Ua hoʻonohonoho maʻalahi ʻia kekahi mau pilikia makemakika i hiki ke wehewehe ʻia i kahi keiki, akā paʻakikī loa a noho lākou i nā hanauna o nā makemakika. ʻO kekahi o ia mau pilikia ka mea i kapa ʻia ʻo ka pilikia mamao unit: E kau i nā kiko \(n\) i loko o ka mokulele. Ehia mau hui o nā kiko e hiki ke loaʻa kahi mamao o \(1\) pololei? Hoʻi ka pilikia iā Paul Erdős a ua aʻo ʻia mai ka makahiki 1946. Ua paʻi ʻo OpenAI i kēia manawa ua hōʻole kahi kumu hoʻohālike kūloko i kahi kuhi lōʻihi e pili ana i kēia pilikia.
I ka nānā mua ʻana, kani ka nīnau me he mea lā ʻaʻole ia he mea ʻino. Inā ʻoe e kau i nā kiko \(n\) ma kahi laina pololei, loaʻa iā ʻoe ma kahi o \(n-1\) mau manawa o ka lōʻihi \(1\) . Inā ʻoe e hoʻonohonoho i nā kiko ma ke ʻano he grid, e like me ka pepa kiʻi, loaʻa iā ʻoe ka nui o kēia mau manawa: ma ka ʻaoʻao a me ke ʻano kū pololei ma waena o nā kiko e pili ana. Ua ʻike mua ʻo Erdős i nā kūkulu ʻana i ʻoi aku ka maikaʻi ma mua o ka linear. Eia naʻe, no ka manawa lōʻihi, ua manaʻo ʻia ʻaʻole hiki ke hoʻomaikaʻi nui ʻia kēia. Ma ke ʻano kūhelu, ʻo ke kuhi ʻana ʻo ka nui o ia mau manawa anakahi e ulu wale ana ma kahi o \(n^{1+o(1)}\) , ʻo ia hoʻi, ʻoi aku ka wikiwiki ma mua o \(n\) , akā ʻaʻole me kahi exponent hoʻohui paʻa.
ʻO kēia ke kiko kupanaha: ʻO ke kumu hoʻohālike OpenAI (ʻaʻole i hōʻike ʻia) ʻaʻole i kūkulu i hoʻokahi counterexample hoʻokahi, akā he ʻohana palena ʻole o nā set kiko \(P\) nona ka helu o nā mamao unit ma ka liʻiliʻi \(|P|^{1+\delta}\) , me kahi paʻa \(\delta>0\) . ʻO kahi hoʻomaʻemaʻe ma hope e Will Sawin, e like me OpenAI, e hua mai ana i \(\delta=0{,}014\) . He leo liʻiliʻi kēia, akā he nui loa ia ma ka makemakika: ʻAʻole ia he koena logarithmic, akā he loaʻa polynomial maoli.
Hōʻike kahi laʻana maʻalahi i ka manaʻo kumu. E noʻonoʻo kākou i ka helu paʻakikī
\[
u=\frac{2+i}{2-i}=\frac{3+4i}{5}=\frac35+\frac45i.
\]
He \(1\) ka waiwai paʻa o kēia helu, no ka mea,
\[
\left|\frac35+\frac45i\right|=\sqrt{\left(\frac35\right)^2+\left(\frac45\right)^2}=1.
\]
No laila, inā he kiko ʻo \(x\) ma ka mokulele, a laila ua like ka mamao o \(x\) a me \(x+u\) me \(1\) . ʻO ke kikoʻī, no ka laʻana, aia nā kiko ma...
\[
0
\quad\text{und}\quad
\frac35+\frac45i
\]
hoʻokahi wale nō anakahi ke kaʻawale. No laila, ʻo kahi hōʻike helu-kumumanaʻo e hua mai ana i kahi kuhikuhi geometric o ka lōʻihi \(1\) . ʻAʻole kēia ka hōʻoia kūʻē nui, akā ʻo ia ka mana liʻiliʻi o ka hana hoʻopunipuni: ʻaʻole wale kekahi e nānā no nā mamao anakahi ākea a me ke kū pololei e like me ka grid, akā no nā kuhikuhi he nui i hana ʻia ma ka arithmetic, nona nā mea āpau ka lōʻihi pololei \(1\) .
Hoʻohana ke kūkulu maoli i nā mea hana ikaika loa. Ma kahi o ka hana wale ʻana me nā helu helu Gaussian \(\mathbb{Z}[i]\) , hoʻohana ka hōʻoia i nā kahua helu algebraic paʻakikī \(K=L(i)\) . Ma laila, he nui nā mea o ke ʻano
\[
u=\frac{\alpha}{c(\alpha)}
\]
i kūkulu ʻia, kahi e hoʻokani ai \(c\) i ke kuleana o ka hoʻohuihui paʻakikī. ʻO ka hopena koʻikoʻi: ma waena o nā hoʻokomo paʻakikī pili, ʻo kēia mau \(u\) he nui o \(1\) . No laila he mau moho lākou no nā kuhikuhi ʻokoʻa like ʻole.
Ma nā huaʻōlelo koʻikoʻi loa, ʻaʻole like ka counterexample maoli me kahi kiʻi liʻiliʻi a nani me nā kiko he ʻumi, akā, e like me kahi pūʻulu nui o nā kiko i kūkulu ʻia ma ke ʻano helu. Lawe ʻoe i kahi grid kiʻekiʻe-dimensional, ʻokiʻoki i nā kiko kūpono mai ia mea, a laila hoʻohuli i kēia mau mea i ka mokulele maʻamau. Ma ka hōʻailona hōʻoia, ua hōʻike ʻia kēia ma ke ʻano maʻamau ...
\[
P_j=\pi_1\big((y+\Lambda_j)\cap W\big),
\]
I nā huaʻōlelo ʻē aʻe: Lawe kekahi i nā kiko mai kahi grid i hoʻololi ʻia \(y+\Lambda_j\) , kaupalena iā lākou e kahi ʻāpana \(W\) , a hoʻolālā iā lākou me \(\pi_1\) ma luna o kahi hoʻonohonoho paʻakikī, i.e., ma luna o \(\mathbb{C}\cong\mathbb{R}^2\) . ʻO nā ʻokoʻa he nui ma waena o kēia mau kiko a laila pololei ia mau mea \(u\) me ka nui \(1\) . No laila, ma hope o ka hoʻolālā ʻana, lilo lākou i mau mamao ʻoiaʻiʻo ma ka mokulele.
ʻO kahi hoʻohālikelike ʻike maka penei: Hoʻohana ka grid maʻamau i kekahi mau kuhikuhi maʻalahi, e like me ka ʻākau, hema, luna, a me lalo. Eia naʻe, hoʻopuka ka hana hou i kahi helu nui o nā kuhikuhi huna i loaʻa mai ke kumumanaʻo helu algebraic. Hoʻokahi anakahi pololei kēlā me kēia kuhikuhi. No ka mea he nui nā kuhikuhi like ʻole a he nui nā kiko e pili ana iā lākou, ʻoi aku ka nui o ka mamao anakahi ma mua o ka manaʻo kahiko e ʻae ai.
He mea kūpono hoʻi ke ʻano o ka hōʻoia. Wahi a OpenAI, ua loaʻa ka hopena ma ke ʻano kūʻokoʻa e kahi kumu hoʻohālike noʻonoʻo laulā, ʻaʻole e kahi ʻōnaehana makemakika i hoʻomaʻamaʻa kūikawā ʻia no kēia pilikia. A laila ua nānā hou ʻia ka hōʻoia ma loko a ma waho a ua hoʻololi ʻia i loko o kahi ʻano hiki ke heluhelu ʻia e ke kanaka. Hoʻomaopopo pū nā memo e pili pū ana mai nā makemakika o waho ʻaʻole kēia he mana hana aunoa o kahi ʻano i ʻike ʻia, akā he pilina i manaʻo ʻole ʻia ma waena o ke geometry discrete a me ke kumumanaʻo helu algebraic.
ʻO kēia paha ke kumu maoli no ke ʻano hoihoi o kēia hopena. ʻAʻole ia e pili ana i kahi AI e hoʻoponopono pololei ana i kahi pilikia. E pili ana ia i ka loaʻa ʻana o kahi ala i ʻike ʻole ʻia e nā poʻe he nui: ke hoʻoponopono ʻana i kahi pilikia geometric e pili ana i nā mamao ma kahi mokulele me nā mea hana hohonu mai ke kumumanaʻo helu. ʻAʻole kēia e hoʻolilo i ka makemakika i mea kanaka ʻole. Akā hāʻawi ia iā ia i kahi hoa paio hou, ikaika loa.