1. Gabatarwa
Hanyoyin barbashi suna wakiltar wani muhimmin nau'in algorithms a cikin lissafin kimiyya tare da aikace-aikace daga lissafin ruwa zuwa kwaikwayoyin kwayoyin halitta. Duk da yawan amfani da su, ƙarfin lissafin ka'idar su bai kasance ba har sai wannan binciken. Wannan binciken yana haɗa gibin tsakanin hanyoyin barbashi na aiki da kimiyyar kwamfuta ta ka'ida ta hanyar nazarin matsayinsu a cikin tsarin Chomsky da kuma tantance cikakkiyar Turing.
Binciken ya magance tambayoyi biyu masu mahimmanci: (1) Muna iya ƙuntata hanyoyin barbashi har yaushe yayin da muke riƙe da cikakkiyar Turing? (2) Wadanne ƙuntatawa mafi ƙanƙanta ke haifar da asarar ƙarfin Turing? Waɗannan tambayoyin suna da tasiri mai zurfi ga fahimtar iyakokin ka'idar algorithms na kwaikwayo.
2. Tsarin Ka'ida
2.1 Hanyoyin Barbashi a matsayin Atomata
Ana fassara hanyoyin barbashi a matsayin atomata na lissafi bisa ga ma'anar lissafin hukuma. Kowane barbashi yana wakiltar naúrar lissafi tare da yanayin ciki, kuma hulɗar tsakanin barbashi suna ayyana canjin yanayi. Wannan fassarar tana ba da damar amfani da kayan aikin ka'idar atomata don nazarin ƙarfin lissafi.
Samfurin atomon ya ƙunshi:
- Jihohin barbashi: $S = \{s_1, s_2, ..., s_n\}$
- Dokokin hulɗa: $R: S \times S \rightarrow S$
- Ayyukan juyin halitta: $E: S \rightarrow S$
- Gudanar da yanayin duniya
2.2 Ma'anar Hukuma
Ma'anar hukuma ta bi tsarin lissafi da aka kafa a cikin aikin da ya gabata [10], inda aka ayyana hanyar barbashi a matsayin tuple:
$PM = (P, G, N, U, E)$ inda:
- $P$: Saitin barbashi tare da jihohin mutum ɗaya
- $G$: Masu canji na duniya
- $N$: Ayyukan unguwa da ke ayyana hulɗa
- $U$: Aikin sabuntawa don jihohin barbashi
- $E$: Aikin juyin halitta don masu canji na duniya
3. Binciken Cikakkiyar Turing
3.1 Sharuɗɗan Wadatacce
Binciken ya tabbatar da saiti biyu na sharuɗɗan wadatacce waɗanda a ƙarƙashinsu hanyoyin barbashi suka kasance cikakke na Turing:
- Rufaffen Mai Canji na Duniya: Lokacin da aikin juyin halitta $E$ zai iya rufaffen na'urar Turing ta duniya a cikin masu canji na duniya, tsarin yana riƙe da cikakkiyar Turing ba tare da la'akari da iyakokin hulɗar barbashi ba.
- Lissafi Rarraba: Lokacin da barbashi zasu iya haɗin gwiwar kwaikwayon sel na tef da canjin yanayi ta hanyar haɗin gwiwar hulɗa, ko da tare da iyakacin iyawar mutum ɗaya.
Hujjar ta ƙunshi gina raguwa bayyananne daga sanannun tsarin Turing-cikakke zuwa aiwatar da hanyoyin barbashi.
3.2 Ƙuntatawa da Ake Bukata
Binciken ya gano takamaiman ƙuntatawa waɗanda ke haifar da asarar ƙarfin Turing:
- Barbashi na Yanayi Iyaka: Lokacin da barbashi ke da sararin jihohi masu iyaka ba tare da samun damar ƙwaƙwalwar ajiya na waje ba
- Hulɗar Gida Kacal: Lokacin da hulɗar ke da ƙaƙƙarfan gida ba tare da hanyoyin haɗin gwiwar duniya ba
- Juyin Halitta Tabbacce: Lokacin da aikin juyin halitta ya rasa ikon reshe na sharadi
Waɗannan ƙuntatawa suna rage hanyoyin barbashi zuwa ƙarfin lissafi na atomata iyaka ko atomata turawa a cikin tsarin Chomsky.
4. Aiwatar da Fasaha
4.1 Tsarin Lissafi
Binciken ƙarfin lissafi yana amfani da gine-ginen ka'idar harshe na hukuma. An ayyana aikin canjin yanayi don hulɗar barbashi kamar haka:
$\delta(p_i, p_j, g) \rightarrow (p_i', p_j', g')$
inda $p_i, p_j$ jihohin barbashi ne, $g$ yanayin duniya ne, kuma masu canji na farko suna wakiltar jihohin da aka sabunta.
Kwaikwayon na'urar Turing yana buƙatar rufaffen alamomin tef $\Gamma$ da jihohi $Q$ cikin jihohin barbashi:
$encode: \Gamma \times Q \times \mathbb{Z} \rightarrow S$
inda $\mathbb{Z}$ ke wakiltar bayanin matsayin tef.
4.2 Hanyoyin Canjin Matsayi
Hanyoyin barbashi suna aiwatar da canje-canjen na'urar Turing ta hanyar haɗin gwiwar hulɗar barbashi. Kowane mataki na lissafi yana buƙatar:
- Gano unguwa: $N(p) = \{q \in P : d(p,q) < r\}$
- Musayar yanayi: Barbashi suna raba bayanan tef da kai da aka rufaffe
- Yanke shawara gama gari: Barbashi suna ƙididdige yanayi na gaba ta hanyoyin yarjejeniya
- Daidaituwar duniya: Aikin juyin halitta yana daidaita kammala mataki
5. Sakamako da Abubuwan Da Ke Tattare Da Shi
5.1 Iyakokin Lissafi
Binciken ya kafa takamaiman iyakoki a cikin sararin ƙira na hanyoyin barbashi:
Saitunan Turing Cikakke
- Mai canji na duniya zai iya adana bayanai kowane
- Aikin juyin halitta yana goyan bayan aiwatarwa na sharadi
- Barbashi na iya samun damar yanayin duniya
- An halasta ƙirƙirar barbashi mara iyaka
Saitunan Ba Turing Cikakke Ba
- Hulɗar gida kawai kawai
- Sararin yanayin barbashi mai iyaka
- Sabuntawa tabbatacce, mara ƙwaƙwalwa
- Ƙididdigar barbashi mai iyaka
5.2 Binciken Ƙarfin Kwaikwayo
Binciken ya bayyana cewa yawancin aiwatar da hanyoyin barbashi na aiki a cikin lissafin kimiyya suna aiki ƙasa da cikakkiyar Turing saboda:
- Ƙuntatawa ingantaccen aiki
- Bukatun kwanciyar hankali na lamba
- Iyakokin lissafi layi daya
- Zato na samfurin jiki
Wannan yana bayyana dalilin da yasa kwaikwayoyin barbashi, duk da ƙarfinsu don takamaiman yankuna, ba su nuna iyawar lissafi gabaɗaya ba.
6. Misalin Tsarin Bincike
Nazarin Shari'a: Binciken Kwaikwayon Ruwa na SPH
Yi la'akari da aiwatar da Smoothed Particle Hydrodynamics (SPH) don lissafin ruwa. Ta amfani da tsarin bincike daga wannan binciken:
Kima Ƙarfin Lissafi:
- Wakilcin Matsayi: Jihohin barbashi sun haɗa da matsayi, gudun, yawa, matsa lamba (vector mai girma iyaka)
- Dokokin Hulɗa: An gudanar da su ta hanyar rarrabuwar lissafin Navier-Stokes ta hanyar ayyukan kernel: $A_i = \sum_j m_j \frac{A_j}{\rho_j} W(|r_i - r_j|, h)$
- Masu Canji na Duniya: Matakin lokaci, sharuɗɗan iyaka, ƙa'idodin duniya (ƙaramin ajiya)
- Aikin Juyin Halitta: Tsarin haɗin kai na lokaci (misali, Verlet, Runge-Kutta)
Sakamakon Bincike: Wannan aiwatar da SPH ba cikakke na Turing ba ne saboda:
- Jihohin barbashi suna da ƙayyadaddun girma, masu iyaka
- Hulɗar gida ne kawai kuma na ilimin lissafi
- Masu canji na duniya ba za su iya adana shirye-shirye kowane ba
- Aikin juyin halitta yana aiwatar da ƙayyadaddun algorithms na lamba
Gyara don Cikakkiyar Turing: Don sanya wannan aiwatar da SPH ta zama cikakke na Turing yayin da ake riƙe da iyawar kwaikwayon ruwa:
- Tsawaita jihohin barbashi tare da ƙarin "lissafi" bits
- Aiwatar da dokokin hulɗa na sharadi bisa yanayin lissafi
- Yi amfani da masu canji na duniya don adana umarnin shirye-shirye
- Gyara aikin juyin halitta don fassara shirye-shiryen da aka adana
Wannan misalin yana nuna yadda za a iya amfani da tsarin don bincika hanyoyin barbashi na yanzu da kuma jagorantar gyare-gyare don buƙatun ƙarfin lissafi daban-daban.
7. Aikace-aikace da Jagorori na Gaba
Tushen ka'idar da aka kafa a cikin wannan binciken ya buɗe jagorori masu ban sha'awa da yawa:
Tsarin Kwaikwayo-Lissafi na Hybrid: Haɓaka hanyoyin barbashi waɗanda zasu iya canzawa da sauri tsakanin yanayin kwaikwayon jiki da yanayin lissafi gabaɗaya, yana ba da damar kwaikwayo masu daidaitawa waɗanda zasu iya yin bincike a cikin wuri.
Kayan Aikin Tabbatarwa na Hukuma: Ƙirƙirar kayan aikin atomatik don tabbatar da ƙarfin lissafi na kwaikwayoyin tushen barbashi, kama da yadda masu duba samfura ke tabbatar da tsarin software. Wannan zai iya hana cikakkiyar Turing da ba a yi niyya ba a cikin kwaikwayoyi masu mahimmanci na aminci.
Gine-ginen Lissafi na Tushen Halitta: Aiwatar da ƙa'idodin hanyar barbashi zuwa sabbin gine-ginen kwamfuta, musamman a cikin tsarin rarraba da na mutum-mutumi na mutum-mutumi inda raka'o'in mutum ɗaya ke da iyakacin iyawa amma halayyar gama gari tana nuna ƙarfin lissafi.
Tsarin Ilimi: Yin amfani da hanyoyin barbashi a matsayin kayan aikin koyarwa don koyar da ra'ayoyin ka'idar lissafi ta hanyar kwaikwayo na gani, masu hulɗa waɗanda ke nuna ƙa'idodin ka'idar atomata a cikin aiki.
Hanyoyin Barbashi na Quantum: Tsawaita tsarin zuwa tsarin barbashi na quantum, bincika ƙarfin lissafi na kwaikwayoyin quantum da alaƙarsu da ka'idar atomata na quantum.
8. Nassoshi
- Chomsky, N. (1956). Samfura uku don bayanin harshe. Canjin IRE akan Ka'idar Bayanai.
- Turing, A. M. (1936). Akan lambobi masu lissafi, tare da aikace-aikace ga matsalar Entscheidungs. Proceedings na London Mathematical Society.
- Church, A. (1936). Matsala maras warwarewa na ka'idar lamba ta farko. Jaridar Kimiyyar Lissafi ta Amurka.
- Veldhuizen, T. L. (2003). Samfuran C++ sun cika Turing. Rahoton Fasaha na Jami'ar Indiana.
- Berlekamp, E. R., Conway, J. H., & Guy, R. K. (1982). Hanyoyin Cin Nasara don Wasannin Lissafin ku.
- Cook, M. (2004). Duniya a cikin atomata na tantanin halitta na farko. Tsarin Hadaddun.
- Adleman, L. M. (1994). Lissafin kwayoyin halitta na mafita ga matsalolin haɗawa. Kimiyya.
- Church, G. M., Gao, Y., & Kosuri, S. (2012). Ajiyar bayanai ta dijital ta gaba a cikin DNA. Kimiyya.
- Pahlke, J., & Sbalzarini, I. F. (2023). Ma'anar lissafi na hanyoyin barbashi. Jaridar Lissafin Lissafi.
- Lucy, L. B. (1977). Hanyar lamba zuwa gwajin hasashen fission. Jaridar Astronomical.
- Gingold, R. A., & Monaghan, J. J. (1977). Smoothed particle hydrodynamics: ka'ida da aikace-aikace ga taurari marasa siffa. Sanarwar Rana na Royal Astronomical Society.
- Degond, P., & Mas-Gallic, S. (1989). Hanyar barbashi mai nauyi don lissafin juyawa-diffusion. Lissafin Lissafi.
- Schrader, B., et al. (2010). Musanya Ƙarfin Barbashi Mai Gyara Rarrabuwa. Jaridar Lissafin Lissafi.
- Isola, P., et al. (2017). Fassarar Hotuna-zuwa-Hoto tare da Cibiyoyin Adawa na Sharadi. CVPR. // Nassoshi na waje don kwatanta hanyar lissafi
- OpenAI. (2023). Rahoton Fasaha na GPT-4. // Nassoshi na waje don tsarin lissafi na zamani
- Hukumar Tarayyar Turai. (2021). Ƙayyadaddun Fasaha na Shirin Duniya. // Nassoshi na waje don buƙatun kwaikwayo masu girma
Binciken Kwararre: Ƙarfin Lissafi a cikin Hanyoyin Barbashi
Fahimta ta Asali: Wannan takarda tana ba da gaskiya mai mahimmanci amma sau da yawa ana yin watsi da ita: hanyoyin barbashi da ke tafiyar da komai daga hasashen yanayi zuwa gano magunguna, a cikin mafi yawan nau'ikan su na ka'ida, suna da ƙarfin lissafi kamar na'urar kwamfuta ta duniya. Marubutan ba kawai suna tabbatar da wani abu mai ban sha'awa ba ne; suna fallasa ɓangaren lissafi da ba a taɓa shi ba a cikin kayan aikinmu na kwaikwayo da aka fi amincewa da su. Wannan yana sanya hanyoyin barbashi a cikin lig ɗaya na ka'ida kamar harsunan shirye-shirye (C++, Python) da tsarin hadaddun kamar Wasan Rayuwa na Conway, kamar yadda aka ambata a cikin takarda kuma aka tabbatar da su ta hanyar ayyukan tushe a ka'idar atomata [1, 2]. Ƙimar gaske ba shine cewa ya kamata mu gudanar da Word akan kwaikwayon SPH ba, amma dole ne yanzu mu fahimci yanayin da kwaikwayonmu suka daina zama masu ƙididdigewa kawai suka fara zama kwamfutoci.
Kwararar Hankali & Ƙarfuka: Hujjar an gina ta da kyau. Na farko, sun kafa hanyoyin barbashi a cikin ma'anar lissafi mai ƙarfi daga Pahlke & Sbalzarini [10], suna sake sanya barbashi a matsayin jihohin atomata da ƙwayoyin hulɗa a matsayin dokokin canji. Wannan ƙayyadaddun shine ginshiƙin takarda. Ƙarfin yana cikin bincikensa na bangarori biyu: bai kawai tabbatar da cikakkiyar Turing ta hanyar haɗa na'urar Turing a cikin yanayin duniya ba (hujja mai rauni), amma yana neman iyakoki na wannan iko. Gano takamaiman ƙuntatawa—jihohin barbashi masu iyaka, hulɗar gida kawai kawai, juyin halitta tabbatacce—waɗanda ke rage tsarin zuwa atomata mai iyaka shine mafi mahimmancin gudummawar takarda. Wannan yana haifar da taswirar sararin ƙira na aiki ga injiniyoyi. Haɗin kai zuwa ƙayyadaddun matsayi na lissafi, kamar tsarin Chomsky, yana ba da ƙarfin hankali nan take ga masu ka'ida.
Kurakurai & Gibin Mai Mahimmanci: Binciken, duk da yake yana da inganci a ka'ida, yana aiki a cikin kogon gaskiyar jiki. Yana ɗaukar ƙididdigar barbashi da ƙwaƙwalwar ajiya a matsayin albarkatu na zahiri, mai yuwuwa mara iyaka. A aikace, kamar yadda aka gani a cikin manyan shirye-shirye kamar Duniya ta EU [16], kowane byte da FLOP ana takara da su. Zato na "ƙwaƙwalwar ajiya mara iyaka" wanda ke ba da cikakkiyar Turing shine zato ɗaya wanda ke raba na'urar Turing ta ka'ida da kwamfutarka. Takardar ta yarda yawancin aiwatar da aiki sun kasa cikakkiyar Turing saboda ƙuntatawa aiki, amma ba ta ƙididdige wannan gibin ba. Ana buƙatar ƙarin bits nawa a kowane barbashi don duniya lissafi? Menene kudin sama na asymptotic? Bugu da ƙari, binciken ya kauce wa abubuwan da ke tattare da matsalar tsayawa. Idan kwaikwayon ruwa ya cika Turing, shin za mu iya tabbatar cewa zai ƙare? Wannan yana da sakamako mai zurfi ga sarƙoƙin kimiyyar kwamfuta masu sarrafa kansa, masu gudana.
Fahimta Mai Aiki & Jagora na Gaba: Ga masu aiki, wannan aikin alamar gargaɗi ce da kuma littafin ƙira. Gargadi: Ku kasance masu sane da cewa ƙara "kawai ƙarin fasali ɗaya" ga manajan yanayin duniya na kwaikwayonku zai iya sanya shi cikakke na Turing ba da gangan ba, yana gabatar da rashin yanke hukunci a cikin binciken lamba da aka riga aka yi hasashe. Littafin ƙira: Yi amfani da ƙuntatawa da aka gano (misali, tilasta sabuntawa mai iyaka, na gida kawai) a matsayin lissafin bincike don hana cikakkiyar Turing da gangan don kwanciyar hankali da tabbaci. Nan gaba yana cikin tsarin haɗin gwiwar, haɗin gwiwa. Ka yi tunanin samfurin yanayi na zamani inda kashi 99.9% na barbashi ke gudanar da ƙuntataccen, lissafin da ba Turing-cikakke ba don inganci, amma tsarin sarrafawa na musamman na "barbashi masu sarrafawa" za a iya sake tsara su cikin sauri zuwa atomata Turing-cikakke don gudanar da hadaddun tsare-tsare na daidaitawa akan-tafi, wanda aka yi wahayi ta hanyar iyawar daidaitawa da ake gani a cikin samfuran AI na zamani [15]. Mataki na gaba shine gina masu tarawa da kayan aikin tabbatarwa na hukuma waɗanda za su iya bincika lambobin hanyar barbashi (kamar manyan lambobin SPH ko kwayoyin halitta) da kuma tabbatar da matsayinsu akan bakan ƙarfin lissafi, tabbatar da cewa suna da ikon da suke buƙata kawai—kuma ba fiye da haka ba.