMBUJPO TJNQMZ DSFBUFE GSFTI DPOTUBOUT GPS λUFSNT BOE EJE OPU DIBSBDUFSJTF UIFTF DPOTUBOUT GVSUIFS 'PS FYBNQMF XIJMF USZJOH UP QSPWF UIF 5)' QSPCMFN i?7U+QMDny-+QMD2+im`2-U \ (6, 0Q = 0Q) , 5 (S, 0Q = 0Q-Z, 0Q) , U  U S ! U 4= ! U 6 ! 0i`m2 V ! Z V V %  U 6 ! U 4= ! Z ! U 6 ! 07Hb2 V V V % U 6 ! Z V V VVX UIF PVUQVU PG UIF QSFWJPVT USBOTMBUJPO JODMVEFT BYJPNT TVDI BT  7Q7U99-tBQK-U   H2QGBiUH2QhBUH2QiUH2QhBUbEknauj-H2Q6iUH2Q6iUQ-QV-QVV-  H2QhBU#bi`asyasy-H2Q6iUQ-QVVV-QVV VVX ĦF FODPEFE UZQF H2Q6iUQ-QV PO MJOF  JOEJDBUFT UIBU UIF DPOTUBOU #bi`asyasy JT PG UZQF o → o ĦF OBNF #bi`asyasy JT EFSJWFE GSPN TFSJBMJTJOH UIF UFSN λSX0. 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Xr Xx Xy ! Xr Xx Xy (8Xr : a! a! o, Xx : a, Xy : a. Xr Xx Xy ! Xr Xx Xy) = False (8Xr : a! a! o, Xx : a, Xy : a. Xr Xx Xy ! Xr Xx Xy) = False (8SY3 : a, SY5 : a. sK1Xr SY3 SY5 ! sK1Xr SY3 SY5) = False (8SY5 : a. sK1Xr sK2SY3 SY5 ! sK1Xr sK2SY3 SY5) = False (sK1Xr sK2SY3 sK3SY 5 ! sK1Xr sK2SY3 sK3SY 5) = False A A B B False = True False = True False negate conjecture unfold def extcnf forall neg extcnf forall neg extcnf forall neg split conjecture solved all splits B Ħ FCFHJOOJOHBOE FOE TFHN FOUTPG-Əƙ**ÙTQSPPGGPS4&7?Ħ FTQMJUUJOHSFMBUFE TVCQSPPGT IBQQFO UP CFJEFOUJDBMĦ JTTVCQSPPGJTTIPX O CFMPX (sK1Xr sK2SY3 sK3SY5 ! sK1Xr sK2SY3 sK3SY5) = False (¬(sK1Xr sK2SY3 sK3SY5 ! sK1Xr sK2SY3 sK3SY5)) = True (¬(sK1Xr sK2SY3 sK3SY5 ! sK1Xr sK2SY3 sK3SY5)) = True (¬(¬sK1Xr sK2SY3 sK3SY5 _ sK1Xr sK2SY3 sK3SY5)) = True (¬sK1Xr sK2SY3 sK3SY5 _ sK1Xr sK2SY3 sK3SY5) = False (¬sK1Xr sK2SY3 sK3SY5) = False sK1Xr sK2SY3 sK3SY5 = True sK1Xr sK2SY3 sK3SY5 = False (False = True ) polarity switch clause copy unfold def extcnf not pos extcnf or neg extcnf not neg extcnf or neg fo atp e A B C Ħ FTQMJUUJOH TVCQSPPG8 FPCUBJO UIFGVMMQSPPGPG4&7? CZ JOMJOJOH UIJTTVCQSPPGJO B  'JHVSF7JTVBMJTBUJPO PG-Əƙ**ÙTQSPPGPGQSPCMFN 4&7? 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UIF TUZMF PG TQMJUUJOH JNQMFNFOUFE JO & "T BO JOGFSFODF SVMF UIJT GPSN PG TQMJUUJOH DPVME CF EFTDSJCFE BT GPMMPXT C ∨D Free(C) ∩ Free(D) = ∅ d GSFTI (C ∨ d) & (D ∨ ¬d)  3FDBMM UIBU UIF EBTIFE MJOF JOEJDBUFT UIBU UIF JOGFSFODF QSFTFSWFT TBUJTǪBCJMJUZ BOE UIF EPVCMF MJOF JOEJDBUFT UIBU PQFSBUJPOBMMZ UIF SVMF EFMFUFT UIF QSFNJTF GSPN UIF TFBSDI TQBDF BOE BEET UIF DPODMVTJPOT UP UIF TFBSDI TQBDF ĦF ØÙ PQFSBUPS IFSF JOEJDBUFT UIBU UIF UXP DMBVTFT BSF DPOKPJOFE XJUI FBDI PUIFS ĦBU JT UIF UXP DMBVTFT OFFE UP CF TBUJTǪFE UPHFUIFS 0QFSBUJPOBMMZ UIJT NFBOT UIBU UIF QSFNJTF DMBVTF C ∨ D JT SFXSJUUFO JOUP UIF UXP DPODMVTJPO DMBVTFT C ∨ d BOE D ∨ ¬d  *O & UIF BCPWF SVMF JT DBMMFE JOUSPEVDF EFǮOJUJPO 4USVDUVSBMMZ JU JT MJLF B DPOWFSTF PG SFTPMVUJPO TJODF JU JOUSPEVDFT DPNQMFNFOUBSZ BUPNT JO UIF DMBVTFT UIBU BSF QSP EVDFE CZ UIF TQMJU /PUF UIBU d JT B GSFTI QSPQPTJUJPOBM MFUUFS UIBU JT VTFE UP SFMBUF UIF UXP QBSUT PG UIF TQMJU *U JT B EFǮOJUJPO TZNCPM BOE JU DPOTJTUT PG B GSFTI DPOTUBOU 3FDBMM UIBU UIF TFBSDI TQBDF IBT OPU CFFO QBSUJUJPOFE ĦFSFGPSF DBSF NVTU CF UBLFO UP BWPJE SFTPMWJOH UIF UXP TQMJUT CFGPSF UIFZ BSF SFEVDFE UP TJOHMFUPOT DPOUBJOJOH d BOE ¬d SFTQFDUJWFMZ ĦBU JT JG CPUI C BOE D UVSO PVU UP CF VOTBUJTǪBCMF XF XJMM CF MFĤ XJUI UIF TJOHMFUPO DMBVTFT {d} BOE {¬d} UP XIJDI VOJU SFTPMVUJPO DBO CF BQQMJFE ĦF JOUSPEVDF EFǮOJUJPO SVMF JT DPOTJTUFODZQSFTFSWJOH TJODF à *G UIFSF JT OP NPEFM TBUJTGZJOH C ∨D UIFO UIFSF JT OP #PPMFBO WBMVBUJPO PG d UIBU DBO TBUJTGZ CPUI C ∨ d BOE D ∨ ¬d à *G UIFSF JT B NPEFM TBUJTGZJOH C ∨D UIFO Ö *G C JT TBUJTǪFE CZ UIF NPEFM UIFO UIF DPODMVTJPO JT TBUJTǪBCMF JO BO FYUFOEFE NPEFM JO XIJDI d = False Ö *G D JT TBUJTǪFE CZ UIF NPEFM UIFO UIF DPODMVTJPO JT TBUJTǪBCMF JO BO FYUFOEFE NPEFM JO XIJDI d = True *G CPUIC BOED BSF TBUJTǪFE CZ UIF NPEFM UIFO UIF WBMVBUJPO PG d EPFT OPU NBUUFS× BOZ WBMVBUJPO PG d XPVME TBUJTGZ CPUI C ∨ d BOE D ∨ ¬d & IBT BOPUIFS SVMF DBMMFE BQQMZ EFǮOJUJPO XIJDI JT GPSNBMJTFE JO UIF GPMMPXJOH XBZ σ(C ∨ d) & (C ∨D) Free(C) ∩ Free(D) = ∅ σ(C ∨ d) & (D ∨ ¬d) XIFSF d JT B QSFWJPVTMZJOUSPEVDFE EFǪOJUJPO TZNCPM σ JT B SFOBNJOH PG WBSJBCMFT BOE C EPFT OPU DPOUBJO PUIFS EFǪOJUJPO TZNCPMT /BUVSBMMZ TJODF σ JT B SFOBNJOH PG WBSJBCMFT JU EPFT OPU BǩFDU EFǪOJUJPO TZNCPMT *OUVJUJWFMZ UIJT SVMF SFEVDFT UIF  TJ[F PG B DMBVTF XJUIPVU BǩFDUJOH UIF TBUJTǪBCJMJUZ PS VOTBUJTǪBCJMJUZ PG UIF DMBVTF TFU 0QFSBUJPOBMMZ UIJT SVMF SFXSJUFT UIF DMBVTF C ∨ D JOUP UIF TNBMMFS DMBVTF D ∨ ¬d /PUF UIBU XF DPVME SFDPWFS UIF PSJHJOBM DMBVTF NPEVMP SFOBNJOH PG GSFF WBSJBCMFT CZ SFTPMWJOH D ∨ ¬d XJUI σ(C ∨ d) ĦF BQQMZ EFǮOJUJPO SVMF JT DPOTJTUFODZQSFTFSWJOH TJODF à *G UIFSF JT OP NPEFM TBUJTGZJOH CPUI σ(C ∨ d) BOE C ∨D UIFO JU NVTU NFBO UIBU JO BMM NPEFMT C = False *U BMTP NFBOT UIBU JO BMM NPEFMT d = False PS D = False *U DBOOPU CF UIBU d = False JO BMM NPEFMT UIFSFGPSF JU NVTU CF UIBU D = False ĦFO JSSFTQFDUJWF PG σ UIFSF JT OP WBMVBUJPO PG d UIBU DBO TBUJTGZ CPUI σ(C ∨ d) BOE (D ∨ ¬d) TJODF BU MFBTU POF PG UIFN JT OPU TBUJTǪFE à *G UIFSF JT B NPEFM TBUJTGZJOH CPUI σ(C ∨ d) BOE C ∨D UIFO Ö *GC JT TBUJTǪFE CZ UIF NPEFM UIFO UIF DPODMVTJPO JT TBUJTǪBCMF JO B NPEFM JO XIJDI d = False Ö *GD JT TBUJTǪFE CZ UIF NPEFM UIFO UIF DPODMVTJPO JT TBUJTǪBCMF JO B NPEFM JO XIJDI d = True "T CFGPSF JG CPUI C BOE D BSF TBUJTǪFE CZ UIF NPEFM UIFO UIF WBMVBUJPO PG d EPFT OPU NBUUFS×BOZ WBMVBUJPO PG d XPVME TBUJTGZ CPUI σ(C ∨ d) BOE D ∨ ¬d  0UIFS SVMFT 4PNF SVMFT UIBU BQQFBS JO DPOTJTUFODZQSFTFSWJOH DBMDVMJ BMTP IBQQFO UP CF WBMJEJUZ QSFTFSWJOH &YBNQMFT JODMVEF SFTPMVUJPO GBDUPSJOH BOE QSJNJUJWF TVCTUJUVUJPO JO IJHIFSPSEFS SFTPMVUJPO DBMDVMJ #FO[N»MMFS  BOE UIF α β BOE γ SVMFT JO UBCMFBV DBMDVMJ 'JUUJOH   ĦFTF SVMFT DBO SFBEJMZ CF VTFE JO WBMJEJUZQSFTFSWJOH DBMDVMJ XJUIPVU BOZ USBOTGPSNBUJPO 8F XJMM CF JOUFSFTUFE JO QSFTFSWJOH UIF WBMJEJUZQSFTFSWJOH RVBMJUZ PG TVDI SVMFT ĦBU JT XF XJMM TUVEZ USBOTGPSNBUJPOT UIBU NBQ DPOTJTUFODZQSFTFSWJOH SVMFT JOUP WBMJEJUZQSFTFSWJOH POFT BOE NBQ WBMJEJUZQSFTFSWJOH SVMFT JOUP WBMJEJUZQSFTFSWJOH POFT   &NCFEEJOH SFGVUBUJPO QSPPGT JO *TBCFMMF 6Q UP OPX UIJT DIBQUFS IBT QSFTFOUFE QSJPS XPSL JO BVUPNBUFE UIFPSFN QSPWJOH 8F XJMM OPX TUBSU QSPCJOH UIF QSPCMFN PG FNCFEEJOH SFGVUBUJPO QSPPGT JO *TBCFMMF UP ǪOE B XBZ PG USBOTMBUJOH QSPPGT HFOFSBUFE CZ BO "51 TZTUFN JOUP *TBCFMMF)0- ĦF QVSQPTF PG UIJT TFDUJPO JT UP FYQMPSF TPNF JEFBT GPS FNCFEEJOH QSPPGT CFGPSF EFWFMPQJOH POF PG UIFN JO UIF OFYU TFDUJPO "O "51 TZTUFN JNQMFNFOUT BO JOGFSFODF QJQFMJOF DPOTJTUJOH PG B QSFQSPDFTTJOH TUBHF GPMMPXFE CZ UIF NBJO MPPQ TUBHF #PUI PG UIFTF TUBHFT DPOUBJO DPOTJTUFODZ QSFTFSWJOH JOGFSFODFT 8F XPVME MJLF UP USBOTMBUF UIF PVUQVU QSPPG PG UIJT QJQFMJOF JOUP *TBCFMMF)0- XIJDI JNQMFNFOUT B WBMJEJUZQSFTFSWJOH DBMDVMVT 6MUJNBUFMZ UIF QSPPG QSPEVDFE CZ UIF "51 TZTUFN TIPVME CF NBQQFE JOUP B UIFPSFN PG *T BCFMMF)0- )PX DBO XF HP BCPVU UIJT  *EFBT GPS FNCFEEJOH *O UIJT TFDUJPO XF DPOTJEFS TPNF XBZT PG FNCFEEJOH &ÙT DBMDVMVT JO *TBCFMMF 6MUJ NBUFMZ XF XBOU UP BMMPX B QSPPG GPVOE CZ UIF GPSNFS UP CF WBMJEBUFE CZ UIF MBUUFS $PNQBSFE UP &ÙT DBMDVMVT -Əƙ**ÙT QSPPG TZTUFN JT TJNQMFS JO UIF TFOTF UIBU JU EPFT OPU EJTUJOHVJTI CFUXFFO QSFQSPDFTTJOH BOE NBJOMPPQ TUBHFT ĦJT JT CFDBVTF DMBVTJǪDBUJPO DBO PDDVS BU BSCJUSBSZ QPJOUT EVSJOH QSPPG TFBSDI JO )0- VOMJLF JO '0- XIFSF JU POMZ UBLFT QMBDF BU UIF TUBSU #PUI &ÙT QSFQSPDFTTJOH BOE NBJOMPPQ TUBHFT DPOUBJO DPOTJTUFODZQSFTFSWJOH JOGFSFODFT *O UIF JOGFSFODF TZTUFN PG UIF QSF QSPDFTTJOH TUBHF XF IBWF 5TFJUJOTUZMF DMBVTJǪDBUJPO BOE 4LPMFNJTBUJPO XIJMF JO UIF NBJOMPPQÙT JOGFSFODF TZTUFN XF IBWF TQMJUUJOH &ÙT DMBVTJǪDBUJPO QSPDFEVSF JT CBTFE PO UIBU EFTDSJCFE CZ /POOFOHBSU BOE 8FJEFOCBDI   8F XJMM UIFSFGPSF GPDVT PO UBDLMJOH & JO UIJT TFDUJPO TJODF JU JT UIF NPSF DIBM MFOHJOH PG UIF UXP -FU VT DPODFOUSBUF PO IBOEMJOH DMBVTJǪDBUJPO JO *TBCFMMF)0- UP FNVMBUF &ÙT DMBVTJǪDBUJPO 8F DPVME BQQSPBDI UIJT CZ ǪSTU DBSSZJOH PVU B WBMJEJUZ QSFTFSWJOH DMBVTJǪDBUJPO JO *TBCFMMF)0- BOE UIFO QSPWJOH UIBU UIF SFTVMU JT FRVJW BMFOU UP UIF 5TFJUJOTUZMF SFTVMU GPVOE CZ & BT EFTDSJCFE OFYU -FU F CF UIF PSJHJOBM GPSNVMB T(¬F ) CF UIF 5TFJUJODMBVTJǪFE GPSN PG UIF OFHBUFE DPOKFDUVSF BOEV(¬F ) CF UIF JǩDMBVTJǪFE GPSN "T EFTDSJCFE JO 4FDUJPO  DPNQVUJOH V(¬F ) JT NVDI NPSF FYQFOTJWF JO HFOFSBM UIBO DPNQVUJOH T(¬F ) )PXFWFS JU TIPVME CF UIF DBTF  UIBU F JT WBMJE Jǩ ¬V(¬F ) JT WBMJE Jǩ ¬T(¬F ) JT WBMJE. 0VS HPBM JT UP QSPWF F JO *TBCFMMF)0- TUBSUJOH XJUI B QSPPG PG ¬T(¬F ) QSPWJEFE CZ & 4JODF V(·) JT WBMJEJUZQSFTFSWJOH XF DBO QSPWF UIBU ¬V(¬F ) = ¬¬F JO *T BCFMMF)0- TJODF XF BSF VTJOH MPHJDBM FRVJWBMFODFT 3FDBMM UIBU F = ¬¬F JO & BOE *TBCFMMF)0- BOE &ÙT QSPPG JT EPOF CZ SFGVUBUJPO UIFSFGPSF XF XJMM TFFL UP QSPWF ¬¬F JO PSEFS UP QSPWF F  6TJOH UIJT GBDU JO PSEFS UP QSPWF F JU SFNBJOT UP TIPX UIBU ¬V(¬F ) VTJOH UIF QSPPG PG ¬T(¬F ) QSPWJEFE CZ &ÙT NBJO MPPQ 4JODF XF BSF GPDVTTFE PO QSPWJOH UIF FRVJWBMFODF CFUXFFO¬T(¬F ) BOE¬V(¬F ) MFU VT BTTVNF UIBU &ÙT QSPPG PG ¬T(¬F ) DBO CF SFDPOTUSVDUFE JO *TBCFMMF)0- ĦFO JU SFNBJOT UP TIPX JO *TBCFMMF)0- UIBU T(¬F ) = V(¬F ) 4JODF GSPN &ÙT SFDPO TUSVDUFE QSPPG XF LOPX UIBU ¬T(¬F ) JT WBMJE JU SFNBJOT UP TIPX UIBU ¬V(¬F ) JT WBMJE *U JT OPU JNNFEJBUFMZ DMFBS IPX UP CVJME UIJT QSPPG GSPN UIF BWBJMBCMF JOGPSNB UJPO 6TJOH QSPPG TFBSDI XPVME BEE NPSF DPNQMFYJUZ PO UPQ PG UIF JǩDMBVTJǪDBUJPO BOE UIJT JT VOEFTJSBCMF ĦJT BQQSPBDI XPVME OPU CF WFSZ BQQFBMJOH *U UVSOT PVU UIBU XF EP OPU OFFE UP SFMZ PO V(·) BOE XF DBO TPMFMZ SFMZ PO UIF QSPPG PG ¬T(¬F ) UP QSPWF F WJB B QSPPGUIFPSFUJDBM FNCFEEJOH #FGPSF TUVEZJOH UIJT JO NPSF EFUBJM MFU VT CSJFǫZ MPPL BU BMUFSOBUJWF BQQSPBDIFT %FFQMZ FNCFEEJOH & JO *TBCFMMF)0- ĦJT XPVME JOWPMWF FNCFEEJOH &ÙT QSPPG TZTUFN JO *TBCFMMF)0- TJNJMBS UP UIF XPSL EPOF CZ 3JEHF BOE .BSHFUTPO  UP FNCFE BOE WFSJGZ B QSPPG QSPDFEVSF GPS '0- JO *TBCFMMF)0- ĦF QSPPG TZTUFN PG & DPOTJTUT PG UIF SVMFT SFMBUFE UP QSFQSPDFTTJOH DMBVTJǪDBUJPO BOE 4LPMFNJTBUJPO BT XFMM BT UIPTF VTFE CZ UIF NBJO MPPQ 6TJOH UIJT FNCFEEJOH XF DPVME UIFO JOUFSQSFU &ÙT QSPPGT JO *TBCFMMF)0- CVU OPU BT OBUJWF UIFPSFNT PG 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*TBCFMMF& ĦJT DPVME TFSWF BT B DIFDLFS GPS &ÙT QSPPGT 8F DPVME UIFO BEE NFUBUIF PSFUJDBM GVODUJPOT UP USBOTGFS *TBCFMMF& UIFPSFNT JOUP *TBCFMMF)0- *U JT OPU JNNF EJBUFMZ DMFBS IPX UP EFǪOF UIFTF NFUBUIFPSFUJDBM GVODUJPOT "O JEFB XPVME JOWPMWF EFTDSJCJOH UIFN BT UIFPSFN FNCFEEJOHT #VU UIFO XF DPVME FTDIFX DSFBUJOH *T BCFMMF& BOE NBQ UIF QSPPGT GSPN & EJSFDUMZ UP *TBCFMMF)0- XIFSF UIFZ XPVME CF DIFDLFE BOZXBZ 1SPPGUIFPSFUJDBM FNCFEEJOH ĦJT JT UIF BQQSPBDI UIBU XF XJMM EFWFMPQ *U JT DPODFQUVBMMZ DMFBO BOE JOWPMWFT EFǪOJOH BO BCTUSBDU NBQQJOH UIBU BMMPXT VT UP FNCFE B MPHJD UIBU SFMJFT PO BO JTTBUJTǮBCMF KVEHFNFOU JOUP B MPHJD XIPTF KVEHFNFOU JT JTWBMJE ĦJT XPVME CF B QSPPG USBOTGPSNBUJPO BQQMJFE VOJGPSNMZ BDSPTT UIF QSPPG JO UIF TFOTF UIBU UIF TBNF USBOTGPSNBUJPO JT BQQMJFE UP FBDI JOGFSFODF TUFQ ĦJT NBLFT JU NVDI FBTJFS UP EFǪOF BOE BSHVF BCPVU *U XPVME OFFE NFUBUIFPSFUJDBM KVTUJǪDBUJPO CVU TIPVME BMMPX GPS BO FBTJFS JNQMFNFOUBUJPO UIBO UIF QSFWJPVT JEFBT ĦF JNQMFNFOUBUJPO DPVME UBSHFU *TBCFMMF)0- EJSFDUMZ SBUIFS UIBO ǪSTU HPJOH UP *TBCFMMF& BOE UIFO NBQQJOH UP *TBCFMMF)0- ĦJT JEFB XJMM CF EFWFMPQFE OFYU  5BCMFBV FNCFEEJOH ĦJT TFDUJPO EFTDSJCFT IPX UP FNCFE B DMBTT PG DPOTJTUFODZQSFTFSWJOH QSPPG DBMDVMJ JOUP B DMBTT PG WBMJEJUZQSFTFSWJOH QSPPG DBMDVMJ ĦJT JT POF PG UIF DPOUSJCVUJPOT PG UIJT DIBQUFS " QSPUPUZQF JNQMFNFOUBUJPO JT EFTDSJCFE BU UIF FOE PG UIF TFDUJPO *O UIJT TFDUJPO XF XJMM VTF *TBCFMMFTUZMF OPUBUJPO EFTDSJCFE JO 4FDUJPO  UP NBLF NFUBJOGFSFODF BOE NFUBCJOEJOH FYQMJDJU 8F XJMM FYUFOE UIJT OPUBUJPO XJUI B NFUBMFWFM FYJTUFOUJBM RVBOUJǪFS EFOPUFE CZ UIF TZNCPM ∨  ĦJT XJMM CF VTFE JO DPOTJTUFODZQSFTFSWJOH DBMDVMJ TVDI BT UIPTF PG & BOE -Əƙ** CVU OPU JO  WBMJEJUZQSFTFSWJOH DBMDVMJ TVDI BT UIBU PG *TBCFMMF)0-  ĦJT OPUBUJPO XJMM CF VTFE POMZ UP NBLF NFUBMFWFM SFBTPOJOH NPSF QSFDJTF XIFO SFBTPOJOH BCPVU DPOTJTUFODZ QSFTFSWJOH TZTUFNT ĦBU JT UIF TZNCPM ∨ XJMM OPU CF JOUFSQSFUFE CZ *TBCFMMF OPS EPFT JU BQQFBS FYQMJDJUMZ JO UIF QSPPGT QSPEVDFE CZ "51 TZTUFNT 8F XJMM OFWFS VTF UIF TZNCPMT ∧ BOE ∨ JO UIF TBNF GPSNVMB *OTUFBE PG HJWJOH UIF TFNBOUJDT PG ∨ WJB JOGFSFODF SVMFT BT EPOF CZ 1BVMTPO  GPS ∧ XF XJMM HJWF BO JOUVJUJWF EFǪ OJUJPO PG ∨ IFSF BOE XJMM EFGFOE JUT VTF JO FWFSZ SVMF JO XIJDI JU BQQFBST ĦJT JT TVǬDJFOU GPS PVS QVSQPTFT UIFSF JT OP OFFE UP EFǪOF B TQFDJǪD DBMDVMVT PS FYQMPSF B DPOTJTUFODZQSFTFSWJOH DPVOUFSQBSU UP *TBCFMMFÙT WBMJEJUZQSFTFSWJOH DBMDVMVT ĦF SPVHI NFBOJOH PG B GPSNVMB ∨ (x : τ). 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F (x) JT GPS BMM x UZQFE τ XF IBWF UIBU F (x) = True "T CFGPSF XF XJMM OPU SFQSFTFOU PS SFGFS UP UZQF JOGPSNBUJPO XIFO UIJT JT OPU JNQPSUBOU %FǪOJUJPO  DMPTFE DMPTFE NPEVMP TDPQFE DPOTUBOUT  " GPSNVMB JT DMPTFE JG BMM JUT WBSJBCMFT BSF PCKFDUMFWFM CPVOE ħBU JT UIFZ BSF OPU CPVOE CZ ∧ PS ∨  " GPSNVMB JT DMPTFE NPEVMP B TJHOBUVSF FYUFOTJPO JG UIF PCKFDUMFWFM GSFF WBSJBCMFT BSF CPVOE CZ ∧ PS ∨ BU UIF NFUBMFWFM  ħF TFU PG ∧ CPVOE PS ∨ CPVOE TZNCPMT UIBU BSF JO TDPQF JO B TVCUFSN JT DBMMFE UIF TJHOBUVSF FYUFOTJPO SFMBUJWF UP XIJDI UIBU TVCUFSN JT JOUFSQSFUFE ħF WBSJBCMFT UIBU BSF CPVOE BU UIF NFUBMFWFM BSF DBMMFE TDPQFE DPOTUBOUT 'PS FYBNQMF ∃x.x JT DMPTFE XIFSFBT ∨ f. ∃x.fx JT DMPTFE NPEVMP BOE UIF TJH OBUVSF FYUFOTJPO DPOTJTUT PG f  ĦF GPSNVMB ∃x. ∨ f.fx JT NBMGPSNFE XF DBOOPU IBWF PCKFDUMFWFM RVBOUJǪFST XIPTF TDPQF JODMVEFT NFUBMFWFM CJOEJOH PDDVSSFODFT /PUBUJPO  8F XSJUF T1[T2] XIFSF Ti∈{1,2} SBOHF PWFS PCKFDUMPHJD GPSNVMBT UP JO EJDBUF UIBU T2 PDDVST JO T1 *G T2 JT BO PCKFDUMFWFM WBSJBCMF UIFO JU NFBOT UIBU T1[T2] JT OPU DMPTFE 'SPN IFSF PO XIFOFWFS XF TQFBL PG B SVMF A1 . . . An An+1 JO B DPOTJTUFODZQSFTFSWJOH DBMDVMVT XF JNQMJDJUMZ JODMVEF UIF GPMMPXJOH WBSJBCMF DMPTVSF DPOTUSBJOUT A1 . . . An∨ κ⃗ ∨ κ⃗′. An+1  ĦBU JT A1, . . . , An+1 BSF JO UIF TDPQF PG ∨ κ⃗ BOE An+1 JT JO UIF TDPQF PG ∨ κ⃗, κ⃗′ 8F XJMM BMTP DPOTJEFS NPSF HFOFSBM DPOTJTUFODZQSFTFSWJOH SVMFT IBWJOH NVMUJQMF DPODMVTJPOT 6TJOH UIJT OPUBUJPO UIF 4LPMFNJTBUJPO SVMF EFTDSJCFE JO 4FDUJPO  JT XSJUUFO ∃x. F [x]∨ c. F [c] 8F IBWF TJNQMZ NPWFE UIF FYJTUFOUJBM RVBOUJǪFS JOUP UIF NFUB MBOHVBHF ĦF NFBO JOH PG UIF JOGFSFODF SVMF JT FYBDUMZ BT XIFO JU XBT XSJUUFO XJUIPVU VTJOH ∨  6TJOH UIF ∨ TZNCPM BMMPXT VT UP CF NPSF QSFDJTF BCPVU UIF TDPQF PG ∨ CPVOE TZNCPMT BOE UIJT XJMM CF DSVDJBM JO UIF USBOTGPSNBUJPO EFTDSJCFE JO %FǪOJUJPO  :PV NJHIU IBWF OPUJDFE UIBU UIF TJEFDPOEJUJPO PG UIF 4LPMFNJTBUJPO SVMF SF RVJSJOH c UP CF GSFTI XBT PNJUUFE ĦF TJEFDPOEJUJPO JT FTTFOUJBM GPS TPVOE SFBTPO JOH EVSJOH QSPPG TFBSDI CVU QSPPG USBOTMBUJPO EPFT OPU DIFDL GPS TPVOEOFTT "OZ VOTPVOE SFBTPOJOH XJMM CF DBVHIU BU UIF DIFDLJOH TUBHF XIFO UIF USBOTMBUFE QSPPG JT QSPDFTTFE CZ B EJǩFSFOU UPPM *O UIJT DBTF JG c XFSF OPU B GSFTI TZNCPM UIFO XIFO XF BQQMZ UIF ∨ CJOEJOH XF NJHIU IBWF DBQUVSFE GSFF WBSJBCMFT MFBEJOH UP VOTPVOE SFBTPOJOH ĦJT XJMM CF DBVHIU EVSJOH QSPPG DIFDLJOH 8F XJMM QSPWF UIBU UIF QSPPG USBOTGPSNBUJPO XF XJMM VTF XJMM QSFTFSWF UIF SFGVUBUJPO NFBOJOH PG UIF QSPPG×CPUI JG UIF QSPPG JT DPSSFDU BOE BMTP JG JU JT JODPSSFDU ĦF ∨ FOSJDIFE SVMFT BSF BQQMJFE JO UIF TBNF GBTIJPO BT UIF PSJHJOBM SVMFT UIF SVMFT BSF BQQMJFE CZ BO "51 TZTUFN VOUJM B DPOUSBEJDUJPO JT FODPVOUFSFE PS UIF DMBVTF TFU JT TBUVSBUFE &BDI UJNF XF BQQMZ B SVMF IBWJOH B ∨ JO UIF DPODMVTJPO XF TXJUDI UP TFBSDIJOH GPS B NPEFM SFMBUJWF UP BO FYUFOEFE TJHOBUVSF 8IFO UIF TDPQF PG ∨ FYUFOET BDSPTT UIF XIPMF SVMF JU NFBOT UIBU XF BSF TFBSDIJOH XJUIJO UIF TQBDF PG NPEFMT FYUFOEJOH UIF TBNF TJHOBUVSF 8F XJMM TFF BO FYBNQMF PG CPUI OFYU 8F OPX USBOTMBUF UIF TUBDLMFTT TQMJUUJOH SVMFT GSPN 4FDUJPO  JOUP UIJT OPUB UJPO ĦJT JOWPMWFT VTJOH EJǩFSFOU CJOEJOH MPDBUJPOT×JOEJDBUJOH B EJǩFSFOU TDPQF GPS UIF CPVOE TZNCPMT ĦF JOUSPEVDF EFǮOJUJPO SVMF JT XSJUUFO C ∨D∨ d. (C ∨ d) & (D ∨ ¬d) BOE UIF BQQMZ EFǮOJUJPO SVMF JT XSJUUFO σ(d ∨ C) & (C ∨D)∨ d σ(d ∨ C) & (¬d ∨D)  /PUF UIBU UIF TDPQF PG ∨ d FYUFOET POMZ BDSPTT UIF SVMFÙT DPODMVTJPO JO JOUSPEVDF EFǮOJUJPO XIFSFBT JU FYUFOET BDSPTT UIF FOUJSF SVMF JO BQQMZ EFǮOJUJPO "T NFOUJPOFE FBSMJFS JO UIF ǪSTU DBTF XF BSF NBLJOH QSFDJTF UIBU JO UIF DPODMVTJPO MJOF XF BSF MPPLJOH JO B TQBDF PG NPEFMT XIFSF d JT JOUFSQSFUFE *O UIF TFDPOE DBTF XF BSF DPOUJOVJOH UP MPPL JO B TQBDF PG NPEFMT XIFSF d JT JOUFSQSFUFE×NPSFPWFS UIF JOUFS QSFUBUJPO PG d JT OPU DIBOHJOH %FTQJUF UIF JUBMJD TDSJQU XJUIJO UIF PCKFDUGPSNVMBT UIF TZNCPM d CFIBWFT BT B DPOTUBOU 6TJOH UIF DPOTUBOU OPUBUJPO d XPVME TFFN JODPOTJTUFOU TJODF XF EP OPU VTVBMMZ CJOE DPOTUBOUT "T NFOUJPOFE FBSMJFS XF DBO ESPQ NFOUJPO PG TJEFDPOEJUJPOT XIFO TUBUJOH SVMFT JO UIF DPOUFYU PG USBOTMBUJOH QSPPGT×CVU OPU XIFO CVJMEJOH QSPPGT :PV NJHIU SFDBMM UIBU JO UIF PSJHJOBM GPSNVMBUJPO PG UIF BCPWF SVMFT XF VTFE UXP MJOFT UP TFQBSBUF UIF QSFNJTFT GSPN UIF DPODMVTJPO CVU JO UIF BCPWF WFSTJPO XF POMZ VTFE B TJOHMF MJOF " EPVCMF MJOF DG QBHF  JOEJDBUFT UIBU PQFSBUJPOBMMZ UIF SVMF EFMFUFT UIF QSFNJTFT BOE SFQMBDFT UIFN XJUI UIF DPODMVTJPO 4JODF XF IBWF BMSFBEZ BSHVFE UIBU UIF SVMFT QSFTFSWF DPOTJTUFODZ DG QBHF  XF DBO CF BTTVSFE UIBU DIBOHJOH UIF SVMFÙT PQFSBUJPOBM CFIBWJPVS UP OPU EFMFUF UIF QSFNJTFT EPFT OPU BǩFDU TPVOEOFTT PS DPNQMFUFOFTT *U EPFT BǩFDU QFSGPSNBODF EVSJOH QSPPG TFBSDI CVU XF BSF BTTVNJOH UIBU B QSPPG IBT BMSFBEZ CFFO GPVOE BOE PVS UBTL JT UP NBQ JU GSPN POF DBMDVMVT UP BOPUIFS 8F XJMM OPX EFTDSJCF TVDI B NBQQJOH 3FDBMM UIBU SFGVUBUJPO DBMDVMJ QSPWF SFTVMUT PG UIF GPSN |= φ −→ False BOE UIBU SVMFT JO TVDI DBMDVMJ BSF DPOTJTUFODZQSFTFSWJOH CVU OPU HFOFSBMMZ WBMJEJUZQSFTFSWJOH *O UIF BSHVNFOU UIBU GPMMPXT UIF QSFDJTF SVMFT PG B DBMDVMVT EP OPU BMXBZT NBUUFS× XF TUVEZ QSPQFSUJFT PG B DBMDVMVT BT B XIPMF CZ SFMZJOH PO JUT TPVOEOFTT BOE DPN QMFUFOFTT UP BTTVSF VT UIBU BOZ JODPOTJTUFOU TFU PG GPSNVMBT DBO CF SFGVUFE *O XIBU GPMMPXT MFU UIF TZNCPM C UIF 4DIXBCBDIFS MFUUFS Û$Ü SFQSFTFOU BO BS CJUSBSZ TPVOE BOE DPNQMFUF SFGVUBUJPO DBMDVMVT XIPTF JOGFSFODF SVMFT DBO CF BEF RVBUFMZ GPSNBMJTFE BT *TBCFMMF UFSNT -FU V UIF 4DIXBCBDIFS MFUUFS Û7Ü SFQSFTFOU B DBMDVMVT XIPTF SVMFT QSFTFSWF WBMJEJUZ BOE XIJDI DBO CF GPSNBMJTFE BT *TBCFMMF UFSNT *TBCFMMF)0- JT TVDI BT DBMDVMVT 8F UBLF TPVOEOFTT BOE DPNQMFUFOFTT UP CF SFMBUJWF UP FYUFOTJPOBM )FOLJO NPEFMT DG 4FDUJPO  CVU TFWFSBM PG UIF QSPQFSUJFT EFTDSJCFE CFMPX IPME JO B CSPBE DMBTT PG DMBTTJDBM MPHJDT 8F TUBSU XJUI B CBTJD MFNNB UIBU FTUBCMJTIFT B DPSSFTQPOEFODF CFUXFFO UXP GBNJMJFT PG DBMDVMJ ĦJT DPOTUSBJOU NBZ FYDMVEF FTPUFSJD GPSNBMJTBUJPOT CVU JU JT XFBL FOPVHI UP JODMVEF DBMDVMJ TVDI BT UIPTF PG -Əƙ** BOE &  -FNNB  3FGVUBUJPO DPSSFTQPOEFODF  'PS BMM TFUT PG TFOUFODFT Φ Φ ⊢C ! Jǭ ⊢V Φ −→ False 1SPPG TLFUDI ĦF TFNBOUJD FYQBOTJPOT PG Φ ⊢C ! BOE ⊢V Φ −→ False BSF JEFOUJ DBM 1SPPG JO 4FDUJPO $ ▹ 5P USBOTMBUF QSPPGT GSPN -Əƙ** PS & JOUP QSPPGT JO *TBCFMMF)0- XF XPVME POMZ VTF UIF ÛPOMZ JGÜ EJSFDUJPO PG -FNNB  #VU UIF MFNNB JT OPU EJSFDUMZ VTBCMF TJODF JU EPFT OPU TIPX VT IPX UP USBOTMBUF B QSPPG GSPN ⊢C UP ⊢V 8F OPX EFǪOF B USBOTMBUJPO UIBU NBQT JOGFSFODFT GSPN ⊢C UP ⊢V BOE UIBU XJMM CF VTFE UP NBQ QSPPGT CFUXFFO UIF UXP TZTUFNT *OUVJUJWFMZ UIF USBOTMBUJPO GPSNBMJTFT UIF DPOTJTUFODZ QSFTFSWBUJPO QSPQFSUZ PG B ⊢CSVMF XIJDI XF SFQSFTFOU VTJOH UIF HFOFSBM OPUBUJPO A1 . . . An B1 . . . Bm /PUF UIBU UIF QSFNJTFT A1 . . . An BSF JNQMJDJUMZ DPOKPJOFE XIJMF UIF DPODMVTJPOT B1 . . . Bm BSF JNQMJDJUMZ EJTKPJOFE ĦBU JT BMM UIF QSFNJTFT NVTU IPME JO B NPEFM GPS JU UP CF OFDFTTBSZ UIBU BU MFBTU POF PG UIF DPODMVTJPOT IPMET JO B NPEFM ĦF TZNCPMT Ai BOE Bj GPS FBDI 1 ≤ i ≤ n BOE 1 ≤ j ≤ m SBOHF PWFS HFOFSBM DMBVTFT BT EFTDSJCFE JO 4FDUJPO  0VS USBOTGPSNBUJPO XJMM QSPEVDF BOPUIFS SVMF XIJDI TBZT JG UIFSF JT OP NPEFM TBUJTGZJOH BU MFBTU POF PG UIF GPMMPXJOH TFUT PG DMBVTFT {A1, . . . , An, B1} {A1, . . . , An, B2}  {A1, . . . , An, Bm} UIFO UIFSF JT OP NPEFM TBUJTGZJOH A1, . . . , An ĦJT GPMMPXT IPX B SFGVUBUJPOCBTFE "51 TZTUFN PQFSBUFT FYUFOEJOH B TFU PG DMBVTFT XJUI UIFJS EFSJWBUJPOT JO B DPOTJTUFODZ QSFTFSWJOH XBZ ĦF USBOTGPSNFE SVMF JT BENJTTJCMF JO⊢V JF JU JT WBMJEJUZQSFTFSWJOH  ĦJT JT BMTP UIF JOGPSNBM BSHVNFOU PG -FNNB  XIJDI WBMJEBUFT UIJT USBOTGPS NBUJPO %FǪOJUJPO  $POUSBOFHBUJPO  (JWFO B SVMF r A1 . . . An∨ κ⃗ ∨ κ⃗1.B1 . . . ∨ κ⃗m.Bm  XIFSF FBDI ∨ κ⃗, κ⃗i.Bi JT DMPTFE NPEVMP XF USBOTGPSN JU JOUP r⊥ A1 . . . An ∧ κ⃗, κ⃗1. ⎡⎢⎢⎣ A1, . . . , An, B1  False ⎤⎥⎥⎦ . . . ∧ κ⃗, κ⃗m. ⎡⎢⎢⎣ A1, . . . , An, Bm  False ⎤⎥⎥⎦∧ κ⃗ False *O %FǪOJUJPO  UIF SFTVMU r⊥ JT DMPTFE NPEVMP "MTP TJODF Ø ∧ Ù JT B CJOEFS OFTUFE CJOEJOHT DBO BMXBZT CF SFOBNFE UP BWPJE WBSJBCMF DBQUVSF -FU Σ CF UIF TFU PG DPOTUBOUT JO UIF TJHOBUVSF 8F DBO BMXBZT FOTVSF UIBU Σ∩ κ = ∅ Σ∩ κi = ∅ BOE κ ∩ κi = ∅ -FNNB  "TTVNJOH ⊢C UP CF TPVOE BOE ⊢V UP CF DPNQMFUF UIFO GPS FBDI JOGFSFODF r ∈ ⊢C r JT DPOTJTUFODZQSFTFSWJOH Jǭ r⊥ JT WBMJEJUZQSFTFSWJOH 1SPPG TLFUDI 6TJOH TFNBOUJD SFBTPOJOH XF DBO TIPX r BOE r⊥ UP CF FRVJWBMFOU XJUIJO B SFGVUBUJPO TFUUJOH QSPWJEFE UIBU r JT TPVOE 1SPPG JO 4FDUJPO $ ▹ -FU UIF TZNCPM π SBOHF PWFS QSPPGT 8F XSJUF π ∈ ⊢C JG π JT FODPEFE JO B DPOTJTUFODZQSFTFSWJOH DBMDVMVT 1SPPGT DPOTJTU PG ǪOJUF DIBJOT PG JOTUBODFT PG JOGFS FODF SVMFT (r1, . . . , rn) JO UIBU DBMDVMVT 8F XSJUF Φ ⊢C,π F UP NFBO UIBU GSPN UIF TFU PG TFOUFODFT Φ XF DBO EFSJWF TFOUFODF F VTJOH DBMDVMVT ⊢C BOE UIBU π JT TVDI B EFSJWBUJPO /PUF UIBU JO PVS TFUUJOH C ∈ {C,V} 8F DBO OPX QSPWF B DPOTUSVDUJWF WFSTJPO PG -FNNB  " QSPPG π DBO CF NBQQFE JOUP π⊥ CZ NBQQJOH FBDI SVMF r FMFNFOUXJTF UP r⊥ -FNNB  3FGVUBUJPO DPSSFTQPOEFODF Ö DPOTUSVDUJWF  'PS FWFSZ QSPPG π ∈ ⊢C BOE GPS FWFSZ TFU PG TFOUFODFT Φ XF IBWF Φ ⊢C,π ! Jǭ ⊢V,π⊥ Φ −→ False 1SPPG TLFUDI *OEVDUJPO PO EFSJWBUJPOT π 1SPPG JO 4FDUJPO $ ▹ 4USVDUVSBMMZ %FǪOJUJPO  NBQT ⊢CTUZMF TZTUFNT JOUP MFĤIBOEFE UBCMFBV TZT UFNT UIF JOGFSFODFT PG XIJDI DBO CF BENJUUFE JO ⊢VTUZMF TZTUFNT *O PVS TFUUJOH UIF UBCMFBV TZTUFN JT TIBMMPXMZ FNCFEEFE JO *TBCFMMF)0- ĦF SFDPOTUSVDUFE GSBH NFOU PG UIF TFBSDI TQBDF GPSNT B UBCMFBV CSBODI UIJT CSBODIFT GVSUIFS XIFO EJT KVODUJPOT BSF QSPDFTTFE XJUI FBDI EJTKVODU GPSNJOH B OFX CSBODI "T JT OPSNBM XJUI UBCMFBV TZTUFNT UIF QSFDFEJOH FMFNFOUT PG UIF CSBODI BSF TIBSFE CZ BMM TVCTF RVFOU CSBODIFT  8IFO XF SFDPOTUSVDU B ⊢CSFGVUBUJPO JOUP B ⊢VQSPPG WJB %FǪOJUJPO  BOE -FNNB  XF BSF CFJOH MFTT FǬDJFOU UIBO XIFO VTJOH B UBCMFBV DBMDVMVT EJSFDUMZ CFDBVTF UIF MBUUFS EFMFUFT SFEVOEBOU GPSNVMBT GSPN UIF CSBODIFT 4QFDJǪDBMMZ XIFO BO αSVMF JT BQQMJFE UIF PSJHJOBM DPOKVODUJPO JT EFMFUFE /PUF UIBU XF POMZ EFMFUF B GPSNVMB XIFO BO FRVJWBMFOU TFU PG GPSNVMBT JT BEEFE UP UIF CSBODI BOZUIJOH XFBLFS XPVME HFOFSBMMZ JOUSPEVDF JODPNQMFUFOFTT *U TIPVME CF QPTTJCMF UP NPEJGZ %FǪOJUJPO  BOE TVCTFRVFOUMZ -FNNB  UP BMMPX UIJT EFMFUJPO ĦJT JT BO PQUJNJTBUJPO XIJDI NBLFT UIF SFDPOTUSVDUJPO CFIBWF MJLF B UBCMFBV QSPPG CVU XF EP OPU QVSTVF UIJT IFSF ĦF TQBDF DPNQMFYJUZ PG UIJT NFUIPE JT RVBESBUJD JO UIF TJ[F PG UIF QSPPG *G |Φ| JT UIF TJ[F PG UIF JOJUJBM TFU PG GPSNVMBT UIBU XFSF QSPWFE UP CF JODPOTJTUFOU CZ UIF "51 TZTUFN BOE π JT UIF QSPPG QSPEVDFE CZ UIF "51 TZTUFN UIFO UIF USBOTMBUFE QSPPG XJMM IBWF TJ[F n∑ i=1 ( |Φ|+ i∑ j=1 |πj| ) XIFSF n JT UIF OVNCFS PG JOGFSFODFT JO π BOE |πj| JT UIF TJ[F PG JOGFSFODF j JO π JF UIF TJ[F PG JUT IZQPUIFTJT BOE DPODMVTJPO GPSNVMBT  8F HFU UIJT TQBDF CFIBWJPVS TJODF UIF USBOTMBUJPO DPQJFT UIF FOUJSF CSBODI VQ UP UIBU QPJOU JO UIF QSPPG BU FBDI TUFQ JO UIF QSPPG ĦJT DBO CF TFFO GSPN %FǪOJUJPO  BOE BMTP GSPN UIF FYBN QMF JNQMFNFOUBUJPO JO 4FDUJPO  ĦF UJNF DPNQMFYJUZ GPS USBOTMBUJPO JT BMTP RVBESBUJD UIF BMHPSJUIN XJUIJO UIF QSPPG PG -FNNB  POMZ TUJUDIFT UPHFUIFS UIF USBOTMBUFE JOGFSFODFT ĦJT DBO BMTP CF TFFO JO UIF USBOTMBUFE FYBNQMFT JO 4FD UJPO  /PUF UIBU UIFTF BSF UIF DPNQMFYJUZ FTUJNBUFT GPS USBOTMBUJOH B QSPPG DIFDL JOH B QSPPG DBO CF TVCKFDU UP EJǩFSFOU CPVOET JG TFBSDI PS VOJǪDBUJPO BSF JOWPMWFE  *TBCFMMF FODPEJOH PG UBCMFBV FNCFEEJOHT ĦJT TFDUJPO GPDVTFT PO TQFDJǪD JOTUBODFT PG -FNNB  UP BSHVF UIBU QSPPGT GSPN CPUI -Əƙ** BOE & DBO CF FNCFEEFE JO *TBCFMMF)0- "T QBSU PG UIJT BSHVNFOU XF XJMM BMTP MPPL BU FYBNQMFT PG DPOTJTUFODZQSFTFSWJOH SVMFT GSPN UIF DBMDVMJ PG -Əƙ** BOE & BOE FYQMBJO IPX UIFZ BSF USBOTGPSNFE CZ %FǪOJUJPO  " QSPUP UZQF JNQMFNFOUBUJPO PG UIJT USBOTGPSNBUJPO JT EFTDSJCFE JO 4FDUJPO  JODMVEJOH FYBNQMF USBOTMBUJPOT PG & QSPPGT   -Əƙ** *O QSJODJQMF XF TIPVME CF BCMF UP NBQ BOZ QSPPG FODPEFE JO -Əƙ**ÙT DBMDVMVT JOUP UIBU PG *TBCFMMF)0- $PSPMMBSZ  *G -ƓƝ** JT TPVOE BOE *TBCFMMF)0- JT DPNQMFUF XJUI SFTQFDU UP )FOLJO NPEFMT UIFO FWFSZ UIFPSFN QSPWFE CZ -ƓƝ** JT BMTP B UIFPSFN PG *TBCFMMF)0- 1SPPG TLFUDIĦF QSPPG GPMMPXT GSPN -FNNB  UIF DPOUFOU PG XIJDI EFTDSJCFT IPX UP NBQ B -Əƙ** QSPPG JOUP BO *TBCFMMF)0- QSPPG ▹ -FNNB  JT QBSBNFUSJD JO UIF QSPPG DBMDVMJ VTFE 8F DBO BMTP WFSJGZ UIF MFNNBÙT VTF CZ DIFDLJOH UIBU GPS FBDI PG UIF SVMFT JO -Əƙ**ÙT DBMDVMVT UIF USBOTGPS NBUJPO ZJFMET *TBCFMMF)0-BENJTTJCMF SVMFT ĦJT JT EPOF JO 4FDUJPO $ 8F OPX MPPL BU TPNF SVMFT GSPN -Əƙ**ÙT DBMDVMVT BOE UIF GPSNBMJTBUJPO PG UIFJS UBCMFBV FODPEJOH JO *TBCFMMF)0- BT NFUBUIFPSFNT .FUBUIFPSFNT BSF FTTFOUJBMMZ BENJTTJCMF SVMFT 8F IBWF DIFDLFE UIBU UIF DPOUSBOFHBUJPO r⊥ PG FBDI -Əƙ** SVMF r JT WBMJE JO *TBCFMMF)0- ĦJT JT EFTDSJCFE JO -FNNB $ BOE JO BO *TBCFMMF)0- TDSJQU UIBU JT JODMVEFE JO 4FDUJPO # 'VSUIFSNPSF 4FDUJPO # QSPWJEFT BO FYBNQMF QSPPG UIBU XBT USBOTMBUFE VTJOH UIJT NFUIPE 3FTPMVUJPO -FUC,D CF NFUBWBSJBCMFT SBOHJOH PWFS -Əƙ** TVCDMBVTFT BOE A,B CF NFUBWBSJBCMFT SBOHJOH PWFS GPSNVMBT " TVCDMBVTF JT TJNQMZ B DPMMFDUJPO PG GPS NVMBT UIBU DBO GPSN B DMBVTF -Əƙ**ÙT SFTPMVUJPO SVMF { C,+ A } { D,− B } { C,D,− A = B } XIPTF USBOTGPSNFE %FǪOJUJPO  GPSN JT C ∨ + A D ∨ − B C ∨ + A D ∨ − B C ∨D ∨ − A = B  False False  DBO CF FODPEFE JO *TBCFMMF)0- BT⎡⎢⎢⎢⎢⎢⎢⎣ C ∨ + A , D ∨ − B ,⎡⎢⎣ C ∨ + A , D ∨ − B , C ∨D ∨ − A = B ⎤⎥⎦ =⇒ False ⎤⎥⎥⎥⎥⎥⎥⎦ =⇒ False. "T CFGPSF UIF -Əƙ** MJUFSBM + A JT FODPEFE JO *TBCFMMF)0- BT A = True BOE UIF MJUFSBM − A JT FODPEFE JO *TBCFMMF)0- BT A = False -Əƙ** TVC DMBVTFT BSF FODPEFE BT EJTKVODUJWF )0- GPSNVMBT /PUF UIBU UIF SVMF DBO CF TJNQMJǪFE UP SFNPWF SFEVOEBOU QSFNJTFT UP HJWF⎡⎢⎢⎣ C ∨ + A , D ∨ − B ,[ C ∨D ∨ − A = B ] =⇒ False ⎤⎥⎥⎦ =⇒ False. *O XIBU GPMMPXT XF XJMM OPU HFOFSBMMZ DBSSZ PVU UIJT TJNQMJǪDBUJPO TUFQ 4JNQMJǪDB UJPO JT QPTTJCMF VOMFTT UIFSF JT B ∧ CJOEJOH PDDVSSJOH JOTJEF UIF SVMF "O FYBNQMF PG UIJT JT HJWFO GVSUIFS EPXO PO QBHF  /PUF UIBU UIF MBDL PG TJNQMJǪDBUJPO BMTP BǩFDUT UIF TQBDF DPNQMFYJUZ PG UIF USBOTMBUJPO BT EFTDSJCFE JO UIF QSFWJPVT TFDUJPO 4LPMFNJTBUJPO ĦF -Əƙ** 4LPMFNJTBUJPO SVMF { C,− ∀A } sk GSFTI DPOTUBOU{ C,− A sk } BOE XIJDI DBO CF SFXSJUUFO VTJOH PVS ∨ OPUBUJPO BT { C,− ∀A } ∨ sk . { C,− A sk } BOE XIPTF USBOTGPSNFE %FǪOJUJPO  GPSN JT  "T NFOUJPOFE FBSMJFS PO QBHF  XF DBO ESPQ NFOUJPO PG TJEFDPOEJUJPOT XIFO TUBUJOH SVMFT JO UIF DPOUFYU PG USBOTMBUJOH QSPPGT  C ∨ − ∀A ∧ sk . ⎡⎢⎢⎣ C ∨ − ∀A , C ∨ − A sk  False ⎤⎥⎥⎦ False DBO CF FODPEFE JO *TBCFMMF)0- BT⎡⎢⎣ C ∨ − ∀A ,∧ sk . [ C ∨ − ∀A , C ∨ − A sk ] =⇒ False ⎤⎥⎦ =⇒ False. ĦF ∨ CPVOE sk JO UIF ⊢CTUZMF DBMDVMVT CFDPNFT ∧ CPVOE XIFO NBQQFE JOUP UIF ⊢V DBMDVMVT ĦJT JT EFUBJMFE JO UIF QSPPG PG -FNNB  *OUVJUJWFMZ UIF ∨ CPVOE sk DBO EFOPUF BOZ FMFNFOU PG UIF EPNBJO UIBU DBO TBUJTGZ UIF DMBVTF TFU XIFSFBT UIF ∧ CPVOE sk JT VTFE JO B TFUUJOH XIFSF BOZ EFOPUBUJPO PG sk NVTU VMUJNBUFMZ SFTVMU JO B SFGVUBUJPO 4QMJUUJOH 3FDBMM UIBU -Əƙ** JNQMFNFOUT TQMJUUJOH UISPVHI TFBSDITQBDF QBSUJUJPO JOH EFTDSJCFE JO 4FDUJPO  ĦF TQMJUUJOH SVMF IBT UXP DPODMVTJPOT CPUI PG XIJDI BSF VMUJNBUFMZ FYQFDUFE UP EFSJWF False UIVT SFGVUJOH UIF PSJHJOBM EJTKVODUJPO )BWJOH B SVMF XJUI NVMUJQMF DPODMVTJPOT JT OPU B QSPCMFN TJODF %FǪOJUJPO  IBOEMFT TVDI SVMFT 8IFO USBOTGPSNFE UIJT SVMF CFDPNFT + A ∨ B ⎡⎢⎢⎣ + A ∨B , + A  False ⎤⎥⎥⎦ ⎡⎢⎢⎣ + A ∨ B , + B  False ⎤⎥⎥⎦ False BOE XIJDI DBO CF FODPEFE JO *TBCFMMF BT⎡⎢⎢⎢⎣ + A ∨ B ,[ + A ∨ B ,+ A ] =⇒ False,[ + A ∨ B ,+ B ] =⇒ False ⎤⎥⎥⎥⎦ =⇒ False. "T PCTFSWFE CFGPSF XF DBO TJNQMJGZ UIJT GPSNVMB UP SFNPWF UIF SFEVOEBOU+ A ∨ B JO UIF JOOFS SVMFT ĦJT XJMM ZJFME B TQFDJBMJTBUJPO PG UIF EJTKVODUJPOFMJNJOBUJPO  SVMF GSPN UIF /BUVSBM %FEVDUJPO DBMDVMVT CVU TQFDJBMJTFE UP IBWF UIF DPODMVTJPO GPSNVMB False ĦJT XJMM CF SFWJTJUFE JO 4FDUJPO   & *O QSJODJQMF XF TIPVME CF BCMF UP NBQ BOZ QSPPG FODPEFE JO &ÙT DBMDVMVT JOUP UIBU PG *TBCFMMF)0- $PSPMMBSZ  *G & JT TPVOE BOE *TBCFMMF)0- JT DPNQMFUF XJUI SFTQFDU UP )FOLJO NPE FMT UIFO FWFSZ UIFPSFN QSPWFE CZ & JT BMTP B UIFPSFN PG *TBCFMMF)0- 1SPPG TLFUDIĦF QSPPG GPMMPXT GSPN -FNNB  UIF DPOUFOU PG XIJDI EFTDSJCFT IPX UP NBQ BO & QSPPG JOUP BO *TBCFMMF)0- QSPPG ▹ "T XJUI -Əƙ** JO UIF QSFWJPVT TFDUJPO XF OPX MPPL BU TPNF SVMFT GSPN &ÙT DBMDVMVT BOE UIF GPSNBMJTBUJPO PG UIFJS UBCMFBV FODPEJOH JO *TBCFMMF)0- BT NFUB UIFPSFNT 0SEFSJOH DSJUFSJB BSF B LFZ GFBUVSF PG QSPPG TFBSDI JO TVQFSQPTJUJPO DBM DVMJ TVDI BT UIBU JNQMFNFOUFE CZ & ĦJT JT OPU B DPODFSO JO UIF TUZMF PG QSPPG USBOTMBUJPO TUVEJFE IFSF TJODF XF BSF GPDVTTJOH PO QSPPGT OPU PO QSPPG TFBSDI UIFSF GPSF XF DBO JHOPSF UIF TJEFDPOEJUJPOT SFMBUJOH UP PSEFSJOH 4QMJUUJOH & TQMJUT B EJTKVODUJPO CZ JOUSPEVDJOH OFX TZNCPMT BT EFTDSJCFE JO 4FD UJPO  ĦF USBOTGPSNFE SVMF JT FODPEFE JO *TBCFMMF BT⎡⎢⎢⎢⎢⎣ C ∨D, ∧ d. ⎡⎢⎣ C ∨D,C ∨ d, D ∨ ¬d ⎤⎥⎦ =⇒ False ⎤⎥⎥⎥⎥⎦ =⇒ False. ĦF BQQMZ EFǮOJUJPO SVMF JT USBOTGPSNFE BT GPMMPXT ∧ d. ⎡⎢⎢⎢⎢⎢⎢⎣ d ∨C, C ∨D, ∧ d. ⎡⎢⎣ d ∨C,C ∨D, ¬d ∨D ⎤⎥⎦ =⇒ False ⎤⎥⎥⎥⎥⎥⎥⎦ =⇒ False. #PUI TDIFNFT XFSF WFSJǪFE JO *TBCFMMF)0- UP CF WBMJEJUZQSFTFSWJOH /PUF UIBU XF DPVME TJNQMJGZ UIF GPSNVMB CZ SFNPWJOH UIF SFEVOEBOUC∨D GSPN UIF JOOFS SVMF  CVU XF DBOOPU SFNPWF FJUIFS PG UIF DMBVTFT NFOUJPOJOH d JO UIF JOOFS SVMF TJODF UIF TZNCPM d JT CFJOH CPVOE CZ UIBU JOOFS SVMF $MBVTJǪDBUJPO &ÙT DMBVTJǪDBUJPO JODMVEJOH 4LPMFNJTBUJPO IBQQFOT EVSJOH QSF QSPDFTTJOH 4DIVM[  g  ĦJT JT VOMJLF JO -Əƙ** XIFSF DMBVTJǪDBUJPO IBQ QFOT EVSJOH UIF NBJO MPPQ ĦF SFBTPO GPS UIJT JT UIBU JO )0- B OPSNBM JF DMBVTJǪFE GPSNVMB MJLF x : o ∨ ¬ y : o DBO CFDPNF OPOOPSNBM GPS JOTUBODF CZ JOTUBOUJBUJOH x UP ∀z. z ∧ y ĦJT LJOE PG JOTUBOUJBUJPO DBOOPU UBLF QMBDF JO '0- &ÙT DMBVTJǪDBUJPO QSPDFEVSF JT CBTFE PO UIBU EFTDSJCFE CZ /POOFOHBSU BOE 8FJ EFOCBDI  CVU IFSF XF UBSHFU UIF TJNQMFS QSPDFEVSF EFTDSJCFE JO 4FDUJPO  BOE IBOEMF 4LPMFNJTBUJPO BT JO -Əƙ** BCPWF 4DIFNBUJDBMMZ UIF USBOTGPSNBUJPO EFTDSJCFE JO 4FDUJPO  UBLFT B GPSNVMB F BOE SFUVSOT B DPOKVODUJPO T(F ) VTJOH B DPMMFDUJPO x⃗ PG GSFTI OBNFT F∨ x⃗. T(F ) 4DIFNBUJDBMMZ UIJT JT USBOTGPSNFE CZ %FǪOJUJPO  BT GPMMPXT⎡⎢⎣ F,∧ x⃗. [ F, T(F ) ] =⇒ False ⎤⎥⎦ =⇒ False -FU VT VTF A ∨ ((B −→ C) ∧D) BT BO FYBNQMF GPSNVMB $POKPJOJOH UIF DPO UFOUT PG UIF TFU T A ∨ ((B −→ C) ∧D) HJWFT x1 ∧ (x1 = (A ∨ x2)) ∧ (x2 = (x3 ∧D)) ∧ (x3 = (B −→ C)) 'PS UIF TBLF PG SFBEBCJMJUZ XF XJMM OPU FYQBOE UIJT JOUP B DPOKVODUJPO PG EJTKVOD UJPOT CVU UIF FYQMBOBUJPO JO 4FDUJPO  TIPXT IPX UP EP UIJT "QQMZJOH %FǪOJ UJPO  XF HFU⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ A ∨ ((B −→ C) ∧D), ∧ x1 x2 x3. ⎡⎢⎢⎢⎢⎢⎢⎣ A ∨ ((B −→ C) ∧D), x1 ∧ (x1 = (A ∨ x2)) ∧ (x2 = (x3 ∧D)) ∧ (x3 = (B −→ C)) ⎤⎥⎥⎥⎥⎥⎥⎦ =⇒ False ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ =⇒ False  XIJDI DBO CF TIPXO UP CF WBMJE  1SPUPUZQF JNQMFNFOUBUJPO *O BEEJUJPO UP TQBDF DPNQMFYJUZ UIFSF JT TPNF JNQMFNFOUBUJPOBM DPNQMFYJUZ JO DVSSFE XIFO JNQMFNFOUJOH UIF QSFWJPVT UFDIOJRVF JO *TBCFMMF /PSNBMMZ "51 TZTUFNT VTF EBUB TUSVDUVSFT BMMPXJOH UIFN UP BEESFTT TQFDJǪD DMBVTFT CVU *TBCFMMF OBUJWFMZ POMZ BMMPXT UIF BEESFTTJOH PG TVCHPBMT×OPU IZQPUIFTFT ĦJT NFBOT UIBU QJDLJOH UIF DMBVTFT UP XIJDI XF XBOU UP BQQMZ BO JOGFSFODF JT OPU B TJNQMF NBUUFS XF NJHIU OFFE UP SFPSEFS UIF DMBVTFT NBLJOH VQ UIF IZQPUIFTFT TJODF XF DBO OPU QPJOU UP UIF SFMFWBOU DMBVTFT EJSFDUMZ 8PSLJOH UISPVHI UIF TFDPOE FYBNQMF JO 4FDUJPO # NBLFT UIJT EJǬDVMUZ BQQSFDJBCMF )BWJOH GFXFS JUFNT JO UIF TFU PG IZQPUIFTFT NBLFT JU FBTJFS UP NBOBHF UIBU TFU NBLJOH UIJT TFU TNBMMFS IBT CFFO TVHHFTUFE BCPWF BT B QPTTJCMF JNQSPWFNFOU 'PSUVOBUFMZ *TBCFMMF BMTP TVQQPSUT GPSXBSE QSPPG×UIBU JT QSPPGT CVJMU CZ DPN CJOJOH FYJTUJOH GBDUT SBUIFS UIBO CZ SFEVDJOH GPSNVMBT JOUP BYJPNT ĦJT TUZMF PG QSPPG BMMPXT IZQPUIFTJT BEESFTTJOH 1SPPGT DBO CF XSJUUFO JO UIJT TUZMF VTJOH UIF *TBS MBOHVBHF 8FO[FM  PS UIF VOEFSMZJOH "1* ĦJT TUZMF PG QSPPG MFOET JU TFMG NPSF UP JNQMFNFOUJOH UIF NFUIPE EFTDSJCFE JO UIJT TFDUJPO DPNQBSFE UP UIF CBDLDIBJOJOH TUZMF PG QSPPG UIBU XBT VTFE JO FYQFSJNFOUT "O FYBNQMF PG TVDI BO FYQFSJNFOU JT TIPXO JO 4FDUJPO # ĦF QSPPG NBQQJOH EFTDSJCFE JO UIJT TFDUJPO XBT JNQMFNFOUFE BT B QSPUPUZQF GPS NBQQJOH QSPPGT QSPEVDFE CZ & JOUP *TBS TDSJQUT *TBCFMMF)0- JT UIFO VTFE UP JNQPSU BOE DIFDL UIFTF QSPPGT BĤFS XIJDI UIFZ DBO CF VTFE JO *TBCFMMF)0- EFWFMPQNFOUT ĦF QSPUPUZQF IBT CFFO NBEF BWBJMBCMF POMJOF /PUF UIBU JO &ÙT DBMDVMVT QSPPGT EP OPU CSBODI TJODF TQMJUUJOH EPFT OPU QBSUJ UJPO UIF TFBSDI TQBDF MJLF JO -Əƙ** #PUI TQMJUUJOH BQQSPBDIFT XFSF EFTDSJCFE JO 4FDUJPO  4JODF QSPPGT EP OPU CSBODI BO & QSPPG DBO CF SFHBSEFE BT B USBDF UISPVHI B HSPXJOH TFU PG DMBVTFT EFSJWFE GSPN UIF PSJHJOBM TFU PG DMBVTFT DPOTJTU JOH PG UIF BYJPNT BOE UIF DPOKFDUVSF TVDI UIBU DPOTJTUFODZ JT QSFTFSWFE 4JODF &ÙT QSPPGT BSF MJOFBS JO UIJT SFTQFDU UIF USBOTMBUJPO XJMM ZJFME B MJOFBS TFSJFT PG JOGFS FODFT JO *TBCFMMF)0- 8F XJMM MPPL BU FYBNQMFT PG UIJT CFMPX ĦF QSPUPUZQF JT JNQMFNFOUFE BT B RVJDL DIFDL PO UIJT NFUIPE $VSSFOUMZ JU IBT OPU CFFO UFTUFE PO B MBSHF TDBMF BOE EPFT OPU IBOEMF ∧ CJOEJOHT ?iiT,ffrrrX;Bi?m#X+QKfMBFbmfT+2BH  8F OPX MPPL BU UXP FYBNQMF QSPPGT USBOTMBUFE CZ UIF QSPUPUZQF ĦF 5151 QSPCMFN 4:/  DPOTJTUT PG UIF DPOKFDUVSF ¬(∃Y. pY ) −→ ∀Y. (∃X. pX) −→ pY &ÙT QSPPG GPS UIJT DPOKFDUVSF JT TIPXO JO 'JHVSF  ĦF SFTVMUJOH *TBS TDSJQU GSPN USBOTMBUJPO CZ UIF QSPUPUZQF JT TIPXO JO 'JHVSF  8F OPX BOBMZTF UIF USBOTMBUFE QSPPG JO 'JHVSF  UP SFMBUF JU UP UIF QSPPG NBQ QJOH EFTDSJCFE FBSMJFS ĦF USBOTMBUFE QSPPG DPOTJTUT PG GPVS QBSUT  1SPCMFN TQFDJǪDBUJPO BOE TUBSU PG QSPPG CZ DPOUSBEJDUJPO MJOFT  1SPPGT BMXBZT TUBSU JO UIF TBNF NBOOFS XIJDI JOWPMWFT TUBUJOH UIF DPOKFDUVSF TUBU JOH UIBU UIF QSPPG XJMM QSPDFFE CZ DPOUSBEJDUJPO BOE OBNJOH UIF OFHBUFE DPOKFDUVSF GPSNVMB GPS MBUFS SFGFSFODF  4QFDJǪDBUJPO PG JOGFSFODF DPODMVTJPOT MJOFT  ĦJT JOWPMWFT VTJOH B IBOEZ NBDSPEFǪOJUJPO EFWJDF UP TIPSUFO UIF MJOFT NBLJOH VQ UIF OFYU TUFQ  5SBOTMBUFE QSPPG MJOFT  ĦJT DPOTJTUT PG UIF PSJHJOBM QSPPGÙT JOGFSFODFT USBOTMBUFE JO UIF NBOOFS EFTDSJCFE FBSMJFS %FǪOJUJPO   ĦFTF JOGFSFODFT BSF DIBJOFE UPHFUIFS UP GPSN UIF USBOTMBUFE QSPPG /PUF UIBU XF BSF VTJOH UIF CMBTU UBDUJD UP WBMJEBUF FBDI TUFQ ĦJT UBDUJD JNQMFNFOUT B UBCMFBVCBTFE QSPPG TFBSDI BOE JUT EFWFMPQNFOU JT EFTDSJCFE JO 4FDUJPO  /PUF IPX UIF ǪSTU GBDU \`R PO MJOF  JT B UBVUPMPHZ &TTFOUJBMMZ JU JT TVC TVNFE CZ UIF UBVUPMPHZ False =⇒ False POF PG JUT IZQPUIFTFT \+nynN JT BO BMJBT PG False "MM QSPPGT USBOTMBUFE VTJOH UIJT NFUIPE TUBSU XJUI TVDI B UBV UPMPHZ BOE QSPDFFE UISPVHI B TFSJFT PG WBMJEJUZQSFTFSWJOH TUFQT UP EFSJWF B DPOUSBEJDUJPO GSPN UIF OFHBUFE DPOKFDUVSF  %FSJWBUJPO PG False GSPN UIF OFHBUFE DPOKFDUVSF MJOF  ĦF PCTFSWBUJPO JO QPJOU  BCPWF JT DFOUSBM UP IPX UIJT NFUIPE XPSLT "MM UIF QSPPGT USBOTMBUFE VTJOH UIJT NFUIPE XJMM IBWF UBVUPMPHJFT BU UIF MFBWFT PG UIF QSPPG USFF BOE XJMM HSBEVBMMZ QSPDFFE CZ WBMJEJUZQSFTFSWJOH TUFQT GSPN UIPTF UBVUPMPHJFT UP EFSJWF B DPOUSBEJDUJPO GSPN UIF OFHBUFE DPOKFDUVSF   7Q7U+nyny- +QMD2+im`2- UUU\(sR),TUsRVV4=5(sR),U\(sk),TUskV4=TUsRVVVV-  7BH2U^fhShS@p8X9XyfS`Q#H2KbfauLfauLNkNYRXT^- T`Qp2ni?BbVVX  7Q7U+nynR- M2;i2/n+QMD2+im`2- UUUU\(sR),TUsRVV4=  5(sR),U\(sk),TUskV4=TUsRVVVVV-  BM72`2M+2UbbmK2nM2;iBQM-(biimbU+i?V)-(+nyny)VVX  7Q7U+nynk- M2;i2/n+QMD2+im`2- U5(sj),UTUsjVUTU2bFRnyVTU2bFknyVVVV-  BM72`2M+2Ub?B7in[mMiQ`b-(biimbUi?KV)-  (BM72`2M+2UbFQH2KBx2-(biimbU2bV)-  (BM72`2M+2Up`B#H2n`2MK2-(biimbUi?KV)-  (BM72`2M+2Ub?B7in[mMiQ`b-(biimbUi?KV)-  (BM72`2M+2U7Q7nMM7-(biimbUi?KV)-(+nynR)V)V)V)V)VVX  +M7U+nynj-M2;i2/n+QMD2+im`2-UTU2bFRnyVV-  BM72`2M+2UbTHBin+QMDmM+i-(biimbUi?KV)-(+nynk)VVX  +M7U+nyn9-M2;i2/n+QMD2+im`2-UTUsRVV-  BM72`2M+2UbTHBin+QMDmM+i-(biimbUi?KV)-(+nynk)VVX  +M7U+nyn8-M2;i2/n+QMD2+im`2-UTU2bFRnyVV- +nynjVX  +M7U+nyne-M2;i2/n+QMD2+im`2-UTUsRVV- +nyn9VX  +M7U+nynd-M2;i2/n+QMD2+im`2-UTU2bFRnyVV- +nyn8VX  +M7U+nyn3-M2;i2/n+QMD2+im`2-UTUsRVV- +nyneVX  +M7U+nynN-M2;i2/n+QMD2+im`2-U07Hb2V-  BM72`2M+2Ub`-(biimbUi?KV)-(+nynd- +nyn3- i?2Q`vU2[mHBivV)V-  (^T`QQ7^)VX 'JHVSF  5151FODPEFE QSPPG QSPEVDFE CZ & GPS UIF 5151 QSPCMFN 4:/    H2KK  b?Qrb ]UUUU\ sR X TUsRVVV@@=U5 sR X UU\ sk X TUskVV@@=TUsRVVVVV]  T`QQ7 U`mH2 ++QMi`V  bbmK2 +nyny , ]UUUUU\ sR X TUsRVVV@@=  U5 sR X UU\ sk X TUskVV@@=TUsRVVVVVV]  b?Qr ]6Hb2]  T`QQ7 @  H2i \+nynR 4 ]UUUUU\ sR X TUsRVVV@@=  U5 sR X UU\ sk X TUskVV@@=TUsRVVVVVV]  H2i \+nynk 4 ]UU5 sj X UTUsjVUTU2bFRnyVTU2bFknyVVVVV]  H2i \+nynj 4 ]UTU2bFRnyVV]  H2i \+nyn9 4 ]UTUsRVV]  H2i \+nyn8 4 ]UTU2bFRnyVV]  H2i \+nyne 4 ]UTUsRVV]  H2i \+nynd 4 ]UTU2bFRnyVV]  H2i \+nyn3 4 ]UTUsRVV]  H2i \+nynN 4 ]U6Hb2V]  ?p2 `R , ](%\+nynRc \+nynkc \+nynjc \+nyn9c \+nyn8c \+nynec \+nyndc  \+nyn3c \+nynN%) 44= 6Hb2] #v #Hbi  7`QK `R ?p2 `k , ](%\+nynRc \+nynkc \+nynjc \+nyn9c \+nyn8c \+nynec  \+nyndc \+nyn3%) 44= 6Hb2] #v #Hbi  7`QK `k ?p2 `j , ](%\+nynRc \+nynkc \+nynjc \+nyn9c \+nyn8c \+nynec  \+nynd%) 44= 6Hb2] #v #Hbi  7`QK `j ?p2 `9 , ](%\+nynRc \+nynkc \+nynjc \+nyn9c \+nyn8c  \+nyne%) 44= 6Hb2] #v #Hbi  7`QK `9 ?p2 `8 , ](%\+nynRc \+nynkc \+nynjc \+nyn9c  \+nyn8%) 44= 6Hb2] #v #Hbi  7`QK `8 ?p2 `e , ](%\+nynRc \+nynkc \+nynjc \+nyn9%) 44= 6Hb2]  #v #Hbi  7`QK `e ?p2 `d , ](%\+nynRc \+nynkc \+nynj%) 44= 6Hb2] #v #Hbi  7`QK `d ?p2 `3 , ](%\+nynRc \+nynk%) 44= 6Hb2] #v #Hbi  7`QK `3 ?p2 `N , ](%\+nynR%) 44= 6Hb2] #v #Hbi  7`QK +nyny b?Qr \i?2bBb #v U`mH2 `NV  [2/  [2/ 'JHVSF  5SBOTMBUJPO JOUP *TBS PG UIF 5151 QSPPG GSPN 'JHVSF   8F OPX MPPL BU B TMJHIUMZ NPSF DPNQMFY QSPCMFN ĦF 5151 QSPCMFN 48$  JT TQFDJǪFE BT GPMMPXT /PUF UIBU UIF QSPCMFNÙT PSJHJOBM GPSNVMBUJPO BMTP JODMVEFT WBSJPVT BYJPNT CVU OPOF PG UIFN JT OFFEFE UP QSPWF UIF DPOKFDUVSF 7Q7U+QR-+QMD2+im`2- U 5 (l) , U bbGBbiUlV 4= 5 (o) , U bbGBbiUoV 4= 5 (q) , U bbGBbiUqV 4= 5 (s) , U bbGBbiUsV 4= U o 54 s % l 54 q %  M2[Uo-MBHV % M2[Us-MBHV V V V V V VVX 1BSU PG JUT USBOTMBUFE QSPPG JT TIPXO JO 'JHVSF  XIJDI IBT B TJNJMBS TUSVDUVSF UP UIF QSFWJPVT QSPPG XF BOBMZTFE ĦF BCCSFWJBUJPO \+nynRR JO 'JHVSF  JT VTFE UP QSPEVDF UIF UBVUPMPHZ BU UIF TUBSU PG UIF WBMJEJUZQSFTFSWJOH TFRVFODF PG JOGFSFODFT  *NQMJDBUJWF FNCFEEJOH ĦF FNCFEEJOH EFTDSJCFE JO UIF QSFWJPVT TFDUJPO JT BQQFBMJOH TJODF JU JOWPMWFT BQ QMZJOH B TJOHMF USBOTGPSNBUJPO VOJGPSNMZ UP BMM JOGFSFODFT ĦJT GBDJMJUBUFT JUT JN QMFNFOUBUJPO 0O UIF PUIFS IBOE UIF SFTVMUJOH USBOTMBUJPO JT OPU FǬDJFOU JO JUT DVSSFOU GPSN *U TVǩFST JO QSBDUJDF CFDBVTF JU EPFT OPU EFMFUF TVCTVNFE GPSNVMBT *O UIJT TFDUJPO XF MPPL BU B EJǩFSFOU BQQSPBDI UIBU XPSLT CFTU JO DBMDVMJ IBWJOH NPTUMZ WBMJEJUZQSFTFSWJOH JOGFSFODFT *OTQFDUJOH -Əƙ**ÙT DBMDVMVT SFWFBMT UIBU POMZ UXP SVMFT BSF OPU WBMJEJUZ QSFTFSWJOH 4LPMFNJTBUJPO BOE TQMJUUJOH .PTU PG UIF SVMFT JO &ÙT DBMDVMVT BSF WBMJEJUZQSFTFSWJOH UPP *G BO JOGFSFODF r ∈ ⊢C JT WBMJEJUZQSFTFSWJOH UIFO JU TIPVME CF BENJTTJCMF JO ⊢V 8F TIPVME CF BCMF UP FODPEF FWFSZ TVDI SVMF EJSFDUMZ JO *TBCFMMF)0- BOE QSPWF JU UP CF B NFUBUIFPSFN 1FSIBQT TVDI BO BSSBOHFNFOU JT OPU TVǬDJFOUMZ XPSUIZ PG   H2KK  b?Qrb ]UU5 sR X UUbbGBbi sRV@@=U5 sk X UUbbGBbi skV@@=  U5 sj X UUbbGBbi sjV@@=  U5 s9 X UUbbGBbi s9V@@=  UUUsk4s9%sR4sjV%UUM2[ sk MBHVVV%UM2[ s9 MBHVVVVVVVVVVV]  T`QQ7 U`mH2 ++QMi`V  bbmK2 +nyny , ]UUU5 sR X UUbbGBbi sRV@@=U5 sk X UUbbGBbi skV@@=  U5 sj X UUbbGBbi sjV@@=U5 s9 X UUbbGBbi s9V@@=  UUUsk4s9%sR4sjV%UUM2[ sk MBHVVV%UM2[ s9 MBHVVVVVVVVVVVV]  b?Qr ]6Hb2]  T`QQ7 @  H2i \+nynR 4 ]UUU5 sR X UUbbGBbi sRV@@=U5 sk X UUbbGBbi skV@@=  U5 sj X UUbbGBbi sjV@@=U5 s9 X UUbbGBbi s9V@@=  UUUsk4s9%sR4sjV%UM2[ sk MBHVV%UM2[ s9 MBHVVVVVVVVVVVV]  H2i \+nynk 4 ]UUUbbGBbi 2bF93nyVUUbbGBbi 2bF9NnyVUUbbGBbi 2bF8ynyV  UUbbGBbi 2bF8RnyVUUU2bF9Nny42bF8Rny2bF93ny42bF8ynyV  UM2[ 2bF9Nny MBHVV  UM2[ 2bF8Rny MBHVVVVVVV]  H2i \+nynj 4 ]U2bF9Nny42bF8RnyV]  H2i \+nyn9 4 ]UUM2[ 2bF8Rny MBHVV]  H2i \+nyn8 4 ]UUM2[ 2bF9Nny MBHVV]  H2i \+nyne 4 ]U2bF8Rny42bF9NnyV]  H2i \+nynd 4 ]UUM2[ 2bF8Rny MBHVV]  H2i \+nyn3 4 ]UUM2[ 2bF9Nny MBHVV]  H2i \+nynN 4 ]U2bF9Nny42bF8RnyV]  H2i \+nynRy 4 ]UUM2[ 2bF8Rny MBHVV]  H2i \+nynRR 4 ]U6Hb2V]  (XXX)  7`QK `Ry ?p2 `RR , ](%\+nynR%) 44= 6Hb2] #v #Hbi  7`QK +nyny b?Qr \i?2bBb #v U`mH2 `RRV  [2/  [2/ 'JHVSF  1BSU PG UIF USBOTMBUJPO JOUP *TBS PG &ÙT 5151 QSPPG GPS 48$  ĦF USBOTMBUFE JOGFSFODFT IBWF CFFO FMJEFE VOEFS UIF TZNCPM <á>  UIF OBNF ØFNCFEEJOHÙ CVU GPS UIF TBLF PG DMBTTJǪDBUJPO XF DBO DPOTJEFS JU BT TVDI -FU VT DBMM UIJT FNCFEEJOH JNQMJDBUJWF CFDBVTF JU SFUBJOT UIF JNQMJDBUJWF OBUVSF PG NPTU PG UIF JOGFSFODF SVMFT ĦJT TUZMF PG FNCFEEJOH PǩFST BQQFBMJOH GFBUVSFT à 8F DBO GPSNBMJTF FBDI WBMJEJUZQSFTFSWJOH r BT B TDIFNF WBMJEBUF JU JO *T BCFMMF)0- QSPWJOH JU UP CF B NFUBUIFPSFN UIFO JOTUBOUJBUF JU UP PCUBJO UIF QSFDJTF JOTUBODF PG r VTFE CZ -Əƙ** $MBVTFT JO -Əƙ** BSF HSPVOE UIFSF GPSF XF EP OPU IBWF GSFF VOJWFSTBM WBSJBCMFT BU UIF MFWFM PG JOGFSFODFT à 4JODF XF EP OPU IBWF GSFF VOJWFSTBM WBSJBCMFT XF DBO BWPJE VTJOH MPHJDBM WBSJ BCMFT XIFO DIBJOJOH SFDPOTUSVDUFE JOGFSFODFT ĦJT JT EFTJSBCMF TJODF IBO EMJOH TVDI WBSJBCMFT SFRVJSFT VT UP SFMZ PO VOJǪDBUJPO XIJDI JT OPU FBTZ UP DPOUSPM PS UVOF .PSFPWFS UIF TDPQF PG TVDI WBSJBCMFT TQBOT BDSPTT CSBODIFT DSFBUJOH EFQFOEFODJFT à "MM JOGFSFODF SVMFT DBO CF SFDPOTUSVDUFE TFQBSBUFMZ JO JTPMBUJPO UIFO CBDL DIBJOFE UPHFUIFS *OEFFE UIF SFDPOTUSVDUJPO PG TFQBSBUF JOGFSFODFT DBO CF QBSBMMFMJTFE TJODF BT NFOUJPOFE BCPWF UIFZ BSF 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KV )- (3 )V VX  i? 7U RR -T H BM -U UU UT Qr 2` b2 i! bE Rn V 4 UT Qr 2` b2 i! bE kn au kV V4 07 H b2 VV -  BM 72 `2 M+ 2U 2t i+ M7 nM Qi nT Qb -( bi i mb Ui ?K V) -( Ry )V VX  i? 7U Rk -T H BM -U UU 07 H b2 V4 0i `m 2V V-  BM 72 `2 M+ 2U 7Q n iT n2 -( bi i mb Ui ?K V) -( N- RR )V VX  i? 7U Rj -T H BM -U 07 H b2 V-  BM 72 `2 M+ 2U bQ Hp 2/ n HH nb TH Bi b- (b QH p2 /n H Hn bT HB ib UD QB M- () V) -( Rk )V VX 'J HV SF   -Ə ƙ **Ù TQ SP PG PG 4& 6  ?   4P GBS XF IBWF QSPDFTTFE UIF MPHJDBM TJHOBUVSF BOE UIF TUSVDUVSF PG UIF QSPPG 8F OPX UVSO PVS BUUFOUJPO UP UIF JOGFSFODFT UIFNTFMWFT ĦF JOGFSFODFT BSF JOUFSQSFUFE SFMBUJWF UP UIF UBSHFU DBMDVMVT UP ZJFME NFUBUIFPSFNT /PUF UIBU UIJT JT EJǩFSFOU GSPN UIF JOUFSQSFUBUJPO PG GPSNVMBT GSPN UIF TPVSDF MPHJD 5P BWPJE DPOGVTJPO XF VTF UIF XPSE FNVMBUJPO UP NFBO UIF JOUFSQSFUBUJPO PG JOGFSFODFT &NVMBUJPO NJHIU CF JNQMFNFOUFE VTJOH TDIFNFT PS UBDUJDT PS DPVME SFMZ PO SFǪOEJOH JOTUFBE 'PS JOTUBODF UIF JOGFSFODF EFTDSJCFE JO MJOF  PG 'JHVSF  DBO CF FNVMBUFE CZ UIF *TBCFMMF)0- NFUBUIFPSFN (sK1A = sK2SY2) = True ∧ (powerset sK1A = powerset sK2SY2) = False False = True XIFSF BT TQFDJǪFE JO UIF 5151 FODPEJOH PG UIF QSPPG UIF ǪSTU QSFNJTF JT DPO USJCVUFE GSPN UIF DPODMVTJPO PG UIF JOGFSFODF MBCFMMFE  PDDVSSJOH JO MJOF  BOE UIF TFDPOE QSFNJTF GSPN UIF JOGFSFODF MBCFMMFE  MJOF   "T TIPXO BCPWF XF DPOKPJO UIF QSFNJTFT UP GPSN B TJOHMF QSFNJTF ĦJT FOTVSFT UIBU BMM JOGFSFODFT IBWF B TJNQMF GPSN DPOTJTUJOH PG B TJOHMF QSFNJTF BOE B TJOHMF DPODMVTJPO ĦF QSPPG UFYU BMTP JOEJDBUFT UIBU UIJT JOGFSFODF XBT NBEF VTJOH & ĦF SFTVMUJOH NFUBUIFPSFN JO *TBCFMMF)0- JT MBCFMMFE  DPOTJTUFOU XJUI UIF OBNF VTFE JO UIF 5151 FODPEJOH PG UIF QSPPG &NVMBUJPO JT EFTDSJCFE JO NPSF EFUBJM JO 4FDUJPO  4P GBS XF IBWF JNQPSUFE BMM PG UIF DPOTUJUVFOU JOGPSNBUJPO GSPN UIF TPVSDF MPHJD JOUP UIF UBSHFU MPHJD 8F OPX OFFE UP DPNCJOF UIF JOGPSNBUJPO UP SFDPOTUSVDU UIF UIFPSFN JO UIF UBSHFU MPHJD ĦF QSPPG HSBQI JT USBWFSTFE UP QSPEVDF B USBDF PG UIF QPTTJCMZ USBOTGPSNFE QSPPG 'PMMPXJOH UIJT USBDF TIPVME BMMPX VT UP SFDPO TUSVDU UIF QSPPG "T XJMM CF EFTDSJCFE MBUFS GPMMPXJOH UIJT USBDF EPFT OPU BMXBZT MFBE UP B TVDDFTTGVM SFDPOTUSVDUJPO 1FSIBQT UIJT USBDF DPVME CF USFBUFE TPNFXIBU BT B QSPPG QMBO #VOEZ   1SPPG QMBOOJOH JT UIF BQQMJDBUJPO PG QMBOOJOH UP QSPPGTFBSDI 1MBOOJOH JOWPMWFT ǪOEJOH B TPMVUJPO JO B TQBDF PG DBOEJEBUFT HJWFO DPOTUSBJOUT EFTDSJCFE JO UFSNT PG B TFU PG PQFSBUPST *O QSPPG QMBOOJOH PQFSBUPST BSF DBMMFE NFUIPET .FUIPET DPOUBJO UBDUJDT XIJDI JO UVSO VTF MPXMFWFM MPHJDBM JO GFSFODFT UP CVJME QSPPGT *O UIJT TFUUJOH USBWFSTJOH UIF QSPPG HSBQI ZJFMET B QMBO PG IPX UP CVJME UIF QSPPG JO UIF UBSHFU TZTUFN ĦF QMBO JT BMNPTU BMXBZT VOGBJMJOH BOE JU TFFNT NPSF BDDVSBUF UP SFGFS UP JU BT B QSPHSBN 'PS UIF FYBNQMF QSPPG BCPWF UIF QSPHSBN DPOTJTUT PG  JOTUSVDUJPOT  (ai2T ]Rj]- ai2T ]Rk]- lM+QMDQBM-  ai2T ]RR]- ai2T ]Ry]- ai2T ]3]- ai2T ]d]-   ai2T ]8]- ai2T ]9]- ai2T ]j]- bbmK2/-  ai2T ]N]- ai2T ]e]-  ai2T ]8]- ai2T ]9]- ai2T ]j]- bbmK2/-  *#QQb2) ĦJT QSPHSBN XJMM CF JOUFSQSFUFE OFYU "U UIF TUBSU PG UIF QSPHSBN UIF DPOKFDUVSF JT JNQMJDJUMZ EPVCMF OFHBUFE ĦBU JT JO *TBCFMMF OPUBUJPO UIF QSPHSBN JO UIF BCPWF FYBNQMF TFUT PVU UP QSPWF (∀A,B. A = B −→ powersetA = powersetB) = False =⇒ False. ĦJT JT PVS QSPPG HPBM BOE JU DPSSFTQPOET UP UIF OFHBUFE DPOKFDUVSF 8F BMXBZT TUBSU XJUI B TJOHMF HPBM BOE SFǪOF JU JOUP TVCHPBMT BT UIF QSPPG QSPHSFTTFT "T XF XJMM TFF CFMPX UIF OVNCFS PG HPBMT DBO WBSZ EVSJOH UIF QSPPG ĦF QSPPG JT DPNQMFUFE XIFO BMM HPBMT IBWF CFFO EJTDIBSHFE /PUF UIBU UIF DPOKFDUVSFE GPSNVMB DBO CF FBTJMZ GPVOE JO UIF 5151 QSPPG TJODF JU JT MBCFMMFE XJUI UIF DPOKFDUVSF SPMF DG 4FDUJPO   *O PVS FYBNQMF UIF DPOKFDUVSF JT TIPXO PO MJOF  8F OPX EFTDSJCF UIF TFNBOUJDT PG UIF DPNNBOET BQQFBSJOH JO UIF QSPHSBN TIPXO FBSMJFS à Ø4UFQ *%Ù BQQMJFT UIF FNVMBUFE JOGFSFODF MBCFMMFE XJUI *% 'PS FYBNQMF XF TBX UIF JOGFSFODF GPS TUFQ  BCPWF XIJDI SFMJFE PO UIF DPODMVTJPOT PG TUFQT  BOE  ĦF FNVMBUJPO PG JOGFSFODFT XJMM CF EFTDSJCFE JO 4FDUJPO  à 6ODPOKPJO TJNQMZ BQQMJFT DPOKVODUJPO FMJNJOBUJPO ĦJT JT OFFEFE TJODF UIF QSPPG HSBQI JT OPU B USFF JO HFOFSBM×JU SFDPNCJOFT 'PS FYBNQMF UIJT DPN NBOE JT BQQMJFE UP TUFQ  GPSNBMJTFE JO *TBCFMMF)0- FBSMJFS UP CSFBL VQ UIF DPOKPJOFE QSFNJTFT JOUP UXP QSFNJTFT ĦJT TUFQ DPOTVNFT POF HPBM BOE QSPEVDFT UXP HPBMT &BDI HPBM SFMBUFT UP B QBUI UP UIF SPPU PG UIF HSBQI ĦF SPPU PG UIF HSBQI DPOTJTUT PG UIF DPOKFDUVSF GPSNVMB à "TTVNFE TJNQMZ BQQMJFT BO JOTUBODF PG UIF BTTVNQUJPO TDIFNF A =⇒ A UP EJTDIBSHF B HPBM 'PS FYBNQMF JO UIF QSPHSBN TIPXO BCPWF CPUI VTFT PG UIJT DPNNBOE BQQMZ UIF UBVUPMPHZ (∀A,B. A = B −→ powerset A = powerset B) = False =⇒ (∀A,B. A = B −→ powerset A = powerset B) = False. 8F XJMM VTF UIF UFSNT HPBM BOE TVCHPBM JOUFSDIBOHFBCMZ  à $BCPPTF UFSNJOBUFT UIF QSPPG *U TVDDFFET JG UIFSF BSF OP NPSF TVCHPBMT " DBCPPTF JT UIF MBTU DBSSJBHF JO B USBJO BOE IFSF JU TIPVME CF UIF MBTU JOTUSVDUJPO JO B QSPHSBN /PUF UIBU JO UIF QSPHSBN TIPXO FBSMJFS MJOFT  BOE  BSF EVQMJDBUFT 8F DPVME FYUSBDU B MFNNB UIBU GVTFT UPHFUIFS UIF DPOUFOUT PG MJOF  JOUP BOPUIFS BENJTTBCMF SVMF UIFO SFQMBDF MJOF  XJUI BO BQQMJDBUJPO PG UIJT MFNNB ĦJT BOE UIF NPEFM XF VTF GPS TVDI QSPHSBNT JT EFTDSJCFE JO 4FDUJPO  'JOBMMZ XF FYFDVUF UIJT QSPHSBN PO UIF EPVCMFOFHBUFE DPOKFDUVSF *G BMM FNVMB UJPO TUFQT BSF TVDDFTTGVM UIFO UIF FYFDVUJPO TIPVME ZJFME BO *TBCFMMF)0- UIFPSFN DPSSFTQPOEJOH UP UIF UIFPSFN QSPWFE CZ -Əƙ**  1BSTJOH BOE JOUFSQSFUBUJPO ĦF JNQMFNFOUBUJPO PG UIJT GSBNFXPSL VTFT UIF *TBCFMMFJOUFHSBUFE 5151 QBSTFS BOE JOUFSQSFUFS EFWFMPQFE EVSJOH QSFWJPVT XPSL 4VMUBOB FU BM   *U JT DPOWFOJFOU UIBU CPUI 5151FODPEFE QSPCMFNT BOE QSPPGT TIBSF UIF TBNF TZOUBY JO 5151 UIF UXP LJOET PG GPSNBM PCKFDUT EJǩFS NBJOMZ JO UIF BOOPUBUJPOT HJWFO UP GPSNVMBT ĦFTF BOOPUBUJPOT DPOUBJO JOGPSNBUJPO SFMBUFE UP UIF JOGFSFODF NBEF CZ B UIFPSFN QSPWFS ĦF QBSTFS DMPTFMZ GPMMPXT UIF HSBNNBS TQFDJǪDBUJPO PG UIF 5151 MBOHVBHF UP QSPEVDF TFRVFODFT PG QBSTFE "OOPUBUFE 'PSNVMBT "'T XFSF EFTDSJCFE JO 4FD UJPO   3FDBMM UIBU BO "' DBO DPOUBJO BEEJUJPOBM JOGPSNBUJPO UP BO FODPEJOH PG B GPSNVMB "OOPUBUJPO JOGPSNBUJPO JT WFSZ VTFGVM XIFO UIF 5151 ǪMF FODPEFT B QSPPG CVU JU JT IBSEMZ FWFS QSFTFOU XIFO UIF 5151 ǪMF FODPEFT B QSPCMFN ĦVT XIFO XF BSF JNQPSUJOH B 5151 QSPCMFN JO *TBCFMMF)0- XF POMZ DBSF BCPVU JOUFS QSFUJOH GPSNVMBT 8IFO JNQPSUJOH QSPPGT XF BEEJUJPOBMMZ JOUFSQSFU JOGFSFODFT×UIJT XJMM CF EFTDSJCFE JO 4FDUJPO  3FDBMM UIBU XF XJMM VTF UIF XPSE FNVMBUJPO UP SFGFS UP UIF JOUFSQSFUBUJPO PG JOGFSFODFT BOE XF XJMM SFTFSWF UIF XPSE JOUFSQSFUBUJPO UP NFBO UIF JOUFSQSFUBUJPO PG GPSNVMBT ĦFSF JT B DPSF TJHOBUVSF UIBU JT BMSFBEZ JOUFSQSFUFE JO *TBCFMMF)0- DPOTJTUJOH PG UIF CBTF UZQFT ι, o EFOPUJOH JOEJWJEVBMT BOE #PPMFBO QSPQPTJUJPOT SFTQFDUJWFMZ BOE UIF UZQF PG GVODUJPOT τ → τ ′ GPS BMM UZQFT τ, τ ′ ĦF TJHOBUVSF BMTP JOUFSQSFUT UIF DPOTUBOUT True, False,¬,∧,∨,−→ BOE ∀τ , ∃τ ,=τ , ετ GPS BMM UZQFT τ  ĦJT DPSF TJHOBUVSF JT FYUFOEFE XJUI UIF TJHOBUVSF JOGPSNBUJPO GSPN UIF 5151 ǪMF EFTDSJCJOH UIF UZQFT BOE DPOTUBOUT VTFE JO UIF FODPEFE QSPCMFN PS QSPPG 5ZQF  8A : ◆, B : ◆. (A = B) ! (powerset A = powerset B) (8A : ◆, B : ◆. (A = B) ! (powerset A = powerset B)) = False (8A : ◆, B : ◆. (A = B) ! (powerset A = powerset B)) = False (8SY2 : ◆. (sKA = SY2) ! (powerset sKA = powerset SY2)) = False ((sKA = sKSY2) ! (powerset sKA = powerset sKSY2)) = False (sKA = sKSY2) = True (powerset sKA = powerset sKSY2) = False (¬(powerset sKA = powerset sKSY2)) = True (sKA = sKSY2) = True (¬(powerset sKA = powerset sKSY2)) = True (powerset sKA = powerset sKSY2) = False False = True False negate conjecture unfold def extcnf forall neg extcnf forall neg standard cnf standard cnf clause copy polarity switch clause copy extcnf not pos fo atp e solved all splits 'JHVSF  (SBQI GPS -Əƙ**ÙT QSPPG PG 4&6?  JOGPSNBUJPO JT OPU JODMVEFE JO VOUZQFE 5151 MBOHVBHFT TVDI BT '0' BOE $/' 8IFO BWBJMBCMF XF VTF UIF UZQF JOGPSNBUJPO PUIFSXJTF XF UZQF BMM BUPNMFWFM UFSNT BT QSPQPTJUJPOT 4VCBUPN MFWFM nBSZ DPOTUBOUT XIFSF n ≥ 0 BSF HJWFO UIF nBSZ GVODUJPO UZQF PWFS ι ĦBU JT UIFZ BSF HJWFO UIF UZQF ι JG n = 0 UIF UZQF ι → ι JG n = 1 ι → ι → ι JG n = 2 BOE TP PO 4JODF UIF VOUZQFE MBOHVBHFT BSF ǪSTUPSEFS BMM DPOTUBOU GVODUJPOT BQQFBS GVMMZBQQMJFE  1SPPG USBOTGPSNBUJPOT *O UIJT TFDUJPO XF MPPL BU UIF BOBMZTJT BOE USBOTGPSNBUJPO PG QSPPGT GPDVTTJOH PO QSPPGT QSPEVDFE CZ -Əƙ** UP GBDJMJUBUF USBOTMBUJOH UIF QSPPGT JOUP *TBCFMMF)0- ĦFSF BSF UISFF QSPPG USBOTGPSNBUJPOT UIBU XFSF GPVOE UP CF VTFGVM GPS QSPDFTTJOH -Əƙ** QSPPGT  &MJNJOBUJOH SFEVOEBOU QBSUT PG UIF QSPPG 0DDBTJPOBMMZ -Əƙ** JODMVEFT SFEVO EBOU DIBJOT PG JOGFSFODFT UIBU EP OPU DPOUSJCVUF UP UIF SFGVUBUJPO 'JHVSF  TIPXT BO FYBNQMF PG B QSPPG XJUI SFEVOEBOU JOGFSFODFT *O UIBU HSBQI OPUF UIBU UIFSF BSF UXP DMBVTFT QSPEVDFE CZ UIF TUBOEBSE@DOG JOGFSFODF PO UIF GPSNVMB (∀Xp : ι→ o. Xp y −→ ∀Xx. cR Xx ∧ cQ y) = False 0OF PG JUT EFSJWBUJWFT (∀Xp : ι → o. Xp y) = True JT DPQJFE UXJDF *UT MFĤTJEF EFSJWBUJWF SFBDIFT B DPOUSBEJDUJPO JOEFQFOEFOUMZ PG JUT PUIFS EFSJWB UJWF .PSFPWFS JUT SJHIUTJEF EFSJWBUJWF SFMJFT PO BOPUIFS CSBODI PG JOGFSFODFT CVU UIJT JT SFEVOEBOU TJODF XF DBO EJSFDUMZ SFGVUF JUT EFSJWBUJWF ∀SV8 : ι → o. SV8 y = True XJUIPVU VTJOH UIF DMBVTFT DPOUSJCVUFE CZ UIF SJHIUNPTU CSBODI 8F XPVME TJNQMZ VTF UIF CJOEJOH SV8 ,→ λ@.False. 8F DBO ǪOE BOE FMJNJOBUF SFEVOEBOU QBSUT PG QSPPGT CZ ǪOEJOH UIF TIPSUFTU QBUI GSPN UIF DPODMVTJPO PG UIF QSPPG UP UIF DPOKFDUVSF ĦJT NVTU IFFE TQMJUUJOH×JU NVTU LFFQ BMM QBUIT UIBU EFSJWF GSPN B TQMJU TJODF FBDI PG UIFN  (8Xp : ◆ ! o. (Xp y) ! 8Xx : ◆. (cR Xx) ^ cQ y) (8Xp : ◆ ! o. (Xp y) ! 8Xx : ◆. (cR Xx) ^ cQ y) = False (8Xp : ◆ ! o. (Xp y) ! 8Xx : ◆. (cR Xx) ^ cQ y) = False (8Xp : ◆ ! o. Xp y) = True (8Xx : ◆. (cR Xx) ^ cQ y) = False cQ y = False (¬ cQ y) = True (8Xp : ◆ ! o. Xp y) = True 8SV1 : ◆ ! o. SV1 y = True False = True False = True (8Xp : ◆ ! o. Xp y) = True 8SV8 : ◆ ! o. SV8 y = True (¬cQ y) = True cQ y = False False = True False negate conjecture unfold def standard cnf clause copy extcnf forall pos standard cnf split conjecture polarity switch clause copy extcnf not pos fo atp e clause copy extcnf forall pos prim subst sim solved all splits 'JHVSF  (SBQI GPS -Əƙ**ÙT QSPPG PG 4:0?  NVTU CF SFGVUFE JOEFQFOEFOUMZ *O UIF FYBNQMF BCPWF XF LFFQ UIF MFĤNPTU QBUI PG UIF HSBQI EJTDBSEJOH UIF PUIFS UXP QBUIT  &YUSBDUJOH TVCQSPPGT SFMBUFE UP TQMJUUJOH "T EFTDSJCFE JO 4FDUJPO  FBDI TVC QSPPG PG B TQMJU ZJFMET B SFGVUBUJPO ĦF TFU PG OPEFT HFOFSBUFE EVSJOH B TVC QSPPG PG B TQMJU JT EJTKPJOU GSPN UIF OPEFT GSPN PUIFS TQMJUT &BDI TVCQSPPG JT VTFE UP DPOTUSVDU B MFNNB UIBU QSPEVDFT B QSFNJTF UP UIF EJTKVODUJPO FMJNJOBUJPO SVMF "O FYBNQMF PG TVDI B QSPPG DBO CF TFFO JO 'JHVSF   4FQBSBUJOH JOTUBOUJBUJPO GSPN PUIFS JOGFSFODFT -Əƙ** TPNFUJNFT PWFSMPBET JO GFSFODFT XJUI JOTUBOUJBUJPO PG DMBVTFMFWFM VOJWFSTBM WBSJBCMFT ĦJT NBLFT JU IBSEFS UP FNVMBUF BO JOGFSFODF 8F DPVME TUSFOHUIFO UIF FNVMBUJPO PG FBDI JOGFSFODF SVMF UP CF XBSZ PG BOZ BEEJUJPOBM JOTUBOUJBUJPO XIJDI NJHIU IBWF UBLFO QMBDF *OTUFBE PG DPNQMJ DBUJOH UIF CFIBWJPVS PG FNVMBUJPO XF DBO VTF B HMPCBM USBOTGPSNBUJPO UP TFQBSBUF PVU JOTUBOUJBUJPO GSPN JOGFSFODFT ĦJT XPVME BMMPX VT UP IBOEMF UIF UXP LJOET PG JOGFSFODF TFQBSBUFMZ ĦF UXP LJOET PG JOGFSFODF SFGFSSFE UP IFSF BSF UIF JOGFSFODFT UIBU TPMFMZ DBSSZ PVU JOTUBOUJBUJPO BOE BMM PUIFS JOGFSFODFT ĦJT USBOTGPSNBUJPO JT VTFE JO PVS JNQMFNFOUBUJPO 4FQBSBUJOH JOTUBOUJBUJPO GSPN PUIFS JOGFSFODFT DSFBUFT BEEJUJPOBM OPEFT JO UIF QSPPG HSBQI×UIF OFX OPEFT DPSSFTQPOE UP UIF JOTUBOUJBUJPO TUFQT ĦF JOTUBOUJBUJPO TUFQT EP OPU BQQFBS FYQMJDJUMZ BT JOGFSFODFT JO -Əƙ**ÙT JNQMF NFOUFE DBMDVMVT TP XF BEEFE B OFX JOUFSOBM JOGFSFODF SVMF UP FODPEF UIJT LJOE PG JOGFSFODF ĦJT SVMF JT DBMMFE CJOE UP NBUDI UIF BOOPUBUJPO VTFE CZ -Əƙ** UP JOEJDBUF TVDI JOTUBOUJBUJPOT " VTFGVM BOBMZTJT UP DBSSZ PVU PO -Əƙ** QSPPGT JOWPMWFT EJTDPWFSJOH 4LPMFNJTB UJPO FRVBUJPOT EFTDSJCFE JO 4FDUJPO   ĦJT JOWPMWFT BOBMZTJOH UIF TZOUBY PG 4LPMFNJTBUJPO TUFQT UP QSPEVDF 4LPMFNJTBUJPO FRVBUJPOT BOE BEEJOH UIFTF BT BY JPNT UP UIF UIFPSZ 0OF NJHIU GFFM KVTUJǪBCMZ TRVFBNJTI BCPVU BEEJOH BYJPNT UP UIF UIFPSZ CVU UIF BYJPNT DPODFSOFE IFSF BSF EFǪOJUJPOBM BYJPNT GPS 4LPMFN DPO TUBOUT /PUF UIBU -Əƙ** QSPPGT HFOFSBMMZ DPOUBJO UIF EFDMBSBUJPOT PG 4LPMFN DPO TUBOUT BT DBO CF TFFO JO 'JHVSF  CVU OPU UIFJS EFǪOJUJPOT ĦFTF EFǪOJUJPOT BSF JNQMJDJU JO UIF QSPPG TP UIJT BOBMZTJT TUFQ FYUSBDUT UIFTF EFǪOJUJPOT UP DPNQMFUF UIF TQFDJǪDBUJPO PG UIF UIFPSZ JO XIJDI UIF QSPPG JT CFJOH DBSSJFE PVU  ĦF MJTU PG PQFSBUJPOT PO QSPPGT EFTDSJCFE JO UIJT TFDUJPO JT OPU FYIBVTUJWF× NPSF DBO CF EPOF 'PS JOTUBODF UIF QSPPG TIPXO JO 'JHVSF  JOWPMWFE TQMJUUJOH CVU -Əƙ** VTFE BO JEFOUJDBM TVCQSPPG GPS CPUI TJEFT PG UIF TQMJU 'JOEJOH JEFOUJDBM TVCQSPPGT BOE SFVTJOH QSFWJPVTMZDPOTUSVDUFE UBDUJDT GPS UIFN XPVME JNQSPWF UIF QFSGPSNBODF PG SFDPOTUSVDUJPO "EEJUJPOBMMZ NPSF BHHSFTTJWF TJNQMJǪDBUJPO PG QSPPGT DPVME CF EPOF UP ǪMUFS PVU OPOQSPEVDUJWF JOGFSFODFT 'PS JOTUBODF JO 'JHVSF  UIF TFRVFODF PG JOGFSFODFT VOGPME@EFG DMBVTF@DPQZ BOE FYUDOG@OPU@QPT VOEPFT UIF FǩFDU PG QPMBSJUZ@TXJUDI "MTP UIF JOGFSFODF CFJOH EPOF CZ TPMWFE@BMM@TQMJUT JT USJWJBM ĦF QSPPG DPVME CF SFEVDFE GSPN OJOF JOGFSFODFT UP GPVS  )ZCSJE QSPPGT 8F DBO HFOFSBMJTF USBOTMBUJOH CFUXFFO UXP QSPPG MBOHVBHFT UP MPPL BU USBOTMBUJOH CFUXFFO B IZCSJE QSPPG MBOHVBHF BOE B UBSHFU QSPPG MBOHVBHF ĦF QSPPGT PG B IZCSJE QSPPG MBOHVBHF NBZ DPOUBJO TVCQSPPGT FODPEFE JO EJǩFSFOU MPHJDT ĦJT JT SFMBUFE UP CPSSPXJOH $FSJPMJ BOE .FTFHVFS  XIFSF UIF UPPMT GPS B MPHJD BSF VTFE UP TPMWF QSPCMFNT JO BOPUIFS MPHJD ĦJT JT VTVBMMZ EPOF GPS UIF MBUUFS UP CFOFǪU GSPN UIF JNQMFNFOUBUJPO PG UIF GPSNFS 'PS JOTUBODF UIJT UFDIOJRVF JT VTFE GPS NPEBM GPSNVMBT CZ )VTUBEU BOE 4DINJEU  CZ USBOTMBUJOH UIFN JOUP '0- ĦF DPNCJOBUJPO PG -Əƙ** BOE & JT BOPUIFS FYBNQMF PG CPSSPXJOH -Əƙ** EFMF HBUFT UP & UIF UBTL PG SFGVUJOH UIF DPOTJTUFODZ PG B TFU PG ǪSTUPSEFS GPSNVMBT ĦFTF GPSNVMBT BSF DPNQVUFE CZ -Əƙ** VTJOH JUT QSPPG DBMDVMVT BOE BSF EFSJWFE GSPN UIF QSPCMFN HJWFO UP -Əƙ** ĦVT QSPPG PCMJHBUJPOT BSF GPSNFE CZ -Əƙ** BOE FODPEFE UP &ÙT JOQVU MBOHVBHF GPS QSPWJOH -FU VT EFǪOF SFMZJOH MPHJD UP NFBO UIF TPVSDF MPHJD CPSSPXJOH PUIFS MPHJDT 'PS FYBNQMF JO UIF -Əƙ** & DPNCJOBUJPO -Əƙ**ÙT JT UIF SFMZJOH MPHJD BOE CPUI -Əƙ**ÙT BOE &ÙT BSF UIF TPVSDF MPHJDT *O UIF QSPPG SFUVSOFE CZ -Əƙ** &ÙT DPOUSJCVUJPO UP UIF QSPPG JT BDLOPXMFEHFE JO UXP XBZT FJUIFS BT B DPNQPVOE JOGFSFODF GP@BUQ@F BT DBO CF TFFO JO 'JHVSF  GPS FYBNQMF PS BT BO FYQBOTJPO PG UIJT JOGFSFODF JOUP UIF QSJNJUJWF SVMFT VTFE CZ & 5SBOTMBUJOH CFUXFFO UXP QSPPG MBOHVBHFT UIF TPVSDF BOE UBSHFU MPHJDT JT BO JOTUBODF PG UIJT NPSF HFOFSBM BQQSPBDI 8IFO USBOTMBUJOH CFUXFFO UXP QSPPG MBO HVBHFT UIF SFMZJOH MPHJD JT UIF POMZ TPVSDF MPHJD #VU BT XF DBO TFF GSPN -Əƙ** & UIFSF DPVME CF NVMUJQMF TPVSDF MPHJDT VTFE JO UIF TPVSDF QSPPG×UIVT GPSNJOH B  8Xx : a, Xy : a. Xy = Xy (8Xx : a, Xy : a. Xy = Xy) = False (8Xx : a, Xy : a. Xy = Xy) = False (8Xy : a. Xy = Xy) = False (sK2Xy = sK2Xy) = False ¬(sK2Xy = sK2Xy) = True ¬(sK2Xy = sK2Xy) = True (sK2Xy = sK2Xy) = False False = True False negate conjecture unfold def extcnf forall neg extcnf forall neg polarity switch clause copy extcnf not pos fo atp e solved all splits 'JHVSF  (SBQI GPS -Əƙ**ÙT QSPPG PG 4&7?  IZCSJE QSPPG )ZCSJE QSPPGT DPVME CF DSVEFMZ UVSOFE JOUP TJOHMFMPHJD QSPPGT CZ FMJEJOH UIF TVCQSPPGT PG BMM MPHJDT FYDFQU GPS UIF SFMZJOH MPHJD VOEFS B TJOHMF JOGFS FODF 'PS FYBNQMF -Əƙ** BMMPXT IJEJOH &ÙT JOGFSFODFT BT B TJOHMF -Əƙ** JOGFSFODF BT DBO CF TFFO JO 'JHVSF  GPS FYBNQMF  ĦJT NBLFT USBOTMBUJPO NPSF EJǬDVMU TJODF JU MBDLT UIF JOGPSNBUJPO DPOUBJOFE XJUIJO UIF TVCQSPPGT BOE XF NVTU SFTPSU UP SFǪOEJOH &ÙT JOGFSFODFT 8F DBO BWPJE UIJT MPTT JG XF IBWF BO FNCFEEJOH PG BMM UIF TVCQSPPGT JOUP UIF UBSHFU MBOHVBHF×GPS FYBNQMF BO FNCFEEJOH PG -Əƙ**ÙT JO GFSFODFT JO *TBCFMMF)0- BOE BO FNCFEEJOH PG &ÙT DBMDVMVT JO *TBCFMMF)0- 8F DPVME UIFO VTF UIF DVU SVMF UP TUJUDI UPHFUIFS UIF USBOTMBUFE QSPPGT 8F XJMM TFF IPX UIF NPEFM EFTDSJCFE JO UIF OFYU TFDUJPO BMMPXT VT UP USBOTMBUF IZCSJE QSPPGT CVU OPUF UIBU UIF JNQMFNFOUBUJPO EFTDSJCFE JO UIJT DIBQUFS POMZ IBOEMFT QSPPGT JO -Əƙ**ÙT DBMDVMVT OPU UIF GVMM -Əƙ** & DPNCJOBUJPO $PNQPTJOH SFDPOTUSVDUFE & QSPPGT XJUI -Əƙ** QSPPGT SFNBJOT GVSUIFS XPSL  $VU NBDIJOFT ĦJT TFDUJPO DPOUBJOT B EFTDSJQUJPO PG BO BCTUSBDU NBDIJOF GPS NBQQJOH QSPPGT GSPN POF MPHJD JOUP BOPUIFS ĦJT NBDIJOF XJMM CF VTFE BT BO BCTUSBDU NPEFM PG QSPPG USBOTMBUJPO BOE XJMM CF WBMJEBUFE CZ QSPPG " TJNJMBS NFUIPE XBT VTFE CZ EF /JWFMMF  UP EFTDSJCF UIF HFOFSBUJPO PG QSPPG UFSNT UIBU WBMJEBUF DMBVTJǪDBUJPOT 1SPPG TFBSDI BCIPST UIF DVU SVMF #FO[N»MMFS FU BM  CVU QSPPG USBOTMBUJPO CFOFǪUT GSPN JU ĦF DVU SVMF DBO CF VTFE UP TQMJDF QSPPGT UPHFUIFS ĦJT JT QBSUJD VMBSMZ VTFGVM TJODF UIF QSPPGT DPVME IBWF CFFO USBOTMBUFE TFQBSBUFMZ GSPN EJǩFSFOU MPHJDT ĦJT QSPDFTT JT TPNFXIBU SFNJOJTDFOU PG QVSJǮDBUJPO JO 4.5 TPMWFST ,STUJD BOE (PFM  XIFSF B GPSNVMB FODPEJOH B TUBUFNFOU TQBOOJOH TFWFSBM UIFPSJFT JT USBOTGPSNFE JOUP B TFSJFT PG GPSNVMBT FBDI FODPEJOH B TUBUFNFOU JO B TJOHMF UIFPSZ BOE FBDI PG XIJDI NVTU CF TBUJTǪBCMF 1VU BCTUSBDUMZ QSPPG USBOTMBUJPO JT B GVODUJPO NBQQJOH QSPPGT GSPN POF MPHJD UP BOPUIFS ĦJT WJFX JT UPP BCTUSBDU UP BMMPX VT UP TUVEZ IPX UP FYUFOE BOE DPNQPTF USBOTMBUJPOT &YUFOTJCJMJUZ BOE DPNQPTJUJPO JT EFTJSBCMF GPS JNQMFNFOUJOH NPSF NBJOUBJOBCMF USBOTMBUJPOT *U BMTP BQQFBST UIBU DPNQPTJUJPO GBDJMJUBUFT JNQMFNFOU JOH USBOTMBUJPOT PG IZCSJE QSPPGT EFTDSJCFE JO UIF QSFWJPVT TFDUJPO )FSF DPNQPTJUJPO EPFT OPU SFGFS UP GVODUJPO DPNQPTJUJPO CVU SBUIFS UP UIF DPNQPTJUJPO PG QSPPGT GSPN EJǩFSFOU DBMDVMJ  " DVU NBDIJOF JT EFǪOFE JO UFSNT PG UXP GFBUVSFT 4UBUF DPOTJTUT PG B UVQMF (ρ, σ, F ) XIFSF ρ JT B TFU PG BTTVNQUJPOT UIBU DBO JODMVEF BYJPNT PG B QBSUJDVMBS UIFPSZ BOE WBMJEBUFE JOGFSFODFT σ JT B TUBDL PG TVCHPBMT BOE F JT UIF HPBM GPSNVMB *OTUSVDUJPOT HJWFO UP UIF NBDIJOF NBZ DPOTJTU PG UIF GPMMPXJOH à Ø1ƜƙƠƏ F Ù TUBUFT UIBU F JT UIF HPBM GPSNVMB " HPBM GPSNVMB NBZ POMZ CF TFU PODF QFS QSPPG à Ø$Ɵƞ rÙ BQQMJFT UIF SVMF r ∈ ρ UP UIF TUBDL PG TVCHPBMT JO UIF NBDIJOFÙT TUBUF ĦJT XJMM CF EFTDSJCFE JO NPSF EFUBJM CFMPX à Ø&ƘƎÙ BTTFSUT UIBU B NBDIJOF JT JO B UFSNJOBM TUBUF " UFSNJOBM TUBUF JT POF XIFSF B HPBM TUBUF IBT CFFO TFU JO UIF QBTU BOE UIF TVCHPBM TUBDL σ JT FNQUZ "O JOTUBODF PG &ƘƎ NBZ PDDVS POMZ PODF JO B QSPHSBN BU UIF FOE -FU UIF TZNCPM ◃ SFQSFTFOU UIF TJOHMFTUFQ USBOTJUJPO SFMBUJPO CFUXFFO TUBUFT 8F XJMM VTF Ø−Ù UP EFTDSJCF BO FNQUZ TUBDL BOE SJHIUBTTPDJBUJWF Ø:Ù UP EFTDSJCF UIF QVTI PQFSBUJPO ĦF TZNCPM ∪1 XJMM CF VTFE UP EFTDSJCF UIF VOJPO CFUXFFO B TFU BOE B TJOHMFUPO UIBU JT A ∪1 B BCCSFWJBUFT A ∪ {B} ĦF GPSNBM TFNBOUJDT PG UIF NBDIJOFÙT JOTUSVDUJPOT BSF BT GPMMPXT 1ƜƙƠƏ F  (ρ, −, True) ◃ (ρ, F, F ) $Ɵƞ r (ρ, B : σ, F ) ◃ (ρ, A1 : . . . :An :σ, F ) XIFSF r ∈ ρ BOE r = A1,...,AnB &ƘƎ (ρ, −, F ) ◃ (ρ, −, F ) %FǪOJUJPO  " DVU QSPHSBN DPOTJTUT PG B ǮOJUF TFRVFODF PG JOTUSVDUJPOT " XFMM GPSNFE DVU QSPHSBN DPOTJTUT PG B TJOHMF 1ƠƝƤƓ JOTUSVDUJPO [FSP PS NPSF $ƣƢ JOTUSVDUJPOT BOE ǮOBMMZ B TJOHMF &Ɯƒ "O JOJUJBM TUBUF DPOTJTUT PG BOZ TUBUF PG UIF GPSN (ρ, −, True) ħBU JT BO JOJUJBM TUBUF TJNQMZ BTTFSUT B WFSZ XFBL TUBUFNFOU True " UFSNJOBM TUBUF DPOTJTUT PG BOZ TUBUF PG UIF GPSN (ρ, −, F ) ħBU JT B UFSNJOBM TUBUF NVTU IBWF BO FNQUZ TVCHPBMTUBDL 8F XJMM VTF UIF TZNCPM ⊢ρ UP EFOPUF UIF ǪOJUF QSPPG TZTUFN FODPEFE JO ρ /PUF UIBU B DVU QSPHSBN XJUIPVU B TJOHMF 1ƜƙƠƏ JOTUSVDUJPO XJMM POMZ QSPWF UIF UBVUPMPHZ  ⊢ρ True " DVU NBDIJOF SVOOJOH B XFMMGPSNFE DVU QSPHSBN DBO HFU TUVDL JO UXP XBZT à XIFO FYFDVUJOH Ø$Ɵƞ rÙ JG r ̸∈ ρ PS JG UIF DPODMVTJPO PG r EPFT OPU NBUDI UIF UPQ FMFNFOU JO σ PS à XIFO FYFDVUJOH &ƘƎ JG UIF NBDIJOF JT OPU JO B UFSNJOBM TUBUF %FǪOJUJPO  " DVU QSPHSBN UIBU EPFT OPU HFU TUVDL JT DBMMFE XFMMHPJOH  7BMJEBUJOH UIF NPEFM 6TF PG UIF NPEFM SFMJFT PO UIF BTTVNQUJPO UIBU ρ DPOUBJOT BMM UIF SVMFT OFFEFE CZ UIF DVU QSPHSBN ĦF ǪOJUF TFU ρ DPOUBJOT B SFTUSJDUFE JOGFSFODF TZTUFN DPOTJTUJOH PG JOGFSFODF SVMFT JO UIF UBSHFU MPHJD ĦJT TFU JT B QBSBNFUFS UP UIF NPEFM BOE UIF HFOFSBUJPO PG UIFTF SVMFT UBLFT QMBDF FYUFSOBMMZ×TVDI BT CZ VTJOH UIF NFUIPET EFTDSJCFE JO UIF QSFWJPVT DIBQUFS UP NBQ UIF TPVSDF JOGFSFODFT JOUP B GPSN UIBU DBO CF WBMJEBUFE JO UIF UBSHFU TZTUFN 1SPWJEFE UIBU B TVJUBCMF ρ FYJTUT UIF NPEFM IBT UIF GPMMPXJOH QSPQFSUJFT -FNNB  1SPWJEFE JU JT HJWFO B XFMMHPJOH DVU QSPHSBN DG %FǮOJUJPO  UIF DVU NBDIJOF IBT UIF GPMMPXJOH JOWBSJBOU JG UIF TVCHPBMT BSF WBMJE UIFO UIF HPBM JT WBMJE UPP 1SPPG TLFUDI ĦF QSPPG QSPDFFET CZ JOEVDUJPO PO UIF MFOHUI PG DVU QSPHSBNT 1SPPG JO 4FDUJPO $ ▹ -FNNB  *G B DVU QSPHSBN SFBDIFT B UFSNJOBM TUBUF UIFO JUT QSPPG HPBM JT WBMJE 1SPPG -FU 1ƜƙƠƏ G . . . CF B UFSNJOBUJOH DVUQSPHSBN G JT UIF QSPPG HPBM " UFSNJOBM TUBUF FODPEFT UIF GPMMPXJOH NFUBUIFPSFN ⊢ρ True ⊢ρ G XIJDI JT FRVJWBMFOU UP Ø⊢ρ GÙ ĦJT XBT EFSJWFE GSPN B WBMJE JOJUJBM TUBUF XIJDI TJNQMZ FODPEFE ⊢ρ True TP CZ -FNNB  G NVTU CF WBMJE $PSPMMBSZ  " XFMMHPJOH DVU QSPHSBN BMXBZT QSPEVDFT B UIFPSFN JO UIF UBSHFU MPHJD .PSFPWFS UIJT DBO CF WFSJǮFE CZ JOTQFDUJOH B QSPPG JO UIF UBSHFU MPHJD  1SPPG " XFMMHPJOH DVU QSPHSBN NVTU FWFOUVBMMZ SFBDI B UFSNJOBM TUBUF ĦF JOWBSJ BOU -FNNB  FOTVSFT UIBU WBMJEJUZ JT QSFTFSWFE BDSPTT UIF NBDIJOFÙT FYFDVUJPO TUBSUJOH XJUI UIF NBDIJOFÙT JOJUJBM TUBUF XIJDI JT DMFBSMZ WBMJE  ĦF SFTVMUJOH UFS NJOBM TUBUF BTTVSFT UIF WBMJEJUZ PG UIF HPBM -FNNB   8F OPX UVSO UP IPX UIF NBDIJOF SFDPOTUSVDUT UIF QSPPG JO UIF UBSHFU MPHJD &TTFOUJBMMZ UIJT JT EFTDSJCFE JO UIF $ƟƞDBTF PG -FNNB  ĦF QSPPG HSPXT GSPN UIF HPBM CZ BQQMZJOH SVMFT GSPN ρ $VU NBDIJOFT BSF JOTQJSFE CZ IPX *TBCFMMF XPSLT XIJDI JO UVSO JT JOTQJSFE CZ FBSMJFS XPSL PO -$' XIJDI JT CBTFE PO CBDLXBSE 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bQK2 G2Qk BM72`2M+2bX V H2KK TQH`Bb2, ]S 44= S 4 h`m2] #v miQ ĦFO UIF FNVMBUPS GPS UIF DMBVTF@DPQZ SVMF TJNQMZ NBQT UP UIF GPMMPXJOH UBDUJD VTJOH *TBCFMMFÙT "1* >1.:PG Ui+ P_1Ga1^ U`i+ !&i?K TQH`Bb2' h>1L^ i+VV ĦJT UBDUJD USJFT UP BQQMZ UIF BTTVNQUJPO UBDUJD i+ XIJDI TVDDFFET XIFO UIF QSFNJTF BOE DPODMVTJPO BSF JEFOUJDBM Ħ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ĦF NPTU JOUFSFTUJOH LJOE PG FNVMBUJPO DPOTJTUT PG B DPNCJOBUJPO PG EJǩFSFOU FNV MBUPST *O FTTFODF UIJT JT BO JNQMFNFOUBUJPO PG BO "51 TZTUFN FNCFEEFE JO B QSPPG USBOTMBUPS *O PVS JNQPSUFS UIJT JNQMFNFOUT -Əƙ**MJLF QSPPG TFBSDI JO *TBCFMMF)0- ĦF JNQMFNFOUBUJPO DPOTJTUT PG B QSFQSPDFTTJOH TUBHF B MPPQ TUBHF BOE B DMFBOVQ TUBHF 1SFQSPDFTTJOH DBO JODMVEF TUSJQQJOH RVBOUJǪFST BOE TJNQMJGZJOH UIF JOGFSFODFÙT DPO DMVTJPO ĦF MPPQ TUBHF DPOTJTUT PG BQQMZJOH WBSJPVT JOGFSFODFT VOUJM UIF JOGFSFODF  CFDPNFT USJWJBM ĦF DMFBOVQ TUBHF DPOTJTUT PG TJNQMJGZJOH UIF JOGFSFODF UP GBDJMJ UBUF NBUDIJOH XJUI B 4LPMFN FRVBUJPO PS SFNPWJOH EVQMJDBUF IZQPUIFTFT ĦF QSPWFS JT QBSBNFUSJTFE CZ B DPOǪHVSBUJPO UIBU FYQMJDBUFT XIJDI TUBHFT TIPVME CF DBSSJFE PVU BOE XJUIJO FBDI TUBHF XIJDI JOGFSFODFT TIPVME CF BUUFNQUFE *O PUIFS XPSET JU JT B WFSZ DPOǪHVSBCMF UIFPSFN QSPWFS "T XJUI BOZ UIFPSFN QSPWFS UIF NBJO MPPQ DBSSJFT PVU NPTU PG UIF SFDPHOJTBCMF MPHJDBM JOGFSFODF *O UIJT QSPWFS UIF NBJO MPPQ DPOUBJOT JOGFSFODFT UIBU FNVMBUF -Əƙ**ÙT JOGFSFODF SVMFT *U NBEF TFOTF UP DPMMFDU UIFTF SVMFT JOUP B TNBMM QSPWFS UIBU DBO CF TQFDJBMJTFE UP DBSSZ PVU TQFDJǪD JOGFSFODFT TJODF WBSJPVT SVMFT TIBSFE TJNJMBS QSFQSPDFTTJOH TUFQT BOE DPEF *O QSJODJQMF JU JT QPTTJCMF UP VTF UIJT TNBMM QSPWFS GPS QSPPGTFBSDI PWFS BMM UIF JOGFSFODF SVMFT CVU UIJT XPVME OPU CF JEFBM TJODF UIF QSPWFS JT OPU BU BMM PQUJNJTFE "MTP UIFSF JT TVǬDJFOU JOGPSNBUJPO JO UIF QSPPGT JO UIF GPSN PG JOUFSNFEJBUF JO GFSFODFT UP BWPJE VOHVJEFE QSPPGTFBSDI  %FQFOEFODJFT 4PNF PG UIF FNVMBUPST NBLF VTF PG UIF CMBTU UBDUJD ĦJT UBDUJD JT EFTDSJCFE JO 4FD UJPO  *U JNQMFNFOUT B UBCMFBV QSPWFS JOUFHSBUFE XJUI *TBCFMMF 1BVMTPO   ĦJT UBDUJD IBT B EFQUI MJNJU UP CPVOE QSPPG TFBSDI BOE JU XBT OPUJDFE EVSJOH FYQFSJNFOUT UIBU UIJT MJNJU DBO NBLF B TJHOJǪDBOU EJǩFSFODF UP TVDDFTT BOE QFSGPS NBODF *G UIF MJNJU JT TFU UPP MPX JU EJNJOJTIFT TVDDFTT *G JU JT TFU UP IJHI JU EFHSBEFT QFSGPSNBODF *TBCFMMF VTFT IJHIFSPSEFS VOJǪDBUJPO EVSJOH QSPPG DIFDLJOH BT JU SFTPMWFT SVMFT JOUP JUT HPBM TUBUF ĦF JNQMFNFOUBUJPO PG VOJǪDBUJPO UPP IBT B TFBSDI CPVOE BOE JU XBT OPUJDFE UIBU UIJT IBT B TJNJMBS JOǫVFODF UP UIF EFQUI CPVOE NFOUJPOFE BCPWF "ĤFS B TNBMM OVNCFS PG FYQFSJNFOUT CPUI UIFTF WBMVFT XFSF TFU UP WBMVFT UIBU BMMPXFE UIF WBMJEBUJPO PG NPTU PG -Əƙ**ÙT QSPPGT GPS 5)' 5151 QSPCMFNT XJUIJO  TFDPOET BO FWBMVBUJPO JT QSPWJEFE JO 4FDUJPO   *U JT WFSZ MJLFMZ UIBU BEEJUJPOBM QSPPGT DPVME CF WBMJEBUFE JG XF JODSFBTF UIFTF QBSBNFUFSTÙ WBMVFT CVU UIJT NJHIU EFHSBEF QFSGPSNBODF JO HFOFSBM 'PS BO FYBNQMF PG BO JOGFSFODF UIBU SFRVJSFT B IJHIFS VOJǪDBUJPOCPVOE UP FN VMBUF DPOTJEFS UIJT JOTUBODF GSPN -Əƙ**ÙT QSPPG GPS 4&7? (sK2SY13X = sK2SY13 Y ) = False (X = Y ) = False  ĦJT JOGFSFODF TIPVME CF USJWJBM UP WBMJEBUF CVU JO UIJT QSFDJTF JOTUBODF UIF TZNCPMT X BOE Y NBQ UP SBUIFS MBSHF UFSNT SFRVJSJOH VT UP SFMBY UIF CPVOET EFTDSJCFE FBSMJFS ĦF TZNCPM X TUBOET GPS UIF UFSN⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ (SV36 : b→ b→ b→ b→ b→ a) (sK1SY12 (λSX0 : b. sK2SY13 (λSX1 : b. sK2SY13 (λSX2 : b. sK2SY13 (λSX3 : b. sK2SY13 (SV36 SX0 SX1 SX2 SX3)))))) (sK9E (λSX0 : b, SX1 : b. sK2SY13 (λSX2 : b. sK2SY13 (λSX3 : b. sK2SY13 (SV36 SX0 SX1 SX2 SX3))))) (sK10E (λSX0 : b, SX1 : b, SX2 : b. sK2SY13 (λSX3 : b. sK2SY13 (SV36 SX0 SX1 SX2 SX3)))) (sK11E (λSX0 : b, SX1 : b, (SX2 : b)SX3 : b. sK2SY13 (SV36 SX0 SX1 SX2 SX3))) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ BOE Y JT λSX0 : b. sK2SY13 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ SV36 (sK9E (λSX1 : b, SX2 : b. sK2SY13 (λSX3 : b. sK2SY13 (λSX4 : b. sK2SY13 (SV36 SX1 SX2 SX3 SX4))))) (sK10E (λSX1 : b, SX2 : b, SX3 : b. sK2SY13 (λSX4 : b. sK2SY13 (SV36 SX1 SX2 SX3 SX4)))) (sK11E (λSX1 : b, SX2 : b, SX3 : b, SX4 : b. sK2SY13 (SV36 SX1 SX2 SX3 SX4)))SX0 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠  )BOEMJOH MBSHF GPSNVMBT -FU VT DPOTJEFS JNQMFNFOUJOH B UBDUJD UIBU WBMJEBUFT UIF JOGFSFODF SVMF A1 . . . Am B1 . . . Bn *G A1, . . . , Am, B1, . . . , Bn JODMVEF MBSHF UFSNT UIFO QFSGPSNBODF JT MJLFMZ UP TVǩFS FWFO JG UIF JOGFSFODF JUTFMG JT WFSZ TJNQMF×BT XF TBX JO UIF QSFWJPVT TFDUJPO ĦF GPMMPXJOH JNQMFNFOUBUJPO UFDIOJRVF XBT GPVOE UP CF VTFGVM UP JNQSPWF QFSGPSNBODF 'JSTU DSFBUF B OFX BCTUSBDU WFSTJPO PG UIF JOGFSFODF A′1("A1#) . . . A′m("Am#) B′1("B1#) . . . B′n("Bn#)  *O UIJT SVMF UIF TZNCPMT "A1#, . . . , "Am#, "B1#, . . . , "Bn# BSF GSFTI WBSJBCMFT ĦF UFSNT A′1, . . . , A′m, B′1, . . . , B′n BSF BCTUSBDUJPOT PG UIF UFSNT A1, . . . , Am, B1, . . . , Bn SFTQFDUJWFMZ VQ UP TPNF EFQUI /PUF UIBU A′1, . . . , B′1, . . . 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Tn , . . .} ĦF QSJN@TVC SVMF JOTUBOUJBUFT WBSJBCMFT JO DJSDVNTUBODFT XIFSF XF DBOOPU VTF VOJǪDBUJPO ĦF JOTUBOUJBUJPO JT DBSSJFE PVU UISPVHI BO IFVSJTUJD FOVNFSBUJPO PG UFSNT *O -Əƙ**ÙT JNQMFNFOUBUJPO UIF SVMF QSJN@TVC JT DPOUSPMMFE CZ B QBSBNFUFS JOEJDBUJOH IPX DPNQMFY UIF TVCTUJUVUFE GPSNVMBT TIPVME CF 0UIFS SVMFT -Əƙ** IBT WBSJPVT PUIFS SVMFT JODMVEJOH SVMFT UP TJNQMJGZ GPSNBMJTB UJPOT BOE BEE TVQQPSU GPS UIF "YJPN PG $IPJDF .BOZ PG UIF PUIFS SVMFT VTFE JO -Əƙ** TFSWF BENJOJTUSBUJWF QVSQPTFT BOE EP OPU BEE JOIFSFOUMZ MPHJDBM TUSFOHUI UP UIF DBMDVMVT 'PS JOTUBODF UIF SVMF QPMBSJUZ@TXJUDI TJNQMZ ǫJQT B MJUFSBMÙT QPMBSJUZ BOE BEETSFNPWFT B QSFǪYJOH OFHBUJPO DMBVTF@DPQZ DSFBUFT B OFX DPQZ PG B DMBVTF BOE VOGPME@EFGT VOGPMET EFǪOJUJPOT ĦFSF JT BO BEEJUJPOBM GBNJMZ PG SVMFT JO -Əƙ** UIBU EPFT BEE MPHJDBM TUSFOHUI ĦF OBNFT PG UIFTF SVMFT BSF QSFǪYFE CZ GP@BUQ@ BOE TVǬYFE CZ UIF OBNF PG B ǪSTU PSEFS UIFPSFN QSPWFS ĦFTF JOGFSFODFT SFQSFTFOU PSBDMF DBMMT UP UIF UIF SFTQFDUJWF UIFPSFN QSPWFST "O FYBNQMF GP@BUQ@F SFQSFTFOUJOH DBMMT UP UIF & UIFPSFN QSPWFS DBO CF TFFO JO 'JHVSF  ĦF CFIBWJPVS PG UIF DMBVTF@DPQZ JOGFSFODF SVMF JT EFTDSJCFE JO 4FDUJPO  NPSF QSFDJTFMZ  "QQFOEJY # .BOVBM TDIFNBUJD DPOUSBOFHBUJPO 5P EFNPOTUSBUF UIF UFDIOJRVFT UIBU XFSF EFTDSJCFE JO $IBQUFS  UIJT TFDUJPO HJWFT TPNF FYBNQMFT PG GPSNBM BOE NFDIBOJTFE QSPPGT ĦF FYBNQMFT BSF TDIFNBUJD JO UIF TFOTF UIBU XF VTF UIF USBOTGPSNBUJPO PO UIF JOGFSFODF SVMFT SBUIFS UIBO PO JO TUBODFT PG JOGFSFODF SVMFT 8F TUBSU CZ QSPWJOH UIF GPSNVMB ∃A.∀P.∃B. (P AB ∨ ¬P B A) JO B ⊢CTUZMF TZTUFN CBTFE PO -Əƙ**ÙT DBMDVMVT UIFO USBOTMBUF UIF QSPPG JOUP B ⊢VTUZMF TZTUFN FNCFEEFE JO *TBCFMMF)0- &BDI MJOF PG UIF QSPPG FYDFQU GPS UIF ǪSTU SFQSFTFOUT BO JOGFSFODF CBTFE PO UIF QSFWJPVT MJOFT ĦF JOGFSFODFT BSF OPU MBCFMMFE XJUI UIF SVMF VTFE UIFTF BSF TJNQMF UP EFUFSNJOF PXJOH UP UIF TJNQMJDJUZ PG UIF HPBM ĦF TJOHMFUPO IZQPUIFTJT JO UIF ǪSTU MJOF DPOTJTUT PG UIF OFHBUFE DPOKFDUVSF ĦF VOEFSMJOFE IZQPUIFTJT PO MJOF  JT UIF POF UIBU QBSUJDJQBUFT JO UIBU JOGFSFODF ĦF Û. . .Ü JOEJDBUF QSFWJPVTMZ EFSJWFE DMBVTFT XIJDI BSF FMJEFE UP BWPJE DMVUUFS  {¬(∃A.∀P.∃B.(P AB ∨ ¬P B A))} ⊢C ¬(∀P.∃B.(P XAB ∨ ¬P BXA))  {. . . ,¬(∀P.∃B.(P XAB ∨ ¬P BXA))} ⊢C ¬(∃B.((skPXA)XAB ∨ ¬(skPXA)BXA))  {. . . ,¬(∃B.((skPXA)XAB ∨ ¬(skPXA)BXA))} ⊢C ¬((skPXA)XAXB ∨ ¬(skPXA)XBXA)  {. . . ,¬((skPXA)XAXB ∨ ¬(skPXA)XBXA)} ⊢C ¬((skPXA)XAXB)  {. . . ,¬((skPXA)XAXB ∨ ¬(skPXA)XBXA)} ⊢C ¬(¬(skPXA)XBXA)   {. . . ,¬((skPXA)XAXB),¬(¬(skPXA)XBXA)} ⊢C (skPXA)XBXA  {. . . ,¬((skPXA)XAXB), (skPXA)XBXA} ⊢C ((skPXA)XAXB) ̸= ((skPXA)XBXA)  {. . . , ((skPXA)XAXB) ̸= ((skPXA)XBXA)} ⊢C ! *O *TBCFMMF)0- XF TUBSU UIF QSPPG CZ BQQMZJOH UIF QSPPGCZDPOUSBEJDUJPO SVMF [[¬P ] =⇒ False] =⇒ P UIFO QSPDFFE BT GPMMPXT )FSF XF UBLF B EJǩFSFOU BQQSPBDI UP UIBU VTFE JO UIF JN QMFNFOUFE QSPUPUZQF EFTDSJCFE JO 4FDUJPO  ĦFSF XF USBOTMBUFE UIF QSPPG CZ DIBJOJOH UPHFUIFS JOTUBODFT PG UIF JOGFSFODF SVMFT )FSF XF EP OPU XPSL XJUI JO TUBODFT XF GPSNBMJTF UIF JOGFSFODF SVMFT BT TDIFNFT BOE XF BQQMZ UIFTF TDIFNFT %VF UP UIF HFOFSBMJUZ PG UIF TDIFNFT XF FOE VQ XJUI MPHJDBM WBSJBCMFT JO UIF USBOT GPSNFE SVMFT -PHJDBM WBSJBCMFT BSF QMBDFIPMEFST PG UIF UFSNT UIBU XPVME TQFDJBMJTF UIF JOGFSFODFT GVSUIFS 8F SFQSFTFOU B MPHJDBM WBSJBCMF VTJOH UIF OPUBUJPO Aˆ(κ⃗) XIFSF A JT UIF WBSJBCMFÙT OBNF BOE κ⃗ JT B MJTU PG TZNCPMT UP XIJDI UIF WBSJBCMF JT BQQMJFE ĦFTF TZNCPMT DPOTJTU PG ∧ CPVOE WBSJBCMFT BOE TFSWF UP FOTVSF TPVOE SFBTPOJOH 1BVMTPO  EFTDSJCFT IPX UIFZ XPSL UIF EFUBJMT BSF OPU JNQPSUBOU IFSF  [¬(∃A.∀P.∃B.(P AB ∨ ¬P B A))] =⇒ False  [ . . . ;¬(∀P.∃B.(P Aˆ()B ∨ ¬P B Aˆ())) ] =⇒ False  [ . . . ;¬(∃B.(P Aˆ()B ∨ ¬PB Aˆ())) ] =⇒ False  [ . . . ;¬(P Aˆ() Bˆ(P) ∨ ¬P Bˆ(P) Aˆ()) ] =⇒ False  [ . . . ;¬(P Aˆ() Bˆ(P));¬(¬P Bˆ(P) Aˆ()) ] =⇒ False  [ . . . ;¬(P Aˆ() Bˆ(P));P Bˆ(P) Aˆ() ] =⇒ False  [ . . . ; (P Aˆ() Bˆ(P)) ̸= (P Bˆ(P)) Aˆ() ] =⇒ False  [. . . ; False] =⇒ False *O TUFQT  UP  UIFSF JT BO JNQMJDJU ∧ P QSFǪYJOH UIF TUBUF XF FMJEF UIJT IFSF CVU XSJUF P UP JOEJDBUF UIBU UIBU TZNCPM JT OPU CPVOE BU UIF PCKFDU MFWFM  # *TBCFMMF)0- FYBNQMFT # )FBEFS i?2Q`v 1tKTH2 BKTQ`ib >PG #2;BM # 5BCMFBV FNCFEEJOH bm#b2+iBQM ]*H+mHmb] H2KK 2ti+M7nQ`nTQb, ](%* % UU % "V 4 h`m2Vc (%* % UU % "V 4 h`m2Vc * % U 4 h`m2V % U" 4 h`m2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2ti+M7nQ`nM2;R, ](%* % UU % "V 4 6Hb2Vc (%* % UU % "V 4 6Hb2Vc * % U 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2ti+M7nQ`nM2;k, ](%* % UU % "V 4 6Hb2Vc (%* % UU % "V 4 6Hb2Vc * % U" 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2ti+M7nMQinTQb, ](%* % UU V 4 h`m2Vc (%* % UU V 4 h`m2Vc * % U 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ  H2KK 2ti+M7nMQinM2;, ](%* % UU V 4 6Hb2Vc (%* % UU V 4 6Hb2Vc * % U 4 h`m2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2ti+M7n7Q`HHnTQb, ](%* % UUHH V 4 h`m2Vc (%* % UUHH V 4 h`m2Vc * % UU sV 4 h`m2V%) 44= 6Hb2%) 44= 6Hb2] ](%* % UU1t V 4 6Hb2Vc (%* % UU1t V 4 6Hb2Vc * % UU sV 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2ti+M7n7Q`HHnM2;, ](%* % UUHH V 4 6Hb2Vc 55 bFX (%* % UUHH V 4 6Hb2Vc * % UU bFV 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2tin7mM+nTQb, ](%* % UUJ 4 LV 4 h`m2Vc (%* % UUJ 4 LV 4 h`m2Vc * % UUUJ sV 4 UL sVV 4 h`m2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2tin#QQH2nTQb, ](%* % UUJ 4 LV 4 h`m2Vc (%* % UUJ 4 LV 4 h`m2Vc * % UUUJ @@= LV  UL @@= JVV 4 h`m2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ  H2KK 2tin7mM+nM2;, ](%* % UUJ 4 LV 4 6Hb2Vc 55 bFX (%* % UUJ 4 LV 4 6Hb2Vc * % UUUJ bFV 4 UL bFVV 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 2tin#QQH2nM2;, ](%* % UUJ 4 LV 4 6Hb2Vc (%* % UUJ 4 LV 4 6Hb2Vc * % UUUJ @@= LV  UL @@= JVV 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK mMBni`Bp, ](%* % UU 4 V 4 6Hb2Vc (%* % UU 4 V 4 6Hb2Vc *%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK mMBn/2+R, ](%* % UU? l 4 ? oV 4 6Hb2Vc (%* % UU? l 4 ? oV 4 6Hb2Vc * % UUl 4 oV 4 6Hb2V %) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK mMBn7H2t`B;B/, ](%* % UU6 l 4 ? oV 4 6Hb2Vc (%* % UU6 l 4 ? oV 4 6Hb2Vc * % U6 4 :V % UU6 l 4 ? oV 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK `2b, ](%* % U 4 h`m2Vc . % U" 4 6Hb2Vc (%* % U 4 h`m2Vc  . % U" 4 6Hb2Vc * % . % UU 4 "V 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ H2KK 7+n`2bi`, ](%U 4 TV % U" 4 TVc (%U 4 TV % U" 4 TVc U 4 TV % UU 4 "V 4 6Hb2V%) 44= 6Hb2%) 44= 6Hb2] #v miQ # )FMQFS DPEF BOE MFNNBT bm#b2+iBQM ]>2HT2`b] H2KK mM7QH/n/27, ]U @@= "V 4 U  % "V] ]U  "V 4 U U  %  "VV] #v miQ JG & 7mM `2bQHp2nmMB7v i?K 4 >1.:PG U`i+ i?K h>1L^ i+V ' H2KK +QHHTb2n7Hb2, ]U6Hb2 % U6Hb2 % SVV 4 U6Hb2 % SV] ]UU6Hb2 % 6Hb2V % SV 4 U6Hb2 % SV] #v miQ # *TBCFMMF)0-SFDPOTUSVDUFE FYBNQMFT bm#b2+iBQM ]1tKTH2b] H2KK ]6Hb2 % U5 SX S %  SV 4 6Hb2 44= 6Hb2] TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7n7Q`HHnM2;' 'V TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7nQ`nM2;R' 'V  TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7nQ`nM2;k' 'V TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7nMQinM2;' 'V TTHv Ui?BMni+ ]6Hb2 % U5 SX S %  SV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % U5 SX S %  SV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UbF %  bFV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UbF %  bFV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UbF %  bFV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % U bFV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % U bFV 4 6Hb2]V TTHv U`mH2 `2b(Q7 ]6Hb2] n ]6Hb2])V TTHv bbmKTiBQM TTHv bbmKTiBQM TTHv UbBKT QMHv, +QHHTb2n7Hb2V TTHv U`mH2 mMBni`BpV TTHv bbmKTiBQM TTHv bbmKTiBQM /QM2 H2KK ]6Hb2 % U5  X \ s X 5 "X U s  " sV @@= U" s   sVV 4 6Hb2 44= 6Hb2] TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7n7Q`HHnM2;' 'V TTHv Ui?BMni+ ]6Hb2 % UGG X 1s sX GG "X  s  " s @@= " s   sV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UGG X 1s sX GG "X  s  " s @@= " s   sV 4 6Hb2]V TTHv U2`mH2 2ti+M7n7Q`HHnTQbV TTHv Ui+iB+ & `2bQHp2nmMB7v !&i?K 2ti+M7n7Q`HHnM2;' 'V TTHv Ui?BMni+ ]6Hb2 % U1s sX GG "X bF s  " s @@= " s  bF sV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UGG "X bF U\s8 bFV  " U\s8 bFV @@= " U\s8 bFV  bF U\s8 bFVV 4 6Hb2]V TTHv Ui?BMni+ ]6Hb2 % UGG "X bF U\s8 bFV  " U\s8 bFV @@= " U\s8 bFV  bF U\s8 bFVV 4 6Hb2]V TTHv UbBKT QMHv, mM7QH/n/27V TTHv U2`mH2 2ti+M7nQ`nM2;RV  TTHv U2`mH2 2ti+M7nQ`nM2;kV TTHv Ui?BMni+ ]6Hb2 % U  U bF U\sR8 bF bFV %  bF U\sR8 bF bFVV %  U bF U\sR8 bF bFV %  bF U\sR8 bF bFVVV 4 6Hb2]V TTHv U2`mH2 2ti+M7nMQinM2;V TTHv U`Qii2ni+ kV TTHv U2`mH2 2ti+M7nMQinM2;V TTHv U`Qii2ni+ @RV TTHv U2`mH2 2ti+M7nQ`nTQbV TTHv U2`mH2 2ti+M7nMQinTQbV TTHv U`Qii2ni+ @RV TTHv U2`mH2 2ti+M7nQ`nM2;RV TTHv U`Qii2ni+ @kV TTHv U2`mH2 2ti+M7nQ`nM2;kV U p2`v mMrB2H/v iQ +QMi`QH `2+QMbi`m+iBQM KMmHHv 7`QK ?2`2 V TTHv miQ /QM2 ĦF QSFTFODF PG MPHJDBM WBSJBCMFT NBEF JU NPSF EJǬDVMU UP DPOUSPM UIF SFDPOTUSVDUJPO× BU MFBTU NBOVBMMZ 6OMJLF UIJT NBOVBM FYQFSJNFOU UIF JNQMFNFOUFE QSPUPUZQF 4FDUJPO  SFMJFT PO JOTUBODFT PG JOGFSFODF SVMFT BOE UIF USBOTMBUJPO UBLFT QMBDF TNPPUIMZ  "QQFOEJY $ 1SPPGT $ $POUSBOFHBUJPO USBOTGPSNBUJPO ĦJT TFDUJPO QSPWJEFT QSPPGT SFMBUFE UP UIF WBMJEJUZ PG UIF DPOUSBOFHBUJPO USBOTGPSNB UJPO %FǪOJUJPO   8F XJMM VTF UIF )FOLJO TFNBOUJD GSBNFXPSL #FO[N»MMFS FU BM  XIJDI JT DPNNPOMZ VTFE GPS UIF GPSNBMJTBUJPO PG IJHIFSPSEFS MPHJD XF VTF "OESFXT   'PS DPOTJTUFODZ XF XJMM VTF UIF OPUBUJPO VTFE CZ #FO[N»MMFS FU BM   /PUF UIBU UIF TZNCPM ≡ TJNQMZ NFBOT FRVBMJUZ BU UIF MFWFM PG USVUI WBMVFT $ -FNNB $ *O UIF QSPPG PG -FNNB  QSPWJEFE JO 4FDUJPO $ XF XJMM SFMZ PO UIF GPMMPXJOH MFNNB BCPVU TJHOBUVSF FYUFOTJPO *OUVJUJWFMZ JU DMBJNT UIBU XF DBO BMXBZT FYUFOE UIF TJHOBUVSF XJUIPVU BǩFDUJOH UIF TFU PG GPSNVMBT UIBU B NPEFM TBUJTǪFT -FNNB $ 'PS BOZ ΣNPEFMM = (D,@, E , v) BOE Σ′NPEFMM′ = (D,@, E ′, v) XIFSF Σ ⊆ Σ′ BOE XIFSF Eϕ(c) = E ′ϕ(c) GPS BMM c ∈ Σ UIF GPMMPXJOH QSPQFSUZ IPMET 'PS BOZ GPSNVMB F XJUI DPOTUBOUT POMZ JO Σ M |= F JǭM′ |= F  1SPPG 8F BSF UP TIPX UIBU v(Eϕ(F )) ≡ v(E ′ϕ(F )) GPS BMM ϕ JG F ∈ wff(Σ) 8F QSPWF UIJT CZ TIPXJOH UIBU Eϕ(F ) = E ′ϕ(F ) GPS BMM ϕ JG F ∈ wff(Σ) 1SPDFFE CZ JOEVDUJPO PO F  à $BTFF JT B WBSJBCMFx 6TJOH UIF EFǪOJUJPO PG FWBMVBUJPO GVODUJPOT #FO[N»MMFS FU BM  3FNBSL  XF IBWF UIBU Eϕ(x) = ϕ(x) = E ′ϕ(x)  à $BTF F JT B DPOTUBOU c 4JODF F ∈ wff(Σ) UIFO JU JT OFDFTTBSZ UIBU c ∈ Σ #Z BTTVNQUJPO XF IBWF Eϕ(c) = E ′ϕ(c) GPS BMM c ∈ Σ à $BTF F JT BO BQQMJDBUJPO t1t2 Eϕ(t1t2) = Eϕ(t1)@Eϕ(t2) = E ′ϕ(t1)@E ′ϕ(t2) = E ′ϕ(t1t2) VTJOH *) à $BTF F JT BO BCTUSBDUJPO λx. t *) HJWFT VT UIBU Eϕ,[a/x](t) = E ′ϕ,[a/x](t) GPS FBDI a ∈ D 6TJOH UIF *) GPS BSCJUSBSZ a XF IBWF Eϕ(λx. t)@a = Eϕ,[a/x](t) = E ′ϕ,[a/x](t) = E ′ϕ(λx. t)@a ĦFSFGPSF Eϕ(λx. t) = E ′ϕ(λx. t) CZ FYUFOTJPOBMJUZ $ -FNNB  1SPPG 3FDBMM UIBU Φ ⊢C ! Jǩ UIFSF JT OP )FOLJO NPEFM M TVDI UIBU M |= Φ 3FDBMM UIBU M XPVME DPOTJTU PG B UVQMF (D,@, E , v) BOE UIBU M |= Φ Jǩ GPS BMM φ ∈ Φ v(E(φ)) ≡ ⊤ 4JODF FBDI φ ∈ Φ JT DMPTFE XF OFFE OPU XPSSZ BCPVU WBSJBCMF BTTJHONFOUT &RVJWBMFOUMZ GPS BMM )FOLJO NPEFMT UIFSF JT B φ ∈ Φ TVDI UIBU v(E(φ)) ≡ ⊥ ĦJT JT FRVJWBMFOU UP TBZJOH UIBU GPS BMM )FOLJO NPEFMT v(E(φ1 ∧ . . . ∧ φn)) ≡ ⊥ XIFSF {φ1, . . . ,φn} = Φ BOE XIFSF XF VTF UIF VTVBM JOUFSQSFUBUJPO PG UIF DPOKVOD UJPO TZNCPM Ø∧Ù #FO[N»MMFS FU BM  'JHVSF   8F XJMM DPOUJOVF UP VTF UIF TZNCPM Φ UP BCCSFWJBUF JUT DPOKVODUJWF GPSN φ1 ∧ . . . ∧ φn v(E(Φ)) ≡ ⊥ IPMET Jǩ GPS BMM )FOLJO NPEFMT v(E(Φ −→ False) ≡ ⊤ VTJOH UIF VTVBM JOUFSQSFUBUJPO PG Ø−→Ù BOE ØFalseÙ ĦBU JT |= Φ −→ False 'SPN UIF BTTVNQUJPO UIBU ⊢V JT DPNQMFUF XF IBWF ⊢V Φ −→ False $ -FNNB  1SPPG 'SPN UIF TPVOEOFTT QSPPG PG ⊢C XF IBWF UIBU FBDI SVMF r ∈ ⊢C QSFTFSWFT TBUJTǪBCJMJUZ 3VMF r DBO CF SFQSFTFOUFE BT B UVQMF PG QPTTJCMZ TFWFSBM QSFNJTFT BOE QPTTJCMZ TFWFSBM DPODMVTJPOT ((H1, . . . , Hn), (C1, . . . , Cm)) &BDI Hi BOE Ci JT B GSFFGPSN GPSNVMB PG )0- #Z UIF TPVOEOFTT PG r JG UIFSF FYJTUT B NPEFM M TVDI UIBU M |= H UIFO UIFSF FYJTUT B NPEFM M′ TVDI UIBU M′ |= H BOE M′ |= Ci GPS TPNF 1 ≤ i ≤ m BOE XIFSF H = (H1, . . . , Hn)  -FU C = (C1, . . . , Cm) 3FDBMM UIBU GPS DPOTJTUFODZQSFTFSWJOH SVMFT XF VTF UIF∨ OPUBUJPO UP CJOE GSFF WBSJBCMFT PDDVSSJOH JO H BOE C ĦJT JT JOUFOEFE UP GPS NBMJTF UIF JEFB UIBU UIF JOUFSQSFUBUJPO PG GSFF UPQMFWFM WBSJBCMFT PDDVSSJOH JO H BOE C CZ M′ JT VODPOTUSBJOFE UIFSFGPSF M′ NVTU TBUJTGZ H BOE C GPS BU MFBTU POF QPTTJCMF JOTUBOUJBUJPO PG UIFJS ∨ CPVOE WBSJBCMFT 'PS DMBSJUZ MFU VT NBLF GSFF UPQ MFWFM WBSJBCMFT FYQMJDJU UISPVHI /PUBUJPO  ĦFO UIF TUBUFNFOU PG rÙT TPVOEOFTT CFDPNFT JG UIFSF FYJTUT B NPEFM M TVDI UIBU M |= ∨ x⃗.H[x⃗] UIFO UIFSF FYJTUT B NPEFMM′ TVDI UIBUM′ |= ∨ x⃗.H[x⃗] BOEM′ |= ∨ y⃗i.Ci[y⃗i] -JĤJOH UIF∨ OPUBUJPO JOUP UIF NFUBMBOHVBHF JF &OHMJTI UIF TPVOEOFTT TUBUFNFOU CFDPNFT JG UIFSF FYJTUT B NPEFM M TVDI UIBU M |= H[x⃗] GPS TPNF x⃗ UIFO UIFSF FYJTUT B NPEFM M′ TVDI UIBU M′ |= H[x⃗] BOE M′ |= Ci[y⃗i] GPS TPNF i, x⃗, y⃗i 5BLJOH UIF DPOUSBQPTJUJWF UIJT JT FRVJWBMFOU UP UIF TUBUFNFOU JG UIFSF EPFT OPU FYJTU B NPEFM M′ TVDI UIBU M′ |= H[x⃗] BOE M′ |= Ci[y⃗i] GPS TPNF i, x⃗, y⃗i UIFO UIFSF EPFT OPU FYJTU B NPEFM M TVDI UIBU M |= H[x⃗] GPS TPNF x⃗ 3FDBMM UIBU GPS BMM φ XF IBWF UIBU |= ¬φ Jǩ UIFSF EPFT OPU FYJTU B NPEFMM TVDI UIBU M |= φ "MTP GPS BMM φ1,φ2 XF IBWF UIBU M |= φ1 ∧ φ2 Jǩ M |= φ1 BOE M |= φ2 #Z VTJOH UIFTF FRVJWBMFODFT BOE QVTIJOH JO UIF OFHBUJPO XF DBO SFXSJUF UIF QSFWJPVT TUBUFNFOU BT JG |= ¬(H[x⃗] ∧ Ci[y⃗i]) UIFO |= ¬H[x⃗] /PX UIJT TIPVME IPME GPS BSCJUSBSZ i, x⃗, y⃗i CFDBVTF XF QVTIFE JO UIF OFHBUJPO /PUF UIBU UIF BOUFDFEFOU BOE DPOTFRVFOU PDDVSSFODFT PG x⃗ PDDVS JO EJǩFSFOU TDPQFT XF NBLF UIJT FYQMJDJU CZ VTJOH UIF ∧ CJOEFS XIFO XF TXJUDI UP VTJOH *TBCFMMF OPUBUJPO 6TJOH ∧ BMTP NBLFT FYQMJDJU UIBU UIF GPSNVMBT BSF FYQFDUFE UP IPME GPS BMM x⃗ BOE x⃗, y⃗i SFTQFDUJWFMZ 4JODF ⊢V JT DPNQMFUF XF TIPVME CF BCMF UP BENJU UIJT JO SVMF GPSN ⊢V ¬(H[x⃗] ∧ C1[y⃗1]) . . . ⊢V ¬(H[x⃗] ∧ Cm[y⃗m]) ⊢V ¬H[x⃗] 6TJOH UIF FRVJWBMFODF (¬A) = (A −→ False) GPMMPXFE CZ UIF JNQMJDBUJPO "MUFSOBUJWFMZ XF DPVME IBWF FYJTUFOUJBMMZDMPTFE UIF PCKFDU GPSNVMB CVU UIJT XPVME JOWPMWF BEEJ UJPOBM TZOUBDUJD NBOJQVMBUJPO JU JT CFTU UP LFFQ UIJOHT TJNQMF BT MPOH BT UIF TFNBOUJDT BSF VOEFSTUPPE BOE QSFTFSWFE &WFO CFUUFS UIJT SVMF JT EFSJWBCMF  JOUSPEVDUJPO SVMF BOE TJNQMJǪDBUJPO XF HFU UIF FRVJWBMFOU SVMF ( ∧ x⃗, y⃗1. ¬(H[x⃗] ∧ C1[y⃗1])) =⇒ . . . =⇒ ( ∧ x⃗, y⃗m. ¬(H[x⃗] ∧ Cm[y⃗m])) =⇒∧ x⃗. H[x⃗] =⇒ False. #Z SFVTJOH UIF FRVJWBMFODF NFOUJPOFE QSFWJPVTMZ UPHFUIFS XJUI UIF FRVJWBMFODF (−→) ≡ (=⇒) XF PCUBJO⎛⎜⎜⎝∧ x⃗, y⃗1. ⎡⎢⎢⎣ H[x⃗] ∧ C1[y⃗1]  False ⎤⎥⎥⎦ ⎞⎟⎟⎠ =⇒ . . . =⇒⎛⎜⎜⎝∧ x⃗, y⃗m. ⎡⎢⎢⎣ H[x⃗] ∧ Cm[y⃗m]  False ⎤⎥⎥⎦ ⎞⎟⎟⎠ =⇒ ∧ x⃗. H[x⃗] =⇒ False XIJDI XIFO SFXSJUUFO VTJOH NPSF DPOWFOUJPOBM OPUBUJPO BOE CZ FMJNJOBUJOH UIF UPQMFWFM JUFSBUFE DPOKVODUJPO JO H UP HJWF H1, . . . , Hn JT FRVJWBMFOU UP ∧ x⃗, y⃗1. φ(1) . . . ∧ x⃗, y⃗m. φ(m) H1[x⃗] . . . Hn[x⃗]∧ x⃗ False XIFSF φ(i) = ⎡⎢⎢⎣ H1[x⃗], . . . , Hn[x⃗], Ci[y⃗i]  False ⎤⎥⎥⎦ ĦF FYQBOEFE GPSNVMB JT FRVJWBMFOU UP r⊥ 4UBSUJOH XJUI UIF BTTVNQUJPO UIBU r JT DPOTJTUFODZQSFTFSWJOH XF IBWF BSSJWFE WJB B TFSJFT PG JǩTUFQT BU B WBMJEJUZQSFTFSWJOH SVMF FRVJWBMFOU UP r⊥  $ -FNNB  1SPPG 8F BSHVF CZ JOEVDUJPO PO EFSJWBUJPOT π $BTF π = () /PUF UIBU π⊥ = π = () " QSPPG π JT FNQUZ JG JU JT JNNFEJBUF UIBU Φ ⊢C,π ! *G B QSPPG JT FNQUZ UIFO JU NVTU CF OFDFTTBSZ UIBU False ∈ Φ ĦF BSHVNFOU JT CZ DPOUSBEJDUJPO BTTVNF UIBU Φ ⊢C ! DBO CF TIPXO JO  TUFQT BOE BTTVNF GPS DPOUSBEJDUJPO UIBU False ̸∈ Φ ĦFO JU DPVME OPU CF USVF UIBU Φ ⊢C ! DBO CF TIPXO JO  TUFQT ĦF BSHVNFOU GPS ⊢V Φ −→ False JT TJNJMBS 'SPN UIF GBDU UIBU False ∈ Φ JU JT TUSBJHIUGPSXBSE UP QSPWF UIF CBTF DBTF $BTF π = (π′, r) ĦBU JT π DPOTJTUT PG UIF TVCQSPPG π′ GPMMPXFE CZ BO BQQMJDBUJPO PG SVMF r 8JUIPVU MPTT PG HFOFSBMJUZ MFU r CF UIF SVMF A1 . . . Am B 8PSLJOH XJUIJO BO FYUFOEFE TJHOBUVSF κ⃗′ UIF JOEVDUJPO IZQPUIFTJT BMMPXT VT UP BTTVNF UIBU UIJT QSPQFSUZ IPMET GPS π′ BOE π′⊥ Φ∪{A1, . . . , Am, B} ⊢C,π′ ! Jǩ ⊢V,π′⊥ ∀κ⃗′. Φ∧A1∧. . .∧Am∧B −→ False 8F BSF UP TIPX UIBU Φ ∪ {A1, . . . , Am} ⊢C,π ! Jǩ ⊢V,π⊥ ∀κ⃗. Φ ∧ A1 ∧ . . . ∧ Am −→ False XIFSF κ⃗ ⊆ κ⃗′ 8F TUBSU XJUI UIF ÛPOMZ JGÜ EJSFDUJPO "TTVNF Φ ∪ {A1, . . . , Am} ⊢C,π ! UP TIPX ⊢V,π⊥ ∀κ⃗. Φ ∧ A1 ∧ . . . ∧ Am −→ False. /PUF UIBU π DPOTJTUT PG π′ GPMMPXFE CZ r BOE UIBU JG Φ∪ {A1, . . . , Am} ⊢C,π ! UIFO Φ ∪ {A1, . . . , Am, B} ⊢C,π′ ! CZ UIF TPVOEOFTT PG r 8F DBO VTF UIJT UP JOGFS WJB UIF *) UIBU ⊢V,π′⊥ ∀κ⃗′. Φ∧A1∧ . . .∧Am∧B −→ False 8F VTF UIJT GBDU BT B IZQPUIFTJT UP r⊥ XIJDI XF PCUBJO WJB -FNNB  UPHFUIFS XJUI JUT BENJTTJCJMJUZ JO ⊢V BOE JOUSPEVDF DPOOFDUJWFT UP EFSJWF ⊢V,π⊥ ∀κ⃗. Φ ∧ A1 ∧ . . . ∧ Am −→ False  'PS UIF ÛJGÜ EJSFDUJPO BTTVNF ⊢V,π⊥ ∀κ⃗. Φ ∧ A1 ∧ . . . ∧ Am −→ False UP TIPX Φ ∪ {A1, . . . , Am} ⊢C,π ! "T CFGPSF UIF QSPPG π SFMJFT PO B TVCQSPPG π′ XIJDI QSPWFT ⊢V,π′⊥ ∀κ⃗′. Φ ∧ A1 ∧ . . . ∧ Am ∧ B −→ False 6TJOH UIF *) XF EFSJWF Φ ∪ {A1, . . . , Am, B} ⊢C,π′ ! #Z UIF TPVOEOFTT PG r JG S = Φ ∪ {A1, . . . , Am, B} JT JODPOTJTUFOU UIFO TP JT S\B ĦFSFGPSF XF DPODMVEF UIBU Φ ∪ {A1, . . . , Am} ⊢C,π ! $ $PSPMMBSZ  ĦJT DPSPMMBSZ GPMMPXT GSPN -FNNB  CVU XF DBO JNQSPWF PVS DPOǪEFODF JO UIF BSHVNFOU CZ DIFDLJOH UIF DPOUSBOFHBUJPO USBOTGPSNBUJPO PO FBDI -Əƙ** SVMF UP FOTVSF UIBU UIF SFTVMU JT EFSJWBCMF JO *TBCFMMF)0- 8F DBO UIVT QSPWF UIF DPSPMMBSZÙT TUBUFNFOU JO BOPUIFS XBZ XF DBO TIPX IPX DIBJOJOH UPHFUIFS UIF DPOUSBOFHBUFE -Əƙ** SVMFT JO *TBCFMMF)0- ZJFMET B WBMJE *T BCFMMF)0- QSPPG *O B TFOTF UIJT JOTUBOUJBUFT UIF BSHVNFOU NBEF JO -FNNB  JO POF EJSFDUJPO UIFO VOGPMET JU JO -FNNB  8F XJMM NBLF SFGFSFODF UP UIF DPOTUJUVFOU SVMFT PG -Əƙ**ÙT DBMDVMVT EFTDSJCFE JO 4FDUJPO " 8F XJMM GPDVT PO SVMFT UIBU SFMBUF UP OPSNBMJTBUJPO FYUFOTJPOBMJUZ VOJǪDBUJPO BOE SFTPMVUJPO XF XJMM JHOPSF SVMFT UIBU DBSSZ PVU B MPHJTUJDBM GVODUJPO TVDI BT DPQZJOH DMBVTFT SFOBNJOH WBSJBCMFT FUD  %FǪOJUJPO $ ħF DPSF SVMFT PG -ƓƝ**ÙT DBMDVMVT BSF UIPTF UIBU CFMPOH UP POF PG UIF GPMMPXJOH GBNJMJFT {normalisation, extensionality, unification, resolution} 8F XJMM ǪSTU TFFL UP SFEVDF B TFU PG GPSNVMBT Φ BOE B DPSSFTQPOEJOH EFSJWBUJPO π JOUP BO FRVJTBUJTǪBCMF TFU PG GPSNVMBT Φ′ BOE DPSSFTQPOEJOH EFSJWBUJPO π′ ĦJT JT EPOF UP KVTUJGZ XIZ XF DBO GPDVT TPMFMZ PO UIF DPSF SVMFT PG -Əƙ** ĦBU JT π′ DPOTJTUT POMZ PG JOTUBODFT PG DPSF SVMFT 3FMZJOH PO DPSF SVMFT NJHIU TFFN UP JNQPTF SFTUSJDUJPOT PO UIF JOQVU GPSNVMBTΦ CVU UIF OFYU MFNNB XJMM EJTQFM UIJT CZ TIPXJOH UIBU UIF JOQVU GPSNVMBT DBO CF USBOTMBUFE JOUP BO FRVJTBUJTǪBCMF TFU PG GPSNVMBT Φ′ -FNNB $ 'PS FBDI EFSJWBUJPO π JO -ƓƝ** GPS Φ UIFSF FYJTUT B EFSJWBUJPO π′ JO -ƓƝ** GPSΦ′ TVDI UIBU π′ DPOTJTUT PG JOGFSFODFT NBEF VQ TPMFMZ PG DPSF SVMFT BOE UIBU Φ JT TBUJTǮBCMF Jǭ Φ′ JT TBUJTǮBCMF  1SPPG 8F OFFE UP TIPX UIBU FBDI JOQVU QSPCMFN DBO CF SFQMBDFE XJUI BO FRVJTBUJTǪ BCMF JOQVU QSPCMFN BOE UIBU BSCJUSBSZ EFSJWBUJPOT GSPN UIF GPSNFS 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