Fundamentals of Undecidability in Computational Theory
Short Description
MTH 401 Theory of Computation Project Report by Arcchit Jain, Minnie Kabra, Oajasavee K. Mourya and Ved Gupta...
Description
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?$ Xj Qpjym> Qpjym> Xfmpm hjms djx mvlsxs od ogbjplxfk xj dh rfmxfmp o godb}obm ls ok+ `lb}j}s$ Qpjji > Rm pmh}cm xfls sqpj`gmk sqpj`gmk ipjk Qjsx s Cjppmsqjdhmdcm Qpj`gmk$ Qpj`gmk$ S}qqjsm S}qqjsm rm cod' ld iocx' hmclhm xfm godb}obm uBwB ls o CIB odh G& G &B, ls ok`lb}j}s{$ ?3
Blymd ο? ' $ $ $ ' οk ' ΰ ? ' $ $ $ ' ΰk > Cjdsxp}cx xfm ijggjrldb CIB B ; &Y' Σ' P ' S , rfmpm Y ; uS' S ? ' S 0 {' P ; uS ← S ? w S 0 ' S ? ← ο? Sσ ? w ο0 Sσ 0 w $ $ $ w οk Sσ k ' S 0 ← ΰ ? Sσ ? w $ $ $ w ΰ k Sσ k { &rfmpm σl opm dmr cfopocxmps ohhmh xj xfm ogqfo`mx' m$b$' σl ; l,$ Li xfm godb}obm ls ok`lb}j}s' xfmd xfmpm ls o hmplyoxljd ji sjkm sxpldb r ld xrj hlfimpmdx hlfimpmdx ro|s$ ro|s$ Rlxfj}x Rlxfj}x gjss ji bmdmpoglx| bmdmpoglx|'' gmx }s s}qqjsm s}qqjsm xfox xfm hmplyoxljds hmplyoxljds `jxf sxopx rlxf xfm p}gm S ← S ? ' pmohldb xfm dmr cfopocxmps `ocerophs }dxlg xfm| mdh koems koems s}pm xfmpm cod jdg| `m jdm hmplyoxljd' hmplyoxljd' sj xfox“s djx qjssl`gm$ Fmdcm' Fmdcm' rm smm xfox xfm jdg| ok`lb}lx| cod cjkm ipjk jdm S ? odh jdm S 0 ‖sxopx“$ @}x xfmd' xoeldb xfm s}`sxpldb ji r }q xj xfm `mblddldb ji xfm dmr cfopocxmps' rm foym o sjg}xljd xj xfm QCQ &sldcm xfm sxpldbs ji ldhlcms }smh oixmp xfjsm qjldxs koxcf,$ Slklgopg|' li xfmpm ls dj ok`lb}lx|' xfmd xfm QCQ coddjx `m sjgymh' sldcm o sjg}xljd rj}gh lkqg| od ok`lb}lx| xfox n}sx ijggjrs S ⇐ S ? ⇐− οσ odh S ⇐ S 0 ⇐− ΰ σ ' rfmpm ο ; ΰ opm sxpldbs ji koxcfldb ο“s odh ΰ “s “s &sldcm xfm σ “s koxcf,$ koxcf,$
Fmdcm' rm“ym pmh}cmh xj QCQ' odh sldcm xfox“s }dhmclho`gm' rm foym sfjrd xfox xfmpm coddjx mvlsx od| ogbjplxfk xj dh rfmxfmp o blymd godb}obm ls ok`lb}j}s$ 0$ Xj Qpjym> Qpjym> Blymd xrj CIGs' xfm qpj`gmk ji hmclhldb rfmxfmp xfm CIGs opm hlsnjldx jp djx ls }dhmclho`gm$ Qpjji > Xj sjgym xfls qpj`gmk rm rlgg pm}lpm o gmkko os blymd `mgjr>+ Gmkko> Gmkko> Xfm smx ji yoglh cjkq}xoxljds ji o x}pldb kocfldm K ls xfm ldxmpsmcxljd ji xrj CIG“s G? odh G0 ' odh bpokkops ijp xfmsm CIG“s cod `m mfimcxlymg| cjdsxp}cxmh ipjk K $ K $ Gmx B? odhB0 `m xfm xrj blymd bpokkops' odh rm foym xj cfmce rfmxfmp rm cod so| xfox G&B? , ∬ G&B0 , ls mkqx| mkqx| jp djx$ ]sldb ]sldb xfm gmkko gmkko rm cod cjdsxp cjdsxp}cx }cx ipjk K bpokkdmps B? odhB0 s}cf xfox G&B? , ∬ G&B0 , ls xfm smx ji yoglh cjkq}xoxljds ji K $ K $ Li xfmpm ls od ogbjplxfk O xj xmgg rfmxfmp xfm ldxmpsmcxljd ji xfm godb}obms ji xrj CIB“s ls mkqx|' rm cod cjdsxp}cx od ogbjplxfk xj @ xj xmgg rfmxfmp G&K , K , ; Ω ijp op`lxpop| XK K $ K $ Slkqg| hmslbd @ xj cjdsxp}cx B? odh B0 ipjk K os ld gmkko' ?8
xfmd oqqg| Ogbjplxfk O xj xmgg rfmxfmp G&B G&B? , ∬ G&B0 , ls mkqx|$ Li xfm ldxmpsmcxlj ls mkqx|' xfmd xfmpm opm dj yoglh cjkq}xoxljds ji K' sj G&K , K , ; Ω$ Li xfm ldxmpsmcxljd ls djx mkqx|' G&K , ; Ω$ Xfox ls' xfm qpj`gmk ji mkqxldmss ijp p$m$ smxs pmh}cms xj xfm qpj`gmk ji ldxmpsmcxljd ijp CIB“s$ Ogbjplxfk @ coddjx mvlsx' fjrmymp' sldcm G&K , K , ; Ω ls }dhmclho`gm `| Plcm Xfmj+ pmk$ Xfmpmijpm O hjms djx mvlsx' sj lx ls }dhmclho`gm rfmxfmp xfm ldxmpsmcxljd ji xrj CIG“s ls mkqx|$ Qpjym> Ijp o blymd CIG' lx ls }dhmclho`gm xj cfmce rfmxfmp G ; Σ− 3$ Xj Qpjym> Qpjji > Gmx }s s}qqjsm s}qqjsm xfox xfm qpj`gm qpj`gmk k ji hmclhmld hmclhmldbb G ; Σ− ls hmclho`gm$ hmclho`gm$ Xoem − G ; &G&B? , ∬ G&B0 ,, $ Lx kmods xfm qpj`gm qpj`gmk k ji hmclhm hmclhmldb ldb G&B? , ∬ G&B0 , ; Ω ls hmclho` hmclho`gm$ gm$ @}x `| qpmylj} qpmylj}ss pms}gx pms}gx'' lx ls }dhmcl }dhmclho` ho`gm$ gm$ Fmdcm Fmdcm j}p oss}kqxl oss}kqxljd jd ros − rpjdb$ Xf}s' lx ls }dhmclho`gm ijp o CIB B rfmxfmp G&B, ; Σ $ 8$ Xj Qpjym> Qpjym> Ijp od| blymd CIB' lx ls }dhmclho`gm xj cfmce rfmxfmp lxs godb}obm m}og xj o pmb}gop smx: Qpjji > Rm edjr xfox Σ− ls o pmb}gop pmb}gop smx$ smx$ Sj li rm cod hmclhm hmclhm rfmxfmp rfmxfmp o godb}o godb}obm bm G ls m}og xj o pmb}gop smx' xfmd rm cod hmclhm xfox rfmxfmp G ; Σ− `}x rm foym qpjymh qpjymh qpmylj}sg| qpmylj}sg| xfox xfls qpj`gmk ls }dhmclho`gm }dhmclho`gm$$ =$ Xj Qpjym> Qpjym> Ijp od| xrj blymd CIGs G? odh G0 ' cfmce li G? ; G0 Qpjji > Gmx G? ; G&B? , odh G0 ; G&B0 ,$ Ilv B0 `m xfm bpokkop bmdmpoxldb Σ− ' rfmpm Σ ls xfm xmpkldog s|k`jg ogqfo`mx ji B? $ Xfmd lx ls m}lyogmdx xj hmclhm rfmxfmp G? ; Σ − ' rflcf ls }dhmclho`gm$ Fmdcm xfm blymd qpj`gmk ls }dhmclho`gm$ 9$ Xj qpjym> qpjym> Blymd xrj CIGs G? odhG0 ' lx ls }dhmclho`gm xj cfmce li G0 ⊉ G? Qpjji > Xfm qpjji ls sokm os xfm o`jym qpjji$ 2$ Xj Qpjym> Qpjym> Blymd o pmb}gop smx P odh o CIG' G' lx ls }dhmclho`gm xj cfmce li P li P ⊉ G Qpjji > Xoem oem P ; Σ− xfmd xfm xfmjpmk ls pmh}cmh xj jdm ji xfm qpmylj}sg| smmd qpjjis$ Fmdcm lx ls ogsj }dhmclho`gm$
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Cfoqxmp 8
Fogxldb Qpj`gmk Hmdlxl Hmdlxljd jd Fogxld Fogxldb b Qpj` Qpj`gmk gmk Blymd o hmscplqxljd ji od op`lxpop| cjkq}xmp qpjbpok' hmclhm rfmxfmp xfm qpjbpok dlsfms p}ddldb jp cjdxld}ms xj p}d ijpmymp$ Xfls ls m}lyogmdx xj xfm qpj`gmk ji hmclhldb' blymd o qpjbpok odh od ldq}x' rfmxfmp xfm qpjbpok rlgg mymdx}ogg| fogx rfmd p}d rlxf xfox ldq}x' jp rlgg p}d ijpmymp$ Ld ? xj hmclhm rfmxfmp rfmxfmp op`lxpop| qpjbpoks rj}gh fogx jp gjjq$ Rm foym foym o i}gg+ffmhbmh i}gg+ffmhbmh djxoxljd ijp ogbjplxfks> ogbjplxfks> X}pldb kocfldms$ kocfldms$ Rm opm xf}s pmoh| xj hmdm o pmc}pslymg| md}kmpo`gm godb}obm' odh qpjym xfox lx ls qopxlogg| hmclho`gm$ Gmx F ; u&K' r, > X}pldb kocfldm K fogxs jd ldq}x sxpldb r {$ Djxlcm psx xfox F ls pmc}pslymg| pmc}pslymg| md}kmpo`gm md}kmpo`gm>> Jd ldq}x &K' & K' r,' }dlympsog X}pldb kocfldm ] fogxs qpmclsmg| rfmd xfm ldq}x ls ld F $ I}pxfmpkjpm' li F li F ls pmc}pslym' xfmd mymp| pmc}pslymg| md}kmpo`gm godb}obm ls pmc}pslym$ Ld jxfmp rjphs' ogg pmc}pslymg| md}kmpo`gm godb}obms opm ogsj hmclho`gm li odh jdg| li F ls pmc}pslym$ pmc}pslym$ Ijp s}qqjsm xfox lx ls ldhmmh ldhmmh hmclhmh `| sjkm X}pldb X}pldb kocfldm K 1 $ Xfmd Xfmd blymd od| qopxlc}gop X}pldb kocfldm K rflcf qopxlogg| hmclhms o godb}obm G&K ,' K ,' rm cj}gh hmslbd o X}pldb kocfldm K xfox i}gg| hmclhms G&K ,$ K ,$ Xfmjpmk Xfmjpmk ]dhmclho ]dhmclho`lg `lglx lx| | ji Fogxld Fogxldb b Qpj`gmk Qpj`gmk Gmx OX K ; uK' rwK ls o XK odh K occmqxs r {$ Xfmd OX K ls }dhmclho`gm$ Qpjji Ld o qpjji `| cjdxpohlcxljd lx rlgg `m oss}kmh xfox OX K ls hmclho`gm$ hmclho`gm$ S}qqjsm xfox F ls o hmclhmp ijp OX K ' l$m$' F &K' r, ;
occmqxs li K occmqxs r pmnmcxs pmnmcxs li K hjms djx occmqx r
O dmr X}pldb kocfldm H rlxf F os o s}`pj}xldm rlgg djr `m cjdsxp}cxmh$ Xfm ldq}x xj H ls o hmscplqxljd ji o X}pldb kocfldm K $ K $ Xfls ldijpkoxljd ls smdx xj X}pldb kocfldm F rflcf hmxmpkldms rfox K hjms rfmd xfm ldq}x xj K ls lxs jrd hmscplqxljd$ Jdcm H fos hmxmpkldmh xfls ldijpkoxljd' lx hjms xfm jqqjslxm' l$m$' lx pmnmcxs li K li K occmqxs odh occmqxs li K hjms djx occmqx> H&K , ;
occmqxs li K hjms djx occmqxs K pmnmcxs li K occmqxs K ?2
Ld xfm cosm rfmd H ls p}d rlxf lxs jrd hmscplqxljd xfm ijggjrldb ls j`xoldmh> H&H, ;
occmqxs li H hjms djx occmqxs H pmnmcxs li H occmqxs H
Xfm Fogxldb Qpj`gmk Qpj`gmk ls djdcjkq}xo` djdcjkq}xo`gm' gm' `}x lx ls od lkqjpxodx lkqjpxodx qpj`gmks$ Lx ls }smi}g }smi}g xj edjr li o qpjcmh}pm oqqglcoxljd rlgg xmpkldoxm ld o pmosjdo`gm okj}dx ji xlkm' `}x xfm Fogxldb Fogxldb Qpj`gmk hjms djx odsrmp odsrmp xfox }msxljd$ Lx cjdcmpds xfm }msxljd }msxljd ji rfmxfmp rfmxfmp xfm qpjcmh}pm oqqglcoxljd rlgg xmpkldoxm ld od| dlxm okj}dx ji xlkm' dj koxxmp fjr gjdb lx ls$
Ylp}s hmxmcxljd O ylp}s ls o qpjbpok xfox ldimcxs jxfmp qpjbpoks$ O ylp}s sqpmohs `| cjq|ldb lxs jrd cjhm ldxj xfm cjhm ji jxfmp qpjbpoks' sj rfmd xfjsm qpjbpoks opm mvmc}xmh xfm ylp}s rlgg mvmc}xm$ mvmc}xm$ Ld xfls koddmp' xfm ylp}s sqpmohs sqpmohs xj ldimcx kjpm odh kjpm qpjbpoks$ qpjbpoks$ O x|qlcog x|qlcog ylp}s ogsj ldcg}hms o koglclj}s qo|gjoh sj rfmd lx mvmc}xms ld ohhlxljd xj ldimcxldb jxfmp qpjbpoks lx ogsj qmpijpks sjkm hokobldb &cjpp}qxldb hoxo gms, jp oddj|ldb &qjqqldb }q kmssobms, `mfoyljp$ Xfm Ls+Ylp}s Qpj`gmk ls xj hmxmpkldm li o qpjcmh}pm sqmclcoxljd cjdxolds o ylp}s> Ldq}x> O sqmclcoxljd ji o Q|xfjd qpjbpok$ J}xq}x> Li xfm mvqpmssljd cjdxolds o ylp}s &o cjhm ipobkmdx xfox rlgg ldimcx jxfmp gms, j}xq}x Xp}m$ Jxfmprlsm' j}xq}x Iogsm$ Rm hmkjdsxpoxm xfm Ls+Ylp}s Qpj`gmk ls djdcjkq}xo`gm' rm sfjr fjr xj hmdm o fogxs ogbjplxfk blymd o f|qjxfmxlcog lsYlp}s ogbjplxfk$ Sldcm rm edjr fogxs ls djdcjkq}xo`gm' xfls sfjrs xfmpm ls dj lsYlp}s ogbjplxfk$
?6
Oss}km ldimcxIlgms ls o qpjcmh}pm xfox ldimcxs gms' sj xfm pms}gx ji myog}oxldb lsY lsY lp}s lp}s&‖ldimcxIlgms &‖ldimcxIlgms&, &, , ls Xp}m$ Rm cj}gh hmdm fogxs os> hmi fogxs& fogxs& q, q, > pmx}pd lsY lsY lp}s lp}s& q% q% 5 ldimcxIlgms&, ldimcxIlgms&, , Xfls rjpes os gjdb os xfm qpjbpok sqmclmh `| q hjms djx mvfl`lx xfm gm+ldimcxldb `mfoylj `mfoyljp$ p$ Li lx hjms' q cj}gh ldimcx o gm odh dmymp xmpkldoxm' odh fogxs rj}gh qpjh}cm xfm rpjdb rpjdb j}xq}x j}xq}x$$ Xj sjgym sjgym xfls rm dmmh dmmh xj flhm flhm xfm qpldxld qpldxldbb `mfoylj `mfoyljpp ji xfm jplbld jplbldog og qpjbpok$ qpjbpok$ O pj}bf hmdlxljd ji gm+ldimcxldb gm+ldimcxldb `mfoyljp rj}gh rj}gh `m xj cjdslhmp od| rplxm xj od mvmc}xo`gm mvmc}xo`gm gm xj `m od ldimcxljd$ ldimcxljd$ Xj oyjlh od| gm ldimcxljds ldimcxljds ld xfm sqmclc qpjbpok' qpjbpok' rm pmqgocm ogg qpjcmh}pms xfox rplxm xj gms rlxf qpjcmh}pms xfox rplxm xj sfohjr cjqlms ji xfmsm gms$ Ijp mvokqgm' rm cj}gh hj xfls `| cpmoxldb o dmr xmkqjpop| hlpmcxjp| hlpmcxjp| odh qpmqmdh xfox qoxf xj ogg gm dokms$ Rm cogg xfls &oss}kmh, qpjcmh}pm' sodh@jv' sodh@jv' sldcm lx xpodsijpks xfm jplbldog qpjbpok sqmclcoxljd ldxj jdm xfox rj}gh mvmc}xm ld o qpjxmcxmh sodh@jv$ sodh@jv$ hmi fogxs& fogxs& q, q, > lsY lsY lp}s lp}s&sodh@jv& sodh@jv& q,% q,% 5 ldimcxIlgms&, ldimcxIlgms&, , Sldcm rm edjr xfmpm ls dj ogbjplxfk xfox sjgyms xfm Fogxldb Qpj`gmk' xfls qpjyms xfox xfmpm ls dj ogbjplxfk xfox sjgyms xfm Ls+Ylp}s qpj`gmk$ Ylp}s scoddmps s}cf os S|kodxmc“s Djpxjd OdxlYlp}s oxxmkqx xj sjgym xfm Ls+Ylp}s Qpj`gmk' Qpj`gmk' `}x lxs djd+cjkq}xo djd+cjkq}xo`lglx `lglx| | kmods xfm| opm hjjkmh xj ogro|s ogro|s iolg$ Ylp}s scoddmps scoddmps hmxmcx edjrd ylp}sms `| scoddldb gms ijp sxpldbs xfox koxcf slbdox}pms ld o hoxo`osm ji edjrd ylp}sms$ Os gjdb os xfm slbdox}pm hoxo`osm ls ipm}mdxg| }qhoxmh xfm| ko| `m o`gm xj hmxmcx c}ppmdxg| sqpmohldb ylp}sms' `}x xfls oqqpjocf coddjx hmxmcx o dmr ylp}s xfox rlgg djx koxcf xfm slbdox}pm ji o qpmylj}sg| edjrd ylp}s$ Sjqflsxlcoxmh ylp}s scoddmps mkqgj| kjpm ohyodcmh xmcfdl}ms xj oxxmkqx xj hmxmcx cjkqgmv ylp}sms s}cf os kmxokjpqflc ylp}sms xfox ogxmp xfmlp jrd cjhm os xfm| qpjqoboxm xj oyjlh hmxmcxljd$ @}x' `mco}sm xfm bmdmpog Ls+Ylp}s Qpj`gmk ls djdcjkq}xo`gm' rm edjr xfox lx ls lkqjssl`gm xj cpmoxm o qpjbpok xfox ogro|s xmpkldoxms odh xfox ogro|s cjppmcxg| hmxmpkldms li od ldq}x qpjcmh}pm sqmclcoxljd ls o ylp}s$ ?<
Cfoqxmp =
Flg`mpx“s Xmdxf Qpj`gmk Xfm sxoxmkmdx ji Flg`mpx“s xmdxf qpj`gmk ls os ijggjrs > Blymd o Hljqfodxldm m}oxljd rlxf od| d}k`mp ji }dedjrd }odxlxlms odh rlxf poxljdog poxljdog ldxmbpog ldxmbpog d}kmplco d}kmplcogg cjmfficlmdx jmfficlmdxs> s> Xj hmylsm hmylsm o qpjc qpjcmss mss occj occjph phldb ldb xj rflcf lx cod cod `m hmxmpkldmh ld o dlxm d}k`mp ji jqmpoxljds rfmxfmp xfm m}oxljd ls sjgyo`gm ld poxljdog ldxmbmps $ Xfm }dhmclho`lglx| ji xfm mvlsxmdcm ji ldxmbpog pjjxs x}pdmh j}x xj `m xfm cjdsm}mdcm ji o hmmq m}lyogmdcm$ Hmdlxljd Hmdlxljd Hljqfodxl Hljqfodxldm dm Qpmhlcoxms Qpmhlcoxms odh Pmgoxlj Pmgoxljds ds O qpmhlcoxm Q jd De ls coggmh hljqfodxldm lfi xfmpm ls od d ∆ D odh o qjg|djklog q rlxf ldxmbmp cjmfficlmdxs ld e % d yoplo`gms' s}cf xfox Q & Q &v, ⇑⇐ ∃| ∆ Dd > q&v' | , ; 1$ O smx jp pmgoxljd S ⊀ De ls coggmh hljqfodxldm lfi v ∆ S ls o hljqfodxldm qpmhlcoxm qpmhlcoxm $ O i}dcxljd i > De ← D ls coggmh hljqfodxldm lfi lxs bpoqf &i &i &&v,' v, ⊀ De%? ls o hljqfodxldm smx $ Xfm hmdlxljd oggjrs ijp qjg|djklogs qjg|djklogs ji op`lxpop| op`lxpop| &xfj}bf dlxm, hmbpmm$ hmbpmm$ Hljqfodxl Hljqfodxldm dm qpj`gmks foym imrmp m}oxljds xfod }dedjrd yoplo`gms odh ldyjgym dhldb ldxmbmps xfox rjpe cjppmcxg| ijp ogg m}oxljds$
01
Xfm ijggjrldb xplce `| Sejgmk' Sejgmk ' sfjrs xfox rm cod xpohm xfm hmbpmm rlxf xfm d}k`mp ji yoplo`gms xj xfm mvxmdx xfox dogg| rm ko| pmsxplcx j}psmgyms xj qjg|djklogs ji hmbpmm ox kjsx ij}p$ Gmkko Li S ⊀ De ls o hljqfodxldm smx' xfmd xfmpm ls od k ∆ D odh o qjg|djklog ld k % e yoplo`gms rlxf ldxmbmp cjmfficlmdxs odh ji hmbpmm ox kjsx ij}p s}cf xfox S ; uv ∆ e k D w ∃ ~ ∆ D > &v' ~ , ; 1{$ Qpjji > @| oss}kqx oss}kqxljd ljd xfmpm xfmpm ls o qjg|djk qjg|djklog log q rlxf ldxmbmp cjmfficlmdxs s}cf xfox S ; cjdsxp}cx }cxljd ljd ji xfm dmr qjg|djklog qjg|djklog ls xfmd uv ∆ De w ∃ v| ∆ Dd > q&v' |, ; 1{$ Xfm cjdsxp hjdm pmc}pslym pmc}pslymg|> g|> ijp ogg kjdjklogs ld q rflcf foym hmbpmm gopbmp xfod xrj' ldxpjh}cm dmr yoplo`gms }? ' }0 ' $ $ $ hmdmh os o qpjh}cx ji xfm psx xrj yoplo`gms ji xfm cjppmsqjdhldb kjdjklog$ kjdjklog$ Ldsmpxldb Ldsmpxldb xfm dmr yoplo`gms yoplo`gms xfmd gmohs xj o dmr qjg|djklog qjg|djklog q? &v'|'}, v'|'}, rfjsm kovlkog hmbpmm ls jdm gmss xfod xfox ji q odh q&v' |, ; q? &v'|'}, v'|'}, li rm lkqjsm xfm hmdldb cjdsxpoldxs ijp xfm }ls$ Lxmpoxldb xfls qpjcmh}pm' rm cod j`xold o sm}mdcm ji qjg|djklog ld kjpm odh kjpm yoplo`gms yoplo`gms rflcf mymdx} mymdx}ogg| ogg| ls ox kjsx }ohpoxlc }ohpoxlc ld ogg yoplo`gms$ yoplo`gms$ S}qqjsm qd ls xfls xfls }ohpo }ohpoxlc xlc qjg|djklog qjg|djklog$$ Xfm lkqjsmh lkqjsmh cjdsxp cjdsxpold oldxs xs jd xfm dmr yoplo`g oplo`gms ms rflcf rflcf b}opodxmm xfox q&v' | , ; qd &v ' | ' } ' $ $ $, $, cod djr `m ijpk}goxmh ld xmpks ji xfm mvlsxmdcm ji ldxmbpog pjjxs ji }ohpoxlc qjg|djklogs rlxf ldxmbmp cjmfficlmdxs$ Hmdjxm xfjsm qjg|djklogs `| c? ' c0 ' $ $ $$ $$ Xfox ls' li rm foym ijp ldsxodcm }? >; v0 |2 ' }0 >; }? v0 ' xfmd rm hmdm c? &}? ' v0 ' |2 , >; }? ∐ v0 |2 odh c0 &}0 ' }? ' v0 , >; }0 ∐ }? v0 $ Ld xfls ro|' rm ocflmym xfox q&v' |, ; 1 lfi ∃} > q0d % Σl cl0 ; 1' rfmpm rm hmdjxm `| } xfm cjggmcxljd ji ogg yoplo`gms ohhmh xj v odh |$ Fmdcm' xfm qjg|djklog >; q0d % Σl cl0 gmohs xj xfm sj}bfx `l}ohpoxlc cfopocxmpl~oxljd ji xfm hljqfodxldm smx S $ Qpjqjslxljd Mymp| hljqfodxldm smx ls pmc}pslymg| md}kmpo`gm$ Qpjji > Gmx q&v' | , `m o cfopocxmpl~ldb cfopocxmpl~ldb qjg|djklog qjg|djklog ijp xfm hljqfodxldm hljqfodxldm smx$ Xfm sxoxmkmdx sxoxmkmdx ijggjrs ipjk j`smpyldb xfox q&v' |, ; 1 ls o qplklxlym pmc}pslym qpmhlcoxm rflcf rm cogg Q & Q &v' |, odh ipjk pmcoggldb xfox o pmc}pslymg| md}kmpo`gm smx S ls mvocxg| jdm ijp rflcf xfmpm ls o qplklxlym pmc}pslym qpmhlcoxm ijp rflcf S ; uvw∃ | > Q & Q &v' | ,{$
0?
O `oslc qpjqmpx| ji pmc}pslymg| pmc}pslymg| md}kmpo`gm md}kmpo`gm smxs ls xfox xfm cgoss ls cgjsmh r$p$x$ }dljds odh ldxmpsmcxljds$ Xfls ls moslg| smmd xj fjgh ogsj ijp hljqfodxldm smxs> Qpjqjslxljd> Qpjqjslxljd> Xfm cgoss ji hljqfodxldm hljqfodxldm qpmhlc qpmhlcoxms oxms ls cgjsmh cgjsmh r$p$x$ r$p$x$ &l, cjdn}dcxljd' cjdn}dcxljd' &ll, hlsn}dcxljd odh &lll, xfm }sm ji mvlsxmdxlog }odxlmps$ Qpjji > Gmx }qqmp cosm Q s `m hljqfodxldm qpmhlcoxms odh gjrmp cosm qs xfmlp cfopocxmpl~ldb qjg|djklogs$ Xfmd
Q ? &v, ∢ Q 0 &v, Q ? &v, ∦ Q 0 &v, v0 > Q & Q &v? ' v0 ,
⇑⇐ ⇑⇐ ⇑⇐
∃ |? ' |0 > q? &v' |? ,0 % q0 &v' |0 ,0 ; 1 ∃ |? ' |0 > q? &v' |? , q0 &v' |0 , ; 1' ∃ |' v0 > q&v? ' v0 ' |, ; 1
Djr' rm cod bj xfpj}bf xfm qpjji ji xfm iocx xfox mymp| X}pldb cjkq}xo`gm i}dcxljd ls pmc}ps pmc}psly lym$ m$ Sldcm Sldcm xfm mdcjhldb mdcjhldb ji od op`lxp op`lxpop| op| X}pld X}pldbb koc kocfld fldmm ld xmpks xmpks ji pmc}ps pmc}psly lymm i}dcxljds }sms jdg| i}dcxljds ijp rflcf rm djr edjrd xfox xfm| opm hljqfodxldm' jdm cod rlxf o glxxgm `lx ji mfijpx smm xfox xfm qpmhlcoxm ‗xfm X}pldb kocfldm fogxs‛ ls o hljqfodxldm qpmhlcoxm$ Ijggjrldb xfmsm gldms gmohs xj xrj pmkopeo`gm cjdsm}mdcms> Xfmjpmk Mymp| pmc}pslymg| md}kmpo`gm smx ls hljqfodxldm$ Xfmjpmk Xfmpm ls od d ∆ D odh o qjg|djklog q rlxf ldxmbmp cjmfficlmdxs s}cf xfox ijp od| pmc}pslymg| md}kmpo`gm smx S ⊉ D xfmpm mvlsxs od s ∆ D sj xfox S ; uv ∆ D w ∃ | ∆ Dd > q&s'v'|, s'v'|, ; 1{$ Hmdjxldb `| &d' &d' h, xfm d}k`mp ji yoplo`gms odh xfm kovlkog hmbpmm ji o qjg|djklog' xfmd xfmpm opm }dlympsog qjg|djklogs edjrd ijp &d' & d' h, ; &=6' &=6' 8, &djxm xfox xfm qjssl`lglx| ji foyldb h ; 8 ijggjrs ipjk Sejgmk“s xplce, xj &d' &d' h, ; & qpj`gmk>
00
Cjpjggop| Gmx Q Gmx Q `m xfm cgoss ji qjg|djklogs rlxf ldxmbmp cjmfficlmdxs odh ji hmbpmm ox kjsx ij}p$ Xfmd' ?$ Xfmp Xfmpm ls dj ogbjpl ogbjplxfk xfk rflcf rflcf }qjd }qjd ldq}x ldq}x ji od| mgmkmd mgmkmdx x q ∆ Q hmclhms rfmxfmp jp djx q fos od ldxmbpog pjjx' odh 0$ xfmpm xfmpm ls dj ogbjplxfk ogbjplxfk rflcf }qjd ldq}x ji od| mgmkmdx q mgmkmdx q ∆ Q hmclhms rfmxfmp jp djx q fos o djd+dmboxlym ldxmbpog pjjx$
Qpjji > Oss}km Oss}km xfmpm xfmpm rj}gh rj}gh `m od ogbjplxf ogbjplxfk k ijp hmclhl hmclhldb db ldxmbpo ldxmbpogg pjjxs$ pjjxs$ Xfmd Xfmd xfmpm rj}gh `m jdm ijp djd+dmboxlym ldxmbpog pjjxs os rmgg' sldcm rm cod mvqgjlx xfm Gobpodbm ij}p s}opm xfmjpmk xfmjpmk xj xfm mdh xfox 1 ∆ q&Dd , ⇘ 1 ∆ q &^ 8d ,$ Xfmd ijp od| hljqfodxldm smx S ; uv ∆ De w ∃ | ∆ Dk > q&v' | , ; 1{' xfm f|qjxfmxlcog ogbjplxfk cj}gh `m }smh ld jphmp xj hmclhm v ∆ S ijp od| v$ Ld jxfm jxfmpp rjph rjphs' s' mymp mymp| | hljqfodxldm smx rj}gh `m o pmc}pslym smx$ Fjrmymp' rm edjrd xfox xfmpm opm djd+pmc}pslym smxs rlxfld xfm pmc}pslymg| md}kmpo`gm smxs$ Od sldcm xfm goxxmp opm mvocxg| xfm hljqfodxldm smxs' xfm oss}kqxljd ji s}cf od ogbjplxfk gmohs xj o cjdxpohlcxljd$ Xfm iocx xfox rm cod pmsxplcx j}psmgyms xj hmbpmm ox kjsx ij}p ijggjrs ipjk Sejgmk“s gmkko$ Rflgm ijp qjg|djklogs rlxf kovlkog hmbpmm xrj' xfmpm mvlsxs s}cf od ogbjplxfk' xfm cosm ji kovlkog hmbpmm xfpmm ls sxlgg jqmd$ Slklgopg| Slklgopg|'' ijp poxljdog &poxfmp &poxfmp xfod ldxmbpog, ldxmbpog, pjjxs' pjjxs' hmclho hmclho`lgl `lglx| x| ls od jqmd qpj`gm qpj`gmk$ k$ Ijp pmog pjjxs' jd xfm jxfmp jxfmp fodh' o pms}gx pms}gx ji Xopsel lkqglms xfox xfm qpj`gmk xfmd `mcjkms hmclho`gm$ Os rm rlgg qpjym ld xfm mvmpclsm' jdm cod mvxmdh xfm o`jym }dhmclho`lglx| pms}gx ld xfm ijggjrldb ijggjrldb hlpmcxljd> hlpmcxljd> gmx C `m od| smx ji cophldog d}k`mps ≨ D1 rflcf ls dmlxfmp mkqx| djp hjms lx cjdxold cjdxold ogg s}cf s}cf cophld cophldog og d}k`m d}k`mps$ ps$ Xfmd Xfmd xfm }msxljd }msxljd ji rfmxfm rfmxfmpp jp djx xfm d}k`mp ji djd+dmboxlym ldxmbpog pjjxs ji o qjg|djklog ls ld C x}pds j}x xj `m }dhmclho`gm os rmgg$ rmgg$ Xfm qpjji ls o pmh}cxljd pmh}cxljd ipjk C ; u1{ + xfm }dhmclho`lglx| ji Flg`mpx“s xmdxf qpj`gmk$
03
Qpjqjslxljd Xfmpm ls o qjg|djklog qjg|djklog &|? ' |0 ' $ $ $ ' |d ' v, rlxf ldxmbmp cjmfficlmdxs s}cf xfox xfm qjslxlym ldxmbmps ld lxs podbm opm mvocxg| ogg qplkm d}k`mps ld xfm smdsm xfox
&Dd%? , ∬ D u1{ ; xfm smx ji ogg qplkms$ qplkms$ Qpjji > Qplkms Qplkms ijpk o pmc}pslym pmc}pslymg| g| md}kmpo`gm md}kmpo`gm odh xf}s xf}s hljqfodxl hljqfodxldm dm smx S $ Xfls Xfls lkqglms lkqglms xfox xfmpm ls o qjg|djklog q rlxf ldxmbmp cjmfficlmdxs s}cf xfox S ; uv ∆ D w ∃ |Dd > q& q&v' |, ; 1{$ Hmdld Hmdldbb &|' v, >; v&? ∐ q& q&v' | ,0 , xfmd blyms xfm sj}bfx qjg|djklog sldcm xfls ls qjslxlym lfi q fos o pjjx ld rflcf cosm ldhmmh &|' v, xoems jd xfm yog}m ji xfm cjppmsqjdhldb qplkm$ Ijggjrldb xfm pmkope pmbophldb }dlympsog qjg|djklogs' rm j`xold xfox ijp qplkm d}k`mp qpjh}cldb qpjh}cldb qjg|djk q jg|djklogs logs xmd yoplo`gms oplo`gms s}fficm$ O slklgop cjdsxp}cxljd cjdsxp}cxljd gmohs xj xfm ijggjrldb> ijggjrldb> Qpjqjslxljd Gmx i > D ← D `m od| qopxlog pmc}pslym i}dcxljd$ Xfmpm mvlsxs o qjg|djklog rlxf ldxmbmp cjmfficlmdxs s}cf xfox ijp ogg v' | ∆ D> | ; i & i &v, ⇘ ∃ v1 ' $ $ $ ' vd ∆ D > | ; &v' v1 ' $ $ $ ' vd ,$ Qpjji > Xfm bpoqf ji i ji i ls pmc}pslymg| md}kmpo`gm odh xf}s hljqfodxldm$ Sj | ; i & i &v, fjghs lfi ijp o cmpxold qjg|djklog q rm foym ∃ v1 ' $ $ $ ' vd > &? ∐ q& q&v1 ' $ $ $ ' vd ' v,0 , 4 1 ∢ v1 ; | $ Xfls Xfls ld x}pd x}pd ls m}ly m}lyogm ogmdx dx xj ∃ v1 ' $ $ $ ' vd > &v1 % ?,&? ∐ q&v1 ' $ $ $ ' vd ' v,0 , ; | % ?$ Xfmpmi Xfmpmijpm jpm xfm sj}bfx sj}bfx qjg|djk qjg|djklog log cod `m hmdmh hmdmh os &v' v1 ' $ $ $ ' vd, >; &v &v1 % ?,&? ∐ q& q&v1 ' $ $ $ ' vd ' v,0 , ∐ ?$ Oqqglcoxljd ji Sjg}xljd O qopxlc}gopg| ijpk ji B¦jhmg“s jhmg“s ldcjkqgmxmdmss xfmjpmk ls ogsj o cjdsm}mdcm ji xfm Koxl|osmylcf*KPHQ Xfmjpmk> Gmx q&o' v? ' $ $ $ ' ve , ; 1' qpjylhm o Hljqfodxldm Hljqfodxldm hmdlxljd hmdlxljd ji o djd+cjkq}xo djd+cjkq}xo`gm `gm smx$ Gmx O `m od ogbjplxfk xfox j}xq}xs o sm}mdcm ji dox}pog d}k`mps s}cf xfox xfm cjppmsqjdhldb m}oxljd q&d' v? ' $ $ $ ' ve , ; 1'fo 1'foss dj sjg}x sjg}xlj ljds ds ld dox} dox}po pogg d}k`m }k`mps ps$$ Xfmd Xfmd xfmpm xfmpm ls o d}k`mpd d}k`mpd1 rflcf ls djx j}xq}x `| O rflgm ld iocx xfm m}oxljd q&d' v? ' $ $ $ ' ve , ; 1' fos dj sjg}xljds ld dox}pog d}k`mps$
08
Xj smm xfox xfm xfmjpmk ls xp}m' lx s}fficms xj djxlcm xfox li xfmpm rmpm dj s}cf d}k`mp d1 ' jdm cj}gh ogbjplxfklcogg| xmsx kmk`mpsflq ji o d}k`mp d ld xfls djd+cjkq}xo`gm smx `| slk}gxodmj}sg| p}ddldb xfm ogbjplxfk O xj smm rfmxfmp d ls j}xq}x rflgm ogsj cfmceldb ogg qjssl`gm e+x}qgms ji dox}pog d}k`mps smmeldb o sjg}xljd ji xfm m}oxljd q&d' v? ' $ $ $ ' ve , ; 1$ Rm ko| ossjcloxm od ogbjplxfk O rlxf od| ji xfm }s}og ijpkog s|sxmks s}cf os Qmodj Oplxfkmxlc jp ^ I C `| gmxxldb lx s|sxmkoxlcogg| bmdmpoxm cjdsm}mdcms ji xfm ovljks odh xfmd j}xq}x o d}k`mp d rfmdmymp o smdxmdcm ji xfm ijpk q&d' v? ' v0 ' $ $ $ $ $ve , ; 1[ ∃v? ' v0 ' $ $ $ ve U q& ls bmdmpoxmh$ bmdmpoxmh$ Xfmd xfm xfmjpmk xmggs }s xfox mlxfmp o iogsm sxoxmkmd sxoxmkmdxx ji xfls ijpk ls qpjymh qpjymh jp o xp}m jdm pmkolds }dqpjymh ld xfm s|sxmk ld }msxljd$
0=
Cfoqxmp 9
Rodb“s Xlgm Qpj`gmk S}qqjsm S}qqjsm rm rodx rodx xj cjym cjympp xfm qgodm rlxf hmcjpoxm hmcjpoxmh h s}opm s}opm xlgms ji xfm sokm sl~m$ Xlgms Xlgms opm xj `m cfjsm cfjsmd d ipjk ipjk o dlxm dlxm d}k`mp d}k`mp ji x|qms$ x|qms$ Xfmpm Xfmpm opm }d`j}d }d`j}dhmh hmh xlgms ji mocf mocf x|qm x|qm oyolgo`gm oyolgo`gm$$ H}m xj xfm hmcjpoxljds' hmcjpoxljds' fjrmymp' fjrmymp' xfmpm opm gjcog cjdsxpoldxs cjdsxpoldxs jd rflcf rflcf xlgms cod `m q}x dmvx xj mocf jxfmp' ijp xfm xlgldb xj gjje oqqmogldb$ Ls lx qjssl`gm xj cjymp xfm rfjgm qgodm rlxf xlgms ji blymd x|qms: Fjr li rm pm}lpm o cmpxold xlgm xj `m }smh ox gmosx jdcm: Cod xfm| `m }smh xj xlgm o dlxm pmcxodb}gop pmcxodb}gop opmo' opmo' rlxf o cmpxold `j}dhop| cjdhlxljd: cjdhlxljd: Lx x}pds j}x xfox xfmsm qpj`gmks xfm ro| ijpk}goxmh `| Foj R odb odb opm ogg }dhmclho`gm$ Ld ? Gmx K `m o X}pldb }pldb Koc Kocfld fldmm + od ldsxodcm ldsxodcm ji xfm Fogxldb Fogxldb Qpj`gmk$ Qpj`gmk$ Rm }sm xfm xlgm smx X K K ' cjdsxp}cxmh o`jym' odh o xlgm smx X P ' xj cjdsxp}cx o dmr smx ji xlgms X xfox cod xlgm xfm qgodm' li odh jdg| li' K jd mkqx| ldq}x dmymp fogxs$ Mocf xlgm ld X ls `oslcogg| o qolp &s' &s' x,' rfmpm s ∆ X P ' odh X ls mlxfmp ld X K K jp ls ji od o}vlglop| x|qm' xj qoss ldijpkoxljd ogjdb o ympxlcog jp fjpl~jdxog gldm$ Xfm xrj cjkqjdmdxs opm ldxmpqpmxmh os go|mps ji xfm xlgm$ Gmx s1 ∆ X P `m od| xlgm xfox pmqpmsmdxs xfm gjrmp+gmix cjpdmp cjpdmp ji o cjddmc cjddmcxmh xmh pmbljd$ pmbljd$ Ijp od| s}cf s}cf xlgm' rm qgocm qgocm o xlgm &s &s1 ' x1 , ld X ' X ' rfmpm x1 ls xfm kopemh xlgm ji X K hmdmh o`jym$ o`jym$ Xfls Xfls ls xfm jdg| jdg| qolp ld rflcf x1 oqqmops' K os hmdmh odh ls s}qqjsmh xj xplbbmp xfm sxopx ji slk}goxljd ld mocf cjddmcxmh pmbljd$ Od| jxfmp xlgm X ∆ X K K Wux1 { ls qolpmh rlxf xlgms s ∆ X P xfox pmqpmsmdx o cmgg i}gg| ldslhm o cjddmcxmh pmbljd$ Gmx }s so| o pjr &jp cjg}kd, ji o pmbljd ls ipmm li lx hjms djx qoss xfpj}bf o fjgm$ Jxfmprlsm' xfm pjr &jp cjg}kd, ls `gjcemh `| xfm fjgm$ Xfm ldxmpsmcxljd ji o ipmm pjr odh ipmm ipmm cjg}kd cjg}kd ls o ipmm cmgg$ Jdg| Jdg| ipmm cmggs opm }smh ijp slk}goxl slk}goxljd$ jd$ O cmgg cmgg xfox xfox `mgjdbs `mgjdbs xj o `gjcemh pjr &pmsq$ &pmsq$ o `gjcemh cjg}kd, cjg}kd, slkqg| qossms xfm ldijpkoxljd ldijpkoxljd ogjdb xfm mhbm ji 06
xfm `gjceldb fjgm$ Kjpm cgmopg|' o cmgg rfjsm gjrmp &jp }qqmp, mhbm xj}cfms o fjgm slkqg| qossms xfm cjgj}p ji lxs gmix mhbm xj xfm plbfx' odh oses lxs }qqmp &pmsq$ gjrmp, dmlbf`jp xj hj xfm sokm$ Slklgopg| Slklgopg|'' o cmgg rfjsm gmix &pmsq$ plbfx, plbfx, mhbm xj}cfms o fjgm qossms xfm cjgj}p ji lxs gjrmp mhbm xj xfm }qqmp mhbm odh oses lxs plbfx &pmsq$ gmix, dmlbf`jp xj hj xfm sokm$ Djr' lx sfj}gh `m cgmop xfox xfm o`jym+h mscpl`mh xlgms cod xlgm o cjddmcxmh pmbljd C ji xfm hmcjpoxljd' li odh jdg| li' XK cod xlgm o s}opm rlxf xfm sokm opmo os C' qpjylhmh x1 ls }smh ld xfm gjrmp+gmix cjpdmp$ Sldcm xfm mdxlpm hmcjpoxljd qpjylhms cjddmcxmh pmbljds ji op`lxpoplg| gopbm dmx opmo' }sldb xfm Mvxmdsljd Gmkko' rm cjdcg}hm xfox xfm qgodm cod `m xlgmh `| xfm xlgms ld X ' X ' li odh jdg| li' X K cod xlgm od }qqmp+plbfx }opxmp ji xfm qgodm rlxf x1 ld xfm gjrmp+gmix cjpdmp$ Xfls cjkqgmxms xfm qpjji$
0<
Cfoqxmp 2
@}s| @moymp O `}s| `moymp ls o X}pldb kocfldm xfox oxxolds xfm kovlk}k ‗jqmpoxljdog `}s|dmss‛ okjdb ogg xfm X}pldb kocfldms ld o cmpxold cgoss$ Xfm X}pldb kocfldms ld xfls cgoss k}sx kmmx cmpxold hmslbd sqmclcoxljds odh opm pm}lpmh xj mymdx}ogg| fogx oixmp `mldb sxopxmh rlxf o `gode xoqm$ O `}s| `moymp `moymp i}dcxljd }odxlms }odxlms xfmsm }qqmp glklxs jd o blymd blymd x|qm ji‗jqmpoxljdog `}s|dmss‛' odh ls o djdcjkq}xo`gm i}dcxljd$ Ld iocx' o `}s| `moymp i}dcxljd cod `m sfjrd xj bpjr iosxmp os|kqxjxlcogg| xfod hjms od| cjkq}xo`gm i}dcxljd$ Xj }dhmpsxodh `}s| `moymp i}dcxljd rm dmmh xj koem imr cfodbms ld j}p x}pldb kocfldm odh lxs i}dcxljdldb$ Sj rm hmdm j}p x}pldb kocfldm os ijggjrs> X}pldb kocfldm fmpm' ls o hmylcm cjdslsxldb ji ?$ o xoqm' X ; u1' ?{ odh rm so| xfox xfm cmgg ls `gode li lx cjdxolds o ~mpj' 0$ o pmoh*rplxm pmoh*rplxm fmoh' x|qlcogg| x|qlcogg| 3$ o dlxm smx ji ldxmpdog ldxmpdog sxoxms sxoxms _ ; u1' $ $ $ ' d{ 8$ o glsx ji ldsxp}cxljds' ldsxp}cxljds' x|qlcogg| x|qlcogg| rfmpm uP' G{ opm xfm hlpmcxljds hlpmcxljds xfm fmoh cod kjym$ kjym$ Rm rlgg sjkmxlkms sjkmxlkms rplxm K &d, li rm rodx xj koem xfm d}k`mp ji ldxmpdog sxoxms mvqglclx$ Mvokqgm ?> O XK K &e%?, rflcf rplxms e jdms jdxj o `gode xoqm odh xfmd fogxs o`jym &?' &1' xfm gmixkjsx jdm$ K > &1' &1' x, ← &?' x % ?' G, ijp x ; 1' $ $ $ ' e ∐ ? odh K > &1' &1' e, ← &1' e %? %?'' P,$
31
Rm rlgg }sm xfm cjdymdxljd xfox xfm ldlxlog sxoxm ls ; 1 odh xfm gosx sxoxm ; d ls xfm fogxldb fogxldb sxoxm + xfm jdg| sxoxm sxoxm }qjd rflcf rflcf xfm kocfld kocfldmm fogxs$ fogxs$ Sldcm Sldcm xfls pms}gxs pms}gxs ld d ocxlym ocxlym sxoxms' sxoxms' xfm kocfldm ls coggmh od d+sxoxm d+sxoxm X}pldb X}pldb kocfldm$ Koxfmkoxlcog Koxfmkoxlcogg| g|'' xfm smx ji ldsxp}cxljds cfopocxmpl~ldb o X}pldb kocfldm ls o koq' K > X Ò _ ∐← X Ò _ Ò uP' G{ Ld jphmp xj xoge o`j}x X}pldb kocfldms os hmylcms rflcf cjkq}xm i}dcxljds ji xfm ijpk i > De ∐← D' rm dmmh xj sqmcli| sjkm cjdymdxljds o`j}x fjr ldq}x odh j}xq}x opm pmqpmsmdxmh$ Rm rlgg }sm }dop| mdcjhldb ijp `jxf ji xfmk$ Xfox ls' o d}k`mp v ∆ D rlgg `m pmqpmsmdxmh `| v % ? cjdsmc}xlym ?s jd xfm xoqm rlxf xfm pmsx ji xfm xoqm `gode &m$b$' 0 rj}gh cjppmsqjdh xj 1 $ $ $ ???1 $ $ $ 1,$ Slklgopg|' &v &v? ' $ $ $ ' ve , ∆ De rlgg `m pmqpmsmdxmh `| e s}cf s}cf `gjces ji ?s smqopo smqopoxmh xmh `| sldbgm sldbgm ~mpjs &m$b$ &1' &1' 0, rj}gh `m 1 $ $ $ 1?1???1 $ $ $ 1,$ O &d, X}pldb kocfldm K i ls xfmd solh xj cjkq}xm xfm i}dcxljd i > De ∐← D lfi xfm kocfldm sxopxldb rlxf xfm fmoh qgocmh jd xfm gmixkjsx ? ji xfm }dop| mdcjhldb ji v ji v ∆ De mymdx}ogg| ∆ hjk& fogxs jd xfm gmixkjsx ? ji xfm mdcjhmh i&v, li v li v ∆ hjk& hjk&i , i , odh lx dmymp fogxs li v li v hjk&i ,$ i ,$ Hmdlxljd O i}dcxljd i > De ∐← D ls coggmh X}pldb cjkq}xo`gm lfi xfmpm ls od d+sxoxm X}pldb kocfldm &XK, ijp sjkm dlxm d rflcf cjkq}xms i ld xfm smdsm xfox xfm X}pldb kocfldm fogxs ijp mymp| ldq}x v ∆ hjk& hjk&i , i , rlxf xfm xoqm mymdx}ogg| pmqpmsmdxldb i&v, odh ∆ hjk& lx hjmsd“x fogx li v hjk&i ,$ i ,$ Fmpm' Fmpm' ldq}x odh j}xq}x opm mdcjhmh ld xfm o`jym sqmclmh sqmclmh }dop| ro| odh ox xfm sxopx odh &qjxmdxlog, mdh ji xfm cjkq}xoxljd xfm fmoh ji xfm XK sfj}gh `m qjslxljdmh o`jym xfm gmixkjsx djd+`gode s|k`jg ji xfm xoqm$ Mvokqg Mvokqgm m 0> xfm s}ccms s}ccmssjp sjp i}dcxl i}dcxljd jd s&v,;v s&v,;v%? %? cod `m cjk cjkq}x q}xmh mh `| xfm ijggj ijggjrld rldbb 0+ &0, sxoxm X}pldb kocfldm K v%? $&?' &?' 1, ← &?' 1' P,' &1' &1' 1, ← &?' ?' G,' &?' &?' ?, ← &?' ?' G,' &1' &1' ?, ← &?' &?' &?' &1' &1' 0' P,$ Mvokqg Mvokqgm m 3> xfm ~mpj i}dcxljd ~&v,;1 cod `m cjkq}xmh `| o 0+sxoxm XK> &?' &?' 1, ← &1' &1' 1' P,' &1' &1' 1, ← &?' ?' P,' &1' &1' ?, ← &1' 0' G,$ &?' &1' &=,
0v > &1' &1' Mvokqgm 8> xfm ijggjrldb =+sxoxm XK K 0v lkqgmkmdxs v ← &1' 1, ← &1' 3' P,' &?' &?' 1, ← &?' &?' &?' &?' &1' &1' ?' G,' &1' &1' ?, ← &?' 0' P,' &?' &?' ?, ← &?' ?' G,' &1' &1' 0, ← &?' 1' P,' &?' &?' 0, ← &?' 0' P,' &1' &1' 3, ← &1' &1' &?' &1' &1' 3' G,' &?' &?' 3, ← &1' 8' G,' &1' &1' 8, ← &1' =' P,' &?' &?' 8, ← &?' 8' G,$
3?
Oixmp hmdldb xfm dmr x}pldb kocfldm' rm djr hmdm cjdcoxmdoxljd ji xrj x}pldb &d , &d , kocfld koc fldms$ ms$ Gmx K i i odh K b b `m xrj XKs rlxf di ' db ldxmpdog ldxmpdog sxoxms' sxoxms' cjkq}xldb cjkq}xldb &d %db ,
i}dcxljds i odh b pmsqmcxlymg|$ Xfmd rm cod o hmdm o dmr &d &di % db ,∐sxoxm XK K bi i ylo
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Lxs ocxljd rlgg `m s}cf xfox lx psx cjkq}xms i & i &v, odh xfmd }sms xfm pms}gxldb j}xq}x os od ldq}x ijp b$ Fmdcm' K bi cjkq}xms xfm cjdcoxmdoxlj cjdcoxmdoxljd d cjppmsqjdhldb cjppmsqjdhldb xj v ← i &v,, b &i & bi cjkq}xms ijp rflcf rm rlgg ogsj rplxm bi & bi &v,$ Djxm Djxm xfox xfox xfm qjssl`lglx qjssl`lglx| | ji cjdcoxmdo cjdcoxmdoxldb xldb xrj xrj XKs ld xfls ro| `}lghs }q jd j}p pm}lpmkmdxs xfox xfm j}xq}x ji o cjkq}xoxljd fos xj `m mdcjhmh ld }dop| jd xfm xoqm odh xfox xfm XK &li mymp, fogxs rlxf xfm fmoh qjslxljdmh jd xfm gmixkjsx ?$ Djr rm kjym jd xj hmdldb xfm `}s| `moymp i}dcxljd$ Gmx }s osslbd o d}k`mp @ &K , ∆ D xj mymp| X}pldb kocfldm K `| cjdslhmpldb lxs `mfoyljp rfmd p}d jd od ldlxlogg| `gode xoqm$ Rm smx @ &K , K , >; 1 li K li K dmymp fogxs odh @ &K , K , >; ` li lx fogxs odh xfm xjxog d}k`mp ji &djx dmcmssoplg| cjdsmc}xlym, ?s mymdx}ogg| rplxxmd jd xfm xoqm ls `$ @osmh @osmh jd xfls xfls rm cod hmdm xfm `}s| `moymp i}dcxljd @@ > D ← D @@& @@ &d, >; kovu@ &K , K ,wK ∆ uK &d, {{ @@& @@ &d, ls hmdmh os xfm gopbmsx d}k`mp ji ?s mymdx}ogg| rplxxmd jd od ldlxlogg| `gode xoqm `| od| od| d+sxoxm d+sxoxm XK rflcf rflcf fogxs$ fogxs$ Djxm Djxm xfox xfox xfm i}dcxljd i}dcxljd ls rmgg+ rmgg+hm hmdmh dmh sldcm sldcm xfm kovlk}k ls xoemd jymp o dlxm smx$ Gmkko @}s| @moymp i}dcxljd ls kjdjxjdlcogg| ldcpmosldb> @@& @@ &d % ?, 4 @@& @@ &d, ijp ogg d ∆ D$ Qpjji Hmdjxm xfm XK rflcf ocflmyms @@& @@ &d, `| K &d, $ @o @osm smh h jd xfls rm cod cod hmdm hmdm o &d % ?,∐sxoxm XK K &d%?, rfjsm ldsxp}cxljds m}og xfjsm ji K &d, ijp ogg ldxmpdog sxoxms &?' 7 d odh rflcf ld ohhlxljd ijggjrs xfm p}gm &x' &x' d, ← &?' d % ? ∐ x' P,$ @| cjdsxp}c cjdsxp}cxljd xljd @@& @@ &d, % ? ; @ &K &d%?, , ≨ @@& @@ &d % ?,$
30
Xfls gmohs }s xj o cjkkjd qpjqmpx| ji ogg X}pldb cjkq}xo`gm i}dcxljds xfox xfm| coddjx bpjr iosxmp xfod @@$ @@ $ Xfmjpmk Gmx i > D ∐← D `m od| i}dcxljd rflcf ls X}pldb cjkq}xo`gm `| o e+sxoxm XK K i i &v, 7 @@& @@ &v,$ i $ Xfmd ijp ogg v 4 0e % ?3 ijp rflcf i ls hmdmh rm foym i & Qpjji Rm }xlgl~m cjdcoxmdoxljd ji xfm o`jym hlsc}ssmh mvokqgms odh xfmd hmdm o &e &e % d % 6,∐sxoxm XK &e,
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K i &0d%?, >; K i K v%? K 0v K &d%?, rlxf K &d%?, `mldb xfm XK rflcf rplxms d cjdsmc}xlym ?s$ P}ddldb K i &0d%?, jd xfm `gode xoqm xfmd qpjh}cms i &0 i &0d d % ?, cjdsmc}xlym jdms `mijpm fogxldb$ Xf}s' i &0 i &0d d % ?, ≨ @@& @@ &e % d % 6,$ Kjpmjymp' kjdjxjdlclx| ji @@ lkqglms @@& @@ &e % d % 6, 7 @@&0 @@ &0d d % ?, li e % 2 7 d$ jp' i &0 i &0d d % ?, 7 @ @ &0d &0d % ?, li e % 2 7 d$ Slklgopog| li rm cjdsxp}cx o &e &e % d % 9,∐sxoxm XK &e ,
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K i &0d, >; K i K 0v K &d%?, rfmpm mymp| mymp| cjkqjdmdx cjkqjdmdx ls sokm os hmdmh o`jym$ ]sldb xfm kjdjxjdlclx| kjdjxjdlclx| ji @@ rm j`xold' i &0 i &0d d, 7 @@&0 @@ &0d d, li e % 9 7 d$ Ijp i & i &v, 7 @@& @@ &v, xj fjgh xp}m' v cod mlxfmp mlxfmp `m mymd mymd odh xoem xoem yog}ms og}ms ?8'?9'?6$$ ?8'?9'?6$$$$ jp v cod `m jhh odh xoem xoem yog}ms ?='?2'? xj smm li rfmxfmp rfmxfmp xrj gjjqs opm ipmmg| fjkjxjqlc$ Xfm Cjdn}boc| qpj`gmk ls }dhmclho`gm$ Xfm cjdn}boc| qpj`gmk ijp op`lxpop| qpmsmdxo+ xljds &odh djx n}sx jdm vmh qpmsmdxoxljd, hjms pmh}cm xj xfm rjph qpj`gmk ijp op`lxpop| qpmsmd qpmsmdxoxl xoxljds jds'' sldcm sldcm `jxf opm m}lyogm m}lyogmdx dx xj xfm fogxldb fogxldb qpj`gm qpj`gmk$ k$ Xfox Xfox ls' xfm fogxldb fogxldb qpj`gmk iokj}sg| pmh}cms xj xfm rjph qpj`gmk' rflcf pmh}cms xj xfm cjdn}boc| qpj`gmk os `}x xfls cgmopg| pmh}cms xj xfm fogxldb qpj`gmk$
Lsjkjpqflsk Qpj`gmk Xfm lsjkjpqflsk qpj`gmk ls xj dh od ogbjplxfk xj hmxmpkldm rfmxfmp xrj dlxm qpmsmd qpmsmdxoxl xoxljds jds blym lsjkjpqfl lsjkjpqflcc bpj}qs bpj}qs$$ Xfm bmjkmxplc bmjkmxplc kjxlyoxl kjxlyoxljd jd ls xj smm li jdm cod 32
ogbjplxfklcogg| hlsxldb}lsf sqocms `osmh jd xfmlp i}dhokmdxog bpj}qs$ Xfm }dhmclho`lglx| ji xfm lsjkjpqflsk qpj`gmk cod `m qpjjimh `| Ohnod+Po`ld xfmjpmk$ Xfls xfmjpmk blyms o bmdmpog cjdsxp}cxljd rflcf cod `m oqqglmh xj od| Kopejy qpjqmpx| Q ji dlxmg| qpmsmdxmh bpj}qs xj qpjji xfm }dhmclho`lglx|$ Ogsj' Ogg yoplmxlms ji sjgyo`gm bpj}qs rlxf }dhmclho`gm rjph qpj`gmk foym }dhmclho`gm lsjkjpqflsk qpj`gmk$
]dhmclho`gm sxoxmkmdxs ld ^IC Oss}kldb xfox ^IC ls cjdslsxmdx' xfm koxfmkoxlcog sxoxmkmdxs kmdxljdmh `mgjr opm qpjyo`g| }dhmclho`gm ld ^IC &xfm ^mpkmgj Ipomdemg ovljks qg}s xfm ovljk ji cfjlcm, >
?$ Smx xfmjp| ji xfm pmog pmog gldm> Lx ls od opmo ji koxfmkoxlcs koxfmkoxlcs cjdcmpdmh cjdcmpdmh rlxf xfm oqqglco+ oqqglco+ xljd ji smx xfmjp| xfmjp| xj osqmcxs ji xfm pmog d}k`mps$ d}k`mps$ Ovljkoxlc Ovljkoxlc smx xfmjp|' xfmjp|' `| B¦ jhmg“s jhmg“s ldcjkqgmxmdm ldcjkqgmxmdmss ss xfmjpmk' xfmjpmk' cjdxolds qpjqjslxljds qpjqjslxljds xfox opm }dhmclho`g }dhmclho`gm$ m$ Ogsj' xfm pmog d}k`mps opm kjsx jixmd ijpkogl~mh }sldb xfm ^mpkmgj Ipomdemg ovljkoxl~oxljd ji smx xfmjp|$ 0$ Ovljks Ovljks ji Cjdsxp Cjdsxp}cx }cxo`l o`lglx glx|> |> Lx ls o qjssl`g qjssl`gmm ovljk ovljk ijp smx xfmjp| xfmjp| ld koxfmk koxfmkoxlc oxlcss xfox ossmpxs xfox mymp| smx ls cjdsxp}cxl`gm$ Lx lkqglms xfm ovljk ji cfjlcm jymp ^IC xfmjp|$ 3$ Cjdxld}} Cjdxld}}k k F|qjxfmsls> F|qjxfmsls> Lx sxoxmh xfox xfmpm ls dj lddlxm smx rlxf o cophldog cophldog d}k`mp d}k`mp `mxrmmd xfox ji xfm ‗skogg‛ lddlxm smx ji ldxmbmps odh xfm ‗gopbm‛ lddlxm smx ji pmog d}k`mps &xfm cjdxld}}k, Lx ros sfjrmh `| B¦ jhmg xfox dj cjdxpohlcxljd rj}gh oplsm li xfm cjdxld}}k f|qjxfmsls jhmg rmpm rmpm ohhmh ohhmh xj cjdym cjdymdx dxljd ljdog og ^IC ^IC smx xfmjp| xfmjp|$$ Fj Fjrm rmym ymp' p' }sldb }sldb o xmcfdl xmcfdl}m }m coggmh ijpcldb' Qo}g Cjfmd qpjymh xfox dj cjdxpohlcxljd rj}gh oplsm li xfm dmboxljd ji xfm cjdxld}}k f|qjxfmsls ros ohhmh xj smx xfmjp|$ Xjbmxfmp' xfm pms}gx ros msxo`glsfmh xfox xfm yoglhlx| ji xfm cjdxld}}k f|qjxfmsls hmqmdhs jd xfm ympsljd ji smx xfmjp| `mldb }smh' odh ls xfmpmijpm }dhmclho`gm$ 8$ Bpj}q Xfmjp|> Xfmjp|> Lx sx}hlms xfm ogbm`polc ogbm`polc sxp}cx}p sxp}cx}pms ms edjrd edjrd os bpj}qs$ Lx ls od }dhmclh+ o`gm xfmjp|$ Xfmpm ls dj cjkq}xo`gm qpjcmss xj hmxmpkldm rfmxfmp o blymd sxoxmkmdx ld xfm psx jphmp godb}obm ji bpj}q xfmjp| ls xp}m ld ogg bpj}qs$ 36
=$ Kmos}p Kmos}pmm Xfmjp|> Xfmjp|> Lx ls o`j}x o`j}x xfm sx}h| sx}h| ji kmo kmos}p s}pms$ ms$ Lx bmdmpogl~ bmdmpogl~ms ms xfm ldx}lxly ldx}lxlymm djxljds djxljds ji gmdbxf gmdbxf'' opmo' opmo' odh yjg}km$ yjg}km$ Xfm mopglmsx mopglmsx odh kjsx kjsx lkqjpxod lkqjpxodxx mvokqg mvokqgms ms opm Njphod kmos}pm odh Gm`msb}m kmos}pm' `}x jxfmp mvokqgms opm @jpmg kmos}pm' qpj`o`lglx| kmos}pm' cjkqgmv kmos}pm' odh Foop kmos}pm$ 9$ Jphmp Xfmjp|> Xfmjp|> Lx ls o `podcf ji koxfmkoxlcs koxfmkoxlcs rflcf ldymsxlbox ldymsxlboxms ms j}p ldx}lxlym ldx}lxlym djxljd ji jphmp jphmp }sldb `ldop| `ldop| pmgoxlj pmgoxljds$ ds$ Lx qpjylh qpjylhms ms o ijpkog ijpkog ipokmrj ipokmrjpe pe ijp hmscpl hmscpl`ld `ldbb sxoxmkmdxs s}cf os ‗xfls ls gmss xfod xfox‛ jp ‗xfls qpmcmhms xfox‛$ 2$ I}dcxljdog }dcxljdog Odog|sls> Odog|sls> Lx ls cjdcmpdmh cjdcmpdmh rlxf lddlxm+hlkmdslj lddlxm+hlkmdsljdog dog ymcxjp sqocms &koldg| i}dcxljd i}dcxljd sqocms, odh koqqldbs `mxrmmd `mxrmmd xfmk$ Xfm sqocms ko| `m ji hlfimpmdx' hlfimpmdx' odh qjssl`g| lddlxm hlkmdsljds$ Xfmsm koqqldbs opm coggmh jqmpoxjps jp' li xfm podbm ls jd xfm pmog gldm jp ld xfm cjkqgmv qgodm' i}dcxljdogs$
Koxplv + Kjpxoglx| Qpj`gmk Cjdslhmp o idlxm smx ji h Ò h koxplcms S ; uK ? ' $ $ $ ' Kd { ⊀ K h &^, rlxf ldxmbmp mdxplms$ Rm cogg S kjpxog lfi xfmpm ls o djd+mkqx| rjph r ∆ u?' $ $ $ d{ ' ji gmdbxf k so|' s}cf xfox ijp xfm cjppmsqjdhldb qpjh}cx ji koxplcms> K r? ² ² ² K rk rk ; 1 Mvokqgm ? Cjdslhmp o smx cjdslsxldb ji xrj koxplcms
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Xfls coddjx `m kjpxog sldcm xfm koxplcms foym djd+~mpj hmxmpkldodx odh xfm hmxmpkl+ dodx ji od| qpjh}cx ls n}sx xfm qpjh}cx ji hmxmpkldodxs$ Mvokqgm 0 Xfm xrj koxplcms
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ijpk o kjpxog smx sldcm xfmlp qpjh}cx ls o dlgqjxmdx koxplv rfjsm s}opm yodlsfms$ @mijpm rm sfjr xfox }dglem ld xfmsm slkqgm mvokqgms xfmpm coddjx `m o bmdmpog pmclqm ijp hmclhldb kjpxoglx|' rm rlgg ldxpjh}cm sjkm xjjgs ijp mdcjhldb rjphs ldxj qpjh}cxs ji koxplcms> Ijp rjphs r ; o? $ $ $ ok jymp xfm ogqfo`mx O >; u?5053{ hmdm od ldnmcxlym koq R & R &r, >; Σek;? oe 8k∐e ipjk O xj D$ Hmdjxm `| wrw xfm gmdbxf ji o rjph odh hmdm o koq ipjk O Ò O qplkm ldxj xfm smx ji 3 Ò 3 ldxmbmp koxplcms `| K & K &}' r, >;
8w}w 1 1 1 8wrw 1 R & R &}, R & R &r, ?
Li rm }sm cjdcoxmdoxljd ji rjphs odh koxplv k}gxlqglcoxljd os `ldop| jqmpoxljds ld xfm K &}' r, ls od ldnmcxlym kjdjlh fjkj+ hjkold odh cjhjkold pmsqmcxlymg|' xfmd &}' &}' r, ← kjpqflsk$ Xfox ls' ld qopxlc}gop K & K &}?' y ?,K ?,K &}0' y 0, ; K & K &}?}0' y ?y 0,$ 0,$ Ld ohhlxljd rm rlgg dmmh xfm koxplv @ >;
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