Ezinye iingxaki zezibalo zenziwe ngokulula ukuze zichazwe kumntwana, kodwa zinzima kangangokuba zithatha izizukulwana zeengcali zezibalo. Enye ingxaki enjalo yingxaki ebizwa ngokuba yi-unit distance: Beka amanqaku \(n\) kwi-plane. Zingaphi iipere zamanqaku ezinokuba nomgama ongu- \(1\) ngqo? Le ngxaki iqala kuPaul Erdős kwaye ifundwe ukusukela ngo-1946. I-OpenAI ngoku ipapashe ukuba imodeli yangaphakathi ayichasanga ingcamango ende malunga nale ngxaki.
Ekuqaleni, umbuzo uvakala ungenabungozi. Ukuba ubeka amanqaku kumgca othe tye, ufumana malunga nezithuba zobude ezi-\( \(n\) \(n-1\) \(1\) Ukuba uhlela amanqaku njengegridi, njengakwiphepha legrafu, ufumana ngaphezulu kwezi zithuba: ngokuthe tye nangokuthe nkqo phakathi kwamanqaku akufutshane. U-Erdős wayesele efumene izakhiwo ezingcono kancinci kunezithe ngqo. Nangona kunjalo, ixesha elide, kwakukholelwa ukuba oku akunakuphuculwa kakhulu. Ngokwesiko, ingcinga yayikukuba inani eliphezulu lala manqanaba eyunithi likhula kuphela malunga ne \(n^{1+o(1)}\) , oko kukuthi, ngokukhawuleza kancinci kune \(n\) , kodwa kungekhona nge-exponent eyongezelelweyo emiselweyo.
Le yingongoma emangalisa kanye: Imodeli ye-OpenAI (engazange ityhileke) yakha kungekuphela nje umzekelo omnye ochaseneyo, kodwa usapho olungenasiphelo lweeseti zamanqaku \(P\) apho inani leeyunithi zomgama ubuncinane liyi- \(|P|^{1+\delta}\) , kunye ne- \(\delta>0\) esisigxina. Uhlaziyo lwamva lukaWill Sawin, ngokutsho kwe-OpenAI, lude luvelise \(\delta=0{,}014\) . Oku kuvakala kuncinci, kodwa kukhulu ngokwezibalo: Akusekho ntsalela ye-logarithmic, kodwa kukufumana inzuzo yokwenyani ye-polynomial.
Umzekelo olula ubonisa ingcamango esisiseko. Makhe siqwalasele inani elintsonkothileyo
\[
u=\frac{2+i}{2-i}=\frac{3+4i}{5}=\frac35+\frac45i.
\]
Eli nani linexabiso elipheleleyo le- \(1\) , kuba
\[
\left|\frac35+\frac45i\right|=\sqrt{\left(\frac35\right)^2+\left(\frac45\right)^2}=1.
\]
Ngoko ke, ukuba \(x\) yingongoma ekwiplane, ngoko \(x\) kunye \(x+u\) zahlukene ngokuchanekileyo \(1\) . Ngokukodwa, umzekelo, amanqaku akwi...
\[
0
\quad\text{und}\quad
\frac35+\frac45i
\]
iyunithi enye kuphela eyahlulwe ngokupheleleyo. Intetho yethiyori enamanani ngaloo ndlela ivelisa ulwalathiso lwejometri lobude \(1\) . Oku akukabi yingxaki enkulu, kodwa yinguqulelo encinci yeqhinga: umntu akajongi kuphela imigama yeeyunithi ezithe tye nezithe nkqo njengakwigridi, kodwa nakwimikhombandlela emininzi eyenziwe ngokwezibalo, zonke ezinobude obuchanekileyo \(1\) .
Ulwakhiwo lokwenyani lusebenzisa izixhobo ezinamandla ngakumbi. Endaweni yokusebenza kuphela ngee-Gaussian integers \(\mathbb{Z}[i]\) , ubungqina busebenzisa amasimi amanani e-algebra anzima ngakumbi \(K=L(i)\) . Kukho izinto ezininzi zefom
\[
u=\frac{\alpha}{c(\alpha)}
\]
yakhiwe, apho \(c\) idlala indima yokudibanisa okuntsonkothileyo. Isiphumo esibalulekileyo kukuba: phakathi kokufakwa okuntsonkothileyo okufanelekileyo, ezi \(u\) nganye inobukhulu be \(1\) . Ngoko ke zingabaviwa beendlela ezahlukeneyo zeyunithi.
Ngamazwi arhabaxa, umzekelo ochanekileyo awubonakali njengomfanekiso omncinci, omhle onamachaphaza alishumi, kodwa ufana neqela elikhulu lamachaphaza akhiwe ngokwezibalo. Uthatha igridi enobukhulu obuphezulu, usike amanqaku afanelekileyo kuyo, uze uwabuyisele kwiplani eqhelekileyo. Kwingxelo yobungqina, oku kubonakaliswa ngendlela...
\[
P_j=\pi_1\big((y+\Lambda_j)\cap W\big),
\]
Ngamanye amazwi: Umntu uthatha amanqaku kwigridi etshintshiweyo \(y+\Lambda_j\) , awathintele ngengingqi \(W\) , aze awabonise nge \(\pi_1\) kwi-complex coordinate, oko kukuthi, kwi \(\mathbb{C}\cong\mathbb{R}^2\) . Umahluko omkhulu phakathi kwala manqaku ke ngoko zii \(u\) elements ezinje ngobukhulu \(1\) . Ke ngoko, emva koqikelelo, aba yi-unit distances yokwenyani kwi-plane.
Umzekeliso obonakalayo ngowokuba: Igridi esemgangathweni isebenzisa iindlela ezimbalwa ezilula, ezinje ngasekunene, ngasekhohlo, phezulu, nasezantsi. Nangona kunjalo, ulwakhiwo olutsha luvelisa inani elikhulu leendlela ezifihlakeleyo ezithathwe kwithiyori yamanani e-algebra. Indlela nganye iyunithi enye ubude. Ngenxa yokuba kukho iindlela ezininzi ezinjalo kwaye amanqaku amaninzi ahambelana nazo, umgama weeyunithi uwonke mkhulu kunokuba ingqikelelo endala ingavumela.
Kwakhona kuyaphawuleka apho ubungqina buvela khona. Ngokutsho kwe-OpenAI, isisombululo sifunyenwe ngokuzimela yimodeli yokuqiqa ngokubanzi, kungekhona yinkqubo yezibalo eqeqeshwe ngokukodwa kule ngxaki. Ubungqina babuye bahlaziywa ngaphakathi nangaphandle kwaye baguqulelwa kwimo efundekayo yomntu. Amanqaku ahamba nayo avela kwiingcali zezibalo zangaphandle agxininisa ukuba oku akusiyo nje inguqulelo ezenzekelayo yendlela eyaziwayo, kodwa lunxibelelwano olungalindelekanga phakathi kwe-geometry ehlukeneyo kunye nethiyori yamanani e-algebra.
Mhlawumbi esi sesona sizathu sokuba esi siphumo sibe nomdla kangaka. Asikuko nje ukusombulula ingxaki ngokuchanekileyo nge-AI. Simalunga nokufumana indlela engakhange icace kubantu abaninzi: ukusombulula ingxaki yejometri malunga nemigama kwindawo ethile ngezixhobo ezinzulu ezivela kwithiyori yamanani. Oku akwenzi izibalo zibe yinto engaphantsi kobuntu. Kodwa kuyinika umchasi omtsha, onamandla angakhululekanga.