Rudiments of Presburger Arithmetic

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(1)Rudiments of Presburger Arithmetic Stéphane Demri. To cite this version: Stéphane Demri. Rudiments of Presburger Arithmetic. Master. Paris, France. 2016, pp.44. �hal03188114�. HAL Id: hal-03188114 https://hal.archives-ouvertes.fr/hal-03188114 Submitted on 1 Apr 2021. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(7) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD • v(t + t′ ) = v(t) + v(t′ ) GPS BMM UFSNT t BOE t′ . 'PS JOTUBODF VOEFS UIF WBMVBUJPO v TVDI UIBU v(Y) = 3 BOE v(Z) = 27 XF DBO TUBUF UIBU Z JT HSFBUFS UIBO 2Y + 8 XIJDI DBO CF XSJ॒FO v | (2Y + 8) ≤ Z XIFSF ‫ ۝|۝‬TUBOET GPS UIF TBUJTGBDUJPO SFMBUJPO *OEFFE v(2Y+8) = 14 #Z DPOUSBTU VOEFS UIF WBMVBUJPO v TVDI UIBU v(Y) = 3 BOE v(Z) = 13 Z JT OPU HSFBUFS UIBO 2Y + 8 XIJDI XF XSJUF v ̸| (2Y + 8) ≤ Z )FODF B WBMVBUJPO JT TJNQMZ BO JOUFSQSFUBUJPO GPS WBSJBCMFT BOE UFSNT UIBU DBO NBLF USVF PS GBMTF SFMBUJPOTIJQT CFUXFFO UFSNT "UPNJD GPSNVMBF BSF CVJMU GSPN UFSNT CZ FYQSFTTJOH SFMBUJPOTIJQT CFUXFFO UXP UFSNT BOE UIFZ GPSN UIF NPTU FMFNFOUBSZ JOTUBODFT PG GPSNVMBF 'PS FYBNQMF (2Y + 8) ≤ Z JT BO BUPNJD GPSNVMB BOE JU DBOOPU CF CSPLFO EPXO JOUP TUSJDUMZ TNBMMFS BUPNJD GPSNVMBF "UPNJD GPSNVMBF BSF EF੗OFE BT FYQSFTTJPOT PG UIF GPSN t ≤ t′ 8F BMTP JODMVEF UIF USVUI DPOTUBOUT ⊤ USVF BOE ⊥ GBMTF CFDBVTF UIJT DBO CF VTFGVM TPNFUJNFT य़F USVUI WBMVF PG BO BUPNJD GPSNVMB VOEFS B WBMVBUJPO v JT RVJUF FBTZ UP EF੗OF XIFO ۜ≤‫ ۝‬JT JOUFSQSFUFE BT UIF TUBOEBSE PSEFSJOH v | t ≤ t′ EFG ⇔ v(t) ≤ v(t′ ) /PUF UIBU UIF ੗STU PDDVSSFODF PG ۜ≤‫ ۝‬SFGFST UP B TZNCPM VTFE GPS EF੗OJOH BUPNJD GPSNVMBF XIFSFBT JUT TFDPOE PDDVSSFODF SFGFST UP UIF PSEFSJOH SFMBUJPO PO UIF TFU PG OBUVSBM OVNCFST )FODF UIBU TZNCPM JT DMFBSMZ PWFSMPBEFE CVU UIJT TIPVME OPU DBVTF BOZ DPOGVTJPO JO UIF GPMMPXJOH BOE XF CFIBWF TJNJMBSMZ XJUI PUIFS XFMMVOEFSTUPPE TZNCPMT 'PSNVMBF JO '0(N) DPOUBJOT BUPNJD GPSNVMBF BOE NPSF DPNQMFY GPSNVMBF DBO CF HFOFSBUFE CZ DPNQPTJUJPO XJUI OFHBUJPO XSJ॒FO ¬ BT B VOBSZ DPOOFDUJWF. DPOKVODUJPO XSJ॒FO ∧ BT B CJOBSZ DPOOFDUJWF BOE EJTKVODUJPO XSJ॒FO ∨ BT B CJOBSZ DPOOFDUJWF 'PS FYBNQMF IFSF JT B GPSNVMB UIBU TUBUFT UIBU 2Y + 8 BOE Z IBWF JEFOUJDBM WBMVF ((2Y + 8) ≤ Z) ∧ (Z ≤ (2Y + 8)). #PPMFBO DPOOFDUJWFT TVDI BT OFHBUJPO DPOKVODUJPO PS EJTKVODUJPO BMMPX UP FYQSFTT SJDIFS QSPQFSUJFT UIBO UIPTF CZ UIF BUPNJD GPSNVMBF POMZ *O UIF GPMMPXJOH XF VTF BCCSFWJBUJPOT JO PSEFS UP TJNQMJGZ UIF QSFTFOUBUJPO PG GPSNVMBF t BOE t′ BSF BSCJUSBSZ UFSNT t = t′ t < t′ t ≥ t′ t > t′. EFG. = EFG = EFG = EFG =. (t ≤ t′ ) ∧ (t′ ≤ t) t + 1 ≤ t′ t′ ≤ t t′ + 1 ≤ t. 'PSNVMBF DBO CF BMTP PCUBJOFE CZ VTJOH RVBOUJ੗FST TVDI BT ‫ ۝∃۝‬FYJTUFOUJBM RVBO UJ੗FS BOE ‫ ۝∀۝‬VOJWFSTBM RVBOUJ੗FS 8F IBWF TFFO UIBU (2Y + 8) ≤ Z JT BO BUPNJD GPSNVMB TUBUJOH B QSPQFSUZ CFUXFO Y BOE Z 8F DBO CVJME B GPSNVMB CZ JOUSPEVDJOH BO FYJTUFOUJBM RVBOUJ੗DBUJPO QSPWJEJOH UIF GPSNVMB ∃ Z (2Y + 8) ≤ Z UIBU TUBUFT B QSPQFSUZ BCPVU UIF WBSJBCMF Y OBNFMZ UIBU UIFSF FYJTUT B WBMVF GPS Z GPS XIJDI (2Y + 8) ≤ Z IPMET USVF 0CTFSWF UIBU UIF RVBOUJ੗FS ‫ ۝∃۝‬JT GPMMPXFE CZ B WBSJBCMF.

(8) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . IFSF Z PWFS XIJDI UIF RVBOUJ੗DBUJPO JT QFSGPSNFE *U XPVME OPU NBLF TFOTF UP VTF B RVBOUJ੗FS UIBU JT OPU GPMMPXFE CZ B WBSJBCMF य़JT QSPDFTT PG CVJMEJOH GPSNV MBF DBO CF SFQFBUFE GPS JOTUBODF CZ BEEJOH B VOJWFSTBM RVBOUJ੗DBUJPO MFBEJOH UP B GPSNVMB TVDI BT ∀ Y (∃ Z ((2Y + 8) ≤ Z)) य़JT MB॒FS FYQSFTTJPO TUBUFT B QSPQFSUZ BCPVU UIF TUSVDUVSF ⟨N, +, ≤⟩ SBUIFS UIBO B QSPQFSUZ BCPVU WBSJBCMFT GPS FWFSZ WBMVF GPS Y UIFSF JT B WBMVF GPS Z UIBU TBUJT੗FT (2Y + 8) ≤ Z XIJDI JT B QSF॒Z WBMJE TUBUFNFOU PO UIF TUSVDUVSF ⟨N, +, ≤⟩ 4P GPSNVMBF JO '0(N) BSF TZOUBDUJD PCKFDUT CVJMU GSPN BUPNJD GPSNVMBF PCUBJOFE CZ DPNCJOJOH #PPMFBO DPOOFDUJWFT TVDI BT ¬ ∧ BOE ∨ BOE RVBOUJ੗FST TVDI BT ∃ BOE ∀ य़F BCJMJUZ UP DPNCJOF BUPNJD GPSNVMBF XJUI MPHJDBM DPOOFDUJWFT UP HFOFSBUF NPSF DPNQMFY GPSNVMBF FYQMBJOT XIZ 1SFTCVSHFS BSJUINFUJD JT B WFSZ SJDI MBOHVBHF NPSF BCPVU JU DBO CF GPVOE JO UIF GPMMPXJOH 5P TVN VQ GPSNVMBF VTVBMMZ XSJ॒FO XJUI UIF TZNCPMT ϕ ψ PS χ QPTTJCMZ EFDPSBUFE XJUI FYQPOFOUT PS TVCTDSJQUT BSF EF੗OFE CZ UIF HSBNNBS CFMPX ϕ ::= ⊤ | ⊥ | t ≤ t′ | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ∃Y ϕ | ∀Y ϕ XIFSF t BOE t′ BSF UFSNT BOE Y ∈ 7"3 #Z XBZ PG FYBNQMF IFSF JT B GPSNVMB TUBUJOH UIBU UIFSF JT BO JO੗OJUF OVNCFS PG NVMUJQMFT PG UISFF ∀ Y (∃ Z (Z > Y) ∧ (∃ [ (Z = 3[))). य़F TFNBOUJDT GPS GPSNVMBF JO '0(N) JT NBJOMZ EF੗OFE XJUI UIF IFMQ PG UIF TBUJTGBDUJPO SFMBUJPO CFMPX XSJ॒FO | UIBU EFUFSNJOFT UIF DPOEJUJPOT GPS UIF TBU JTGBDUJPO PG B GPSNVMB VOEFS B HJWFO WBMVBUJPO 8F XSJUF v | ϕ UP EFOPUF UIBU UIF GPSNVMB ϕ IPMET USVF VOEFS UIF WBMVBUJPO v 8F IBWF BMSFBEZ TFFO IPX UP DPNQVUF UIF USVUI WBMVF GPS v | t ≤ t′ JF XIFO BUPNJD GPSNVMBF BSF JOWPMWFE #FMPX XF FYUFOE UIF EF੗OJUJPO PG | UP BMM GPSNVMBF CZ UBLJOH BEWBOUBHF PG UIF EFDPNQPTJUJPO JO UFSNT PG TVCGPSNVMBF • v | ⊤ ⇔ USVF EFG. • v |⊥ ⇔ GBMTF EFG. EFG. • v | t ≤ t′ ⇔ v(t) ≤ v(t′ ) • v | ¬ϕ ⇔ OPU v | ϕ EFG. • v | ϕ ∧ ϕ′ ⇔ v | ϕ BOE v | ϕ′ EFG. • v | ϕ ∨ ϕ′ ⇔ v | ϕ PS v | ϕ′ EFG. • v | ∃ Y ϕ ⇔ UIFSF JT n ∈ N TVDI UIBU v[Y 0→ n] | ϕ XIFSF v[Y 0→ n] JT FRVBM UP v FYDFQU UIBU Y JT NBQQFE UP n EFG. • v | ∀ Y ϕ ⇔ GPS FWFSZ n ∈ N XF IBWF v[Y 0→ n] | ϕ EFG.

(9) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. *U JT UIF QMBDF XIFSF UIF TZNCPMT ¬ ∧ ∨ ∃ BOE ∀ BSF TFNBOUJDBMMZ CPVOE BOE JU JT FBTZ UP WFSJGZ UIBU JOEFFE ¬ JT VOEFSTUPPE BT B OFHBUJPO PQFSBUPS ∃ JT VOEFSTUPPE BT BO FYJTUFOUJBM RVBOUJ੗DBUJPO FUD *U JT BMTP UIF JOUFSQSFUBUJPO PG UIF MPHJDBM DPOOFDUJWFT UIBU HVBSBOUFFT UIBU v | t = t′ XIFSF ‫۝‬t = t′ ‫ ۝‬JT BO BCCSFWJBUJPO J੖ v(t) = v(t′ ) NBLJOH TFOTF UP UIF DIPJDF PG UIF BCCSFWJBUJPO " TJNJMBS SFBTPOJOH DBO CF EPOF XJUI UIF PUIFS BCCSFWJBUJPOT JOUSPEVDFE FBSMJFS .PSF TIPSUDVUT XJMM CF JOUSPEVDFE BOE VTFE JO UIJT EPDVNFOU CVU CFGPSF EPJOH TP MFU VT QSFTFOU B TJNQMF FYBNQMF &YBNQMF -FU ϕUPU CF UIF GPSNVMB TUBUJOH UIBU ⟨N, <⟩ JT B MJOFBSMZ PSEFSFE TFU BCCSFWJBUJPOT BSF VTFE EFG. ϕUPU = ∀ Y ∀ Z ((Y = Z) ∨ (Y < Z) ∨ (Y > Z)). *U JT XPSUI OPUJOH UIBU GPS FWFSZ WBMVBUJPO v XF IBWF v | (Y = Z) ∨ (Y < Z) ∨ (Y > Z), XIJDI JT UIF LFZ BSHVNFOU UP TFF UIBU GPS FWFSZ WBMVBUJPO v XF IBWF v | ϕUPU . य़F TBUJTGBDUJPO SFMBUJPO JT EF੗OFE JO TVDI B XBZ UIBU GPSNVMBF UIBU BSF TZO UBDUJDBMMZ EJ੖FSFOU DBO CF TBUJT੗FE CZ FYBDUMZ UIF TBNF WBMVBUJPOT 'PS JOTUBODF ∃ Y ϕ BOE ¬∀ Y ¬ϕ DBO CF TIPXO UP CF TBUJT੗FE CZ FYBDUMZ UIF TBNF WBMVBUJPOT XIBUFWFS UIF GPSNVMB ϕ JT य़JT XJUOFTTFT UIF TUBOEBSE EVBMJUZ CFUXFFO FYJTUFOUJBM RVBOUJ੗DBUJPO BOE VOJWFSTBM RVBOUJ੗DBUJPO 4JNJMBSMZ ϕ1 ∧ ϕ2 BOE ¬(¬ϕ1 ∨ ¬ϕ2 ) BSF TBUJT੗FE CZ FYBDUMZ UIF TBNF WBMVBUJPOT LOPXO BT POF PG %F .PSHBO‫۝‬T MBXT *O B TFOTF UIJT JMMVTUSBUFT B TMJHIU SFEVOEBODZ JO PVS DIPJDF GPS MPHJDBM DPOOFD UJWFT CVU JT IBT UIF HSFBU BEWBOUBHF UP VTF NPSF DPODJTF GPSNVMBF XJUI IPQFGVMMZ B NPSF TUSBJHIUGPSXBSE VOEFSTUBOEJOH य़FTF PCTFSWBUJPOT NPUJWBUF UIF GPMMPX EFG JOH EF੗OJUJPO ϕ BOE ψ BSF FRVJWBMFOU JO '0(N) ⇔ GPS FWFSZ WBMVBUJPO v XF IBWF v | ϕ J੖ v | ψ 4P ∃ Y ϕ BOE ¬∀ Y ¬ϕ BSF FRVJWBMFOU GPSNVMBF 4JNJMBSMZ ∀ Y ∃ Z (Z < Y) BOE ∀ Y ∃ Z (Y ≤ Z) DBO CF TIPXO BT OPOFRVJWBMFOU 8F IBWF EF੗OFE GPSNBMMZ IPX UP DPOTUSVDU UFSNT BUPNJD GPSNVMBF BOE GPS NVMBF BOE UIFSF JT B DMFBS EJTDJQMJOF UP HFOFSBUF TVDI TZOUBDUJD PCKFDUT )PXFWFS JO TPNF QMBDFT XF NJHIU CF JODMJOFE UP XSJUF FYQSFTTJPOT B CJU EJ੖FSFOUMZ JO PS EFS UP DPNQMZ XJUI TUBOEBSE NBUIFNBUJDBM BOE MPHJDBM QSBDUJDF PS UP VTF TIPSUDVUT BWPJEJOH MFOHUIZ EFWFMPQNFOUT #FMPX XF MJTU TUBOEBSE BCCSFWJBUJPOT • " GPSNVMB PG UIF GPSN ∀ Y1 · · · ∀ Yn ϕ JT BMTP XSJ॒FO ∀ Y1 , . . . , Yn ϕ. • 8F NBLF VTF PG TUBOEBSE #PPMFBO DPOOFDUJWFT UIBU DBO CF EF੗OFE GSPN OFHBUJPO DPOKVODUJPO BOE EJTKVODUJPO TVDI BT NBUFSJBM JNQMJDBUJPO BOE FRVJW BMFODF.

(10) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . – य़F FYQSFTTJPO ϕ ⇒ ψ JT BO BCCSFWJBUJPO GPS UIF GPSNVMB (¬ϕ) ∨ ψ *U JT FBTZ UP DIFDL UIBU GPS FWFSZ WBMVBUJPO v v | ϕ ⇒ ψ J੖ v | ϕ JNQMJFT v | ψ – "OBMPHPVTMZ UIF FYQSFTTJPO ϕ ⇔ ψ JT BO BCCSFWJBUJPO GPS UIF GPSNVMB (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ) /PUF UIBU GPS FWFSZ WBMVBUJPO v v | ϕ ⇔ ψ J੖ v | ϕ J੖ v | ψ 4P ϕ BOE ψ BSF FRVJWBMFOU GPSNVMBF XIFOFWFS GPS FWFSZ WBMVBUJPO v XF IBWF v | ϕ ⇔ ψ. • "OPUIFS VTFGVM BCCSFWJBUJPO DPOTJTUT JO XSJUJOH ∀≤k Y ϕ. XJUI TPNF k ≥ 0 JO QMBDF PG ∀ Y (Y ≤ k) ⇒ ϕ. य़F JOUFOEFE TFNBOUJDT PG TVDI B DPOTUSVDUJPO JT JOEFFE DMFBS TJODF GPS FWFSZ WBMVBUJPO v v | ∀≤k Y ϕ J੖ GPS FWFSZ n ≤ k XF IBWF v[Y 0→ n] | ϕ 4JNJMBSMZ XF XSJUF ∃≤k Y ϕ XJUI TPNF k ≥ 0 JO QMBDF PG ∃ Y (Y ≤ k) ∧ ϕ. • 4USJDUMZ TQFBLJOH UIF DPOTUSVDUJPO PG UIF UFSNT EPFT OPU BMMPX UP VTF OFHB UJWF JOUFHFST TVDI BT JO UIF FYQSFTTJPO −2Y+3Z−8 *OEFFE UIF TVNNBUJPO PG WBSJBCMFT BOE UIF TVNNBUJPO PG UIF DPOTUBOU POF EP OPU QFSNJU UP PCUBJO OFHBUJWF DPFਖ਼DJFOUT )PXFWFS JO BUPNJD GPSNVMBF XF TIBMM QFSNJU UIF VTF PG OFHBUJWF DPFਖ਼DJFOUT CFDBVTF JU JT JOEFFE FRVJWBMFOU UP BO FYQSFTTJPO SF TQFDUJOH TUSJDUMZ PVS TZOUBY 'PS JOTUBODF UIF FYQSFTTJPO −2Y+3Z−8 ≤ 5Y JT VOEFSTUPPE BT UIF BUPNJD GPSNVMB 3Z ≤ 7Y+8 4JNJMBSMZ UFSNT BSF VOEFS TUPPE NPEVMP BTTPDJBUJWJUZ BOE DPNNVUBUJWJUZ GPS JOTUBODF JO B GPSNVMB XF EP OPU EJTUJOHVJTI UIF UFSN 7Y + 8 GSPN 3Y + 8 + 4Y PS 2 + 6Y + 6 + Y *O UIF GPMMPXJOH UIF WFSZ QSFTFOUBUJPO PG TQFDJ੗D BUPNJD GPSNVMBF BOE UFSNT JT HVJEFE CZ UIF NPTU IBOEZ XBZ UP FYQSFTT UIFN • -FU VT DPODMVEF UIJT QBSU BCPVU BCCSFWJBUJPOT CZ JOUSPEVDJOH B TIPSUDVU UP FYQSFTT NPEVMP DPOTUSBJOUT 8F IBWF TFFO UIBU UIF GPSNVMB ∃ Z (Y = 3Z) TUBUFT UIBU Y JT JOUFSQSFUFE CZ B NVMUJQMF PG UISFF *U JT DMFBSMZ NPSF DPOWF OJFOU JG XF DPVME TJNQMZ XSJUF Y ≡3 0 य़BU JT UIF SFBTPO XIZ XF JOUSPEVDF UIF GPMMPXJOH TIPSUDVU 8F XSJUF t ≡k t′ XIFSF t BOE t′ BSF BSCJUSBSZ UFSNT BOE k ≥ 0 UP EFOPUF UIF GPSNVMB ∃ Y (t = kY + t′ ) ∨ (t′ = kY + t). XIFSF Y JT B OFX WBSJBCMF OPU PDDVSSJOH JO t BOE t′ /PUF UIBU UIF DBTFT XIFO k UBLFT UIF WBMVF ≡0 DPSSFTQPOET UP UIF FRVBMJUZ BOE t ≡1 t′ DPSSFTQPOET UP ⊤ .PEVMP DPOTUSBJOUT BSF OPU POMZ IBOEZ NBDSPT CVU UIFZ XJMM QMBZ BMTP BO JNQPSUBOU SPMF JO 4FDUJPO XIFO EFBMJOH XJUI RVBOUJ੗FS FMJNJOBUJPO य़F DPNCJOBUJPO PG BMM UIFTF BCCSFWJBUJPOT BMMPXT VT UP XSJUF UIF FYQSFTTJPO CFMPX BT B GPSNVMB GSPN '0(N) ∀ Y, Z (−2Y + 9 ≡4 Z + 1) ⇔ (−Z ≡4 2Y − 8).

(11) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD . %५३९ॹ९ॵॴ ॶॸॵ२ॲ५ॳॹ. य़F NBJO EFDJTJPO QSPCMFN GPS '0(N) JT UP EFUFSNJOF XIFUIFS B GPSNVMB BENJUT B WBMVBUJPO UIBU TBUJT੗FT JU य़F BUPNJD GPSNVMB Y + 3 ≤ Z BENJUT TVDI B WBMVBUJPO v GPS JOTUBODF CZ SFRVJSJOH UIBU v(Y) = 12 BOE v(Z) = 2100 य़F JOUFSQSFUBUJPO PG UIF WBSJBCMFT EJ੖FSFOU GSPN Y BOE Z JT OPU SFBMMZ SFMFWBOU UP DIFDL XIFUIFS B WBMVBUJPO TBUJT੗FT Y + 3 ≤ Z /PUF BMTP UIBU UIF BUPNJD GPSNVMB DBO DFSUBJOMZ CF TBUJT੗FE CZ B WBMVBUJPO GPS XIJDI UIF WBMVFT GPS v(Y) BOE v(Z) BSF NVDI TNBMMFS 4P UIF TBUJTटBCJMJUZ QSPCMFN GPS 1SFTCVSHFS BSJUINFUJD JT B EFDJTJPO QSPCMFN UIBU UBLFT BT JOQVU B GPSNVMB ϕ BOE BTLT XIFUIFS UIFSF JT B WBMVBUJPO v TVDI UIBU v | ϕ *G TVDI B WBMVBUJPO FYJTUT XF TBZ UIBU ϕ JT TBUJTटBCMF #Z XBZ PG FYBNQMF MFU VT DPOTJEFS UIF GPSNVMB ϕFYQ CFMPX EFG. ϕFYQ = (Y1 ≥ 2) ∧ (Y2 ≥ 2Y1 ) ∧ · · · ∧ (Yn ≥ 2Yn−1 ). *U JT DMFBSMZ TBUJT੗BCMF CVU BOZ WBMVBUJPO UIBU TBUJT੗FT JU SFRVJSFT BU MFBTU UIBU UIF JOUFSQSFUBUJPO PG Yn JT HSFBUFS UIBO 2n FWFO UIPVHI UIF GPSNVMB JT POMZ PG TJ[F O(n) 4BUJT੗BCJMJUZ DPNFT XJUI JUT EVBM DPODFQU LOPXO BT WBMJEJUZ TP UIBU UIF FYJTUFO UJBM RVBOUJ੗DBUJPO PWFS WBMVBUJPOT JO UIF EF੗OJUJPO GPS UIF TBUJT੗BCJMJUZ QSPCMFN JT SFQMBDFE CZ B VOJWFSTBM RVBOUJ੗DBUJPO य़F WBMJEJUZ QSPCMFN GPS 1SFTCVSHFS BSJUI NFUJD JT B EFDJTJPO QSPCMFN UIBU UBLFT BT JOQVU B GPSNVMB ϕ BOE BTLT XIFUIFS GPS FWFSZ WBMVBUJPO v XF IBWF v | ϕ *G ϕ JT TBUJT੗FE CZ FWFSZ WBMVBUJPO XF TBZ UIBU ϕ JT WBMJE य़F BCPWF GPSNVMB ϕFYQ JT DMFBSMZ OPU WBMJE TJODF BOZ WBMVBUJPO TVDI UIBU Y1 JT JOUFSQSFUFE CZ [FSP GBMTJ੗FT ϕFYQ #Z DPOUSBTU UIF GPSNVMB CFMPX JT WBMJE (Y1 ≥ 2 ∧ Y2 ≥ 2Y1 ∧ · · · ∧ Yn ≥ 2Yn−1 ) ⇒ Yn ≥ 2n .. &WFO UIPVHI WBMJEJUZ BOE TBUJT੗BCJMJUZ QSPCMFNT BSF ੗STUDMBTT QSPCMFNT GPS '0 N CFMPX XF FYQMBJO UIF DMPTF SFMBUJPOTIJQT CFUXFFO UIFN TP UIBU JO UIF TF RVFM XF EP OPU OFFE UP SFBMMZ DPOTJEFS WBMJEJUZ BU BMM 'PS JOTUBODF JU JT TUSBJHIU GPSXBSE UIBU FWFSZ WBMJE GPSNVMB JT TBUJT੗BCMF TJODF B WBMJE GPSNVMB JT TBUJT੗FE CZ FWFSZ WBMVBUJPO 8F IBWF BMTP TFFO UIBU TBUJT੗BCJMJUZ EPFT OPU JNQMZ WBMJEJUZ TFF FH UIF GPSNVMB ϕFYQ XIJDI TIPVME OPU DPNF BT B TVSQSJTF TJODF TBUJT੗B CJMJUZ BTLT GPS UIF TBUJTGBDUJPO CZ B TJOHMF WBMVBUJPO य़FSF BSF BMTP GPSNVMBF GPS XIJDI TBUJT੗BCJMJUZ JT FRVJWBMFOU UP WBMJEJUZ BOE JU JT UIF QVSQPTF PG UIF GPMMPXJOH QBSBHSBQI "O PDDVSSFODF PG UIF WBSJBCMF Y JO UIF GPSNVMB ϕ JT GSFF JG JU EPFT OPU PDDVS JO UIF TDPQF PG FJUIFS ∃ Y PS ∀ Y 'PS JOTUBODF BMM UIF PDDVSSFODFT PG UIF WBSJBCMFT JO ϕFYQ BSF GSFF TJODF JU EPFT OPU DPOUBJO BOZ RVBOUJ੗DBUJPO 0UIFSXJTF UIF PD DVSSFODF JT CPVOE 'PS JOTUBODF JO Y1 < Y2 BMM UIF PDDVSSFODFT PG UIF WBSJBCMFT BSF GSFF *O (∃ Y1 , Y2 (Y1 < Y2 )) ∧ Y1 < Y2 FBDI WBSJBCMF BNPOH Y1 Y2 IBT B GSFF PDDVSSFODF BOE B CPVOE PDDVSSFODF (JWFO B GPSNVMB ϕ XF XSJUF f ree(ϕ) UP EFOPUF UIF GSFF WBSJBCMFT PDDVSSJOH JO ϕ.

(12) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . -FNNB CFMPX TUBUFT IPX UP SFEVDF TJNQMZ BO JOTUBODF PG UIF WBMJEJUZ QSPC MFN UP BO JOTUBODF PG UIF TBUJT੗BCJMJUZ QSPCMFN BOE UIF PUIFS XBZ BSPVOE फ़BO UJ੗DBUJPOT JO GPSNVMBF JO UIF PCKFDU MBOHVBHF BSF VTFE UP QFSGPSN RVBOUJ੗DBUJPOT PWFS WBMVBUJPOT JO UIF NFUBMBOHVBHF -FNNB -FU ϕ CF B GPSNVMB XIPTF GSFF WBSJBCMFT BSF BNPOH Y1 Yn ࡋF QSPQPTJUJPOT CFMPX BSF FRVJWBMFOU * ϕ JT WBMJE ** ∀ Y1 , . . . , Yn ϕ JT WBMJE. *** ∀ Y1 , . . . , Yn ϕ JT TBUJTटBCMF. *7 ∀ Y1 , . . . , Yn ϕ JT FRVJWBMFOU UP ⊤ 4JNJMBSMZ UIF QSPQPTJUJPOT CFMPX BSF FRVJWBMFOU 7 ϕ JT TBUJTटBCMF 7* ∃ Y1 , . . . , Yn ϕ JT WBMJE. 7** ∃ Y1 , . . . , Yn ϕ JT TBUJTटBCMF. 7*** ∃ Y1 , . . . , Yn ϕ JT FRVJWBMFOU UP ⊤. य़F QSPPG PG -FNNB JT MFॏ BT BO FYFSDJTF BOE JU VTFT UIF EVBMJUZ CFUXFFO UIF RVBOUJ੗FST ∀ BOE ∃ य़BU JT XIZ JO UIF TFRVFM XF BSF NBJOMZ JOUFSFTUFE JO UIF TBUJT੗BCJMJUZ QSPCMFN .PSFPWFS XIFO B GPSNVMB IBT OP GSFF WBSJBCMF PDDVS SFODFT TBUJT੗BCJMJUZ JT FRVJWBMFOU UP WBMJEJUZ BOE UIFSFGPSF UIF EJ੖FSFODF JT FWFO MFTT SFMFWBOU 8F XSJUF ϕ(Y1 , . . . , Yn ) UP EFOPUF B GPSNVMB XJUI GSFF WBSJBCMFT BNPOH {Y1 , . . . , Yn } GPS TPNF n ≥ 1 य़F GPSNVMB EF੗OFT UIF GPMMPXJOH TFU PG nUVQMFT EFG. !ϕ(Y1 , . . . , Yn )" = {⟨v(Y1 ), . . . , v(Yn )⟩ ∈ Nn : v | ϕ}.. य़F TFU !ϕ(Y1 , . . . , Yn )" DPOUBJOT BMM UIF UVQMFT UIBU NBLF USVF UIF GPSNVMB ϕ CZ JHOPSJOH UIF JSSFMFWBOU JOUFSQSFUBUJPO PG UIF CPVOE WBSJBCMFT BOE CZ ੗YJOH BO BSCJUSBSZ PSEFSJOH CFUXFFO UIF WBSJBCMFT *O UIBU SFTQFDU VTJOH UIF OPUBUJPO ϕ(Y1 , . . . , Yn ) IBT UIF BEWBOUBHF UP QSPWJEF BO PSEFSJOH PG UIF WBSJBCMFT 8IFO Y1 , . . . , Yn JT LOPXO XJUIPVU BNCJHVJUZ UIF TFU !ϕ(Y1 , . . . , Yn )" JT TJNQMZ EFOPUFE CZ !ϕ" 'PS JOTUBODF !Y1 < Y2 " = {⟨n, n′ ⟩ ∈ N2 : n < n′ } 4JNJMBSMZ UIF TFU PG PEE OBUVSBM OVNCFST DBO CF EF੗OFE CZ UIF GPSNVMB CFMPX ∃Z Y = Z + Z + 1.. य़F TFU {0} DBO CF EF੗OFE CZ UIF GPSNVMB Y = Y + Y 4JNJMBSMZ UIF TFU {1} JT EF੗OFE CZ ∃ Z (Z = Z + Z ∧ Y = Z + 1).

(13) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD Y2

(14)

(15) ; Y1 . Y1

(16)

(17) q1. Y1

(18)

(19) ; Y2

(20)

(21). q2. Y1

(22)

(23) ; Y2

(24)

(25). q3. Y1

(26)

(27) ; Y2

(28)

(29). Y2

(30)

(31) q4. 'JHVSF " TJNQMF DPVOUFS NBDIJOF. .PSFPWFS TBUJT੗BCJMJUZ PG ϕ JT FRVJWBMFOU UP UIF OPOFNQUJOFTT PG !ϕ" 4JNJ MBSMZ ϕ JT WBMJE J੖ !ϕ" = Nn -FU VT DPODMVEF UIJT TFDUJPO CZ B EF੗OJUJPO GPS TFUT JO Nn UIBU DBO CF EF੗OFE GSPN GPSNVMBF JO '0(N) %F॑OJUJPO 1SFTCVSHFS TFU -FU ϕ CF B GPSNVMB ϕ(Y1 , . . . , Yn ) XJUI n ≥ 1 GSFF WBSJBCMFT BNPOH Y1 Yn 8F TBZ UIBU !ϕ" JT B 1SFTCVSHFS TFU. य़F DIBSBDUFSJ[BUJPO PG 1SFTCVSHFS TFUT CZ NFBOT EJ੖FSFOU GSPN UIF VTF PG '0(N) JT EJTDVTTFE JO 4FDUJPO *U JT DMFBS UIBU 1SFTCVSHFS TFUT BSF DMPTFE VO EFS JOUFSTFDUJPO VOJPO BOE DPNQMFNFOUBUJPO CFDBVTF PG UIF QSFTFODF PG #PPMFBO DPOOFDUJWFT JO UIF MPHJDBM GPSNBMJTN '0(N) #Z XBZ PG FYBNQMF MFU VT DPOTJEFS UIF DPVOUFS NBDIJOF QSFTFOUFE JO 'JHVSF XJUI UXP DPVOUFST BOE GPVS DPOUSPM TUBUFT ∗ 8F XSJUF Xi UP EFOPUF UIF TFU {⟨n, m⟩ | ⟨q1 , 0, 0⟩ − → ⟨qi , n, m⟩} GPS FWFSZ i ∈ [1, 4] 0OF DBO TIPX UIBU BMM UIF TFUT X1 X4 JODMVEFE JO N2 BSF 1SFTCVSHFS TFUT NPSF QSFDJTFMZ X1 X2 X3 X4. = = = =. !Y1 !Y2 !Y2 !Y1. = Y2 = 0" = 1 ∧ Y1 ≥ 1" ≥ 2 ∧ Y1 + Y2 ≥ 4" ≥ 1 ∧ Y2 ≥ 3 ∧ Y1 + Y2 ≥ 6". "O BMUFSOBUJWF GPSNVMB XJUI RVBOUJ੗FST UP EF੗OF X4 JT QSPWJEFE CFMPX 'PS FBDI TFMGMPPQ B RVBOUJ੗FS WBSJBCMF JT JOUSPEVDFE UIBU DPSSFTQPOET UP UIF OVNCFS PG UJNFT UIF TFMGMPPQ JT WJTJUFE ∃ [1 , [2 , [3 (Y1 = 3 + [1 − [2 ) ∧ (Y2 = 3 + [2 + [3 ) ∧ (2 + [1 − [2 ≥ 0). . 'ॸ१७ॳ५ॴॺॹ. *O UIJT TFDUJPO XF JOUSPEVDF TFWFSBM TZOUBDUJD GSBHNFOUT PG '0(N) CZ SFTUSJDUJOH UIF VTF PG TZOUBDUJD SFTPVSDFT PS CZ JNQPTJOH DPOTUSBJOUT PO UIF GPSN PG UIF UFSNT GPSNVMBF FUD 'JSTU XF OFFE UP QSPWJEF B GFX NPSF FYQMBOBUJPOT BCPVU UIF GPSNBM EJ੖FSFODFT CFUXFFO BCCSFWJBUJPOT BOE CVJMUJO QSFEJDBUF TZNCPMT PS BUPNJD GPSNVMBF "CPWF UIF TZOUBY GPS UFSNT BUPNJD GPSNVMBF BOE GPSNVMBF JO '0(N) IBT CFFO FYUFOEFE JO PSEFS UP BENJU TFWFSBM BCCSFWJBUJPOT UIBU BSF IFMQGVM UP XSJUF GPSNVMBF 'PS.

(32) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . JOTUBODF (Y = Z) ⇒ (Z = Y) VTF BCCSFWJBUJPOT BOE TUSJDUMZ TQFBLJOH TIPVME CF VOEFSTUPPE BT UIF GPSNVMB CFMPX ¬((Y ≤ Z) ∧ (Z ≤ Y)) ∨ ((Z ≤ Y) ∧ (Y ≤ Z)).. *OEFFE ‫ ۝⇒۝‬JT OPU B CVJMUJO MPHJDBM DPOOFDUJWF XIFSFBT ‫ ۝=۝‬JT OPU UIF CVJMUJO FRVBMJUZ QSFEJDBUF TZNCPM UIFTF TZNCPMT BSF VTFE GPS UIF TBLF PG DPODJTFOFTT BOE TJNQMJDJUZ *U JT BMTP QPTTJCMF UP DPOTJEFS UIFTF TZNCPMT BT QSJNJUJWF JO UIF TZOUBY CZ FYUFOEJOH UIF TFU PG MPHJDBM DPOOFDUJWFT PS CZ FYUFOEJOH UIF TFU PG BUPNJD GPSNVMBF य़JT XPVME OPU NBLF BOZ FTTFOUJBM EJ੖FSFODF XJUI UIF DVSSFOU WFSTJPO TJODF NBUFSJBM JNQMJDBUJPO PS FRVBMJUJFT DBO CF FYQSFTTFE JO JU #Z DPOUSBTU UIF VTF PG OFX CVJMUJO TZNCPMT XPVME NBLF B EJ੖FSFODF XIFO GSBHNFOUT PG '0 N BSF JOWPMWFE CFDBVTF XF EP OPU IBWF BDDFTT UP UIF GVMM QPXFS PG UIF MPHJDBM GPSNBMJTN #Z XBZ PG FYBNQMF MFU VT DPOTJEFS UIF FYQSFTTJPO t ≡k t′ UIBU JT B TIPSUDVU GPS ∃ Y (t = kY + t′ ) ∨ (t′ = kY + t).. &WFO UIPVHI t ≡k t′ EPFT OPU NBLF BQQBSFOU UIF QSFTFODF PG RVBOUJ੗FST JU EPFT DPOUBJO BO FYJTUFOUJBM RVBOUJ੗DBUJPO )PXFWFS JU JT BMTP QPTTJCMF UP FYUFOE UIF TFU PG GPSNVMBF CZ DPOTJEFSJOH t ≡k t′ BT B OFX QSJNJUJWF BUPNJD GPSNVMB XJUI EFG UIF GPMMPXJOH TFNBOUJDT v | t ≡k t′ ⇔ v(t) JT FRVBM UP v(t′ ) NPEVMP k *O UIBU MB॒FS DBTF t ≡k t′ JT RVBOUJ੗FSGSFF #FMPX XF EF੗OF GSBHNFOUT PG '0(N) CZ JNQPTJOH TZOUBDUJD SFTUSJDUJPOT CVU BU UIF TBNF UJNF XF BEE OFX QSJNJUJWF QSFEJDBUF TZNCPMT 6OMFTT PUIFSXJTF TUBUFE XF BTTVNF UIBU UIF TFU PG QSJNJUJWF BUPNJD GPSNVMBF JT FYUFOEFE BT GPMMPXT 1"'. ⊤ | ⊥ | t ≤ t′ | t ≡k t′ | t = t′ | t < t′ | t ≥ t′ | t > t′. XJUI k ≥ 2 य़F TFNBOUJDT PG UIF OFX CJOBSZ QSFEJDBUF TZNCPMT JT XJUIPVU BOZ EFG TVSQSJTF GPS JOTUBODF v | t > t′ ⇔ v(t) > v(t′ ) 8F LOPX UIBU BEEJOH UIFTF TZNCPMT EPFT OPU JODSFBTF UIF FYQSFTTJWF QPXFS PG '0 N TJODF XF XFSF BCMF UP ੗OE FRVJWBMFOU GPSNVMBF JO UIF PSJHJOBM WFSTJPO PG '0 N 'SPN OPX PO XF BTTVNF UIBU '0(N) DPOUBJOT BMM UIF BCPWFNFOUJPOFE BUPNJD GPSNVMBF %F॑OJUJPO फ़BOUJ੗FSGSFF GSBHNFOU " GPSNVMB ϕ JT RVBOUJ੗FSGSFF ⇔ ϕ JT B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF JF XJUIPVU RVBOUJ੗FST EFG. 'PS JOTUBODF (Y + Z ≡5 [) ∨ (Z > 23) JT B RVBOUJ੗FSGSFF GPSNVMB UIBU DPOUBJOT FYBDUMZ GPVS PDDVSSSFODFT PG GSFF WBSJBCMFT *U JT DMFBS UIBU JO B RVBOUJ੗FSGSFF GPSNVMB BMM UIF WBSJBCMF PDDVSSFODFT BSF GSFF %F॑OJUJPO -JOFBS GSBHNFOU " GPSNVMB ϕ JT JO UIF MJOFBS GSBHNFOU ⇔ ϕ JT B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG POF PG UIF GPSNT CFMPX EFG. ⊤ | ⊥ | t ≤ t′ | t = t′ | t < t′ | t ≥ t′ | t > t′. -*/.

(33) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. 4P ϕ JT JO UIF MJOFBS GSBHNFOU PG '0 N XIFO JU JT JO UIF RVBOUJ੗FSGSFF GSBH NFOU BOE JU EPFT OPU DPOUBJO BOZ QFSJPEJDJUZ DPOTUSBJOU %F॑OJUJPO %J੖FSFODF GSBHNFOU " GPSNVMB ϕ JT JO UIF EJ੖FSFODF GSBHNFOU EFG ⇔ ϕ CFMPOHT UP UIF MJOFBS GSBHNFOU BOE UIF UFSNT BSF FJUIFS PG UIF GPSN Y + k PS k 'PS JOTUBODF ¬(Y = Z + 8) ∧ Z ≥ 7 CFMPOHT UP UIF EJ੖FSFODF GSBHNFOU XIFSFBT 2Y = 6 PS Y + Z ≥ 3 EP OPU *OEFFE UIF EJ੖FSFODF GSBHNFOU DBO POMZ DPNQBSF B WBSJBCMF PS B EJञFSFODF CFUXFFO UXP WBSJBCMFT XJUI B DPOTUBOU WBMVF -FU VT DPODMVEF UIF QSFTFOUBUJPO PG GSBHNFOUT CZ JOUSPEVDJOH B GSBHNFOU PG '0 N JO XIJDI RVBOUJ੗DBUJPOT BSF QFSGPSNFE BU UIF CFHJOOJOH PG UIF GPSNVMB %F॑OJUJPO 1SFOFY OPSNBM GPSN " GPSNVMB ϕ JO '0(N) JT JO QSFOFY OPSNBM GPSN XIFOFWFS JU JT PG UIF GPSN CFMPX Q1 Y1 · · · Qn Yn ψ XIFSF ψ JT JO UIF MJOFBS GSBHNFOU BOE {Q1 , . . . , Qn } ⊆ {∃, ∀} Q1 · · · Qn JT DBMMFE UIF RVBOUJटFS QSFटY BOE JUT RVBOUJ੗FS BMUFSOBUJPO JT UIF OVNCFS PG QPTJUJPOT i TVDI UIBU Qi BOE Qi+1 BSF EJ੖FSFOU &WFO UIPVHI GPSNVMBF JO QSFOFY OPSNBM GPSN TUSPOHMZ SFTUSJDU UIF VTF PG RVBO UJ੗FST POF DBO TIPX UIBU FWFSZ GPSNVMB JO '0 N JT FRVJWBMFOU UP B GPSNVMB JO QSFOFY OPSNBM GPSN य़F QSPPG JT MFॏ BT BO FYFSDJTF BOE JU JT WFSZ TJNJMBS UP UIF QSPPG PG UIF TJNJMBS QSPQFSUZ GPS DMBTTJDBM QSFEJDBUF MPHJD " ੗STU TUFQ JO UIF QSPPG DPOTJTUT JO TIPXJOH UIBU XJUIPVU BOZ MPTT PG HFOFSBMJUZ XF DBO BTTVNF UIBU UXP EJTUJODU RVBOUJ੗FS PDDVSSFODFT TVDI BT Q Y BOE Q′ Y′ JO UIF PSJHJOBM GPSNVMB WFS J੗FT UIBU Y JT B WBSJBCMF EJ੖FSFOU GSPN Y′ %VBMJUZ CFUXFFO RVBOUJ੗FST BT XFMM BT EVBMJUZ CFUXFFO MPHJDBM DPOOFDUJWFT OFFE BMTP UP CF VTFE -FU VT DPOTJEFS UIF GPSNVMB ϕ CFMPX ¬(∃ Y Y ≥ 3) ∨ (∀ Z Z ≥ 4). ϕ JT FRVJWBMFOU UP UIF GPMMPXJOH GPSNVMB JO QSFOFY OPSNBM GPSN ∀ Y ∀ Z (¬(Y ≥ 3) ∨ Z ≥ 4). %F॑OJUJPO &YUFOEFE QSFOFY OPSNBM GPSN " GPSNVMB ϕ JO '0(N) JT JO FY UFOEFE QSFOFY OPSNBM GPSN XIFOFWFS JU JT PG UIF GPSN CFMPX (Q1 )≤k1 Y1 · · · (Qn )≤kn Yn ψ XIFSF ψ JT JO UIF MJOFBS GSBHNFOU {Q1 , . . . , Qn } ⊆ {∃, ∀} BOE k1 , . . . , kn ∈ N.

(34) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . %५३९४१२ॲ९ॲ९ॺॿ ॵ६ ॺ८५ ॹ१ॺ९ॹ६९१२९ॲ९ॺॿ ॶॸॵ२ॲ५ॳ %FTJHOJOH B EFDJTJPO QSPDFEVSF GPS DIFDLJOH UIF TBUJT੗BCJMJUZ QSPCMFN GPS '0 N. JT LOPXO UP CF B EJਖ਼DVMU UBTL CFDBVTF UIF EPNBJO PG JOUFSQSFUBUJPO GPS WBSJBCMFT JT JO੗OJUF OBNFMZ UIF TFU PG OBUVSBM OVNCFST BOE UIFSFGPSF RVBOUJ੗DBUJPOT BSF QFSGPSNFE PWFS BO JO੗OJUF TFU #Z DPNQBSJTPO EFTJHOJOH B EFDJTJPO QSPDFEVSF GPS QSPQPTJUJPOBM DBMDVMVT JT NVDI FBTJFS CFDBVTF FBDI QSPQPTJUJPOBM WBSJBCMF JT JOUFS QSFUFE CZ FJUIFS USVF PS GBMTF PS FRVJWBMFOUMZ CZ FJUIFS PS BOE UIFSFGPSF UIFSF JT B ੗OJUF TFU PG SFMFWBOU QSPQPTJUJPOBM WBMVBUJPOT GPS DIFDLJOH UIF TBUJT੗BCJMJUZ TUBUVT PG B HJWFO QSPQPTJUJPOBM GPSNVMB .PSFPWFS DIFDLJOH XIFUIFS B QSPQPTJ UJPOBM WBMVBUJPO TBUJT੗FT B QSPQPTJUJPOBM GPSNVMB DBO CF EPOF JO QPMZOPNJBM UJNF XIJDI HVBSBOUFFT UIF FYJTUFODF PG B TJNQMF BMHPSJUIN GPS DIFDLJOH TBUJT੗BCJMJUZ FWFO UIPVHI JU NBZ SFRVJSF FYQPOFOUJBM UJNF JO UIF XPSTU DBTF #Z DPOUSBTU DIFDLJOH XIFUIFS B WBMVBUJPO v TBUJT੗FT B GPSNVMB ϕ FWFO XIFO UIF WBMVBUJPO JT SFTUSJDUFE UP UIF WBSJBCMFT PDDVSSJOH JO ϕ JT BT IBSE BT UIF TBUJT ੗BCJMJUZ QSPCMFN JUTFMG *OEFFE TVQQPTF UIBU ϕ DPOUBJOT UIF WBSJBCMFT Y1 Yn 8F IBWF ϕ JT TBUJT੗BCMF J੖ v0 | ∃ Y1 , . . . , Yn ϕ XIFSF v0 JT B WBMVBUJPO TVDI UIBU BMM UIF WBSJBCMFT GSPN ϕ BSF JOUFSQSFUFE CZ [FSP फ़BOUJ੗FSGSFF GSBHNFOU PG '0(N) BENJUT B TJNQMFS TBUJT੗BCJMJUZ QSPCMFN TJODF RVBOUJ੗DBUJPOT BSF EJTBMMPXFE CVU TUJMM POF OFFET UP EFUFSNJOF XIFUIFS B WBMVB UJPO DBO TBUJTGZ ! UIF GPSNVMB "TTVNF UIBU UFSNT JO RVBOUJ੗FSGSFF GPSNVMBF DBO CF XSJ॒FO BT ( i ai Yi )+k XIFSF UIF ai ‫۝‬T BOE k CFMPOH UP N BOE UIF OBUVSBM OVNCFST BSF FODPEFE JO CJOBSZ य़JT JT JOEFFE B QMBDF XIFSF XF IBWF UP CF QSFDJTF BCPVU UIF FODPEJOHT CVU OPUF UIBU UIF DVSSFOU PQUJPO JT UP FODPEF DPODJTFMZ UIF OBUVSBM OVNCFST य़F TJ[F PG B GPSNVMB XSJ॒FO |ϕ| JT SPVHIMZ EF੗OFE BT UIF OVNCFS PG TZNCPMT PDDVSJOH JO JU XIJDI JT B SFBTPOBCMZ TVDDJODU FODPEJOH ࢊFPSFN -FU ϕ CF B RVBOUJटFSGSFF GPSNVMB XJUI WBSJBCMFT Y1 Yn ࡋF GPSNVMB ϕ JT TBUJTटBCMF Jञ UIFSF JT B WBMVBUJPO TBUJTGZJOH ϕ BOE FBDI WBSJBCMF JT JOUFSQSFUFE CZ B OBUVSBM OVNCFS XIPTF WBMVF JT CPVOEFE CZ 2p(|ϕ|) XIFSF p(·) JT B QPMZOPNJBM JOEFQFOEFOU PG ϕ BOE Y1 Yn य़FPSFN TUBUFT UIBU UP EFUFSNJOF UIF TBUJT੗BCJMJUZ TUBUVT PG B RVBOUJ੗FS GSFF GPSNVMB JU JT TVਖ਼DJFOU UP JOTQFDU WBMVFT JOUFSQSFUJOH WBSJBCMFT CFMPX B CPVOE 2p(|ϕ|) TFF CJCMJPHSBQIJDBM SFGFSFODFT JO 4FDUJPO *U JT QPTTJCMF UP SF੗OF UIJT CPVOE CZ UBLJOH JOUP BDDPVOU JO B NPSF QSFDJTF XBZ UIF OVNCFS PG WBSJBCMFT UIF NBYJNBM TJ[F PG B DPOTUBOU PDDVSSJOH JO ϕ PS UIF OVNCFS PG DPOOFDUJWF PDDVS SFODFT XJUI B DPOKVODUJWF QPMBSJUZ .PSFPWFS UIJT JT TVਖ਼DJFOU UP QSPWF UIF /1 VQQFS CPVOE $PSPMMBSZ ࡋF TBUJTटBCJMJUZ QSPCMFN GPS UIF RVBOUJटFSGSFF GSBHNFOU PG '0(N) JT /1DPNQMFUF /1IBSEOFTT DBO CF FBTJMZ TIPXO CZ EF੗OJOH B TUSBJHIUGPSXBSE SFEVDUJPO GSPN UIF TBUJT੗BCJMJUZ QSPCMFN GPS QSPQPTJUJPOBM DBMDVMVT LOPXO BT 4"5 *OEFFE.

(35) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. HJWFO B GPSNVMB ϕ XJUI QSPQPTJUJPOBM WBSJBCMFT p1 pn POF DBO CVJME B RVBOUJ੗FS GSFF GPSNVMB ϕ′ PCUBJOFE GSPN ϕ CZ SFQMBDJOH pi CZ YOFX = ZOFX TVDI UIBU ϕ JT i i OFX ‫۝‬T BSF OFX BOE UIFZ TBUJT੗BCMF J੖ ϕ′ JT TBUJT੗BCMF य़F WBSJBCMFT YOFX ‫۝‬T BOE Z i i TFSWF UIF QVSQPTF PG FODPEJOH QSPQPTJUJPOBM WBSJBCMFT *O PSEFS UP PCUBJO UIF /1 VQQFS CPVOE JU JT TVਖ਼DJFOU UP HVFTT ⟨α1 , . . . , αn ⟩ ∈ [0, 2p(|ϕ|) ]n BOE UIFO DIFDL UIBU v | ϕ XIFSF v(Yi ) = αi GPS FWFSZ i ∈ [1, n] 4VDI B DIFDL TIPVME CF EPOF JO QPMZOPNJBM UJNF JO UIF TJ[F PG UIF GPSNVMB XIJDI JT UIF DBTF GPS UIF GPMMPXJOH SFBTPOT ⟨α1 , . . . , αn ⟩ JT PG QPMZOPNJBM TJ[F JO |ϕ|. $PNQVUJOH v(t) GPS BOZ UFSN t JO ϕ DBO CF EPOF JO QPMZOPNJBM UJNF JO |ϕ|. 4JNJMBSMZ EFUFSNJOJOH UIF USVUI WBMVF PG BOZ BUPNJD GPSNVMB VOEFS v DBO CF EPOF JO QPMZOPNJBM UJNF JO |ϕ| 'JOBMMZ SFQMBDJOH BMM UIF BUPNJD GPSNVMBF GSPN ϕ CZ FJUIFS ⊤ PS ⊥ EFQFOE JOH PO UIF USVUI WBMVF VOEFS v BOE UIFO TJNQMJGZJOH VTJOH UIF TFNBOUJDT PG #PPMFBO DPOOFDUJWFT MFBET UP ⊤ PS ⊥ BOE UIJT DBO CF EPOF JO QPMZOPNJBM UJNF. )FODF UIF WFSJ੗DBUJPO QIBTJT GPS v | ϕ SFRVJSFT POMZ QPMZOPNJBM UJNF &WFO UIPVHI UIF WBMVFT JO v DBO CF PG FYQPOFOUJBM NBHOJUVEF XJUI SFTQFDU UP UIF TJ[F PG ϕ UIFJS FODPEJOH SFRVJSFT POMZ QPMZOPNJBM TQBDF XIJDI JT VTFGVM UP QFSGPSN UIF ੗OBM DIFDL JO QPMZOPNJBM UJNF -FU VT DPNF CBDL UP PVS PSJHJOBM NBJO RVFTUJPO 8IBU BCPVU UIF EFDJEBCJMJUZ TUBUVT PG UIF TBUJT੗BCJMJUZ QSPCMFN GPS '0(N) *U XPVME CF OJDF JG य़FPSFN DBO CF BEBQUFE UP GVMM 1SFTCVSHFS BSJUINFUJD "DUVBMMZ JU DBO CVU XJUI NVDI IJHIFS VQQFS CPVOET BT TUBUFE MBUFS PO TFF य़FPSFN *OEFFE UIF LFZ BSHVNFOU JT UP TIPX UIBU FWFSZ GPSNVMB ϕ BENJUT BO FRVJWBMFOU RVBOUJ੗FSGSFF GPSNVMB ϕ′ TVDI UIBU ϕ′ DBO CF F੖FDUJWFMZ CVJMU GSPN ϕ &RVJWBMFODF JNQMJFT UIBU UIF TBUJT੗BCJMJUZ TUBUVT PG GPSNVMBF JT JEFOUJDBM 4JODF UIF SFEVDUJPO JT F੖FDUJWF XF DBO UIFO JOWPLF $PSPMMBSZ BOE HFU EFDJEBCJMJUZ ࢊFPSFN %FDJEBCJMJUZ ࡋF TBUJTटBCJMJUZ QSPCMFN GPS '0(N) JT EFDJEBCMF *U TIPVME OPU DPNF BT B TVSQSJTF UIBU UIF FMJNJOBUJPO PG RVBOUJ੗FST JT DPN QVUBUJPOBMMZ FYQFOTJWF BOE UIF DPOTUSVDUJPO PG ϕ′ GSPN ϕ JT EPOF WJB B MFOHUIZ QSPDFTT 4FDUJPO JT BDUVBMMZ EFEJDBUFE UP RVBOUJ੗FS FMJNJOBUJPO 8F IBWF TFFO UIBU EJTKVODUJPOT DBO CF EF੗OFE GSPN DPOKVODUJPOT BOE OFHB UJPOT BOE UIFSFGPSF UIF DPOOFDUJWF GPS EJTKVODUJPO ∨ JT B CVJMUJO DPOOFDUJWF UIBU EPFT OPU BEE BOZ FYQSFTTJWF QPXFS UP '0(N) 4JNJMBSMZ RVBOUJ੗FS FMJNJOBUJPO JNQMJFT UIBU UIF CVJMUJO RVBOUJ੗FST EP OPU BEE BOZ FYQSFTTJWF QPXFS UP '0(N).

(36) 'SPN 'PSNVMBF UP *OUFSQSFUBUJPOT. . XIFO UIF BUPNJD GPSNVMBF BSF PG UIF GPSN FJUIFS t ≤ t′ PS t ≡k t′ *OEFFE QFSJPE JDJUZ DPOTUSBJOUT FOUFS JOUP UIF QMBZ IFSF BOE UIJT JT OFDFTTBSZ BT CSJF੘Z FYQMBJOFE JO 4FDUJPO 4P FWFO UIPVHI VOJWFSTBM BOE FYJTUFOUJBM RVBOUJ੗FST EP OPU BEE BOZ OFX FYQSFTTJWF QPXFS UP UIF RVBOUJ੗FSGSFF GSBHNFOU TVDI CJOEFST BSF VTFGVM UP FYQSFTT DPODJTFMZ QSPQFSUJFT PO OBUVSBM OVNCFST य़JT JT BMTP SF੘FDUFE CZ UIF GBDU UIBU UIF TBUJT੗BCJMJUZ QSPCMFN GPS '0(N) JT DMFBSMZ OPU JO /1 BOE TPNFXIFSF CFUXFFO &ॾॶ5९ॳ५ BOE &ॾॶ4ॶ१३५ .PSF QSFDJTFMZ UIF TBUJT੗BCJMJUZ QSPCMFN GPS '0(N) JT DPNQMFUF GPS UIF DMBTT PG BMUFSOBUJOH 5VSJOH NBDIJOFT XPSLJOH JO EPV CMF FYQPOFOUJBM UJNF XJUI BU NPTU B MJOFBS BNPVOU PG BMUFSOBUJPOT य़JT MPPLT MJLF B TDBSZ TUBUFNFOU CVU UIJT QSPWJEFT B DPNQMFUF DPNQMFYJUZ DIBSBDUFSJ[BUJPO BOE NPTU PG UIF UJNF XF XJMM CF IBQQZ XJUI UIF SFTQFDUJWF CPVOET &ॾॶ5९ॳ५ BOE &ॾॶ4ॶ१३५ 0OF DBO BMTP PCTFSWF UIBU EFDJEBCJMJUZ JT B TUSBJHIUGPSXBSE DPOTFRVFODF PG RVBOUJ੗FS FMJNJOBUJPO BOE UIFSF JT OP OFFE UP JOWPLF UIF /1DPNQMFUFOFTT PG UIF RVBOUJ੗FSGSFF GSBHNFOU *OEFFE MFU ϕ CF B GPSNVMB XIPTF GSFF WBSJBCMFT BSF BNPOH Y1 Yn 8F IBWF TFFO UIBU ϕ JT TBUJT੗BCMF J੖ ∃ Y1 , . . . , Yn ϕ JT FRVJWBMFOU UP ⊤ TFF -FNNB 4BUJT੗BCJMJUZ PG ϕ JT DIFDLFE CZ FMJNJOBUJOH RVBOUJ੗FST JO UIF GPSNVMB ∃ Y1 , . . . , Yn ϕ BOE WFSJGZ UIBU UIF RVBOUJ੗FS FMJNJOBUJPO QSPDFEVSF MFBET UP ⊤ )FODF RVBOUJ੗FS FMJNJOBUJPO JT B TVਖ਼DJFOUMZ TUSPOH QSPQFSUZ UP FOUBJM EFDJEBCJMJUZ य़F QSPDFEVSF GPS FMJNJOBUJOH RVBOUJ੗FST BOE JUT JNQSPWFNFOUT JT BMTP BU UIF IFBSU PG PQUJNBM DPNQMFYJUZ SFTVMUT GPS DIFDLJOH TBUJT੗BCJMJUZ BT TUBUFE CFMPX ࢊFPSFN -FU ϕ CF B TFOUFODF JO '0(N) OP GSFF WBSJBCMFT PG UIF GPSN Q1 Y1 · · · Qs Ys ψ(Y1 , . . . , Ys ) PG MFOHUI n BOE XJUI m RVBOUJटFS BMUFSOBUJPOT TVDI UIBU ψ JT RVBOUJटFSGSFF ࡋFO JO UIF RVBOUJटDBUJPO XF DBO SFTUSJDU PVSTFMWFT UP WBMVFT CPVOEFE CZ w = 2$×n GPS TPNF DPOTUBOU $ ࡋJT NFBOT UIBU ϕ JT TBUJTटBCMF Jञ. [(s+3)m+2 ]. (Q1 )≤w Y1 · · · (Qs )≤w Ys ψ(Y1 , . . . , Ys ) JT TBUJTटBCMF #PVOEJOH UIF RVBOUJ੗DBUJPOT BMMPXT UP EFDJEF UIF TBUJT੗BCJMJUZ TUBUVT PG UIF GPSNVMB CZ SVOOJOH PWFS BMM UIF QPTTJCMF WBMVFT GPS UIF WBSJBCMFT BOE DIFDL IPX UIF NBUSJY GPSNVMB ψ(Y1 , . . . , Ys ) JT TBUJT੗FE " CJU PG DBSF JT OFFEFE CFDBVTF RVBO UJ੗FST BSF FJUIFS FYJTUFOUJBM PS VOJWFSTBM CVU B TJNQMF OPOEFUFSNJOJTUJD BMHPSJUIN DBO FBTJMZ IBOEMF UIBU 7१ॸ९१ॴॺॹ -FU '0(Z) CF UIF WBSJBOU PG '0(N) JO XIJDI WBSJBCMFT BSF JOUFSQSFUFE JO Z BOE RVBOUJ੗DBUJPOT BSF QFSGPSNFE PWFS Z '0(Z) TIBSF XJUI '0(N) BMM JUT OJDF QSPQFSUJFT EFDJEBCJMJUZ RVBOUJ੗FS FMJNJOBUJPO DPNQMFYJUZ FUD.

(37) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. ࢊFPSFN ࡋF TBUJTटBCJMJUZ QSPCMFN GPS '0(Z) JT EFDJEBCMF &YFSDJTF QSFTFOUT B SFEVDUJPO GSPN UIF TBUJT੗BCJMJUZ GPS '0(Z) UP UIF TBUJT ੗BCJMJUZ GPS '0(N) CZ QSPQPTJOH B TJNQMF FODPEJOH PG JOUFHFST XJUIJO UIF OBUVSBM OVNCFST QBSJUZ PG B OBUVSBM OVNCFS EFUFSNJOFT XIFUIFS JU FODPEFT B OFHBUJWF JOUFHFS #FTJEFT JU JT BMTP XPSUI OPUJOH UIBU UIF ੗STUPSEFS UIFPSZ PG ⟨N, ≤, ×⟩ JT EFDJE BCMF UPP LOPXO BT 4LPMFN BSJUINFUJD XIFSFBT UIF ੗STUPSEFS UIFPSZ PG ⟨N, ≤, ×, +⟩ JT VOEFDJEBCMF. . দ१ॴॺ९६९५ॸ &ॲ९ॳ९ॴ१ॺ९ॵॴ. *O UIJT TFDUJPO XF CSJF੘Z FYQMBJO XIBU JT RVBOUJ੗FS FMJNJOBUJPO BOE IPX JU DBO CF QSPWFE GPS '0(N) 3PVHIMZ TQFBLJOH RVBOUJ੗FS FMJNJOBUJPO NFBOT UIBU FWFSZ GPSNVMB JO '0(N) BENJUT BO FRVJWBMFOU GPSNVMB XJUIPVU RVBOUJ੗FST JF RVBO UJ੗FST DBO CF FMJNJOBUFE " RVBOUJ੗FSGSFF GPSNVMB DBO CF WJFXFE BT B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG UIF GPSN t ≤ t′ t ≡k t′ ⊤ PS ⊥ .PSFPWFS XF SFRVJSF UIBU UIF FRVJWBMFOU RVBOUJ੗FSGSFF WBSJBOU IBT OP NPSF GSFF WBSJBCMFT UIBO UIF PSJHJOBM GPSNVMB *O QBSUJDVMBS UIJT NFBOT UIBU JG UIF PSJHJOBM GPSNVMB JT B TFOUFODF JF XJUIPVU GSFF WBSJBCMF PDDVSSFODFT UIFO JUT FRVJWBMFOU GPSNVMB JT FJUIFS ⊤ PS ⊥ 5८५ ॳ१९ॴ ॹॺ१ॺ५ॳ५ॴॺ "U ੗STU HMBODF JU JT SBUIFS EJTUVSCJOH UP EJTDPWFS UIBU UIF RVBOUJ੗FSGSFF GSBHNFOU PG '0(N) JT BT FYQSFTTJWF BT GVMM '0(N) *OEFFE UIJT NJHIU JNQMZ UIBU RVBOUJ੗ DBUJPOT JO '0(N) BSF EVNNZ BOE UIBU UIFZ EP OPU TFSWF BOZ QVSQPTF य़JT JT OPU RVJUF UIF DBTF CFDBVTF FMJNJOBUJOH RVBOUJ੗FST DBO CF DPNQVUBUJPOOBMMZ FY QFOTJWF BOE UIFSFGPSF UIF VTF PG RVBOUJ੗FST JT KVTUJ੗FE CZ UIF OFFE UP XSJUF DPO DJTF GPSNVMBF UP FYQSFTT TJNQMF QSPQFSUJFT य़JT JT EJ੖FSFOU GSPN UIF TJUVBUJPO JO XIJDI EJTKVODUJPO DBO CF FYQSFTTFE CZ OFHBUJPO BOE DPOKVODUJPO " CF॒FS BOBM PHZ XPVME CF UP OPUF UIBU MJOFBSUJNF UFNQPSBM MPHJD -5- BOE ੗STUPSEFS MPHJD PO ωTFRVFODFT BSF FRVBMMZ FYQSFTTJWF MPHJDBM GPSNBMJTNT CZ ,BNQ‫۝‬T य़FPSFN CVU UIF UXP GPSNBMJTNT IBWF EJTUJODU DPODJTFOFTT BOE BENJU TBUJT੗BCJMJUZ QSPCMFNT XJUI WFSZ EJ੖FSFOU DPNQMFYJUZ 14ॶ१३५DPNQMFUFOFTT WFSTVT OPOFMFNFOUBSJUZ JO UIF QSFTFOU DBTF "T B TUBSUFS MFU VT QSPWJEF B GFX FYBNQMFT PG GPSNVMBF JO XIJDI RVBOUJ੗FST DBO CF FBTJMZ FMJNJOBUFE • ∃ Y (Y ≥ 3) JT FRVJWBMFOU UP ⊤. • ∃ [ (Y < [ ∧ [ < Z) JT FRVJWBMFOU UP Y + 2 ≤ Z • ∃ [ (Y < [ ∨ [ < Z) JT FRVJWBMFOU UP ⊤.

(38) फ़BOUJ੗FS &MJNJOBUJPO. . • ∀ [ (Y ≤ [ ⇒ Z ≤ [) JT FRVJWBMFOU UP Z ≤ Y • ∃ [ Y = 2[ JT FRVJWBMFOU UP Y ≡2 0 #FMPX XF QSFTFOU NPSF TZTUFNBUJD XBZT UP QSPEVDF RVBOUJ੗FSGSFF GPSNV MBF *O UIF NFBOUJNF MFU VT TFF XIZ XF OFFE NPEVMP DPOTUSBJOUT JO FRVJWBMFOU RVBOUJ੗FSGSFF GPSNVMBF BॏFS BMM ∃ [ Y = 2[ NJHIU BENJU BOPUIFS RVBOUJ੗FSGSFF GPSNVMB XJUIPVU NPEVMP DPOTUSBJOUT 8F FYQMBJO CFMPX XIZ JU JT JNQPTTJCMF -FU "5(Y) CF UIF TFU PG BUPNJD GPSNVMBF PG UIF GPSN aY + b ≤ a′ Y + b′ , XIFSF a a′ b BOE b′ BSF OBUVSBM OVNCFST #FDBVTF Y JT JOUFSQSFUFE CZ B OBUVSBM OVNCFS FWFSZ BUPNJD GPSNVMB aY + b ≤ a′ Y + b′ PG UIBU GPSN JT FRVJWBMFOU UP BO BUPNJD GPSNVMB PG UIF GPSN FJUIFS ⊤ ⊥ Y ≤ k PS Y ≥ k XJUI k ∈ N 'PS JOTUBODF 3Y + 5 ≤ Y + 8 JT FRVJWBMFOU UP Y ≤ 1 /PX POF DBO TIPX UIBU GPS FWFSZ GPSNVMB ψ UIBU JT B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG UIF GPSN FJUIFS ⊤ " ⊥ PS Y ≤ k !ψ" JT B TFU PG OBUVSBM OVNCFST FRVBM UP B टOJUF VOJPO PG JOUFSWBMT i Ii TVDI UIBU FBDI Ii JT PG UIF GPSN FJUIFS [k1 , k2 ] PS [k1 , +∞[ XJUI k1 , k2 ∈ N "T B DPOTFRVFODF ∃ [ Y = 2[ DBOOPU CF FRVJWBMFOU UP B GPSNVMB ψ PG UIF BCPWF GPSN CFDBVTF " !∃ [ Y = 2[" JT PCWJPVTMZ OPU FRVBM UP B ੗OJUF VOJPO PG JOUFSWBMT PG UIF GPSN i Ii &YFSDJTF JT EFEJDBUFE UP TVDI B QSPPG /PX MFU VT QSPWJEF UIF GPSNBM TUBUFNFOU BCPVU RVBOUJ੗FS FMJNJOBUJPO ࢊFPSFN फ़BOUJ੗FS FMJNJOBUJPO 'PS FWFSZ GPSNVMB ϕ UIFSF FYJTUT BO FRVJW BMFOU RVBOUJटFSGSFF GPSNVMB ϕ′ TVDI UIBU ϕ′ DBO CF FञFDUJWFMZ CVJMU GSPN ϕ .PSF PWFS f ree(ϕ′ ) ⊆ f ree(ϕ) *O PSEFS UP TIPX UIBU RVBOUJ੗FST DBO CF FMJNJOBUFE JU JT TVਖ਼DJFOU UP FTUBCMJTI RVBOUJ੗FS FMJNJOBUJPO GPS GPSNVMBF PG UIF GPSN ∃ Y ψ XIFSF ψ JT B RVBOUJ੗FSGSFF GPSNVMB JF B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG UIF GPSN FJUIFS t ≤ t′ PS t ≡k t′ 8F DBMM UIJT QSPQFSUZ 2&⋆ 'PS FYBNQMF MFU VT DPOTJEFS UIF GPSNVMB ϕ CFMPX ∃ Y (ψ0 (Y) ∧ (∃ Z (ψ1 (Y, Z) ∧ ∃ [ ψ2 (Y, Z, [, [′ )))) XIFSF UIF ψi ‫۝‬T BSF RVBOUJ੗FSGSFF GPSNVMBF *G ∃ [ ψ2 (Y, Z, [, [′ ) JT FRVJWBMFOU UP UIF RVBOUJ੗FSGSFF GPSNVMB ψ2′ (Y, Z, [′ ) UIFO ϕ JT FRVJWBMFOU UP ∃ Y (ψ0 (Y) ∧ (∃ Z (ψ1 (Y, Z) ∧ ψ2′ (Y, Z, [′ )))). *OEFFE UIF FRVJWBMFODF SFMBUJPO GPS GPSNVMBF JT B DPOHSVFODF GPS UIF MPHJDBM DPO OFDUJWF #PPMFBO PQFSBUPST BOE RVBOUJ੗FST "HBJO JG ∃ Z (ψ1 (Y, Z) ∧ ψ2′ (Y, Z, [′ ) JT FRVJWBMFOU UP UIF RVBOUJ੗FSGSFF GPSNVMB ψ1′ (Y, [′ ) UIFO ϕ JT FRVJWBMFOU UP ∃ Y (ψ0 (Y) ∧ ψ1′ (Y, [′ ))..

(39) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. 'JOBMMZ JG ∃ Y (ψ0 (Y)∧ψ1′ (Y, [′ )) JT FRVJWBMFOU UP UIF RVBOUJ੗FSGSFF GPSNVMB ψ0′ ([′ ) UIFO ϕ JT FRVJWBMFOU UP JU BOE ψ0′ ([′ ) JT RVBOUJ੗FSGSFF 4P HJWFO BO BSCJUSBSZ GPSNVMB ϕ JO '0(N) B XBZ UP DPNQVUF BO FRVJWBMFOU RVBOUJ੗FSGSFF GPSNVMB JT UIF GPMMPXJOH 'JSTU SFQMBDF JO ϕ FWFSZ TVCGPSNVMBF PG UIF GPSN ∀ Y ψ CZ ¬ ∃ Y ¬ψ JO PSEFS UP IBWF POMZ FYJTUFOUJBM RVBOUJ੗DBUJPOT QPTTJCMZ OFHBUFE 4BZ XF HFU UIF GPSNVMB ϕ′ *G ϕ′ JT RVBOUJ੗FSGSFF XF BSF EPOF 0UIFSXJTF QJDL BO JOOFSNPTU TVCGPSNVMB PG UIF GPSN ∃ Y χ XIFSF χ JT RVBOUJ੗FS GSFF BOE TVCTUJUVUF JU CZ BO FRVJWBMFOU RVBOUJ੗FSGSFF GPSNVMB UIBOLT UP 2&⋆. MFBEJOH UP UIF OFX GPSNVMB ϕ′′ FRVJWBMFOU UP ϕ य़F OVNCFS PG RVBOUJ੗FST JO ϕ′′ JT TUSJDUMZ MFTT UIBO UIF OVNCFS PG RVBOUJ੗FST JO ϕ′ #Z SFQFBUJOH UIJT QSPDFTT BU NPTU |ϕ| UJNFT XF DBO DPNQVUF B RVBOUJ੗FSGSFF GPSNVMB UIBU JT FRVJWBMFOU UP UIF PSJHJOBM GPSNVMB ϕ *ॴ७ॸ५४९५ॴॺॹ ॺॵ ५ॲ९ॳ९ॴ१ॺ५ ধ१ॴॺ९६९५ॸॹ #FMPX XF FYQMBJO IPX UP FMJNJOBUF RVBOUJ੗FST JO GPSNVMBF PG UIF GPSN VTFE JO 2&⋆ 5P EP TP XF TUBSU CZ B TJNQMF FYBNQMF -FU ϕ = ∃ Y ψ CF B GPSNVMB TVDI UIBU ψ JT B RVBOUJ੗FSGSFF GPSNVMB UIBU JT B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG POF PG UIF GPSNT CFMPX! ⊤ ⊥ t ≤ Y ′ ′ t ≤ t 8F BTTVNF UIBU UIF UFSNT t t BSF PG UIF FYUFOEFE GPSN ( i ai Yi ) + k XIFSF UIF ai ‫۝‬T BOE k CFMPOH UP Z BOE OPU POMZ UP N BOE Y EPFT OPU PDDVS JO UIF UFSNT t PS t′ य़JT NFBOT UIBU XF IBWF JTPMBUFE Y BU POF TJEF PG UIF JOFRVBMJUJFT 8F IBWF TFFO UIBU JU JT IBSNMFTT UP VTF TVDI FYUFOEFE UFSNT TJODF UIJT JT KVTU B TZOUBDUJD DPOWFOUJPO 4P ψ JT B GPSNVMB CVJMU GSPN UIF HSBNNBS CFMPX χ ::= ⊤ | ⊥ | t ≤ Y | t ≤ t′ | ¬χ | χ ∧ χ. †. XIFSF t t′ BSF FYUFOEFE UFSNT XJUIPVU PDDVSSFODFT PG UIF WBSJBCMF Y /PUF UIBU UIFSF JT OP GPSNVMB PG UIF GPSN t ≥ Y TJODF UIBU JT FRVJWBMFOU UP ¬(t + 1 ≤ Y) -FU VT TUBSU CZ OPUJDJOH UIBU BOZ WBSJBCMF WBMVBUJPO v : 7"3 → N DBO CF HFOFSBMJ[FE UP FYUFOEFE UFSNT TVDI UIBU # # EFG v(( ai Yi ) + k) = ( ai v(Yi )) + k. i. i. 5FSNT DBO CF JOUFSQSFUFE CZ WBMVFT JO Z CVU OPU UIF WBSJBCMFT -FU T CF UIF TFU PG UFSNT t PDDVSSJOH JO ψ FYDMVEJOH Y JO BUPNJD GPSNVMBF PG UIF GPSN t ≤ Y BOE QPTTJCMZ BVHNFOUFE XJUI 0 (JWFO B WBMVBUJPO v : 7"3 → N UIFSF JT B UFSN tMFGU ∈ T TVDI UIBU v(tMFGU ) ≤ v(Y) BOE UIFSF JT OP UFSN t JO T TVDI UIBU v(tMFGU ) < v(t) ≤ v(Y) tMFGU JT DBMMFE UIF DMPTFTU MFࡄ UFSN /PUF UIBU tMFGU BMXBZT FYJTUT CFDBVTF 0 JT JO T CVU JU NBZ OPU CF VOJRVF " LFZ PCTFSWBUJPO JT UIBU GPS BOZ WBMVF n ∈ [v(tMFGU ), v(Y)] v BOE v[Y 0→ n] WFSJ੗FT FYBDUMZ UIF TBNF BUPNJD GPSNVMBF GSPN ψ BOE UIFSFGPSF v | ψ J੖ v[Y 0→.

(40) फ़BOUJ੗FS &MJNJOBUJPO. . n] | ψ $POTFRVFOUMZ GPS UIF TBUJTGBDUJPO PG ϕ XF DBO BMXBZT BTTVNF UIBU Y JT FRVBM UP TPNF UFSN t XJUI t ∈ T #FMPX XF XSJUF ψ(Y ← t) UP EFOPUF UIF GPSNVMB PCUBJOFE GSPN ψ CZ SFQMBDJOH FWFSZ PDDVSSFODF PG Y CZ UIF UFSN t /PX POF DBO TIPX UIBU ϕ JT FRVJWBMFOU UP UIF EJTKVODUJPO CFMPX $ ψ(Y ← t) ∧ t ≥ 0. t∈T. /PUF UIBU UIF EJTKVODUJPO DBO CF DPNQVUFE JO QPMZOPNJBM UJNF JO UIF TJ[F PG ϕ *U JT BMTP XPSUI PCTFSWJOH UIBU UIF FYJTUFOUJBM RVBOUJ੗DBUJPO JT SFQMBDFE CZ B HFOFS BMJ[FE EJTKVODUJPO XIJDI JT UIF XBZ JU TIPVME CF DPODFQUVBMMZ &YBNQMF #Z XBZ PG FYBNQMF MFU VT DPOTJEFS UIF GPSNVMB ∃ [ (Y < [ ∧ [ < Z). UIBU IBT CFFO BMSFBEZ DPOTJEFSFE BU UIF CFHJOOJOH PG 4FDUJPO 'JSTU XF TMJHIUMZ VQEBUF JU UP HFU BO FRVJWBMFOU GPSNVMB ϕ = ∃ [ (Y + 1 ≤ [ ∧ ¬(Z ≤ [)). य़F TFU PG UFSNT T JT FRVBM UP {Y + 1, Z, 0} 8F HFU UIF GPMMPXJOH EJTKVODUJPO ⊤. ⊤. % &' ( % &' ( (Y + 1 ≥ 0 ∧ Y + 1 ≤ Y + 1 ∧¬(Z ≤ Y + 1))∨ ⊤. % &' ( (Z ≥ 0 ∧Y + 1 ≤ Z ∧ ¬(Z ≤ Z))∨ ' (% & ⊥. (0 ≥ 0 ∧ Y + 1 ≤ 0 ∧¬(Z ≤ 0)) ' (% & ' (% & ⊤. ⊥. XIJDI JT FRVJWBMFOU UP ¬(Z ≤ Y + 1) य़JT GPSNVMB JT FRVJWBMFOU UP Y + 2 ≤ Z. -FU VT FYUFOE B MJ॒MF CJU XIBU XF IBWF EPOF XJUI GPSNVMBF PG UIF GPSN ∃ Y ψ XIFO ψ SFTQFDUT † *OEFFE RVBOUJ੗FSGSFF GPSNVMBF BSF NVDI NPSF HFOFSBM 'PS JOTUBODF JOFRVBMJUJFT NBZ JOWPMWF UIF UFSNT aY XJUI a > 1 -FU VT DPOTJEFS ϕ = ∃ Y ψ XIFO ψ GPMMPXT UIF HSBNNBS CFMPX χ ::= ⊤ | ⊥ | t ≤ aY | t ≤ t′ | ¬χ | χ ∧ χ. ††. XIFSF t t′ BSF FYUFOEFE UFSNT XJUIPVU PDDVSSFODFT PG UIF WBSJBCMF Y BOE a ≥ 1 -FU l CF UIF MFBTU DPNNPO NVMUJQMF MDN PG BMM UIF DPFਖ਼DJFOUT PDDVSSJOH JO GSPOU UIF WBSJBCMF Y JO ψ BOE MFU ψ ′ UP EFOPUF UIF GPSNVMB PCUBJOFE GSPN ψ CZ SFQMBDJOH FWFSZ BUPNJD GPSNVMB PG UIF GPSN t ≤ aY CZ t × al ≤ lY .VMUJQMZJOH CZ l ′′ a FBDI TJEF PG BO JOFRVBMJUZ QSPWJEFT BO FRVJWBMFOU GPSNVMB /PX MFU ψ CF UIF.

(41) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. GPSNVMB PCUBJOFE GSPN ψ ′ CZ SFQMBDJOH FWFSZ PDDVSSFODF PG lY CZ Y *U JT FBTZ UP TIPX UIBU ϕ BOE ∃ Y (Y ≡l 0) ∧ ψ ′′ BSF FRVJWBMFOU )FODF XF DPOTJEFS ϕ = ∃ Y ψ XIFO ψ GPMMPXT UIF HSBNNBS CFMPX χ ::= ⊤ | ⊥ | t ≡k t′ | Y ≡k t | t ≤ Y | t ≤ t′ | ¬χ | χ ∧ χ. †††. XIFSF t t′ BSF FYUFOEFE UFSNT XJUIPVU PDDVSSFODFT PG UIF WBSJBCMF Y BOE k ≥ 1 य़JT DMFBSMZ BMMPXT VT UP EFBM XJUI UIF GPSNVMB (Y ≡l 0) ∧ ψ ′′ EFTDSJCFE BCPWF फ़BOUJ੗FSGSFF GPSNVMBF SFTQFDUJOH ††† BSF BMNPTU PG UIF HFOFSBM GPSN FYDFQU UIBU NPEVMP DPOTUSBJOUT PS JOFRVBMJUJFT NBZ JOWPMWF UIF UFSNT aY XJUI a > 1 XIJDI JT EJTBMMPXFE JO ††† 4VDI JOFRVBMJUJFT IBWF CFFO DPOTJEFSFE FBSMJFS CVU MFU VT DPOTJEFS NPEVMP DPOTUSBJOUT PG UIF GPSN aY ≡k t *G l JT UIF MDN PG BMM UIF DPFਖ਼DJFOUT PDDVSSJOH JO GSPOU UIF WBSJBCMF Y aY ≡k t JT FRVJWBMFOU UP lY ≡(k× l ) al t a BOE UIFSFGPSF XF DBO QSPDFFE BT BCPWF /PX MFU VT FYQMBJO IPX UP DPNQVUF B RVBOUJ੗FSGSFF GPSNVMB FRVJWBMFOU UP ∃ Y ψ XIFO ψ CFMPOHT UP ††† XF LOPX OPX UIBU JU FOUBJMT 2&⋆ BOE UIFSFGPSF RVBOUJ੗FS FMJNJOBUJPO GPS '0(N) BT XFMM BT JUT EFDJEBCJMJUZ -FU l′ CF UIF MFBTU DPNNPO NVMUJQMF PG BMM UIF DPFਖ਼DJFOUT PDDVSSJOH JO NPEVMP DPOTUSBJOUT JO ψ य़F EFWFMPQNFOUT CFMPX BSF PCUBJOFE CZ TMJHIUMZ BEBQUJOH XIBU IBT CFFO EPOF GPS UIF DBTF XJUI † BOE CZ SFDBMMJOH UIBU GPS BMM n, n′ ∈ N n ≡l′ n′ J੖ n ≡k1 n′ BOE BOE n ≡kβ n′ XIJDI JT B LFZ QSPQFSUZ UP EFBM XJUI NPEVMP DPOTUSBJOUT )FSFJO l′ JT UIF MDN PG UIF WBMVFT k1 kβ "EEJUJPOBMMZ HJWFO B WBMVBUJPO v EF੗OJOH B UFSN tMFGU ∈ T TFF EF੗OJUJPO BCPWF UIFO GPS BOZ WBMVF n ∈ {m ∈ [v(tMFGU ), v(Y)] : m ≡l′ v(Y)} v BOE v[Y 0→ n] WFSJ੗FT FYBDUMZ UIF TBNF BUPNJD GPSNVMBF GSPN ψ BOE UIFSFGPSF v | ψ J੖ v[Y 0→ n] | ψ $POTFRVFOUMZ GPS UIF TBUJTGBDUJPO PG ϕ XF DBO BMXBZT BTTVNF UIBU Y JT FRVBM UP TPNF UFSN t + j XJUI t ∈ T BOE j ∈ [0, l′ − 1] /PX POF DBO TIPX UIBU ϕ JT FRVJWBMFOU UP UIF EJTKVODUJPO CFMPX $. t∈T,j∈[0,l′ −1]. ψ(Y ← t + j) ∧ (t + j ≥ 0). "HBJO UIF WFSZ JEFB GPS FMJNJOBUJOH UIF RVBOUJ੗DBUJPO PWFS UIF WBSJBCMF Y DPOTJTUT JO DPOTJEFSJOH B MJNJUFE BNPVOU PG WBMVFT PCUBJOFE GSPN UFSNT UIBU EP OPU DPOUBJO Y BOE UP VTF B EJTKVODUJPO JOTUFBE PG B ੗STUDMBTT RVBOUJ੗DBUJPO 8F XJTI UP FNQIBTJ[F UIBU UIJT XBZ UP FMJNJOBUF RVBOUJ੗FST JT OPU DPNQMFUFMZ PQUJNBM FWFO UIPVHI UIF FYQPOFOUJBM CMPXVQ JT VOBWPJEBCMF JO GVMM HFOFSBMJUZ. CVU JU IBT BMMPXFE VT UP QSFTFOU UIF NBJO JEFBT GPS UIF QSPPG PG 2&⋆ &YBNQMF -FU VT DPOTJEFS UIF GPSNVMB ∃ [ Y = 2[ *O PSEFS UP BQQMZ UIF BCPWF NFUIPE MFU VT TUBSU CZ SFXSJUJOH UIF GPSNVMB JO PSEFS UP SFNPWF UIF FRVBMJUZ QSFEJDBUF ϕ = ∃ [ (Y ≤ 2[) ∧ (¬(Y + 1 ≤ 2[))..

(42) "VUPNBUBCBTFE "QQSPBDI GPS 1SFTCVSHFS "SJUINFUJD. . -FU VT ੗STU FMJNJOBUF UIF DPFਖ਼DJFOU JO GSPOU PG [ MFBEJOH UP UIF GPSNVMB ϕ = ∃ [ ([ ≡2 0) ∧ (Y ≤ [) ∧ (¬(Y + 1 ≤ [)). य़F TFU PG UFSNT T JT {0, Y, Y + 1} 4P l′ = 2 BOE OPX XF DBO JOUSPEVDF UIF HFOFSBMJ[FE EJTKVODUJPO FRVJWBMFOU UP ϕ ⊤. ⊤ ⊤ % &' ( % &' ( % &' ( [0 ≥ 0 ∧(0 ≡2 0) ∧ (Y ≤ 0) ∧ (¬(Y + 1 ≤ 0))]∨. [1 ≥ 0 ∧(1 ≡2 0) ∧ (Y ≤ 1) ∧ (¬Y + 1 ≤ 1)]∨ ' (% & ' (% & ⊤. ⊥. ⊤. ⊤ ⊤ % &' ( % &' ( % &' ( [Y ≥ 0 ∧(Y ≡2 0) ∧ (Y ≤ Y) ∧ (¬(Y + 1 ≤ Y))]∨. [Y + 1 ≥ 0 ∧(Y + 1 ≡2 0) ∧ (Y ≤ Y + 1) ∧ (¬Y + 1 ≤ Y + 1)]∨ ' (% & ' (% & ⊤. ⊥. [Y + 1 ≥ 0 ∧(Y + 1 ≡2 0) ∧ (Y ≤ Y + 1) ∧ (¬(Y + 1 ≤ Y + 1))]∨ ' (% & ' (% & ⊤. ⊥. [Y + 2 ≥ 0 ∧(Y + 2 ≡2 0) ∧ (Y ≤ Y + 2) ∧ (¬(Y + 1 ≤ Y + 2))] ' (% & ' (% & ⊤. ⊥. XIJDI JT FRVJWBMFOU UP (Y ≤ 0) ∨ (Y ≡2 0) BOE UIFSFGPSF JT FRVJWBMFOU UP Y ≡2 0 य़JT TPVOET BT B MPU PG F੖PSUT GPS BO PCWJPVT PVUDPNF CVU JU IBT CFFO EPOF CZ GPMMPXJOH B TZTUFNBUJD EFDJTJPO QSPDFEVSF UIBU DBO CF UIFO BVUPNBUFE. . "ॻॺॵॳ१ॺ१ং२१ॹ५४ "ॶॶॸॵ१३८ ६ॵॸ 1ॸ५ॹ२ॻॸ७५ॸ "ॸ९ॺ८ॳ५ॺ९३. *O UIF QSFWJPVT TFDUJPO XF IBWF TFFO UIBU UIF TBUJT੗BCJMJUZ QSPCMFN GPS 1SFTCVSHFS BSJUINFUJD JT EFDJEBCMF *O UIJT TFDUJPO XF TIBMM JOGPSNBMMZ EFTDSJCF UIF EFDJE BCJMJUZ PG TBUJT੗BCJMJUZ QSPCMFN GPS 1SFTCVSHFS BSJUINFUJD CZ USBOTMBUJPO JOUP UIF OPOFNQUJOFTT QSPCMFN GPS ੗OJUFTUBUF BVUPNBUB य़JT TFDUJPO JT FTTFOUJBMMZ CPS SPXFE GSPN %FNSJ BOE 1PJUSFOBVE GPMMPXJOH UIF EFWFMPQNFOUT JO #PVEFU BOE $PNPO य़F VTF PG BVUPNBUB GPS MPHJDBM EFDJTJPO QSPCMFNT HPFT CBDL UP #¨ VDIJ B BOE XF TIBMM QSPWJEF CFMPX UIF BQQSPBDI CZ BVUPNBUJD TUSVD UVSFT EFWFMPQFE JO #PVEFU BOE $PNPO TFF BMTP 8PMQFS BOE #PJHFMPU . य़F TFNJOBM QBQFS #¨ VDIJ C EFTDSJCFT IPX UP VTF UIF BVUPNBUBCBTFE BQQSPBDI UP EFBM XJUI 1SFTCVSHFS BSJUINFUJD 0G DPVSTF PUIFS EFDJTJPO QSPDF EVSFT FYJTU GPS 1SFTCVSHFS BSJUINFUJD GPS JOTUBODF RVBOUJ੗FS FMJNJOBUJPO NFUIPE GSPN 3FEEZ BOE -PWFMBOE JNQSPWFT UIF NFUIPE EFWFMPQQFE JO $PPQFS #FGPSF QSFTFOUJOH UIF QSJODJQMFT PG UIF BVUPNBUBCBTFE BQQSPBDI GPS 1SFT CVSHFS BSJUINFUJD MFU VT NFOUJPO UIBU JO HFOFSBM UIF BVUPNBUBCBTFE BQQSPBDI DPOTJTUT JO SFEVDJOH MPHJDBM QSPCMFNT JOUP BVUPNBUBCBTFE EFDJTJPO QSPCMFNT JO.

(43) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. PSEFS UP UBLF BEWBOUBHF PG LOPXO SFTVMUT GSPN BVUPNBUB UIFPSZ "MUFSOBUJWFMZ UIJT DBO CF WJFXFE BT B NFBOT UP USBOTGPSN EFDMBSBUJWF TUBUFNFOUT UZQJDBMMZ GPS NVMBF JOUP PQFSBUJPOBM EFWJDFT UZQJDBMMZ BVUPNBUB XJUI TPNFUJNFT SVEJNFOUBSZ DPNQVUBUJPOBM QPXFS य़F NPTU TUBOEBSE UBSHFU QSPCMFNT JO BVUPNBUB VTFE JO UIJT BQQSPBDI JT UIF OPOFNQUJOFTT QSPCMFN UIBU DIFDLT XIFUIFS BO BVUPNBUPO BENJUT BU MFBTU POF BDDFQUJOH DPNQVUBUJPO " QJPOFFSJOH XPSL CZ #¨ VDIJ #¨ VDIJ B TIPXT UIBU #¨ VDIJ BVUPNBUB BSF FRVJWBMFOU UP GPSNVMBF JO NPOBEJD TFDPOE PSEFS MPHJD .40 PWFS ⟨N, <⟩ NPEFMT PG B GPSNVMB CVJMU PWFS UIF TFDPOEPSEFS WBSJBCMFT P1 , . . . , PN BSF ωTFRVFODFT PWFS UIF BMQIBCFU P({P1 , . . . , PN }) *O GVMM HFOFSBMJUZ UIF GPMMPXJOH BSF B GFX EFTJSBCMF QSPQFSUJFT PG UIF BQQSPBDI • य़F SFEVDUJPO TIPVME CF DPODFQUVBMMZ TJNQMF BQBSU GSPN CFJOH TFNBOUJDBMMZ GBJUIGVM • य़F DPNQVUBUJPOBM DPNQMFYJUZ PG UIF BVUPNBUBCBTFE UBSHFU QSPCMFN TIPVME CF XFMMDIBSBDUFSJTFE *O UIBU XBZ B DPNQMFYJUZ VQQFS CPVOE JT PCUBJOFE UP TPMWF UIF TPVSDF MPHJDBM QSPCMFN • -BTU CVU OPU MFBTU UIF SFEVDUJPO TIPVME QSFGFSBCMZ BMMPX UIF PQUJNBM DPN QMFYJUZ GPS UIF TPVSDF MPHJDBM QSPCMFN UP CF PCUBJOFE *O UIF DPOTUSVDUJPO QSPWJEFE CFMPX UIJT MBTU QSPQFSUZ JT OPU TBUJT੗FE. 8F IBWF TFFO UIBU FBDI 1SFTCVSHFS GPSNVMB ϕ XJUI n ≥ 1 GSFF WBSJBCMFT EF ੗OFT B TVCTFU PG Nn OBNFMZ !ϕ" ⊆ Nn UIBU DPSSFTQPOET UP UIF TFU PG WBSJBCMF WBMVBUJPOT UIBU NBLF ϕ USVF 'PS JOTUBODF !Y = Z + [" = {⟨k1 , k2 , k3 ⟩ ∈ N3 : k1 = k2 + k3 } य़F BVUPNBUBCBTFE BQQSPBDI GPS 1SFTCVSHFS BSJUINFUJD DPOTJTUT PG SFQSFTFOUJOH UIF UVQMFT JO !ϕ" CZ B SFHVMBS MBOHVBHF UIBU DBO CF F੖FDUJWFMZ EF੗OFE GPS JOTUBODF XJUI UIF IFMQ PG B ੗OJUFTUBUF BVUPNBUPO *O UIBU XBZ TBUJT ੗BCJMJUZ PG ϕ XIJDI JT FRVJWBMFOU UP UIF OPOFNQUJOFTT PG !ϕ" CFDPNFT FRVJWBMFOU UP UIF OPOFNQUJOFTTT PG B ੗OJUFTUBUF BVUPNBUPO XIJDI JT BO FBTZ QSPCMFN UP TPMWF PODF UIF BVUPNBUPO JT CVJMU *O PSEFS UP EF੗OF SFHVMBS MBOHVBHFT ੗STU XF OFFE UP TQFDJGZ IPX OBUVSBM OVNCFST BOE UVQMFT PG OBUVSBM OVNFST BSF FODPEFE CZ XPSET PWFS B ੗OJUF BMQIBCFU /VNFSPVT PQUJPOT BSF QPTTJCMF TFF FH -FSPVY ,MBFEULF B. BOE CFMPX XF BEPQU B TJNQMF BOE TUBOEBSE FODPEJOH JO XIJDI OBUVSBM OVNCFST BSF WJFXFE BT ੗OJUF XPSET PWFS UIF BMQIBCFU {0, 1} CZ VTJOH B CJOBSZ SFQSFTFOUBUJPO JO XIJDI UIF MFBTU TJHOJ੗DBOU CJU JT ੗STU 8F BEPQU B SFQSFTFOUBUJPO PG OBUVSBM OVNCFST UIBU JT OPU VOJRVF GPS JOTUBODF UIF OVNCFS ੗WF DBO CF FODPEFE CZ PS CZ 5VQMFT PG OBUVSBM OVNCFST PG EJNFOTJPO n BSF SFQSFTFOUFE CZ ੗OJUF XPSET PWFS UIF BMQIBCFU {0, 1}n CZ VTJOH BO FRVBM MFOHUI SFQSFTFOUBUJPO ) 5 * GPS FBDI OVNCFS QBEEJOH 5ZQJDBMMZ UIF QBJS 8 DBO CF SFQSFTFOUFE CZ UIF XPSE ) 1 *) 0 *) 1 *) 0 *) 0 * 0. 0. 0. 1. 0. PWFS UIF BMQIBCFU {0, 1}2 4P XF JOUSPEVDF UIF NBQ f : N → P({0, 1}∗ ) TVDI EFG EFG UIBU f(0) = 0∗ BOE GPS k > 0 f(k) = bk · 0∗ XIFSF bk JT UIF TIPSUFTU CJOBSZ.

(44) "VUPNBUBCBTFE "QQSPBDI GPS 1SFTCVSHFS "SJUINFUJD. . SFQSFTFOUBUJPO PG k MFBTU TJHOJ੗DBOU CJU ੗STU य़F NBQ f JT FYUFOEFE UP TVCTFUT PG N JO UIF PCWJPVT XBZ BT XFMM BT UP nUVQMFT PG OBUVSBM OVNCFST XJUI BMQIBCFU {0, 1}n TVDI UIBU f(Y) ⊆ P(({0, 1}n )∗ ) XJUI Y ∈ Nn BOE C ∈ f(Y) J੖ GPS i ∈ [1, n] UIF QSPKFDUJPO PG C PO UIF iUI SPX CFMPOHT UP f(Y(i)) य़F NBQ f JT UZQJDBMMZ B TUBUF FODPEJOH TDIFNB JO UIF TFOTF PG -FHBZ (JWFO B 1SFTCVSHFS GPSNVMB ϕ XJUI n ≥ 1 GSFF WBSJBCMFT BOE B ੗OJUFTUBUF BVUPNBUPO A PWFS UIF BMQIBCFU {0, 1}n XF XSJUF ϕ ≈ A XIFOFWFS -(A) = f(!ϕ") ࢊFPSFN TFF FH #PVEFU BOE $PNPO . (JWFO B 1SFTCVSHFS GPSNVMB ϕ XF DBO CVJME B टOJUFTUBUF BVUPNBUPO Aϕ TVDI UIBU ϕ ≈ Aϕ . 8F BMTP IBWF !ϕ" ⊆ !ψ" J੖ -(Aϕ ) ⊆ -(Aψ ) TFF FH -FHBZ य़FPSFN . य़F ੗OJUFTUBUF BVUPNBUPO Aϕ DBO CF CVJMU SFDVSTJWFMZ PWFS UIF TUSVDUVSF PG ϕ 'PS JOTUBODF DPOKVODUJPO JT IBOEMFE CZ UIF QSPEVDU DPOTUSVDUJPO FYJTUFO UJBM RVBOUJ੗FS JT IBOEMFE CZ QSPKFDUJPO OFHBUJPO JT IBOEMFE CZ UIF DPNQMFNFOU DPOTUSVDUJPO TFF EFUBJMT CFMPX /FWFSUIFMFTT B DSVEF DPNQMFYJUZ BOBMZTJT PG UIF DPOTUSVDUJPO PG Aϕ SFWFBMT B OPOFMFNFOUBSZ XPSTUDBTF DPNQMFYJUZ *OEFFE GPS FWFSZ OFHBUJPO B DPNQMFNFOUBUJPO OFFET UP CF PQFSBUFE )PXFWFS EFWFMPQNFOUT SFMBUFE UP UIF PQUJNBM TJ[F PG BVUPNBUB DBO CF GPVOE JO ,MBFEULF C य़F SFDVSTJWF EF੗OJUJPO JT CBTFE PO UIF GPMMPXJOH QSPQFSUJFT -FU ϕ BOE ψ CF 1SFTCVSHFS GPSNVMBF XJUI GSFF WBSJBCMFT Y1 Yn DPOKVODUJPO *G ϕ ≈ A BOE ψ ≈ B UIFO ϕ ∧ ψ ≈ A ∩ B XIFSF ∩ JT UIF QSPEVDU DPOTUSVDUJPO DPNQVUJOH JOUFSTFDUJPO OFHBUJPO *G ϕ ≈ A UIFO ¬ϕ ≈ A XIFSF · QFSGPSNT DPNQMFNFOUBUJPO XIJDI NBZ DBVTF BO FYQPOFOUJBM CMPXVQ RVBOUJटDBUJPO *G ϕ ≈ A UIFO ∃ Yn ϕ ≈ A′ XIFSF A′ JT CVJMU PWFS UIF BMQIBCFU C. {0, 1}n−1 CZ GPSHF॒JOH UIF nUI DPNQPOFOU 5ZQJDBMMZ q − → q ′ JO A′ XIFO C′. FWFS UIFSF JT B USBOTJUJPO q − → q ′ JO A TVDI UIBU C BOE C′ BHSFF PO UIF n − 1 ੗STU CJU WBMVFT *O UIF BCPWF DPOTUSVDUJPO XF BTTVNFE UIBU ϕ BOE ψ TIBSF UIF TBNF TFU PG GSFF WBSJBCMFT XIJDI EPFT OPU BMXBZT IPME USVF GPS BSCJUSBSZ GPSNVMBF *G JU JT OPU UIF DBTF ϕ ≈ A BOE ψ ≈ B UIFO XF QFSGPSN BO PQFSBUJPO UIBU DPOTJTUT PG BEEJOH EVNNZ CJUT 'PS JOTUBODF TVQQPTF UIBU ϕ DPOUBJOT UIF GSFF WBSJBCMFT Y1 , . . . , Yn 8F DBO CVJME UIF BVUPNBUPO A′ PWFS UIF BMQIBCFU {0, 1}n+1 PCUBJOFE CZ BEEJOH C. UIF (n + 1)UI DPNQPOFOU 5ZQJDBMMZ q → − q ′ JO A′ XIFOFWFS UIFSF JT B USBOTJUJPO C′. q− → q ′ JO A TVDI UIBU C BOE C′ BHSFF PO UIF n ੗STU CJU WBMVFT *U SFNBJOT UP EFBM XJUI BUPNJD GPSNVMBF UP BDIJFWF UIF JOEVDUJWF CVJMEJOH PG UIF BVUPNBUPO य़F QSPPG PG UIFPSFN JT DMFBSMZ CBTFE PO UIF BCPWF DPOTUSVDUJPOT CVU XF OFFE UP DPNQMFUF UIF BSHVNFOU JO PSEFS UP EFBM XJUI BUPNJD GPSNVMBF 8JUIPVU.

(45) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD ⎛. ⎞⎛ ⎞⎛ ⎞ 0 1 1 ⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 0 ⎠ 0 0 1. ⎛. ⎞ 0 ⎝ 1 ⎠ 1. ⎛. ⎞⎛ ⎞⎛ ⎞ 1 0 0 ⎝ 1 ⎠ ⎝ 0 ⎠ ⎝ 1 ⎠ 1 1 0. q1. q2 ⎛. ⎞ 1 ⎝ 0 ⎠ 0. 'JHVSF "VUPNBUPO GPS Y1 = Y2 + Y3 /. 0. 0 0. /. 0 1. /. 0. q1. 1 1. 0. q2 /. 1 0. 0. 'JHVSF "VUPNBUPO GPS Y1 = Y2 + Y2. BOZ MPTT PG HFOFSBMJUZ XF DBO SFTUSJDU PVSTFMWFT UP FRVBMJUJFT PG UIF GPSN Y = Z + [ BU UIF DPTU PG JOUSPEVDJOH OFX WBSJBCMFT JO PSEFS UP EFBM XJUI TVNT NBEF PG NPSF UIBO UXP WBSJBCMFT 4VDI B SFTUSJDUJPO JT POMZ IFMQGVM UP TJNQMJGZ UIF QSFTFOUBUJPO PG UIF NFUIPE CVU JU NBLFT TFOTF UP DPOTJEFS UIF GVMM MBOHVBHF XJUI MJOFBS DPOTUSBJOUT JO PSEFS UP PQUJNJ[F UIF SFEVDUJPO UP BVUPNBUB TFF FH #PJHFMPU BOE 8PMQFS #PVEFU BOE $PNPO य़F WBSJBCMFT JO Y = Z + [ BSF OPU OFDFTTBSJMZ EJTUJODU य़F BVUPNBUPO GPS Y1 = Y2 +Y3 JT EFTDSJCFE JO 'JHVSF XIFSF q1 JT UIF JOJUJBM TUBUF BT XFMM BT UIF ੗OBM TUBUF य़F TUBUF q1 SFQSFTFOUT B DBSSZPWFS PG 0 XIFSFBT UIF TUBUF q2 SFQSFTFOUT B DBSSZPWFS PG 1 8F DBO DIFDL UIBU (Y1 = Y2 + Y3 ) ≈ A य़F 'JHVSF EFTDSJCFT UIF BVUPNBUPO GPS Y1 = Y2 + Y2 #Z QSPKFDUJPO UIF FODPEJOH GPS ∃ Y2 (Y1 = Y2 + Y2 ) JT QSFTFOUFE JO 'JHVSF /PUF UIBU UIF BVUPNBUPO SFDPHOJ[FT QSFDJFMZ UIF FODPEJOHT PG FWFO OVNCFST ). 0 q1. *. ) ). 0. 1. *. ). 1. *. q2. *. 'JHVSF &ODPEJOH GPS ∃ Y2 (Y1 = Y2 + Y2 ).

(46) 4FNJMJOFBS 4FUT. . य़F BVUPNBUBCBTFE BQQSPBDI GPS 1SFTCVSHFS BSJUINFUJD DBO CF FYUFOEFE UP SJDIFS UIFPSJFT TVDI BT ⟨R, Z, +, ≤⟩ TFF FH #PJHFMPU BOE 8PMQFS PS DBO CF SF੗OFE CZ QSPWJEJOH PUIFS SFEVDUJPOT TFF FH -FSPVY ,MBFEULF C 4DIVFMF BOE 4DIOFJEFS "O PWFSWJFX PG BVUPNBUBCBTFE EFDJTJPO QSPDF EVSFT GPS 1SFTCVSHFS BSJUINFUJD BOE SFMBUFE GPSNBMJTNT DBO CF GPVOE JO ,MBFEULF C *O %VSBOE(BTTFMJO BOE )BCFSNFIM JU JT BMTP TIPXO UIBU UIF USBOTMB UJPO GSPN GPSNVMBF JO '0(N) UP ੗OJUFTUBUF BVUPNBUB DBO CF EPOF JO USJQMF FYQP OFOUJBM UJNF XIFO UIF SFEVDUJPO GSPN #PVEFU BOE $PNPO UIBU JT TJNQMFS DPODFQUVBMMZ XPVME MFBE UP B OPOFMFNFOUBSZ CPVOE TJODF FBDI OFHBUJPO NBZ DBVTF BO FYQPOFOUJBM CMPXVQ "VUPNBUB DPOTUSVDUJPOT GPS GSBHNFOUT PG '0(N) BSF BMTP TUVEJFE JO %VSBOE(BTTFMJO BOE )BCFSNFIM . 4५ॳ९ॲ९ॴ५१ॸ 4५ॺॹ *O UIJT TFDUJPO XF QSFTFOU B TUBOEBSE DIBSBDUFSJ[BUJPO GPS UIF 1SFTCVSHFS TFUT JF GPS UIF TFUT UIBU BSF FRVBM UP !ϕ(Y1 , . . . , Yd )" GPS TPNF GPSNVMB JO '0(N) XJUI GSFF WBSJBCMFT JODMVEFE JO Y1 Yd d ≥ 1 #FGPSF QSFTFOUJOH UIF DPSSFTQPOEFODF MFU VT EFBM XJUI UIF POFWBSJBCMF DBTF UIBU XJMM CF UIFO FYUFOEFE . 'ॵॸॳॻॲ१५ ॽ९ॺ८ १ॺ ॳॵॹॺ ॵॴ५ ६ॸ५५ ॼ१ॸ९१२ॲ५. 'PSNVMBF XJUI BU NPTU POF GSFF WBSJBCMF DBO VTF NPSF UIBO POF WBSJBCMF CVU UIPTF WBSJBCMFT OFFE UP CF RVBOUJ੗FE BOE UIFSFGPSF UP CF JO UIF TDPQF PG B RVBOUJ੗FS #Z XBZ PG FYBNQMF MFU VT DPOTJEFS UIF GPSNVMB ϕ(Y) CFMPX (Y ̸= 1 ∧ Y ̸= 2) ∧ (Y = 0 ∨ (Y ≥ 3 ∧ ∃ Z (Y = 3 + 2Z))).. *U JT FBTZ UP TIPX UIBU !ϕ(Y)" = {0} ∪ {3 + 2n : n ≥ 0} BOE UIFSFGPSF BॏFS UIF WBMVF FWFSZ UXP WBMVFT CFMPOHT UP UIF TFU !ϕ(Y)" 4VDI B SFHVMBSJUZ DBO CF TMJHIUMZ HFOFSBMJ[FE MFBEJOH UP UIF EF੗OJUJPO CFMPX. %F॑OJUJPO 6MUJNBUFMZ QFSJPEJD TFU " TFU X ⊆ N JT VMUJNBUFMZ QFSJPEJD EFG ⇔ UIFSF FYJTU N ≥ 0 BOE P ≥ 1 TVDI UIBU GPS BMM n ≥ N XF IBWF n ∈ X J੖ n + P ∈ X. य़F TFU PG FWFO OVNCFST JT VMUJNBUFMZ QFSJPEJD XJUI N = 0 GPS JOTUBODF BOE P = 2 BT XFMM BT UIF TFU PG PEE OVNCFST XJUI N = 1 BOE P = 2 UPP .PSF HFOFSBMMZ POF DBO FBTJMZ TIPX UIBU !Y ≡k k ′ " JT VMUJNBUFMZ QFSJPEJD XJUI N = 0 BOE P = k 6MUJNBUFMZ QFSJPEJD TFUT TBUJTGZ OJDF DMPTVSF QSPQFSUJFT UIBOLT UP UIFJS SFHV MBSJUZ 1SPQPTJUJPO 6MUJNBUFMZ QFSJPEJD TFUT BSF DMPTFE VOEFS VOJPO JOUFSTFDUJPO BOE DPNQMFNFOUBUJPO.

(47) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. 'PS JOTUBODF TVQQPTF UIBU X JT VMUJNBUFMZ QFSJPEJD BOE MFU X = N ! X य़F TUBUFNFOUT CFMPX BSF FRVJWBMFOU GPS n ≥ N • n ∈ X. • n ̸∈ X CZ EF੗OJUJPO PG X. • n + P ̸∈ X X JT VMUJNBUFMZ QFSJPEJD XJUI QBSBNFUFST N BOE P. • n + P ∈ X CZ EF੗OJUJPO PG X . 4P X JT VMUJNBUFMZ QFSJPEJD UPP BOE UIF TBNF QBSBNFUFST N BOE P DBO CF VTFE य़F QSPPG GPS VOJPO BOE JOUFSTFDUJPO JT MFॏ BT BO FYFSDJTF .PSFPWFS VMUJNBUFMZ QFSJPEJD TFUT DBO CF FBTJMZ EF੗OFE CZ GPSNVMBF XJUI BU NPTU POF GSFF WBSJBCMF BT TIPXO CFMPX *OEFFE MFU X CF TVDI B TFU EF੗OFE XJUI UIF IFMQ PG N BOE P UIBU BSF OPU VOJRVF य़F GPSNVMB ϕX CFMPX WFSJ੗FT UIBU !ϕX " = X 1 $ ( Y ̸= k) ∧ [( Y = k)∨ k∈[0,N −1]!X. ((Y ≥ N ) ∧ (∃ Z. $. k∈[0,N −1]∩X. (Y = k + P Z)))]. k∈[N,N +P −1]∩X. य़F DPOWFSTF JT BMTP USVF BT TUBUFE CFMPX. 1SPQPTJUJPO 'PS FWFSZ GPSNVMB ϕ(Y) JO '0(N) XJUI B VOJRVF GSFF WBSJBCMF Y !ϕ" JT BO VMUJNBUFMZ QFSJPEJD TFU. 1SPPG -FU ϕ(Y) CF B GPSNVMB XJUI B VOJRVF GSFF WBSJBCMF BOE ϕ′ CF BO FRVJWB MFOU RVBOUJ੗FSGSFF GPSNVMB CZ य़FPSFN ϕ′ DBO CF TIPXO FRVJWBMFOU UP B #PPMFBO DPNCJOBUJPO PG BUPNJD GPSNVMBF PG POF PG UIF GPSNT CFMPX ⊤ ⊥ Y ≤ k Y ≡k k ′ &BDI BUPNJD GPSNVMB EF੗OFT BO VMUJNBUFMZ QFSJPEJD TFU BOE VMUJNBUFMZ QFSJPEJD TFUT BSF DMPTFE VOEFS VOJPO JOUFSTFDUJPO BOE DPNQMFNFOUBUJPO 4P !ϕ′ " JT VMUJNBUFMZ QFSJPEJD BOE TJODF ϕ BOE ϕ′ BSF FRVJWBMFOU GPSNVMBF !ϕ" JT VMUJNBUFMZ QFSJPEJD UPP. 8IBU IBT CFFO EPOF XJUI VMUJNBUFMZ QFSJPEJD TFUT DBO CF HFOFSBMJ[FE XJUI TFNJMJOFBSJUZ TFUT BOE TFNJMJOFBSJUZ TFUT XJMM BMTP TBUJTGZ OJDF DMPTVSF QSPQFSUJFT B NVTU UP DBQUVSF MPHJDBM DPOOFDUJWFT GSPN '0(N) य़JT JT UIF TVCKFDU PG UIF OFYU TFDUJPO 4५ॳ९ॲ९ॴ५१ॸ ॹ५ॺॹ " MJOFBS TFU X PG EJNFOUJPO d ≥ 1 JT EF੗OFE BT B TVCTFU PG Nd GPS XIJDI UIFSF FYJTUT B CBTJT C ∈ Nd BOE B ੗OJUF TFU PG QFSJPET P = {Q1 , . . . , Qm } ⊆ Nd TVDI UIBU X = {C +. m # i=1. λi Qi : λ1 , . . . , λm ∈ N}..

(48) 4FNJMJOFBS 4FUT. . 'PS FYBNQMF UIF TFU PG FWFO OVNCFST {0 + i × 2 : i ∈ N} JT B MJOFBS TVCTFU PG N EJNFOTJPO 4JNJMBSMZ {1 + i × 2 : i ∈ N} JT B MJOFBS TFU XJUI C = 1 BOE XJUI VOJRVF QFSJPE 2 " TFNJMJOFBS TFU JT EF੗OFE BT B ੗OJUF VOJPO PG MJOFBS TFUT &BDI TFNJMJOFBS TFU DBO CF SFQSFTFOUFE CZ B ੗OJUF TFU PG QBJST PG UIF GPSN ⟨C, P⟩ )FSF JT B MJOFBS TFU PG EJNFOTJPO 2/ 0 / 0 / 0 3 3 2 4 +i× +j× : i, j ∈ N . 4 5 7 3FNBSL 4FNJMJOFBS TFUT HFOFSBMJ[F UIF OPUJPO PG VMUJNBUFMZ QFSJPEJD TFUT JO EJNFOTJPO TJODF FWFSZ VMUJNBUFMZ QFSJPEJD TFU X JT B TFNJMJOFBS TFU PG EJNFOTJPO *OEFFE TVQQPTF UIBU X JT DIBSBDUFSJ[FE CZ UIF QBSBNFUFST N BOE P य़F FRVBMJUZ CFMPX JT FBTZ UP TIPX 4 4 X=( {n}) ∪ ( {n + λP : λ ∈ N}) n∈[0,N −1]∩X. n∈[N,N +P −1]∩X. /PUF UIBU FBDI TJOHMFUPO TFU {n} JT B MJOFBS TFU XJUI OP QFSJPE PS FRVJWBMFOUMZ XJUI BMM QFSJPET FRVBM UP [FSP BT XFMM BT FBDI TFU {n + λP : λ ∈ N} XJUI CBTJT n BOE QFSJPE P 4FNJMJOFBS TFUT TBUJTGZ OJDF DMPTVSF QSPQFSUJFT ࢊFPSFN ࡋF DMBTT PG TFNJMJOFBS TFUT BSF FञFDUJWFMZ DMPTFE VOEFS VOJPO JO UFSTFDUJPO BOE DPNQMFNFOUBUJPO $MPTVSF CZ VOJPO JT JNNFEJBUF GSPN UIF EF੗OJUJPO PG TFNJMJOFBS TFUT य़F DMBTT PG TFNJMJOFBS TFUT IBQQFO UP CF NPSF JOUFSFTUJOH TJODF JU DPSSFTQPOET FYBDUMZ UP UIF 1SFTCVSHFS TFUT TFF य़FPSFN 0CTFSWF UIBU JG X1 BOE X2 BSF TFNJMJOFBS TVCTFUT PG Nd TVDI UIBU X1 = !ϕ1 " BOE X2 = !ϕ2 " UIFO X1 ∩ X2 = !ϕ1 ∧ ψ2 " BOE Nd ! X1 = !¬ϕ1 " 'PS FYBNQMF MFU VT DPOTJEFS UIF GPMMPXJOH TFNJMJOFBS TFU 2/ 0 / 0 / 0 3 3 2 4 +i× +j× | i, j ∈ N . 4 5 7 य़F 1SFTCVSHFS GPSNVMB CFMPX DBO EF੗OF JU BOE JUT QSJODJQMF DBO CF HFOFSBMJ[FE UP BOZ MJOFBS TFU ∃ Z, Z′ (Y1 = 3 + 2Z + 4Z′ ∧ Y2 = 4 + 5Z + 7Z′ ). ࢊFPSFN 4FNJMJOFBS TFUT DPJODJEF XJUI 1SFTCVSHFS TFUT JF * GPS FWFSZ GPSNVMB ϕ(Y1 , . . . , Yd ) XJUI d ≥ 1 !ϕ" JT B TFNJMJOFBS TFU BOE JUT SFQSFTFOUBUJPO DBO CF FञFDUJWFMZ DPNQVUFE. ** GPS FWFSZ TFNJMJOFBS TFU X ⊆ Nd UIFSF JT B GPSNVMB ϕ BOE JU DBO CF DPNQVUFE. TVDI UIBU X = !ϕ".

(49) . $IBQUFS 3VEJNFOUT PG 1SFTCVSHFS "SJUINFUJD. 4JODF DIFDLJOH XIFUIFS B MJOFBS TFU HJWFO BT B CBTJT BOE B ੗OJUF TFU PG QFSJ PET JT FNQUZ DBO CF FBTJMZ DIFDLFE य़FPSFN FOUBJMT UIF EFDJEBCJMJUZ PG UIF TBUJT੗BCJMJUZ QSPCMFN GPS '0(N) 1SPPG TLFUDI * -FU ϕ CF B GPSNVMB XJUI GSFF WBSJBCMFT Y1 , . . . , Yd #Z य़FP SFN UIFSF JT B RVBOUJ੗FSGSFF GPSNVMB ψ FRVJWBMFOU UP ϕ #Z य़FPSFN UIF TFU PG TFNJMJOFBS TFUT JT DMPTFE VOJPO JOUFSTFDUJPO BOE DPNQMFNFOUBUJPO 4P JU SFNBJOT UP TIPX UIBU BUPNJD GPSNVMBF EF੗OF TFNJMJOFBS TFUT UPP XIJDI JT MFॏ BT BO FYFSDJTF ** &WFSZ MJOFBS TFU DBO CF FBTJMZ TIPXO UP CF B 1SFTCVSHFS TFU CZ HFOFSBMJ[JOH UIF BCPWF DPOTUSVDUJPO %JTKVODUJPO JO '0(N) BMMPXT UP EFBM XJUI ੗OJUF VOJPO PG MJOFBS TFUT &YBNQMF -FU VT TIPX UIBU UIF TFU Y = {n2 : n ∈ N} JT OPU B 1SFTCVSHFS TFU CZ TIPXJOH UIBU Y JT OPU TFNJMJOFBS य़F QSPPG JT BE BCTVSEVN 4JODF Y JO੗OJUF !JT EFG m UIFSF BSF C ≥ 0 BOE Q1 , . . . , Qm > 0 m ≥ 1 TVDI UIBU Z = {C + i=1 λi Qi : λ1 , . . . , λm ∈ N} ⊆ Y BOE Z JT JO੗OJUF -FU N ∈ N CF TVDI UIBU N 2 ∈ Z BOE (2N + 1) > Q1 य़F WBMVF N BMXBZT FYJTUT TJODF Z JT JO੗OJUF 4JODF Z JT B MJOFBS TFU XF BMTP IBWF (N 2 + Q1 ) ∈ Z )PXFWFS (N + 1)2 − N 2 = (2N + 1) > Q1 )FODF N 2 < N 2 + Q1 < (N + 1)2 XIJDI MFBET UP B DPOUSBEJDUJPO TJODF UIFSF JT OP TRVBSF WBMVF TUSJDUMZ CFGPSF N 2 BOE (N + 1)2 &YBNQMF -FU VT TIPX UIBU X = {2i : i ∈ N} JT OPU TFNJMJOFBS य़F QSPPG JT BMTP BE BCTVSEVN 4VQQPTF UIBU X JT TFNJMJOFBS 4JODF X JT BO JO੗OJUF TFU UIFSF FYJTU B CBTJT!C ∈ N BOE QFSJPE T Q1 , . . . , Qm ∈ (N ! {0}) m ≥ 1 TVDI UIBU Y = {C + m i=1 λi Qi : λ1 , . . . , λm ∈ N} ⊆ X BOE Y JT JO੗OJUF य़FSF α FYJTUT 2 ∈ Y TVDI UIBU Q1 < 2α #Z EF੗OJUJPO PG Y XF IBWF 2α + Q1 ∈ Y CVU 2α < 2α + Q1 < 2α+1 XIJDI MFBET UP B DPOUSBEJDUJPO &YBNQMF -FU VT DPOTJEFS UIF 7"44 JO 'JHVSF XJUI TJY DPVOUFST BOE UXP DPOUSPM TUBUFT "DUVBMMZ UIF 7"44 XFBLMZ DPNQVUFT NVMUJQMJDBUJPO BOE POF DBO FTUBCMJTI UIF GPMMPXJOH SFTVMU. 56. a b f. 7. ∈ N3 | ∃. 6. 56. c d e. 7. n m p. ⎛. ⎜ ⎜ ⎜ ∈ N3 , ⟨q0 , ⎜ ⎜ ⎝. 7. 0 0 0 0 0 0. ⎞. ⎛. ⎟ ⎜ ⎟ ⎜ ⎟ ∗ ⎜ → ⟨q1 , ⎜ ⎟⟩ − ⎟ ⎜ ⎠ ⎝ :. ∈ N3 : p ≤ n × m .. a b c d e f. ⎞. ⎟: ⎟ ⎟ ⎟⟩ = ⎟ ⎠.