
turinginte thelivu

1937 januvariyil "kanakkukoottaavunna samkhyakalil, avaykku oru prayogam" enna thalakkettode aadyamaayi prasidheekaricha alaan turinginte thelivaanu turinginte thelivu. hilbertinte nishedhathinte randaamathe thelivaayirunnu (sabhayude sidhaanthathinushesham); athaayath, chila poornnamaayum ganithashaasthraparamaaya athe-illa chodyangalkku orikkalum kanakkukoottaliloode utharam nalkaan kazhiyillenna anumaanam; kooduthal saankethikamaayi, chila theerumaana prasnangal "theerumaanikkaanaavaathava" aanenna ardhathil prashnatthinte oro sandarbhathinum thettaaya "athe" allengil "illa" enna utharam nalkunna orotta algoritam illa. turinginte swantham vaakkukalil
"njaan theliyikkunnathu godelinte ariyappedunna phalangalil ninnu thikachum vyathyasthamaan... k ["principia maathamattikka"] ll oru nischitha soothravaakyam u theliyikkanaakumo ennu parayunna pothuvaaya oru reethiyum illennu njaan ippol kanikkum.
turing mattu rund perkkoppam eee thelivu pinthudarnnu. randaamatheyum moonnaamatheyum randuperum aadyathethine aasrayikkunnu. lalithamaaya oru koottam niyamangal anusarikkunna typetiter polulla "computing mesheenukalude" vikasanatheyum thudarnnu oru "saarvathrika computing mesheen" vikasippichedukkunnathu aasrayichirikkunnu.
thelivukalude sangraham.
enshaidungs prashnathinu parihaaramillenna thante thelivil, turing thante anthima thelivilekku nayikkunna rund thelivukalil ninnu munnott pooyi. addehathinte aadya sidhaantham nirtthal prashnathinu ettavum prasakthamaanu, randaamathethu arrie sidhaanthathinu kooduthal prasakthamaanu.
aadya thelivu oru ekapaksheeyamaaya "computing mesheen" aano allayo ennu theerumaanikkaan kazhiyunna oru "computing mesheen" nilavililla (oru poornnasankhya 1,2,3 prathinidheekarikkunnath pole). . .) ennathu "vritham-rahita" (i.iii. athinte nambar bainari aadu infinittathil achadikkunnu): "...parimithamaaya ennam ghattangaliloode ithu cheiyunnathinu namukk pothuvaaya prakriyayilla "(p. 132, "ibid".). turinginte thelivu, "diagonal prakriya" upayogikkunnathaayi thonnumenkilum, vaasthavathil addehathinte yanthrathin (echu ennu vilikkunnu) athintethaaya samkhya kanakkakkaan kazhiyillennu kaanikkunnu, muzhuvan diagonal number (kaantarinte diagonal aargyument) ozhike "vaadathile thettidhaarana b [diagonal nambar] kanakkukoottaavunnathaanne anumaanikkunnathil adangiyirikkunnu" thelivinu kooduthal ganithashaasthram aavasyamilla.
randaamathe thelivu ithu orupakshe vaayanakkaarkk arrie sidhaanthamaayi kooduthal parichithamaann "namukk kooduthal kaanikkan kazhiyum" "s nalkumbol oru yanthram e undaakilla.oru ekapaksheeyamaaya mesheen m-nte d [prograam], m eppozhengilum oru chinnathe print cheythittundo ennu nirnnayikkum (0 parayuka)."
moonnaamathe thelivu "oro computing mesheen m-num anusrithamaayi nammal oru formula un (m) nirmmikkunnu, koodaathe un (m) theliyikkanaakumo ennu nirnnayikkaan oru pothu reethi undengil, m eppozhengilum 0 print cheiyunnundo ennu nirnnayikkaan oru pothu reethi undennu nammal kaanikkunnu".
moonnaamathe thelivinu aadyathe lemma theliyikkaan aupachaarika yukthiyude upayogam aavasyamaanu, thudarnnu randaamathethinte hrasvamaaya vaakk-proof aavasyamaanu
avasaanamaayi, 64 vaakkukalilum chinnangalilum maathram turing "reductio aadu asadtadam" vazhi "hilbert enshaidungs prashnathinu pariharamilla" ennu theliyikkunnu.
aadyathe thelivinte sangraham.
turing churukkezhuthukalude oru koottam srishtichu. nirvachanangalkkayi lekhanathinte avasaanathilulla glossery kaanuka.
chila pradhana vishadeekaranangal
turing thante pepparinte bhooribhaagavum yadhaarthathil thante yanthrangal nirmmikkaan chelavazhichathu avayude sathyaavasthayekkurichu namme bodhyappeduthaanaanu. reductio aadu asadrasam" enna thelivu upayogichathukondu ithu aavashyamaayirunnu. eee thelivinte "kriyaathmaka" swabhaavathinu naam oonnal nalkanam. sarikkum nirmikkaavunna oru yadhaartha yanthram enthaayirikkumennu turing vivarikkunnu. samshayaaspadamaaya oreyoru ghadakam mesheen dii yude nilanilppaan, eee thelivu oduvil asaadhyamaanennu kanikkum.
oru "theerumaanam/nirnaya" yanthrathinte asthithvam avakaashappedunnathil ninnaan turing thelivu aarambhikkunnath. eathenkilum ezz nalkumbol.d (a, c, d, l, r, n, s enna chinnangalude string ";") ithu s aanennu nirnnayikkum.dii (chihna string) oru "computing mesheene" prathinidheekarikkunnu, athu onnukil "vrithaakrithiyilulla"-athinaal "thripthikaramallaatha yuu"-allengil "vritharahitham"-athinaal "thripthikaramaaya ezz".
mesheen dii athinte pravarthanathekkurichu engane pookunnu ennathinekkurichu turing oru abhipraayavum parayunnilla. vaadathinuvendi, chinnangalude string "nannayi roopappettathaano" ennu aadyam nokkaamenn nammal karuthunnu (i.iii. chinnangalude oru scramble mathramalla, oru algoritam roopathil), illengil athu upekshikkuka. appol athu "sarkkil vettayaadal" aayi maarum. ithu cheiyunnathinu orupakshe athu "hyooristiks" (thanthrangal padippichatho padichatho) upayogikkum. thelivukalude aavasyangalkkaayi, eee vishadaamshangal pradhaanamalla.
turing pinneedu addeham echu ennu vilikkunna oru yanthram pinthudarenda algoritam (reethi) vivarikkunnu (marichu ayanja). mesheen echu athil theerumaana-mesheen dii adangiyirikkunnu (athinaal dii ennathu echu-nte "subruitin" aanu). mesheen echu-nte algoritam echu-nte nirddeshangalude pattikayil prakadippikkunnu, allengil orupakshe teppile echu-nte staanderd vivaranathil, saarvathrika mesheen yuu-yumayi samyojippichirikkunnu; turing ithu vyakthamakkunnilla.
eathenkilum" nambar n neey thulyamaaya s aakki parivartthanam cheiyunnathinu mesheen h utharavaadiyaanu.pareekshikkaan upa-yanthram dii-ykkulla dii chihna string. (prograaming bhashayil echu oru ekapaksheeyamaaya "kall" kadannupokunnu.dii "muthal dii vare, dii" thripthikaram "allengil" thripthikaramalla "ennu nalkunnu.) oru kanakk r ("record"?) sookshikkunnathinum mesheen echu utharavaadiyaanu.) vijayakaramaya samkhyakal ("vijayakaramaya" samkhyakalude ennam ennu nammal karuthunnu.dii, aii.iii. r, s-nte ennathekkaal valare kuravaanu.dii pareekshichu, aii.iii. enn). oduvil, echu athinte teppinte oru bhaagath "beetaa-primed" bii "enna diagonal nambar print cheiyunnu. oro "thripthikaramaaya" yanthrathinteyum/samkhyayudeyum "chalanagal" (computer sensil) "simulettu" cheythukondu echu eee bii 'srishtikkunnu; oduvil parishodhanaykk vidheyamaaya eee yanthram/nambar athinte aaraamathe "chithram" (1 allengil 0) ll athukayum echu athu print cheyyukayum cheyyum. simuleshan avasheshikkunna "kuzhappam vrithiyaakkunnathinu" echu utharavaadiyaanu, enn vardhippikkukayum athinte parishodhanakalumaayi munnott pookukayum cheiyunnu, "aadu infinittam".
shradhikkuka echu vettayaadunna eee yanthrangalellaam turingine "computing mesheenukal" ennu vilikkunnu. iva bainari-dasaamsha-samkhyakale turing "figezhs" ennu vilikkunna ananthamaaya pravaahathil kanakkaakkunnu 1 umm 0 umm chinnangal maathram.
aadyathe thelivu vyakthamakkaan oru udaaharanam.
oru udaaharanam mesheen echu 13472 samkhyakal pareekshikkukayum 5 thripthikaramaaya samkhyakal nirmmikkukayum cheythuvennu karuthuka, athaayath.iii. 1 muthal 13472 vareyulla samkhyakale echu s aakki maatti.dii (chihna stringukal) pareekshikkunnathinayi avaa dii ennathilekku pasaki. thathfalamaayi, echu 5 thripthikaramaaya samkhyakal kanakkukoottukayum aadyathethu athinte aadya "kanakk" yilekkum randaamathethu athinte randaam kanakk, moonnaamathethu athinte moonnaam kanakk, naalaamathethu athinte naalaamathe kanakk, anchaamathethu athinte anchaamathe kanakk ennivayilekkum nayikkukayum cheythu. ippol n = 13472, r = 5, b '=. 10011 (udaaharanathinu) enninganeyaanu ennam. echu athinte teppile kuzhappam vrithiyaakkukayum munnott pookukayum cheiyunnu
"h" "n" "= 13473 vardhippikkukayum" "13473" "enna chinnathe string adderled aayi parivartthanam cheyyukayum cheiyunnu". sub mesheen dii adlrd thripthikaramallennu karuthunnuvenkil, echu 5 ll tally record aarr vidunnu. echu n enna samkhya 13474 aayi vardhippikkukayum munnott pookukayum cheyyum. maruvashathu, dii addreldy thripthikaramaanennu karuthunnuvenkil echu r neey 6 aayi vardhippikkum. echu n (veendum) adalardil parivartthanam cheyyum [ithu oru udaaharanam maathramaan, adlardy orupakshe upayogashoonyamaanu] koodaathe eee mesheen-under-testt (yuu "running" adrld) athinte aaraamathe "figure" print cheiyunnathuvare yoonivezhsal mesheen yuu upayogichu "rann" cheyyum.iii. 1 allengil 0. echu eee aaraamathe nambar print cheyyum (udha.jii. 0") athinte teppinte "outtputtu" mekhalayil (udha.jii. bii '=. "100110").
echu kuzhappam vrithiyaakkukayum thudarnnu "enn" enna samkhya 13474 aayi vardhippikkukayum cheiyunnu.
h athinte thanne nambaraaya k-ll athumbol muzhuvan prakriya anaavaranam cheyyappedunnu. nammal nammude udaaharanavumaayi munnott pookum. vijayakaramaya kanakk/record r 12 aanennu karuthuka. manikkoor oduvil athinte thanne mainus 1, i enna nambaril athunnu.iii. n = k-1 = 4335.3214, eee samkhya parajayappettu. appol k = 4355 utppaadippikkaan h n vardhippikkunnu.3215, i.iii. swantham nambar. echu ithine "ell. dii. dii. aarr" aakki maattunnu...dii kaar "athu theerumaana-yanthram dii. yilek kaimarunnu. theerumaana-yanthram dii "thripthikaramaayi" thirike nalkanam (athaayath echu "nirvachanam anusarichu" parisodhana thudarukayum "aadu infinittam", kaaranam ithu "sarkkil-free" aanu). athinaal echu ippol 12 ll ninnu 13 aayi vardhippikkukayum thudarnnu testt-under-testt k enna nambarine athinte s aayi punarroopakalppana cheyyukayum cheiyunnu.dii upayogichu athu anukarikkaan yuu upayogikkunnu. ennaal ithinartham echu athintethaaya chalanagal anukarikkum ennaanu. simuleshan aadyam enthaanu cheyyuka? eee simuleshan k-aca-h onnukil oru puthiya n srishtikkunnu allengil "pazhaya" n muthal 1 vare "reesett cheiyunnu. eee" k-aca-h "onnukil oru puthiya r srishtikkunnu allengil" pazhaya "r muthal 0 vare" reesett cheiyunnu. pazhaya-h "athinte 12-aamm samkhyayil athunnathuvare" puthiya "k-aca-h" pravarthippikkunnu.
ennaal athu orikkalum 13-aamm samkhyayilekku athunnilla; keey-akaa-echu oduvil 4355 ll athunnu.3215, veendum, "k-aca-h" testt aavarthikkanam. "keey-akaa-echu" orikkalum 13-aamm nambaril athilla. echu-mesheen orupakshe blank teppiludaneelam "aadu infinittam" ennathinte pakarppukal maathram print cheiyunnu. ennaal ithu 1-umm 0-umm enna diagonal nambarukal ennennekkumaayi achadikkunna thripthikaravum vrithaakrithiyilullathallaa computing mesheenaan enna prameyathinu virudhamaanu. (n ennathu 1 lake reesett cheyyukayum r ennathu 0 lake reesett cheyyukayum cheythaal nammal athe kaaryam kaanum.)
vaayanakkaaran ithu vishwasikkunnillengil, avarkk theerumaanamedukkunna yanthram dii (stub dii "thripthikaramaayi" nalkum) ennathinu oru "stub" ezhuthaam, thudarnnu thalkshana yanthram echu athinte swantham nambar neridumbol enthaanu sambhavikkunnathenn svayam nokkuka.
randaamathe thelivinte sangraham.
oru pejil thaazhe dairkhyamullathinaal, parisarathu ninnu upasamhaarathilekkulla vazhi avyakthamaanu.
reductio aadu absardam" ennathiloodeyaanu turing munnott pogunnathu. oru yanthram iii undennu addeham avakaashappedunnu, athu ezz nalkumbol.dii (staanderd vivaranam, i.iii. "oru ekapaksheeyamaaya yanthram m-nte "prograam"), m eppozhengilum oru chinnathe print cheythittundo ennu nirnnayikkum (0 ennu parayuka). eee emm oru "computing mesheen" aanennu addeham avakaashappedunnilla.
mesheen iii yude nilanilpp kanakkiledukkumbol, turing inipparayunna reethiyil munnott pookunnu
randaamathe thelivinte vishadaamshangal.
thelivile budhimuttu ghattam 1 aanu. turing thante sookshumamaaya karakaushala pravartthanangal vishadeekarikkunnillenn manassilaakkunnathiloode vaayanakkaarane sahayikkum. (churukkatthil logical oopparettarmaarumaayi ezhuthiya thulyamaaya padaprayogangalkkoppam "asthithwa-", "saarvathrika-oopparettarmaar" enniva thammilulla chila thulyathakalaanu addeham upayogikkunnathu.)
ivide oru udaaharanamunda noorukanakkinu kaarukal niranja oru parking sthalam namukk munnil kaanaamenn karuthuka. paranna (mosham) tayarukalulla kaarukal" ennu tirayan njangal muzhuvan sthalavum chuttikkarangaan theerumaanikkunnu. oru manikooro athil kooduthalo kazhinju mosham tayarukalulla rund kaarukal njangal kandethi.chila kaarukalil mosham tayarukalundenn namukk ippol urappode parayaan kazhiyum". allengil namukk ingane parayaam 'ellah kaarukalkkum nalla tayarukalundennathu shariyalla'. allengil 'ellah kaarukalkkum nalla tayarukal illa' ennathu sariyaanu. namukk mattoru sthalathekku pookaam. ivide naam kandethunnathu "ellah kaarukalkkum nalla tayarukalund" ennaanu.oru kaarinu mosham tayar ulla oru udaaharanam polum illa" ennu nammal paranjekkaam.athinaal, oro kaarinekkurichum vevvere enthengilum parayaan kazhiyumenkil avayellaam koottaayi enthengilum parayaan kazhiyumennu naam kaanunnu.
turing cheiyunnathu ithaanu
m" ll ninnu addeham yanthrangalude oru shekharam srishtikkunnu {"m" 1 ", m" 2 ", m" 3 ", m" 4 ",...mn", "ooronninekkurichum addeham oru vaachakam ezhuthunnu" x "kuranjathu oru 0 engilum print cheyyukayum rund" sathya moolyangal "anuvadikkukayum cheiyunnu, sharee = shoonyamo thetto =: 0:. oro yanthrathinum vaakyathinte sathya moolyam oronnaayi addeham nirnnayikkukayum shoonyathakalude oru string allengil 0:, allengil ivayude chila samyojanam undaakkukayum cheiyunnu. namukk ithupolulla enthengilum labhichekkaam "m" 1 print a0 "= sharee," m "2 print a0" = sharee, "m" 3 print a0 "= sharee," m "4 print a0" = thettu,... mn" ennathum a0 = falls ennu print cheiyunnu. avanu charadu labhikkunnu
emm. enn" enna ennathil ennamatta yanthrangal undengil. maruvashathu oro yanthravum oru "sharee" srishtichirunnenkil teppile padaprayogam ingane ayirikkum
angane turing oro yanthratheyum kurichulla prasthaavanakale vevvere pariganichu avayekkurichu orotta "prasthavana" (string) aakki maatti. eee padaprayogam srishticha yanthram (addeham athine jii ennu vilikkunnu) nalkiyaal, addehathinu athu thante yanthram iii upayogichu pareekshikkaanum athu eppozhengilum 0 uthpaadippikkunnundo ennu nirnnayikkaanum kazhiyum. mukalilulla nammude aadya udaaharanathil athu yadhaarthathil cheiyunnuvennu nammal kaanunnu, athinaal nammude shreniyile ellah m-kalum 0s print cheiyunnillennu namukkariyam. ennaal randaamathe udaaharanam kaanikkunnath, string shoonyamaayathinaal nammude shreniyile oro mn umm 0 srishtichu ennaanu.
turinginu baakkiyullath orotta meettaril ninnu emn-kalude kramam srishtikkunnathinulla oru prakriya srishtikkuka maathramaan.
m" eee paatten print cheiyunnuvennu karuthuka
turing mattoru mesheen eph srishtikkunnu, athu emm edukkukayum aadyathe enn 0-kalle "0-baar" (0) aayi thudarchayaayi parivartthanam cheiyunna emn-kalude oru shreni crunch cheyyukayum cheiyunnu
vishadaamshangal kaanikkathe, eee yanthram yadhaarthathil nirmmikkaavunnathaanenna addeham parayunnu. rund kaaryangalil onnu sambhavikkumennu namukk kaanaan kazhiyum. f-ll 0 ulla yanthrangal theernnupoyekkaam, allengil "poojyangal radhaakkaan" yanthrangal srishtikkaan "ananthamaayi" pokendivannekkaam.
turing ippol iii, eph yanthrangale samyojippichu oru samyojitha yanthram jii aayi samyojippikkunnu. g yadhaartha m ll aarambikkunnu, thudarnnu ellah pingamikalaaya yanthrangalaaya m1, m2 enniva srishtikkaan f upayogikkunnu. ., emm. enn. m ll aarambikkunna oro yanthravum pareekshikkaan g e upayogikkunnu. oru yanthram orikkalum poojyathe print cheiyunnillennu iii kandethukayaanengil, aaa yanthrathin jii print cheiyunnu 0:. oru yanthram 0 print cheiyunnuvennu e kandethukayaanengil (turing parayunnillennu njangal anumaanikkunnu) g print cheiyunnun allengil eee entry ozhivaakkunnu, chathurangal shoonyamaayi avasheshikkunnu. rund karyangal sambhavikkumennu namukk kaanaan kazhiyum.
ippol, nammal e muthal g vare prayogikkumbol enthaanu sambhavikkunnath?
m ananthamaayi 1 print cheiyunnundo ennu nirnnayikkaan namukk athe prakriya prayogikkaan kazhiyum. eee prakriyakal samyojippikkumbol, m 1-umm 0-umm "aadu infinittam" print cheiyunnundo illayo ennu namukk nirnnayikkaan kazhiyum. athinaal m vritharahithamaano ennu nirnnayikkaan namukk oru reethiyundu. thelivu 1 prakaram ithu asaadyamaanu. athinaal e nilavilundenna aadya avakaashavaadam thettaanu e nilavililla.
moonnaamathe thelivinte sangraham.
ivide turing theliyikkunnathu "hilbert enshaidungs prashnathinu pariharamilla" ennaanu. ivide addeham
lemmakal #1, #2 enniva aavasyamulla "iff aandu onli iff" (i.iii. thelivu aavashyappedunna logical equivalence):
turing oru soothravaakyathinte nilanilpp kaanikkunnu, athu phalatthil, "m-nte chila poornna konfigareshanil, 0 tapil drishyamaakunnu" (p. (145). eee soothravaakyam sariyaanu, athaayath, ithu "nirmmaanayogyamaan", ithinekkurichu engane pokanamennu addeham kaanikkunnu.
thudarnnu turing rund lemmakal theliyikkunnu, aadyathethinu ellah kadinaadhvaanavum aavasyamaanu. (randaamathethu aadyathethinte vipareethamaanu.) thudarnnu addeham thante anthima phalam theliyikkaan "reductio aadu absardam" upayogikkunnu
moonnaamathe thelivinte vishadaamshangal.
[vaayanakkar thelivu vishadamaayi padikkaan uddeshikkunnuvenkil, turing nalkia thiruthalukal upayogichu moonnaamathe thelivile pejukalude pakarppukal thiruthanam. (i) yukthi (ii) kurt godelinte prabandham "principia ganithashaasthrathinteyum anubandha samvidaanangaludeyum aupachaarikamaayi anischithamaaya nirddeshangalekkurichu" ennathil vaayanakkar sakthamaaya pashchaathalatthode varanam. godelinte pepparinte sahaayathinaayi avar iii. yodu koodiyaalochikkanam.jii. ernest nagal jaims aarr. neuman, "godels proof", newyork yooniversitti pras, 1958.]
saankethika vishadaamshangal pinthudarunnathinu, vaayanakkaaran "theliyikkaavunna" ennathinte nirvachanam manasilakkukayum pradhanappetta "soochanakal" arinjirikkukayum venam.
"theliyikkaavunna" ennaal, godel enna ardhathil, (i) "eee vaachakam theliyikkaavunnathaan" enna vaachakam nirmmikkaan (prakadippikkaan) svayamsidhaantham thanne shakthamaan, (ii) eathenkilum ekapaksheeyamaaya "nannayi roopappetta" thelivukalil chinnangal svayamsidhavaakyangal, nirvachanangal, nigamanatthinte chinnangalkk pakaramaayi nayikkunnu.
aadyathe soochana "namukk m-nte vivaranam §6-nte aadya staanderd roopathil idaam". sekshan 6 oru "saarvathrika mesheen" yuu tapil mesheen emm-nte valare nirddishta "encoding" vivarikkunnu. turinginte saarvathrika yanthramaaya yuu, encoding skeem ennivayude chila prathyekathakal vaayanakkaaran ariyendathundu.
(i) saarvathrika yanthram ennathu oru "nirddesha pattikayil" vasikkunna "saarvathrika" nirddeshangalude oru koottamaanu. ithil ninnu vyathyasthamayi, yuu tapil, oru "computing mesheen" emm "emm-coode" aayi nilanilkkum. saarvathrika nirddeshangalude pattikaykku tapil a, c, d, 0,1, u, v, w, x, y, z ennee chinnangal achadikkaan kazhiyum. vividha yanthrangalkk eee chinnangal achadikkaan yuu command cheythukondu maathrame parokshamaayi achadikkaan kazhiyoo.
(ii) m-nte "mesheen coode" kurachu aksharangalum ardhaviraamavum, i umm maathram ulkkollunnu.iii. dii, sii, eiii, aarr, ell, enn,; m-nte "coode"-ll evideyum samkhyaaparamaaya "akkangal" (chinnangal) 1 umm 0 umm orikkalum drishyamaakilla. shekharanathil ninnu shoonyamaaya 0,1 enna oru chihnam print cheyyaan m aagrahikkunnuvenkil, avaa print cheyyaan ningalodu parayaan athu inipparayunna kodukalilonnu upayogikkunnu. karyangal kooduthal aashayakkuzhappathilaakkaan, turing eee chinnangale s0, s1, s2, i ennu vilikkunnu.iii.
(iii) oru "computing mesheen", athu nerittu oru table nirmmichathaanenkilum (addehathinte aadya udaaharanagal kaanikkunnath pole), allengil yoonivezhsal mesheen yuu teppile mesheen coode emm enna nilayil, athinte nambar shoonyamaaya tapil (emm-kodinte valathuvashath, orennam undengil) 1s umm 0s umm ennennekkumaayi valathuvashathekku neengunnathaayi print cheiyunnu.
(iv) oru "computing mesheen" yuu + "emm-coode" aanenkil, tapil aadyam "emm-coode" drishyamaakum; tapinu idathu attamund, "emm-coode" avide ninnu aarambhich ithara chathurangalil valathuvashathekku neengunnu. emm-coode avasaanikkumbol (eee emm-kodukal parimithamaaya algorithangalaanenna anumaanam kaaranam), "kanakkukal" ithara chathurangalil 1s, 0s enningane aarambhich ennennekkumaayi valathuvashathekku neengum. emm-kodilum mesheen print cheiyunna "akkangalilum" kanakkukoottalukal evideyaanennu track cheyyaan yuu + "emm-coode" neey sahaayikkunnathin turing (shoonyamaaya) ithara chathurangal ("iii"-"maaykkaavunna"-chathurangal) upayogikkunnu.
(v) oru "samboornna krameekaranam" ennathu teppile ellah chinnangaludeyum achadiyaanu, emm-code athuvare "figureukalum" ulppede, nilavil scan cheiyunna chithrathodoppam (scan cheytha chinnatthinte idathuvashathu oru pooyintar-pratheekathode achadichirikkunnu?). turinginte ardham nammal shariyaayi vyaakhyaanichittundil, ithu valare neenda chinnangalude oru koottamaayirikkum. ennaal muzhuvan emm-code aavarthikkendathundo ennu vyakthamalla; nilavile emm-coode nirddeshathinte printing figure marker upayogichu ellah akkangaludeyum printing maathrame aavashyamulloo).
(vi) turing "emm-coode" (veendum tapil drishyamaakaanulla emm coode) leeii saadhyamaaya vishalamaaya nirddeshangal oru cheria canonical settaayi kurachu, ithinu samanamaaya moonnil onnu {qi sj sk r ql} e.jii. "mesheen nirddesham #qi nadappilaakkukayum chihnam sj scan cheiyunna chathuratthilaanenkil, chihnam scan cheyth valathottu pooyi nirddesha ql" lake pookuka mattu nirddeshangal samaanamaan, "idathu" l-num "chalanamilla" n-num encoding cheiyunnu. qi = Da chinnangalude string upayogichu encode cheythirikkunnathu eee settaanu...a, sj = dc...sii, ezz. keey = dissy...c, r, ql = Da...eiii. oro nirddeshavum ardhaviraamathaal mattonnil ninnu verthirikkappedunnu. udaaharanathinu, {q5, s1 s0l q3} ennaal nirddesham #5: scan cheytha chihnam 0 aanenkil blank print cheyyuka, idathu vashathekku pookuka, thudarnnu nirddeshathilekku pookuka #3. ithu inipparayunna reethiyil encode cheythirikkunnu.
randaamathe soochana turing ennathu godelinte pepparil avatharippicha ashayangal upayogikkunnu, athaayath, yunn (emm) enna soothravaakyathinte (kuranjathu athinte bhaagamenkilum) "godelisation". eee soochana 138 () enna pejile adikkurippaayi maathrame kaanappedunnulloo "r praimukalude oru kramatthe soochippikkunnathu" ("ibid") aanu.) [ivide, paraantheesukalkullile r "uyarthappetta" aanu.] eee "praimukalude kramam" f ^ (n) enna soothravaakyathil prathyakshappedunnu.
moonnaamathe soochana ithu randaamathe soochanaye shakthippeduthunnu. thelivinulla turinginte yadhaartha sramam eee prayogam upayogikkunnu
neratthe pepparil turing eee prayogam upayogichirunnu (p. 138) ennathinartham "u oru noonn-negatteevu poornnasamkhyayaanu" ("ibid") ennaanu.) (aii.iii. oru godel nambar). ennaal, bernez thiruthalukalode, turing eee sameepanam upekshichu (i.iii. n (u)-nte upayogavum "godel nambar" vyakthamaayi drishyamaakunna oreyoru sthalavum addeham f ^ (n) upayogikkunna sthalamaanu.
thelivu ennathukondu enthaanu ardhamaakkunnathu? aadya soochana ardhamaakkunnathu teppile emm-kodinte lalithamaaya parisodhana 0 enna chihnam eppozhengilum yuu + "emm-coode" upayogichu achadichittundo ennu velippeduthilla ennaanu. oru pareekshana yanthram oru nirddeshathe prathinidheekarikkunna chinnangalude stringukalilonnil dissy yude roopam thirayam. ennaal eee nirddesham eppozhengilum nadappaakkappedumo?"entho onnu kandethaan" "coode pravarthippikkanam". " ithu oru yanthram aakam, allengil aupachaarika thelivile varikalakaam, athaayath.iii. lemma <aii. dii. 1>.
randaamatheyum moonnaamatheyum soochanakal ardhamaakkunnathu, athinte adithara godelinte paper aayathinal, thelivu budhimuttaanu ennaanu.
chuvadeyulla udaaharanathil nammal yadhaarthathil oru lalithamaaya "sidhaantham" nirmmikkum-oru cheria poost-turing mesheen prograam "rann ittu". shariyaayi roopakalppana cheytha oru sidhaantham engane mechanical akumennu namukk nokkaam. oru thelivu, thudakkathil oru "proof udaaharanam" cherthu avasaanam enthaanu purathuvarunnathennu kaanunnathiloode nammal cheiyunna sidhaanthathinte oru "testt" aanennu namukk kaanam.
lemmakal #1, #2 enniva aavasyamulla "iff aandu onli iff" (i.iii. thelivu aavashyappedunna logical equivalence):
fraansine udharikkaan
fransen thante pusthakatthil "theliyikkaavunna" kaaryam neratthe nirvachichittundu
athinaal oru "vaachakam" chinnangalude oru charadaanu, oru sidhaantham chinnangalude charadukalude oru charadaanu.
turing inipparayunna chumatala abhimukeekarikkunnu
athinaal "vaakyangalude string" chinnangalude stringukalude stringukalaayirikkum. anuvadaneeyamaaya vyakthigatha chinnangal maathrame godelinte pepparil nirvachichirikkunna chinnangalil ninnu varoo.(inipparayunna udaaharanathil oru "chithrathinu" chuttumulla "" <","> "enniva upayogichu" "chithram" ennathu yanthram scan cheiyunna chihnamaanennu soochippikkunnu).
moonnaamathe thelivu vyakthamakkaan oru udaaharanam.
inipparayunnavayil, turinginte oro 'computing mesheenukalum' oru bainari nambar janarettar/srashtaavaanennu nammal svayam ormmippikkendathundu, athu 'blank tapey' ll pravartthanam aarambikkunnu. shariyaayi nirmmicha, athu ellaayppozhum ananthamaayi akannupookunnu, pakshe athinte nirddeshangal ellaayppozhum parimithamaanu. turinginte thelivukalil, turinginte tapinu oru "idathu atta" undaayirunnu, pakshe valath bhaagathekku ananthamaayi vipuleekarichu. udaaharanathinu, thaazhe koduthirikkunna "yanthram" oru saarvathrika yanthramallennum marichu meshayile nirddeshangalodukoodiya lalithamaaya "samarppita yanthram" aanennum nammal anumanikkum.
njangalude udaaharanam oru turing mesheente "parishkkaricha" poost-turing mesheen modaline adisthaanamaakkiyullathaanu. eee model 0 umm 1 umm chinnangal maathram print cheiyunnu. shoonyamaaya tape ellah bii kalaayi kanakkaakkappedunnu. njangalude parishkaricha modalil 7 poost-turing nirddeshangalil rund nirddeshangal koodi cherkkendathundu. njangal upayogikkunna churukkezhuthukal ivayaanu
r, l, e, p0, p1 ennivayude kaaryangalil mesheen athinte jolie cheythathinushesham samkhyaa kramatthil adutha nirddeshathilekku thudarunnu; pareekshanangal parajayappettaal chaadunnathinu ithu baadhakamaan.
ennaal, hraswathaykkaayi, nammude udaaharanagal moonnu chathurangal maathrame upayogikkoo. iva ellaayppozhum idathu vashath scan cheytha chathuramulla shoonyasthaanangalil ninnu aarambikkum i.iii. bii. bii. bii. 1,0, shoonyamaaya rund chinnangal upayogichu namukk 27 vyathyastha konfigareshanukal undayirikkaam
nammal ivide shradhikkanam, kaaranam oru algoritam (thaalkkaalikamaayi) akkangalkkidayil shoonyathakal avasheshippikkukayum pinneedu thirike vannu enthengilum poorippikkukayum cheyyum. oru algoritam manappoorvvam ithu cheyyaan saadhyathayund. vaasthavathil, turinginte yanthram ithu cheiyunnu-ithu ithara chathurangalil print cheiyunnu, ithu lokkettar chinnangal print cheyyaan kazhiyunna tharatthil akkangalkkidayil shoonyatha avasheshikkunnu.
turing ellaayppozhum ithara chathurangal shoonyamaayi avasheshippichu, athinaal addehathinte yanthrathin oru chithratthinte idathuvashathu oru chihnam sthaapikkaan kazhiyum (allengil yanthram saarvathrika yanthravum scan cheytha chathuravum yadhaarthathil "prograam" ll aanenkil oru aksharam). nammude cheria udaaharanathil nammal ithu upekshichu scan cheytha chinnathinu chuttum chinnangal () thaazhepparayunna reethiyil idum
namukk oru lalithamaaya prograam ezhuthaam
nammal ellaayppozhum shoonyamaaya tapil ninnaan aarambhikkunnathenn oorkkuka. poornnamaaya krameekaranam teppile chinnangal print cheiyunnu, thudarnnu adutha nirddesham
namukk formulayilekk "jump" cherkkaam. ithu cheyyumbol, poornnamaaya konfigareshanil tape chinnangal ulppeduthendathu enthukondaanennu nammal kandethunnu. (yadhaarthathil, thaazhe nammal ithu kooduthal nannayi kaanunnu.) eee cheria prograam moonnu '1' kall valathuvashath print cheyyukayum disha vipareethamaakkukayum idathu printing 0 kall shoonyathayil athunnathuvare neekkukayum cheiyunnu. njangalude mesheen upayogikkunna ellah chinnangalum njangal print cheyyum
ivide avasaanam idathuvashathulla oru shoonyatha "praabalyatthil vannirikkunnu" ennu nammal kandethunnu, athinaal ithu motham konfigareshante bhaagamaayi nammal upekshikkunnu.
nammal nammude jolie shariyaayi cheythathinaal, nammal prarambha vyavasthakal cherkkukayum "sidhaantham evidekkaanu pogunnathu" ennu kaanukayum cheiyunnu. thathfalamaayundaakunna krameekaranam-110-aanu thelivu.
sankeernathakal.
turinginte thelivu niravadhi nirvachanangalal sankeernamaanu, martin davis "cheria saankethika vishadaamshangal" ennum "" ennu vilikkunnathumaayi aashayakkuzhappathilaakunnu...nalkiyirikkunnathupole saankethika vishadaamshangal thettaanu. turing thanne 1938-ll "oru thiruthal" prasidheekarichu "rachayithaav pii. eee pishakukal choondikkaanikkunnathinna ithu nallathaanu ".
prathyekichum, athinte yadhaartha roopathil moonnaamathe thelivu saankethika pishakukalaal moshamaayi nasippikkappettirikku. bernesinte nirddeshangalkkum turinginte thiruthalukalkkum sheshavum saarvathrika yanthrathinte vivaranathil pishakukal nilaninnu. aashayakkuzhappamundaakkunna vidhathil, turinginu thante yadhaartha paper sariyaakkaan kazhiyaathathinal, shareerathinullile chila vaachakangal turinginte thettaaya aadya shramathe soochippikkunnu.
bernesinte thiruthalukal ivide kandethaam; yadhaarthamaayathu "ensheyidangsproblamilu oru prayogathodoppam, kanakkukoottaavunna samkhyakalil" aayi kandethanam. lunden maathamaattikkal sosaittiyude nadapadikal" (2), 43 (1938), 544-546.
turinginte pepparinte oonlain pathippil eee thiruthalukal oru anubandhathil undu; ennirunnaalum, saarvathrika yanthrathile thiruthalukal emil poost nalkia vishakalanathil kandethanam.
thudakkathil, thelivukalude vishadaamshangalil shradha cheluthiya oreyoru ganithashaasthranjan poost aayirunnu (sii. eph. hodges pii. 125)-pradhaanamaayum aadima yanthram polulla pravarthanangalilekku "algoritam" oresamayam kuraykkunnathil addeham athiyathinaal, addeham thelivukalil vyakthiparamaaya thaalparyam prakadippichu. vichithramennu parayatte (orupakshe randaam lokamahaayudham edapettirikkaam) 1947-leeii thyoo enna prashnatthinte aavarthanaparamaaya pariharikkaanaavaathathenna addehathinte prabandhatthile "anubandhathil" ithu vichedikkaan ekadesham patthu varshameduthu.
mattu prasnangal svayam avatharippikkunnu addehathinte "appendix" posttil pepparinte budhimuttinekkurichum thelivukalude "rooparekha swabhaavathe"kkurichum "avabodhajanyamaaya roopathe"kkurichum parokshamaayi abhiprayappettu. posttinu vividha pooyintukal oohikkendivannu
eppozhengilum pathram vaayikkaan shramicha arcum hodgesinte parati manassilakum
turing upayogikkunna padangalude glossery.
1 kanakkukoottaavunna samkhya-oru yanthram (i.iii., oru algoritam polulla parimithamaaya maargangaliloode)
2 meter-parimithamaaya nirddesha table scanning/printing head ulla oru yanthram. m chathurangalayi vibhajichirikkunna oru ananthamaaya tape neekkunnu, oronninum "oru chihnam vahikkaan kazhiyum". mesheen-nirddeshangal inipparayunnavayaanu oru chathuram idathu vashathekku neekkuka, oru chathuram valath vashathekku neekkuka, scan cheytha chathura achadi chinnamaaya p-ll, scan cheytha chathuram maaychukalayuka, chihnam p aanenkil nirddesham AAA cheyyuka, scan cheytha chihnam p allengil nirddesham AAA cheyyuka, scan cheytha chihnam onnumallengil nirddesham AAA cheyyuka, scan cheytha chihnam eathenkilum nirddesham AAA aanenkil nirddesham AAA cheyyuka.
3 computing mesheen-rund tharam chinnangal achadikkunna oru emm, aadya tharam chinnangale "figureukal" ennu vilikkunnu, avaa bainari chinnangalaaya 1 umm 0 umm maathramaan; randaamathe tharatthilulla chinnangal mattethenkilum chinnangalaanu.
4 akkangal-1 umm 0 umm chinnangal, a.keey.eiii. aadya tharatthilulla chinnangal
5 meter-konfigareshan-nirddesha-identifier, onnukil nirddesha pattikayile oru chihnam, allengil saarvathrika yanthrathinte teppile nirddesha-samkhyaye prathinidheekarikkunna chinnangalude oru string (udha.jii. "daaa = <aii. dii. 1>")
randaam tharatthilulla 6 chinnangal-1 umm 0 umm ozhikeyulla eathenkilum chinnangal
7 vrithaakrithiyilulla-oru parajayappetta kanakkukoottal yanthram. athu achadikkunnathil paraajayappedunnu, ananthamaayi, binaryil prathinidheekarikkunna 0 allengil 1 akkangal athu kanakkaakkunna samkhya
8 sarkkil free-oru vijayakaramaya computing mesheen. athu print cheiyunnu, ananthamaayi, binaryil prathinidheekarikkunna 0 allengil 1 akkangal athu kanakkaakkunna samkhya
9 kramam-"yanthram kanakkukoottunna kramam" ennathupole aadya tharam a yude chinnangal.keey.eiii. kanakkukal a.keey.eiii. chinnangal 0 umm 1 umm.
10 kanakkukoottaavunna kramam-oru vritham-rahita yanthram upayogichu kanakkakkaam
11 seey.dii-staanderd vivaranam oru turing mesheen tapil a, c, d, l, r, n ", chinnangalude oru shreni;
12 dii.n-vivarana nambar a s.d oru samkhyayaayi parivartthanam cheyyappedunnu 1 = a, 2 = c, 3 = d, 4 = l, 5 = r, 6 = n, 7 =;
13 meter (enn)-dii ulla oru yanthram.n ennathu "n" enna samkhyayaanu.
14 thripthikaram-a.dii allengil dii.vritham illaatha yanthrathe prathinidheekarikkunna n
15 yuu-"saarvathrika" nirddeshangalude pattika sajjeekarichirikkunna oru yanthram. yuu "nalkiyittundenkil athinte thudakkathil ezz ennu ezhuthiyirikkunna oru tape.chila computing mesheenukalude m, u, m-nte athe kramam kanakkakkum."
16 β '-"beetta-primed": n-th figure (i.iii. n-th kanakkukoottaavunna shreniyude 0 allengil 1) [ithum h-nte kanakkukoottaavunna samkhya, chuvade kaanuka.]
17 yuu-thripthikaramallaatha, i.iii. vrithaakrithiyilulla ezz.dii.
18 seey-thripthikaram, i.iii. sarkkil free ezz.dii.
19 dii-echu adangiyirikkunna oru yanthram (chuvade kaanuka). ezz nalkumbol.eathu computing mesheenteyum dii emm, dii ems pareekshikkum.dii, vrithaakruthiyil "yuu" ennu adayaalappeduthukayum vritham illaathathaanengil "ezz" ennu adayaalappeduthukayum cheyyuka.
20 manikkoor-oru computing mesheen. h b 'kanakkukoottunnu, r umm n umm paripaalikkunnu. h-ll d, u ennivayum n, r enniva paripalikkukayum d-nuu thulyamaaya s nalkukayum cheiyunna oru vyakthamaakkaatha yanthravum (allengil prakriya) adangiyirikkunnu.dii oof enn. e 'b' yude kanakkukal kanakkaakkukayum 'b' yude kanakkukal kootticherkkukayum cheiyunnu.
21 r-vijayakaramaya (vritharahitha) vukalude alavinte oru record allengil kanakk.dii pareekshichu
22 n-1 ll aarambikkunna oru samkhya, oru s aayi parivartthanam cheyyanam.dii bye mesheen iii. e paripaalikkunnu n.
23 keey-oru samkhya. dii.echu.
5 meter-konfigareshan-nirddesha-identifier, onnukil nirddesha pattikayile oru chihnam, allengil saarvathrika yanthrathinte teppile nirddeshathinte ennathe prathinidheekarikkunna chinnangalude string (udha.jii. "Daaaaa = nirddesham #5"). turinginte ezz.oro nirddeshathilum m-konfigareshan randuthavana drishyamaakum, ettavum idathu string "karant instruction" aanu; ettavum valath string adutha nirddeshamaanu.
24 poornnamaaya krameekaranam-scan cheytha chathuratthinte nambar (chithram 1 allengil 0), teppile ellah chinnangaludeyum poornna kramam, emm-konfigareshan (nirddesha-identifier, onnukil oru chihnamo oru samkhyaye prathinidheekarikkunna chinnangalude charado, udha.jii. "nirddeshangal daa = <aii. dii. 1>")
25 rsi (x, y)-"m-nte poornnamaaya konfigareshanil x-ll y squareile chihnam sii aanu;" poornna konfigareshan "ennathu nirvachanam #5 aanu.
26 i (x, y)-"m-nte poornnamaaya krameekaranatthil x-nte chathurashra y scan cheiyunnu"
27 kqm (x)-"m-nte poornnamaaya konfigareshanil x mesheen konfigareshan (nirddesha nambar) qm aanu"
28 f (x, y)-"y ennathu x-nte" udanadi "pingamiyaanu (godel" f "neey pingamiyaayi upayogichathine pinthudarunnu).
29 gram (x, y)-"x ennathu y nuu munbaanu", udanadi aayirikkanamennilla.
30 instaal {qi, sjskl ql} ennathu instaal {qi, sjskrql}, instaal {qi, sjsknql} enniva pole oru churukkapperaan. thaazhe kaanuka.
turing thante nirddeshangal moonnu "canonical roopangalaayi" churukkunnu-onnu idathu, valath, noow-movement. sii, ezz. keey enniva teppile chinnangalaanu.
udaaharanathinu, aadya variyile pravartthanangal psk = print chinnamaaya a, c, d, 0,1, u, v, w, x, y, z,: enna shekharathil ninnullathaanu, thudarnnu tape idathu vashathekku neekkuka.
ivaye addeham ingane churukkipparanju
(n1) qi sj sk l qm
(n2) qi sj sk r qm
(n3) qi sj sk n qm
thelivaayi #3 addeham ivayil aadyathethu "instaal {qi sjsklql}" ennu vilikkukayum muzhuvan mesheen engane ezhuthaamennu kaanikkukayum cheiyunnu.d ennathu logical kanjakshan (logical allengil): eee stringine "discription-oof-m" ennathupole "dess (m)" ennu vilikkunnu.
aii.iii. mesheen 0 print cheiyukayaanenkil, valath aadu infinittamilekku ithara chathurangalil 1 umm 0 umm print cheythaal athinu pattika undayirikkaam (samanamaaya oru udaaharanam peju 119 ll drishyamaakum):
avaa "instaa ()" foomil idukayaanengil nirddeshangal inipparayunnathaayirikkum (oormmikkuka s0 shoonyamaanu, s1 = 0, s2 = 1):
staanderd vivaranathilekkulla kuravu (kall.dii) ingane ayirikkum
ithu pusthakatthile addehathinte udaaharanavumaayi yojikkunnu (oro aksharathinum akkatthinum idayil oru shoonyatha undaakum). yoonivezhsal mesheen yuu "pooyintarukal" sthaapikkunnathinulla sthalangalaayi ithara shoonyamaaya chathurangal upayogikkunnu.