I-Karnaugh Maps: Qedela Umhlahlandlela

Ithebula elibonisa inani ngalinye lesamba esisodwa noma eziningi zengxenye eyodwa noma eziningi yiThebula Leqiniso. Kuyisixazululo sokuthi abafundi benze izibalo zesiphakamiso ezinengqondo noma nge-Boolean Algebra. Nakuba, kukhona enye indlela elingana namathebula eqiniso, kodwa yenza umsebenzi ube lula, obizwa ngokuthi amamephu karnaugh.

Yini i-Karnaugh Maps?

Kuwuhlelo oluvame ukusetshenziswa ukuze kuncishiswe futhi kufiphazwe uhlelo kanye nesikhathi sokusebenza kwezibalo ze-Boolean, ukwenza i-schema semodeli esiholela ekwenzeni imisebenzi emikhulu ngesisho esisodwa se-Boolean.

Ifana kakhulu neThebula Leqiniso, ibala amanani angaboniswa eziguquguqukayo ezihlukahlukene ezinokufakwayo futhi inikeze umphumela kokuphumayo. Ibizwa nangokuthi "imephu-k”, futhi kuchazwa njengochungechunge lwamabhokisi lapho ngalinye lawo linikezwa inombolo kanambambili, ngokuphathelene namanani atholakala kokokufakayo.

Inani lamabhokisi noma amaseli atholakala ku- I-Karnaugh Maps ifana nengqikithi yokwakheka kwamanani akhona kokokufakayo, njengoba kusebenza kuThebula Leqiniso, isethi yamakholomu isetshenziswa, isibonelo, kumephu enamanani amathathu, bese kuthi lapho okubili kukhushulwa. kokuthathu umphumela uyisishiyagalombili (2³=8).

En los I-Karnaugh Maps Amanani kufanele abekwe ngendlela yokuthi amakholomu namabhokisi avundlile ahlale ehlukile yenani elilodwa, ngale ndlela lincishiswe libe ngendlela elula yamanani ayisithupha.

Izici ze-Karnaugh Maps

Kuyindlela enezinhlobonhlobo ze-prototypes ezinikeza ububanzi bokuqukethwe nezinjongo:

• Enye yezindlela ezivame ukusetshenziswa ukunciphisa izibalo ze-algebraic eziphusile.
• Igama elinikezwe yona lithi "Itafula laseKarnaugh"noma"Umdwebo we-Veitch".
• Yaziwa ngendlela eyenziwe lula ngegama layo ngokuthi "I-K-Map noma i-KV-Map".
• Isazi sefiziksi uMaurice Karnaugh kanye nesazi sezibalo esasingaphansi kweBell Laboratories, nguyena ongumdali ngonyaka ka-1950.
• Isebenza ukwenza lula umphumela wezibalo.
• Kungumphumela wesamba noma inyunyana yeminye imiphumela.
• Kuwukwakhiwa kweqembu lama-rectangles.
• Isekelwe ekusebenzeni okuzenzakalelayo.
• Ibhokisi ngalinye lenza umugqa wethebula leqiniso.
• Kuleli thebula inani leqiniso lesimiso esihleliwe libekiwe.
• Ngokuya ngamavelu abanawo kuthebula leqiniso, amanani amayunithi abo angabekwa.
• Kuyithebula elibonisa inhlanganisela yamanani eminye imisebenzi ye- “N"amagugu.
• Yakhiwe ngamabili aphakanyiselwe ku-“N” imigqa (2^N).
• Lapho izikwele ezimbili zihlanganiswa futhi inani elilodwa likhanselwa, lapho izikwele ezine zihlanganiswa amanani amabili ayakhanselwa, ngale ndlela inqubo ilandelwa.
• Ebhokisini ngalinye inani libekwe, okungaba kuphela “0"Noma"1".
• Kuye ngenani elinikezwe umsebenzi ngamunye wekholomu. Isetshenziswa ize ifinyelele amanani ayisithupha.
• Kungenziwa ngemisebenzi enobuncane bezibalo ezimbili zebanga lemiphumela.
• Kungukukhetha ukuthi kutholwe amanani ahlukene, noma ngabe afana.
• Lapho kwenziwa umsebenzi inyunyana yamanani yenziwa, ngendlela efanayo amanani ahlanganisiwe ayasuswa.
• Amabhokisi akhululekile asetshenziswa ngendlela yokuthi, phakathi kwamabhokisi, kungakhathaliseki ukuthi yikuphi isikhundla, anokulinganisa okunengqondo.
• Kulawa mamephu "K", kunamagama athile ahambisanayo, acaciswe njengamapheya awo, anomehluko kokuguquguqukayo.
• Iqoqo ngalinye linquma ukuvezwa komphumela, futhi igama eliphethayo kufanele libe “OR"(yini isamba) wawo wonke amanani omphumela.
• Uma izikwele kumephu ye-K zihlobene, inani lama-minterms liyahlanganiswa, okuholela emandleni enombolo “2".
• Kunconyelwe imisebenzi enamanani aphezulu ayisithupha.

• Lapho amabhokisi etholakala inani elikhulu "1” kuhlangene, ukunqanyulwa kuhlala kunamanani amabili, lapho ayisishiyagalombili ehlanganiswa “1” amanani amathathu kumele asuswe ukuze afike ngesikhathi esinenani elilodwa.
• Imisebenzi ivezwe ngendlela yecanonical.
• Ngale mephu ungakha isifunda sedijithali, esilungele ukusebenza kusuka ku-algebra kuya kugesi.
• Inezinhlobonhlobo eziningi zezinyunyana zama-minterms
  emabalazweni.
• Imephu izoncika enanini lamanani atholwe ekuqaleni komsebenzi.

Injani Indlela Yokwenza i-K-Map

Eshadini le-matrix ungaba nezinqubo ezihlukene ezinikeza impendulo elindelekile, kulokho okulandelayo indlela yokusebenza yale mephu izoboniswa.

Isinyathelo sokuqala

• Okuguquguqukayo okuthathu kufanele kubekwe etafuleni elinengqondo, eliqokwe ngezinhlamvu “ABC".
• Bese usebenzisa i-logic, ezonakekela ukwenza inqubo ukuze uthole umphumela "Y” okudingekayo.
• Umphumela ulungile ngokulandelana kwawo. Inikeza izindleko eziphakeme zokwenziwa kwayo.
• Ngalolu hlobo lwamatafula e-Karnaugh, ukwenziwa lula kuyenziwa futhi indlela yokubeka okuguquguqukayo etafuleni iyathuthukiswa, kutholwe “1” womsebenzi othi “Y” endaweni ehambisanayo.

Isinyathelo sesibili

• Lapha imigqa yamalungu afanayo inikezwa incazelo.
• Njengesampula, umugqa oyisicaba lapho okuguquguqukayo kwabelwe khona okuthi “AB” kunikezwa, futhi kukholamu inani elithi “C”.
• Amanani kufanele anyuswe, lapha okuyi-null kufanele kukhonjiswe ngomugqa engxenyeni engenhla yokuguquguquka noma uphawu lokucaphuna luphinde lusetshenziswe.

Isinyathelo sesithathu

• Amanani abekwe kumephu “ABC” ngokulandelana nenani elinenani eliphakeme lezindleko “Y".
• Inani ngalinye kufanele libe khona, ezindaweni zalo.
• "1” esikhundleni A'BC'; “1” esikhundleni esithi ABC′ kanye “1” endaweni ethi A'BC.
• Lezi zinhlobonhlobo zibizwa ngokuthi ama-minterms.

Isinyathelo Sesine

• Siyaqhubeka nokwenza ukwehlisa nge-k-map.
• Izengezo ezilandelanayo ezinengqondo ziseduze, zisusa amanani engeziwe.
• Ezimweni ezithile, isamba samagama afanele abizwa ngokuthi ama-minterms “Z” ikhipha inani elithi “A”, ngoba yethulwa ngokungeziwe.
• Kulandelwa isenzo se-Boolean logic.
• Ngenqubo elula, kufanele uchaze ukuthi inani kufanele lenziwe ize ngesikhathi sokufinyezwa.
• Ukuze uqedele isamba "Z+X” kuwumphumela wokuhlobana okwenziwe lula kwamanani kuthebula lamanani.

Iyini Inzuzo Ye-Karnaugh Maps?

Ngonyaka ka-1953, owasungula indlela yokusebenza noma indlela yokunciphisa ukusebenza kwakungunjiniyela uMaurice Karnaugh, esebenzisa amashadi noma amatafula, enikeza ezinye izindlela, lapho kunikezwa isibonelo kokulandelayo.

Kumathebula e-Karnaugh kuvunyelwe ukukhetha indlela yokuguqula ithebula leqiniso lemisebenzi ye-Boolean, ngendlela ye-SOP eyenziwe lula. Ngakho-ke, inikeza izinketho zemithetho elula yokwenza ukunciphisa nokugcizelela ekunikezeni ubulula bokwenza indlela.

Ukunikeza ithuba lokuthi indlela ilula futhi akuthathi isikhathi esiningi ukuyenza, okubonisa ukuthi iyasebenza uma iqhathaniswa nezinye izindlela ezinengqondo.

Imithetho yemephu yaseKarnaugh

Ukwakhiwa kwalesi grafu kumele kulawulwe yimithetho njengoba kuchaziwe, ngenxa yalesi sizathu uhlu lwemiyalo okufanele yenziwe ngale njongo ibonisiwe.

Into yokuqala okufanele yenziwe ukuqinisekisa ukuthi ukuphela kwendlela yokugcwalisa amaqembu emigomo ngokuthatha inani elithi “1".

Lawa maqembu angenziwa kuphela abe flat futhi abe nomugqa. Kumele kuqashelwe ukuthi wonke amaqembu kufanele abe ngu-2ⁿ amanani, uzama ukuthi iqembu ngalinye lakhiwe ngezinto eziguquguqukayo (1,2,4, 8,…,2ⁿ⁾ inani lamadijithi ukusuka kwelinye kuye kwelinye.

Ukuze ithebula noma imephu igcine ngokuncishiswa okuhle, ukuqoqa kufanele kuphathwe ngobunkimbinkimbi.

Kufanele uhlale uqaphele futhi ungashiyi eceleni okuguquguqukayo "1”. futhi ivumela ukuqoqwa kwamaqembu "1".

Amaqembu angahlanganiswa nezikwele ezitholakala ekugcineni kwemephu. Inani elincane lamaqembu angahlaziywa kufanele lihlaziywe, konke ngaphansi kwemithetho eshiwo ngenhla.

Yiziphi Izinyathelo Zokuncishiswa Kwemephu yaseKarnaugh?

Ukwenza izinyathelo zalokhu kwehliswa kuthebula le-K, indlela yokusebenza kufanele ilandelwe ngamavelu ahlukene ekuqaleni futhi kunconyiwe ukusetshenziswa kwamanani amabili kuya kwamahlanu. Kungakho lokhu okulandelayo kuchaza umzila wonke okufanele wenziwe ukuze kwehle ngendlela efanele.

Ungawakha kanjani i-Karnaugh Maps?

Okulandelayo, khumbula lokho okuzoboniswa:

• Kufanele babe namafreyimu amaningi wemodi 2ⁿ, ukuba "n” inani lenani.
• Isampula izoshintsha ibe ngu-2 wethebula, izovela kumafreyimu amane, esimweni samanani angu-3 uhlaka luhambisana nesishiyagalombili futhi uma kuyivelu engu-4 khona-ke amafreyimu azoba yishumi nesithupha.
• Ekugcineni ungabona ukuthi imephu izobukeka kanjani mayelana nenani lamanani ekuqaleni.

Ungakwenza kanjani ingabe Amanani Kokufaka Ahlanganisiwe?

Okudingekayo ukuthi ekugcineni kwemephu izigaba ziku-0 no-1 kuye ngokwakheka kwamanani atholakala ekuqaleni.

Esibonelweni semephu enamanani angu-3.

• Amanani ka-A kanye no-B kufanele anamathiselwe phezulu kwendawo engenhla, abe emigqeni eqondile.
• Kule migqa eqondile yemephu kunezingxube okungenzeka zalawa manani angu-2: 00, 01, 11 noma 10.
• Engxenyeni evundlile, kufanele ubeke amanani asele.
• Amanani ka-C, kanye nokungenzeka asho emugqeni ngamunye ukuthi angu-0 noma u-1.
• Kufanele uhlale wazi ukuthi u-0 no-1 wevelu ngayinye zi-odiwe njengoba zikumamephu afinyelelwe.

• Kuyingxenye yenkambiso ukuthi uma ubudlelwano benziwa kwenye imephu, okufanele kuguquke inani lokuguquguquka ngakunye.

Gcwalisa Amanani Okukhiphayo

Amabalazwe e-Kamaugh ngemva kokudalwa kwawo agcwaliswa ngolwazi, okuguquguqukayo kokuphela kweqembu ngalinye lezinto eziguquguqukayo zokuqala.

Kunezinketho ezimbili kuphela, eyodwa ukuthi ithebula leqiniso liyatholakala kanti enye ukuthi incazelo enengqondo yemephu kagesi iyatholakala. Ngokuvamile, itafula leqiniso liyasetshenziswa.

Bese idluliselwa kokuqukethwe okunengqondo, kubalwa kudatha yethebula elidaliwe. Kuleli thebula kufanele ubeke «0» ebhokisini lapho ukwakheka kwamanani okugcina kwenziwa, okuguquguqukayo «0» kuleli thebula futhi «1» ebhokisini eliqukethe ukwakheka kwamanani ekugcineni «1»kulelo tafula.

Uma unenhlanganisela enengqondo, kufanele uqaphele ukwakheka okuhlukene kwamanani wemiphumela, lokhu kwenziwa ngokuphumayo ngomphumela kokuthi «1".

Ngabe iQembu loku-1 lenziwa kanjani?

Lawa manani kumele ahlanganiswe ngokulinganayo, amane kuye kwezine, ayisishiyagalombili kuye kwayisishiyagalombili, njalo njalo. Lapho amaqembu "1» emephini, kufanele wenze amaqoqo ahlukene «1»kwa (2ⁿ⁾, kuyadingeka ukuthi la maqembu abambe zonke1» Lapho kudingekile, akufanele kucatshangelwe ukuthi lawa magugu asengawamanye amaqembu.

Okubalulekile ukuthi la maqembu kumele ahambisane nemithetho, angeke ajoyine ngokudayagonali, abheke phezulu naphansi kuphela.

Ungabuthola Kanjani Ubudlelwano Obusha Obuncishisiwe?

Kutholwa inani leqembu ngalinye elithi «1«, Lona umphumela wesamba salokhu. Imiphumela kumele yengezwe.

Ukuze uthole inani lomsebenzi, iqembu elithi «1«, ngesikhathi esifanayo kufanele kuqinisekiswe ukuthi amanani ashintsha ngenani.

Uma kukhona noma yikuphi ukuguqulwa kwevelu, kusuka ku-(0 kuye ku-1) noma ukusuka ku-(1 kuye ku-0), lelo nani liba ize.

Yiziphi Izizathu Izinguquko ezikhona Isusiwe?

Okwamanje lapho kutholakala okuguquguqukayo okuguquliwe inani lalo, emaqenjini okuthi "1", kwenzeka ukuthi lokhu kuguquguquka kuphindaphindeka izikhathi ezimbalwa, eyodwa ngakolunye uhlangothi futhi enye ngakolunye uhlangothi. Futhi okudingekayo ukunciphisa umsebenzi.

Atholakala Kanjani Amamephu e-Karnaugh?

Ilele ekucatshangweni kwezinhlangothi ezimbili zomsebenzi ozoncishiswa. Uma lokhu kwethulwa njengethebula leqiniso, kulokhu imephu ye-K izovezwa ngendlela efana nethi “2D".

Ngoba itafula lokuqala line "n” amanani futhi ino-2ⁿ amakholomu, imephu ka-K yakhiwe ngamaseli ngokulingana no-2ⁿ. Ukucaciswa kwemephu engu-K kufaka igridi ngayinye ngenombolo kanambambili, ngale ndlela ibhokisi ngalinye elihambisanayo liphinde linikezwe idijithi eyodwa.

Emfanekisweni odlule ungabona isampula yekhodi kanambambili lapho kwethulwa umsebenzi wamanani angu-4. Okuguquguqukayo okunengqondo okubonisiwe (A B C D) ngokulandelanayo ingxenye eyodwa yekhodi kanambambili.

Uma usetshenziswa, awudingi ukuchaza wonke amabhokisi; kwanele ukuhumusha unhlokweni oqondile novundlile ngokulandelana, njengoba kukhonjisiwe.

Uma umbhalo wekhodi kanambambili usuvele usunguliwe, ibhokisi ngalinye linikeza “1” uma kusebenza ngokwamagama e-canonical afanele omsebenzi, futhi uma kungenjalo “0”. Uma lenziwa njengethebula leqiniso, kukhona inketho yokusebenzisa isisho se-canonical ukwethula umsebenzi.

Into elungile ukukhetha leyo equkethe amanani amancane ezinombolo. Kulokhu, kudingekile kuphela ukukhetha indlela ehlanganisa inani elincane lamanani. Udinga nje ukwazi ukuthi zingakanani izinombolo ekuchazeni okunengqondo (zihambisana namakholomu ano"1").

Uma inani lokuhumusha lidlula ifomula futhi kutholwa izinombolo ezingaphansi kwenani elingasizi, khona-ke indlela ye-canonical ye-DNF iyathathwa. Uma ifomu le-CNF lingakhethiwe.

Lapho imephu ye-K isivele yenziwe, ukwenziwa lula kwezinombolo kuqala, uma kungenzeka. Ukwenza amaqembu amabhokisi ahambisana nokuguquguquka "1".

Okulandelayo, i-algorithm ye-DNF canonical decrement algorithm izochazwa kafushane.

Ukwehliswa Komsebenzi Onengqondo Nge-DNF Expression

Lapho usukwenzile amamephu karnaugh eveza imisebenzi enengqondo ku-DNF, inqubo imi kanje.

Into yokuqala okufanele yenziwe ukuqoqwa kwamaqembu ezigebengu aneziguquko “1” kucatshangelwa imithetho:

Amaqembu kufanele akhiwe kuphela ngevelu “1".

Inani lamaseli anenani "1" aseqenjini kufanele libe amandla enombolo "2" Ini (1, 2, 4, 8, 16, …, n).

Ekudalweni kwamaqembu, kufanele kucatshangelwe ukuthi amatafula ayi-toroidal, ngoba izindawo ezikude noma amaphuzu ahlangene: indawo eyeqile ngakwesokunene ihambisana nendawo eyedlulele ngakwesobunxele, ngokufanayo. indlela okwenzeka ngayo ohlangothini olungaphezulu nolungezansi. Njengoba ubona esithombeni esingezansi.

Amaseli anokushintshashintsha "1” kumelwe okungenani abe eqenjini elilodwa.

Okuguquguqukayo "1” ezisebhokisini zingaba ngamaqembu ahlukene.

Inani lamachibi kufanele libe lincane.

Nakuba amaqembu emakhudlwana, ukwehla kuzoba kukhulu, enanini lamagama, kanye nenani lamagama angokoqobo anetemu.

Amaqoqo angahluka ngosayizi.

Uma umsebenzi uthola incazelo enenani “x” lokho akunakuxazululeka. Amabhokisi aseduze kwayo anikezwe inani elithi “x”. Futhi lezi azidingi ukujoyina ichibi, nakuba zingasetshenziswa ukunweba amachibi asevele enziwe.

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