semi-tyoo sisttam

saidhaanthika computer shaasthrathilum ganithashaasthra yukthiyilum charitraparamaayi semi-tyoo sisttam ennu vilikkappedunna string reraiting sisttam (srs) oru (saadhaaranayaayi parimithamaaya) aksharamalayil ninnulla stringukalkku mukalilulla reraiting sistamaanu. aksharamalayile sthira stringukal thammilulla oru bainari relation formula nalkumbol, formula 2 kondu soochippikkunna reerait rools ennu vilikkunnu, niyamangalude idathu, valath vashangal substringukalaayi kaanapedunna ellah stringukalilekkum oru srs reraiting bandham vipuleekarikkunnu, athaayath formula 3, ivide formula 4, formula 5, formula 6, formula 7 enniva stringukalaanu.
oru semi-tyoo sisttam enna ashayam adisthaanaparamaayi oru monoidinte avatharanavumayi poruthappedunnu. athinaal monoidukalkkum grouppukalkkumulla prashnam enna vaakk pariharikkunnathinulla oru swaabhaavika chattakkoodaanu avaa.
oru srsine oru amoorthamaaya reraiting sisttamaayi nerittu nirvachikkaam. oru niyantritha tharam padam punarlekhana samvidaanamaayum ithine kaanaan kazhiyum. oru aupachaarikatha enna nilayil, string reraiting samvidaanangal poortthiyaayi. 1914 leeii oru prabandhatthil string reraiting sisttangalude chittayaya chikitsa avatharippicha norwegian ganithashaasthranjanaaya axel thyoovil ninnaan semi-thyoo enna peru vannath. parimithamayi avatharippicha semigrooppukalkkulla prashnam enna vaakk pariharikkaamenna pratheekshayil thuu eee ashayam avatharippichu. 1947ll maathramaan prashnam anishchithathvathilaanenna theliyikkappettath-eee phalam swathanthramaayi emil posttum eiii. eiii. yum nedi. eiii. markov juuniyar.
nirvachanam.
oru string reraiting sisttam allengil semi-tyoo sisttam oru tupple formula _ 8 aanu. 
bandham samamithiyaanengil, systethe oru thuu sisttam ennu vilikkunnu.
substringukal anusarichu mattiyezhuthaan anuvadikkunnathiloode formula _ 9 leeii mattu stringukalilekku reraiting niyamangal swaabhaavikamaayi vyaapippikkaan kazhiyum. kooduthal aupachaarikamaayi, eathenkilum string formula _ 16-nuu formula _ 9-ll indues cheytha vann-steppu reraiting relation formula _ 14:
formula _ 14 formula _ 9-leeii oru bandhamayathinaal, jody formula _ 24 oru amoortha reraiting sisttathinte nirvachanathinu anuyojyamaanu. vyakthamayum ithu formula _ 14-nte oru upavibhagamaanu. chila rachayithakkal formula _ 14 (e.jii. formula _ 14) athine athil ninnu verthirichariyaan (formula _ 28) kaaranam avarkk pinneedu subscript upekshikkaan kazhiyanamennum ennittum avar prerippikkunna oru ghattam maattiyezhuthalum thammilulla ashayakkuzhappam ozhivaakkaanum avar aagrahikkunnu.
oru semi-tyoo sisttathil namukk oru prarambha string formula _ 29 ll aarambhich oru samayat oru substring-replacement nadatthi aavarthichu veendum ezhuthunnathiloode uthppaadippikkunna stringukalude oru (parimithamo ananthamo) shreni roopappeduthaan kazhiyum
ithupolulla poojyamo athiladhikamo gattangalulla punarlekhanam formula _ 14-nte reflexive traansitteevu closser vazhi pidichedukkunnu, ithu formula _ 32 kondu soochippikkunnu (amoortha punarlekhanam system#basic ashayangal kaanuka). ithine reraiting relation allengil reduction relation ennu vilikkunnu, formula _ 9 induced.
eee ekopanam.
pothuve, oru aksharamalayile stringukalude settu formula _ 9 string konkaattaneshante bainari pravartthanavumaayi (formula _ 35 aayi soochippikkukayum chihnam idunnathiloode gunithamaayi ezhuthukayum cheiyunnu) oru swatantra monoid roopappeduthunnu. oru srsil, reduction relation formula _ 32 monoid ooppareshanumaayi poruthappedunnu, athaayath formula _ 37 ellah stringukalkkum formula _ 18-nuu formula _ 38 soochippikkunnu. formula _ 32 nirvachanam anusarichu oru preorder aayathinal, formula _ 41 oru monoidal preorder roopappeduthunnu.
athupole, formula _ 14-nte reflexive traansitteevu cemetric closher, formula _ 43 (amoortha reraiting system#basic ashayangal kaanuka), oru kongruvans aanu, athaayath ithu oru equivalence relations (nirvachanam anusarichu) aanu, koodaathe ithu string konkattaneshanodum poruthappedunnu. 43 enna bandha soothravaakyathe srishticha tyoo kongruvans ennu vilikkunnu. oru thuu sisttathil, i.iii. samamithiyaanengil, reerait relation formula _ 32, tyoo kongruvans formula _ 43-maayi poruthappedunnu.
ghadaka monoid, monoid avatharanangal.
formula _ 43 oru kongruvans aayathinal, free monoid formula _ 9-nte factor monoid formula _ 48 saadhaarana reethiyil tyoo kongruvans upayogichu namukk nirvachikkaam. oru monoid formula _ 50 formula _ 51 ulla isomorphic aanenkil, semi-tyoo sisttam formula _ 8 neey formula _ 50 nte monoid avatharanam ennu vilikkunnu.
beejaganithatthinte mattu mekhalakalumaayi valare upayogapradamaaya chila bandhangal namukk udan labikkum. udaaharanathinu, ε shoonyamaaya string aayirikkunna {"ab" → ε, "baa" → ε} ennee niyamangalulla {"a", "b"} aksharamala oru janarettarile swatantra grouppinte avatharanamaanu. pakaram niyamangal verum {"ab" → ε} aanenkil, namukk cyclic monoidinte oru avatharanam labikkum.
monoyidukalude avatharanamenna nilayil ardha-tyoo samvidaanangalude praadhaanyam inipparayunnava vazhi kooduthal shakthamaakkunnu
sidhaantham oro monoidinum formula _ 8-nte oru avatharanamundu, athinaal ithu ellaayppozhum oru semi-tyoo sisttam avatharippikkam, orupakshe ananthamaaya aksharamaalayilaayirikkam.
eee saahacharyathil, settu formula _ 55 neey formula _ 50 nte janarettarukalude settu ennum formula _ 1 neey nirvachikkunna bandhangalude formula _ 50 ennum vilikkunnu. monoidukale avayude avatharanathe adisthaanamaakki namukk udan tharamthirikkaam. formula _ 50 ennu vilikkunnu
prashnamenna vaakkinte anishchithathvam.
turing mesheenukalude nirtthal prashnam prashnamenna vaakkinte oru udaaharanamaayi kurachukondu, prashnam (semigrooppukalkku) enna vaakk pothuve theerumaanikkaanaavillenna poost theliyichu.
krithyamaayi paranjaal, oru turing mesheen plus teppinte avasthayude parimithamaaya stringayi poost oru encoding aavishkarichu, eee string encodingle pravartthikkunna oru string reerait sisttam vazhi eee mesheente pravartthanangal nadatthaan kazhiyum. encodinginte aksharamaalaykk teppile chinnangalkk oru koottam aksharangalude formula 63 undu (formula 64 ennaal shoonyam ennaanu ardhamaakkunnathu), turing mesheente avasthakalkku mattoru koottam aksharangalude formula 65, oduvil encodingle prathyeka punk vahikkunna moonnu aksharangalude formula 66 ennivayundu. formula _ 67 umm formula _ 68 umm turing mesheente avabodhajanyamaayi adhika aantharika avasthakalaanu, athu nirtthumbol athu parivartthanam cheiyunnu, athesamayam formula _ 69 teppinte shoonyamallaatha bhaagatthinte avasaanam adayaalappeduthunnu; oru formula _ 69 ll athunna oru mesheen avide oru shoonyathayundennathupole perumaaranam, formula _ 69 adutha cellilaayirunnu. turing mesheen avasthakalude saadhuvaaya encodingukalaaya stringukal oru formula 69 ll aarambikkunnu, thudarnnu poojyamo athiladhikamo chihna aksharangal, thudarnnu krithyamaayi oru aantharika avastha akshara formula 73 (mesheente avasthaye encode cheiyunna), thudarnnu onno athiladhikamo chihna aksharangal, thudarnnu oru avasana formula 69. chihna aksharangal teppinte ulladakkangalil ninnu nereyaanu, aantharika avastha aksharam thalayude sthaanam adayaalappeduthunnu; aantharika avastha aksharathinu sheshamulla chihnam ippol turing mesheente thalaykku thaazheyulla cellilaanu.
yanthram stettu formula _ 75 ll aayirikkukayum chihna formula _ 76 kaanukayum cheyyumbol chihna formula _ 77 ezhuthukayum, valathottu neengukayum, stettu formula _ 78 lekkulla parivartthanangal reerait vazhi nadappilaakkukayum cheiyunna oru parivartthanam
pakaram edathottu neengunna aaa parivartthanam reerait vazhi nadappilaakkunnu
oro chihna formulakkum oru udaaharanam-81 idathuvashathulla aaa sellil. teppinte sandarshicha bhaagatthinte avasaanathilethiyaal, pakaram nammal upayogikkunnu.
oru aksharam kondu charadu neettuka. ellah reraittukalilum oru aantharika stettu lettar formula _ 73 ulppedunnathinaal, saadhuvaaya encodingukalil atharamoru aksharam maathrame adangiyittulloo, oro reeraitum krithyamaayi atharamoru aksharam srishtikkunnu, reerait prakriya krithyamaayi encode cheytha turing mesheente pravartthanatthe pinthudarunnu. string reerait sisttangal turing poortthiyaayathaayi ithu theliyikkunnu.
formula 67, formula 68 ennee rund nirthalaakkiya chinnangal ullathinte kaaranam, ellah nirthalaakkunna turing mesheenukalum oru prathyeka "aantharika" avasthayil mathramalla, ore "motham" avasthayil avasaanikkanamenn njangal aagrahikkunnu ennathaanu. ithinu tape nirtthiya sesham vrithiyaakkendathundu, athinaal formula _ 67 formula _ 69-ll athunnathuvare athil idathu chinnathe bhakshikkunnu, avide athu formula _ 68-lake maarunnu, pakaram athinte valathuvashathulla chinnathe bhakshikkunnu. (eee ghattathil string reerait sisttam oru turing mesheene anukarikkunilla, kaaranam athinu tapil ninnu cellukal neekkam cheyyaan kazhiyilla.) ellah chinnangalum illaathaaya sesham, nammal terminal string formula _ 89 ll ethi.
eee string reerait sisttavumaayi bandhappett formula 90 umm formula 89 umm ore kongruvans classil ulppedunnundo ennu parishodhikkunnathiloode, oru prathyeka motham stettu formula 90 ll aarambhikkumbol nalkiyirikkunna turing mesheen avasaanikkunnundo ennu theerumaanikkunnathinulla oru nadapadikramavum problem enna vaakkinte theerumaana nadapadikramam nalkum. saankethikamaayi, namukk inipparayunnava undu
lemma. formula _ 93 oru nirnnayaka turing mesheenayirikkatte, mukalil vivarichathupole formula _ 93 nadappilaakkunna string reerait sisttam formula _ 1 ayirikkatte. formula 98 aanenkil maathram formula 93 nirtthum (athaayath formula 5 umm formula 89 umm formula 1 nuu yojippaanengil maathram).
formula 5 ll ninnu aarambhikkumbol formula 93 nirthukayaanenkil formula 1 nte nirmmaanatthil ninnu udan thanne formula 1 (formula 102 nte oru thelivu nirmmikkunnathuvare formula 93 pravarthippikkunnu), ennaal formula 108 turing mesheen formula 93 neey pinnott pookaan anuvadikkunnu. ivide formula 93 nirnnaayakamaanennu prasakthamaakunnu, kaaranam appol munnottulla ghattangalellaam savisheshamaan; oru formula _ 108 formula 5 muthal formula 89 vare nadakkuka, avasana pinnoca ghattam athinte kaundarpaarttine oru forwade steppaayi pinthudaranam, athinaal iva randum raddaakkukayum induction vazhi ellah pinnoca ghattangalum atharam nadathatthil ninnu illaathaakkukayum cheyyaam. athinaal formula 93 formula 5 ll ninnu aarambhikkumbol nirthunnillenkil, i.iii.namukk 102 enna soothravaakyam illengil, namukk 98 enna soothravaakyam illa. athinaal, 108 enna soothravaakyam nirnnayikkunnath 93 enna soothravaakyathinte nirthalaakkal prashnathinulla utharam nammodu parayunnu.
eee vaadathinte vyakthamaaya parimiti enthennaal, theerumaanikkaanaavaatha padaprashnam ulla oru semigroup formula _ 120 nirmmikkunnathin, oraalkk aadyam oru turing mesheen formula _ 93-nte vyakthamaaya udaaharanam undaayirikkanam, athinaayi nirthalaakkal prashnam theerumaanikkaanaavilla, ennaal pothuvaaya nirtthal prashnatthinte anishchithathvathinte thelivaayi kandethunna vividha turing mesheenukalkkellam nirtthal prashnam pariharikkunna oru saangalppika turing mesheen gadakamaayi undu, athinaal aaa mesheenukalonnum yadhaarthathil nilanilkkilla; theerumaanaprashnam theerumaanikkaanaavaatha "chila" turing mesheen undennu theliyikkunnathellaam. ennirunnaalum, nirnnayakamaaya nirtthal prashnamulla chila turing mesheenukal undennathinartham oru "saarvathrika" turing mesheente nirtthal prashnam anishchithathvathilaanu (athinu eathu turing mesheen anukarikkaan kazhiyumennathinaal), saarvathrika turing mesheenukalude vyakthamaaya udaaharanagal nirmmichittundu.
mattu aashayangalumayi bandhappeduka.
oru semi-tyoo sisttam oru tame-reraiting sisttam koodiyaanu-idathu, valath vashangalile padangalude athe variable avasaanikkunna monadic padangal (functionukal) ulla onnu, udha.jii. oru padam ruul formula _ 122 ennathu string ruul formula _ 123 nuu thulyamaanu.
oru semi-tyoo sisttam oru prathyeka tharam poost canonical sistamaanu, ennaal oro poost canonical sisttavum oru srs aayi kuraykkaam. rund aupachaarikathakalum turing poornnamaanu, athinaal noam chomskyude aniyanthrithamaaya vyaakaranangalkk thulyamaanu, avaye chilappol "semi-tyoo vyakaranangal" ennu vilikkunnu. aksharamalaye terminalukalaayum noonn-terminalukalaayum verthirikkunnathiloodeyum terminalukalallaathavaykilkil oru prarambha chihnam nischayikkunnathiloodeyum maathrame oru aupachaarika vyakaranam semi-tyoo sisttathil ninnu vyathyasthamaakoo. oru nyoonapaksham ezhuthukaar yadhaarthathil oru semi-tyoo systethe triple formula _ 124 aayi nirvachikkunnu, avide formula _ 125 neey "settu oof axiums" ennu vilikkunnu. ardha-thoonukalude eee "generative" nirvachanathinu keezhil, aniyanthrithamaaya vyakaranam ennathu orotta svayamsidhiyulla oru ardha-thoonukalude samvidaanamaanu, athil oraal aksharamalaye terminalukalaayum noonn-terminalukalaayum vibhajikkukayum svayamsidhathaye oru noonn terminalakkukayum cheiyunnu. aksharamalaye terminalukalaayum noonn-terminalukalaayum vibhajikkunna lalithamaaya artifies sakthamaaya onnaanu; terminalukaludeyum noonn-terminalukaludeyum samyojanathe adisthaanamaakki chomsky shreniyude nirvachanam ithu anuvadikkunnu. aupachaarika bhaashakalude sidhaanthathile nirnayakamaaya oru vikasamaayirunnu ithu.
kwaandam computingil, oru kwaandam thuu sisttam enna ashayam vikasippikkaan kazhiyum.
kwaandam kanakkukoottal aantharikamaayi reversable aayathinal, aksharamala soothravaakyathil _ 55-nuu mukalilulla reraiting niyamangal dvidishayilaayirikkanam (i.iii. adisthaana samvidaanam oru thuu sistamaanu, ardha thuu sistamalla).
aksharamala aksharangalude formula 127-nte oru upavibhaagathil oraalkk oru hilbert space formula 128 attaachucheyyaan kazhiyum, oru substring mattonnil kondupookunna oru reraiting niyamathinu stringukalil gadippichirikkunna hilbert spacinte tensor prodakttil oru unitory ooppareshan nadatthaan kazhiyum; ithu soochippikkunnathu avaa settu formula 129-ll ninnu pratheekangalude ennam samrakshikkunnu ennaanu. classical kesinu samaanamaayi oru kwaandam thuu sisttam kwaandam kanakkukoottalinulla oru saarvathrika computational modalaanennu kaanikkan kazhiyum, athaayath execute cheytha kwaandam pravartthanangal unifom surcuted klaasukalumaayi (e aayirikkumbol bqp-ll ullava polullava) poruthappedunnu.jii. inputt valuppathil pala ghattangalilayi polynomially string reraiting niyamangal avasaanippikkumenn urappu nalkunnu) allengil thulyamaayi oru kwaandam turing mesheen.
charithravum praadhaanyavum.
pothuvaaya ganitha sidhaanthangal aupachaarika bhashayil prakadippikkaan anuvadikkunna, thudarnnu yaanthrikavum yaanthrikavumaaya reethiyil theliyikkukayum parisodhikkukayum cheiyunna, nirddeshaparamaaya yukthi polulla samvidaanangal srishtikkunnathinaayi, yukthiyil adhika nirmmithikal cherkkunnathinulla oru prograaminte bhaagamaayi semi-tyoo samvidaanangal vikasippicheduthu. sidhaantham theliyikkunna pravartthanam oru koottam stringukalil nirvachikkapetta krithrimathvangalude oru koottamaayi kuraykkaanaakumennaayirunu prateeksha. semi-tyoo samvidaanangal aniyanthrithamaaya vyaakaranangalkk isomorphic aanennu pinneedu manassilakki, avaa turing mesheenukalkk isomorphic aanennu ariyappedunnu. eee gaveshana reethi vijayikkukayum ippol ganithashaasthrathinteyum yukthisahamaya sidhaanthangaludeyum thelivukal parisodhikkaan computerukal upayogikkukayum cheyyaam.
alonso charchinte nirddeshaprakaaram, 1947-ll prasidheekaricha oru prabandhatthile emil poost, martin davis prasthaavikkunnath "oru prathyeka prashnam pariharikkaanaavillenn" aadyamaayi theliyichu...classical ganithashaasthrathil ninnulla oru prashnatthinte aadyathe pariharikkaanaavaattha thelivu-eee saahacharyathil semigrooppukalkkulla prashnam enna vaakk."
thelivu swathanthramaayi nalkiyathaayum davis vaadikkunnu. eiii. markov.

