BABU ALGORITHM DA ZAI IYA GANE WASU SIFFOFI MASU GIRMA HUƊU

Wasu tambayoyi ba su da girke-girke. Ba don babu wanda ya same shi tukuna ba ne, a’a saboda an tabbatar da cewa babu algorithm — babu wata hanya ta mataki-mataki, komai tsawonta, da za a gudanar a kan kowace kwamfuta — da zai iya amsa su daidai a kowane lokaci. Aikin Cameron McA. Gordon, a University of Texas at Austin, ya ƙara irin waɗannan tambayoyi guda biyu game da siffofi masu girma huɗu.

Shin wannan siffa ɗaya ce?

Masana lissafi suna kiran waɗannan siffofi 4-manifolds (manifolds masu girma huɗu): wurare (spaces) waɗanda, idan aka duba su daga kusa, suke kama da sararin girma huɗu na yau da kullum a ko’ina. Don a miƙa su ga kwamfuta, takardar ta ɗauka cewa an gina su ne daga guntaye masu lebur da aka manna tare (“piecewise linear”, ko PL), ta yadda za a iya bayyana kowace ɗaya da jerin bayanai mai iyaka. Babbar tambayar a lokacin ita ce: idan aka ba da irin waɗannan siffofi biyu, shin iri ɗaya ne, idan aka yarda a canza su ta hanyar miƙewa ba tare da yankewa ba (continuous deformation)?

A 1958, aikin A. A. Markov ya nuna cewa wannan tambaya ba ta da algorithm. Hujjar Markov tana mayar da matsalar siffa zuwa matsala game da ƙungiyoyin lissafi (groups): akwai ƙungiyoyi, da aka ayyana da ƙa’idoji masu iyaka, waɗanda babu algorithm da zai iya yanke hukunci ko wani haɗin abubuwansu ya yi daidai da identity (abin da ba ya canza komai) — wato “matsalar kalma” (word problem). Daga irin wannan ƙungiya, ana iya gina siffofi da algorithm zai iya bambance su ne kawai ta hanyar warware wannan matsala da ba za ta yiwu ba.

Ƙidaya ƙasa har zuwa bakwai

Hujjar Markov har ta ɗaure wahalar a kan wasu siffofi na musamman. Ɗauki S², saman ƙwallo na yau da kullum, sannan a samar da sakamakon ninkawa (product) S² × S², wata siffa mai girma huɗu. Manna kwafi k nasa tare (wato “jimlar haɗi”, connected sum, da ake rubutawa #k). Ga wani k, babu algorithm da zai iya yanke hukunci ko wata siffa da aka bayar ita ce #k(S² × S²). Wane k?

Takardar ta bibiyi wannan ƙidaya ƙasa. Wata ƙungiya mai matsalar kalma da ba za a iya warware ta ba da ƙa’idoji 12, wadda aka samo a aikin Borisov na 1969, ta ba da kwafi 14. Aikin Gordon na 2022 ya kawo shi zuwa 12. Aikin M. Tancer ya sami wata sabuwar hanya, ta hanyar rarrabe-rarraben Freedman na siffofi masu girma huɗu (Freedman’s classification), kuma ya kai 9 a sigar “topological”, inda ba dole ba ne a gina siffofin daga guntaye masu lebur. Wata sigar farko ta wannan aiki ta kai 8. Sai Kegel, Li da Ren suka gina wata ƙungiya da ba za a iya warwarewa ba mafi ƙanƙanta sosai — masu samarwa (generators) 3 da ƙa’idoji 9 — suka kai 7 a sigar topological da 9 a sigar guntaye masu lebur.

Sabuwar takardar Gordon ta sa hanyar Tancer ta yi aiki ga siffofin da aka gina daga guntaye masu lebur ma, kuma ta tabbatar da cewa:

Babu algorithm da zai iya yanke hukunci ko wani 4-manifold da aka bayar iri ɗaya ne, guntu da guntu, da jimlar haɗin kwafi 7 na S² × S².

Zuciyar hujjar wani ƙamus ne. Ga kowace kalma w a cikin ƙungiyar Kegel, Li da Ren, takardar ta gina wata siffa mai girma huɗu. Idan w ta yi daidai da identity, wani jerin motsi da aka yarda da su a kan ƙa’idojin — motsin Andrews-Curtis (Andrews-Curtis moves), waɗanda ke sake rubuta bayanin ba tare da sauya siffar ba — yana nuna cewa siffar daidai kwafi bakwai ne na S² × S². Idan ba haka ba, siffar tana ɗauke da madaukai (loops) da ba za a iya matse su zuwa wuri ɗaya (point) ba — abin da ba ya taɓa faruwa a cikin kwafi bakwai na S² × S² — don haka ba za ta iya zama waɗannan kwafi bakwai ba. Saboda haka algorithm da zai gane #7(S² × S²) zai warware matsalar kalma, abin da ba zai yiwu ba.

Ƙwallaye a ƙulle

Tambaya ta biyu game da ƙulli (knots) ce. Ƙulli na yau da kullum da’ira ce da aka ɗaure a cikin sararin girma uku, kuma yanke hukunci ko za a iya kwance ƙulli abu ne mai yiwuwa: akwai algorithm, na Haken. Idan aka hau girma ɗaya, ƙwallo mai girma biyu (two-dimensional sphere) zai iya kasancewa a ƙulle a cikin sararin girma huɗu. A girman da suka fi haka, an san cewa matsalar da ta yi daidai da wannan ba ta da algorithm. Ga ƙwallaye masu girma biyu a cikin ƙwallo mai girma huɗu S⁴, babu wanda ya sani.

Takardar Gordon ta tabbatar da cewa a cikin #7(S² × S²), babu algorithm da zai iya yanke hukunci ko ƙwallo mai girma biyu — ko kowane fili (surface), mai alkibla (orientable) ko marar alkibla — ba a ƙulle yake ba. A kan hanya, takardar ta gina wani ƙwallo a cikin wannan siffa wanda sararin da ke kewaye da shi yake da ƙungiya mai matsalar kalma da ba za a iya warwarewa ba, kuma, a cikin kwafi 12 na S² × S², wani ƙwallo wanda ƙungiyarsa ke ɗauke da kwafin kowace ƙungiya da kwamfuta za ta iya bayyanawa.

Babbar kyauta: kwafi sifili

Takardar ta faɗi burin a fili: saukar da yawan kwafin zuwa sifili. Kwafi sifili yana nufin ƙwallo mai girma huɗu S⁴ da kansa. A girma biyar zuwa sama, babu algorithm da zai iya gane ƙwallon; a girma biyu da uku, akwai wanda zai iya. Girma huɗu yana nan a buɗe — haka ma tambayar ko za a iya gano ƙwallaye a ƙulle a cikin S⁴. Kowane kwafi da aka cire yana kusantar da tambayoyin biyu.