Karnaugh map - Wikipedia

Karnaugh map ; Karnaugh map ( ; KM or ; K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of ... Karnaughmap FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch ThisarticlemaybeexpandedwithtexttranslatedfromthecorrespondingarticleinGerman.(February2018)Click[show]forimportanttranslationinstructions. Viewamachine-translatedversionoftheGermanarticle. MachinetranslationlikeDeepLorGoogleTranslateisausefulstartingpointfortranslations,buttranslatorsmustreviseerrorsasnecessaryandconfirmthatthetranslationisaccurate,ratherthansimplycopy-pastingmachine-translatedtextintotheEnglishWikipedia. Consideraddingatopictothistemplate:therearealready9,165articlesinthemaincategory,andspecifying|topic=willaidincategorization. Donottranslatetextthatappearsunreliableorlow-quality.Ifpossible,verifythetextwithreferencesprovidedintheforeign-languagearticle. Youmustprovidecopyrightattributionintheeditsummaryaccompanyingyourtranslationbyprovidinganinterlanguagelinktothesourceofyourtranslation.AmodelattributioneditsummaryisContentinthiseditistranslatedfromtheexistingGermanWikipediaarticleat[[:de:Karnaugh-Veitch-Diagramm]];seeitshistoryforattribution. Youshouldalsoaddthetemplate{{Translated|de|Karnaugh-Veitch-Diagramm}}tothetalkpage. Formoreguidance,seeWikipedia:Translation. GraphicalmethodtosimplifyBooleanexpressions AnexampleKarnaughmap.ThisimageactuallyshowstwoKarnaughmaps:forthefunctionƒ,usingminterms(coloredrectangles)andforitscomplement,usingmaxterms(grayrectangles).Intheimage,E()signifiesasumofminterms,denotedinthearticleas ∑ m i {\displaystyle\summ_{i}} . TheKarnaughmap(KMorK-map)isamethodofsimplifyingBooleanalgebraexpressions.MauriceKarnaughintroduceditin1953[1][2]asarefinementofEdwardW.Veitch's1952Veitchchart,[3][4]whichwasarediscoveryofAllanMarquand's1881logicaldiagram[5]akaMarquanddiagram[4]butwithafocusnowsetonitsutilityforswitchingcircuits.[4]VeitchchartsarethereforealsoknownasMarquand–Veitchdiagrams,[4]andKarnaughmapsasKarnaugh–Veitchmaps(KVmaps). TheKarnaughmapreducestheneedforextensivecalculationsbytakingadvantageofhumans'pattern-recognitioncapability.[1]Italsopermitstherapididentificationandeliminationofpotentialraceconditions. TherequiredBooleanresultsaretransferredfromatruthtableontoatwo-dimensionalgridwhere,inKarnaughmaps,thecellsareorderedinGraycode,[6][4]andeachcellpositionrepresentsonecombinationofinputconditions.Cellsarealsoknownasminterms,whileeachcellvaluerepresentsthecorrespondingoutputvalueofthebooleanfunction.Optimalgroupsof1sor0sareidentified,whichrepresentthetermsofacanonicalformofthelogicintheoriginaltruthtable.[7]ThesetermscanbeusedtowriteaminimalBooleanexpressionrepresentingtherequiredlogic. Karnaughmapsareusedtosimplifyreal-worldlogicrequirementssothattheycanbeimplementedusingaminimumnumberoflogicgates.Asum-of-productsexpression(SOP)canalwaysbeimplementedusingANDgatesfeedingintoanORgate,andaproduct-of-sumsexpression(POS)leadstoORgatesfeedinganANDgate.ThePOSexpressiongivesacomplementofthefunction(ifFisthefunctionsoitscomplementwillbeF').[8]Karnaughmapscanalsobeusedtosimplifylogicexpressionsinsoftwaredesign.Booleanconditions,asusedforexampleinconditionalstatements,cangetverycomplicated,whichmakesthecodedifficulttoreadandtomaintain.Onceminimised,canonicalsum-of-productsandproduct-of-sumsexpressionscanbeimplementeddirectlyusingANDandORlogicoperators.[9] Contents 1Example 1.1Construction 1.2Grouping 1.3Solution 1.4Inverse 1.5Don'tcares 2Racehazards 2.1Elimination 2.22-variablemapexamples 3Relatedgraphicalmethods 4Seealso 5Notes 6References 7Furtherreading 8Externallinks Example[edit] KarnaughmapsareusedtofacilitatethesimplificationofBooleanalgebrafunctions.Forexample,considertheBooleanfunctiondescribedbythefollowingtruthtable. Truthtableofafunction   A B C D f ( A , B , C , D ) {\displaystylef(A,B,C,D)} 0 0 0 0 0 0 1 0 0 0 1 0 2 0 0 1 0 0 3 0 0 1 1 0 4 0 1 0 0 0 5 0 1 0 1 0 6 0 1 1 0 1 7 0 1 1 1 0 8 1 0 0 0 1 9 1 0 0 1 1 10 1 0 1 0 1 11 1 0 1 1 1 12 1 1 0 0 1 13 1 1 0 1 1 14 1 1 1 0 1 15 1 1 1 1 0 FollowingaretwodifferentnotationsdescribingthesamefunctioninunsimplifiedBooleanalgebra,usingtheBooleanvariablesA,B,C,Dandtheirinverses. f ( A , B , C , D ) = ∑ m i , i ∈ { 6 , 8 , 9 , 10 , 11 , 12 , 13 , 14 } {\displaystylef(A,B,C,D)=\sum_{}m_{i},i\in\{6,8,9,10,11,12,13,14\}} where m i {\displaystylem_{i}} arethemintermstomap(i.e.,rowsthathaveoutput1inthetruthtable). f ( A , B , C , D ) = ∏ M i , i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 7 , 15 } {\displaystylef(A,B,C,D)=\prod_{}M_{i},i\in\{0,1,2,3,4,5,7,15\}} where M i {\displaystyleM_{i}} arethemaxtermstomap(i.e.,rowsthathaveoutput0inthetruthtable). K-mapdrawnonatorus,andinaplane.Thedot-markedcellsareadjacent. K-mapconstruction.Insteadoftheoutputvalues(therightmostvaluesinthetruthtable),thisdiagramshowsadecimalrepresentationoftheinputABCD(theleftmostvaluesinthetruthtable),thereforeitisnotaKarnaughmap. Inthreedimensions,onecanbendarectangleintoatorus. Construction[edit] Intheexampleabove,thefourinputvariablescanbecombinedin16differentways,sothetruthtablehas16rows,andtheKarnaughmaphas16positions.TheKarnaughmapisthereforearrangedina4 × 4grid. Therowandcolumnindices(shownacrossthetopanddowntheleftsideoftheKarnaughmap)areorderedinGraycoderatherthanbinarynumericalorder.Graycodeensuresthatonlyonevariablechangesbetweeneachpairofadjacentcells.EachcellofthecompletedKarnaughmapcontainsabinarydigitrepresentingthefunction'soutputforthatcombinationofinputs. Grouping[edit] AftertheKarnaughmaphasbeenconstructed,itisusedtofindoneofthesimplestpossibleforms—acanonicalform—fortheinformationinthetruthtable.Adjacent1sintheKarnaughmaprepresentopportunitiestosimplifytheexpression.Theminterms('minimalterms')forthefinalexpressionarefoundbyencirclinggroupsof1sinthemap.Mintermgroupsmustberectangularandmusthaveanareathatisapoweroftwo(i.e.,1, 2, 4, 8...).Mintermrectanglesshouldbeaslargeaspossiblewithoutcontainingany0s.Groupsmayoverlapinordertomakeeachonelarger.Theoptimalgroupingsintheexamplebelowaremarkedbythegreen,redandbluelines,andtheredandgreengroupsoverlap.Theredgroupisa2 × 2square,thegreengroupisa4 × 1rectangle,andtheoverlapareaisindicatedinbrown. Thecellsareoftendenotedbyashorthandwhichdescribesthelogicalvalueoftheinputsthatthecellcovers.Forexample,ADwouldmeanacellwhichcoversthe2x2areawhereAandDaretrue,i.e.thecellsnumbered13,9,15,11inthediagramabove.Ontheotherhand,ADwouldmeanthecellswhereAistrueandDisfalse(thatis,Distrue). Thegridistoroidallyconnected,whichmeansthatrectangulargroupscanwrapacrosstheedges(seepicture).Cellsontheextremerightareactually'adjacent'tothoseonthefarleft,inthesensethatthecorrespondinginputvaluesonlydifferbyonebit;similarly,soarethoseattheverytopandthoseatthebottom.Therefore,ADcanbeavalidterm—itincludescells12and8atthetop,andwrapstothebottomtoincludecells10and14—asisBD,whichincludesthefourcorners. Solution[edit] DiagramshowingtwoK-maps.TheK-mapforthefunctionf(A,B,C,D)isshownascoloredrectangleswhichcorrespondtominterms.Thebrownregionisanoverlapofthered2×2squareandthegreen4×1rectangle.TheK-mapfortheinverseoffisshownasgrayrectangles,whichcorrespondtomaxterms. OncetheKarnaughmaphasbeenconstructedandtheadjacent1slinkedbyrectangularandsquareboxes,thealgebraicmintermscanbefoundbyexaminingwhichvariablesstaythesamewithineachbox. Fortheredgrouping: Aisthesameandisequalto1throughoutthebox,thereforeitshouldbeincludedinthealgebraicrepresentationoftheredminterm. Bdoesnotmaintainthesamestate(itshiftsfrom1to0),andshouldthereforebeexcluded. Cdoesnotchange.Itisalways0,soitscomplement,NOT-C,shouldbeincluded.Thus,Cshouldbeincluded. Dchanges,soitisexcluded. ThusthefirstmintermintheBooleansum-of-productsexpressionisAC. Forthegreengrouping,AandBmaintainthesamestate,whileCandDchange.Bis0andhastobenegatedbeforeitcanbeincluded.ThesecondtermisthereforeAB.Notethatitisacceptablethatthegreengroupingoverlapswiththeredone. Inthesameway,thebluegroupinggivesthetermBCD. Thesolutionsofeachgroupingarecombined:thenormalformofthecircuitis A C ¯ + A B ¯ + B C D ¯ {\displaystyleA{\overline{C}}+A{\overline{B}}+BC{\overline{D}}} . ThustheKarnaughmaphasguidedasimplificationof f ( A , B , C , D ) = A ¯ B C D ¯ + A B ¯ C ¯ D ¯ + A B ¯ C ¯ D + A B ¯ C D ¯ + A B ¯ C D + A B C ¯ D ¯ + A B C ¯ D + A B C D ¯ = A C ¯ + A B ¯ + B C D ¯ {\displaystyle{\begin{aligned}f(A,B,C,D)={}&{\overline{A}}BC{\overline{D}}+A{\overline{B}}\,{\overline{C}}\,{\overline{D}}+A{\overline{B}}\,{\overline{C}}D+A{\overline{B}}C{\overline{D}}+{}\\&A{\overline{B}}CD+AB{\overline{C}}\,{\overline{D}}+AB{\overline{C}}D+ABC{\overline{D}}\\={}&A{\overline{C}}+A{\overline{B}}+BC{\overline{D}}\end{aligned}}} ItwouldalsohavebeenpossibletoderivethissimplificationbycarefullyapplyingtheaxiomsofBooleanalgebra,butthetimeittakestodothatgrowsexponentiallywiththenumberofterms. Inverse[edit] Theinverseofafunctionissolvedinthesamewaybygroupingthe0sinstead.[nb1] Thethreetermstocovertheinverseareallshownwithgreyboxeswithdifferentcoloredborders: brown:AB gold:AC blue:BCD Thisyieldstheinverse: f ( A , B , C , D ) ¯ = A ¯ B ¯ + A ¯ C ¯ + B C D {\displaystyle{\overline{f(A,B,C,D)}}={\overline{A}}\,{\overline{B}}+{\overline{A}}\,{\overline{C}}+BCD} ThroughtheuseofDeMorgan'slaws,theproductofsumscanbedetermined: f ( A , B , C , D ) = f ( A , B , C , D ) ¯ ¯ = A ¯ B ¯ + A ¯ C ¯ + B C D ¯ = ( A ¯ B ¯ ¯ ) ( A ¯ C ¯ ¯ ) ( B C D ¯ ) = ( A + B ) ( A + C ) ( B ¯ + C ¯ + D ¯ ) {\displaystyle{\begin{aligned}f(A,B,C,D)&={\overline{\overline{f(A,B,C,D)}}}\\&={\overline{{\overline{A}}\,{\overline{B}}+{\overline{A}}\,{\overline{C}}+BCD}}\\&=\left({\overline{{\overline{A}}\,{\overline{B}}}}\right)\left({\overline{{\overline{A}}\,{\overline{C}}}}\right)\left({\overline{BCD}}\right)\\&=\left(A+B\right)\left(A+C\right)\left({\overline{B}}+{\overline{C}}+{\overline{D}}\right)\end{aligned}}} Don'tcares[edit] Thevalueof f ( A , B , C , D ) {\displaystylef(A,B,C,D)} forABCD=1111isreplacedbya"don'tcare".Thisremovesthegreentermcompletelyandallowstheredtermtobelarger.Italsoallowsblueinversetermtoshiftandbecomelarger Karnaughmapsalsoalloweasierminimizationsoffunctionswhosetruthtablesinclude"don'tcare"conditions.A"don'tcare"conditionisacombinationofinputsforwhichthedesignerdoesn'tcarewhattheoutputis.Therefore,"don'tcare"conditionscaneitherbeincludedinorexcludedfromanyrectangulargroup,whichevermakesitlarger.TheyareusuallyindicatedonthemapwithadashorX. Theexampleontherightisthesameastheexampleabovebutwiththevalueoff(1,1,1,1)replacedbya"don'tcare".Thisallowstheredtermtoexpandallthewaydownand,thus,removesthegreentermcompletely. Thisyieldsthenewminimumequation: f ( A , B , C , D ) = A + B C D ¯ {\displaystylef(A,B,C,D)=A+BC{\overline{D}}} NotethatthefirsttermisjustA,notAC.Inthiscase,thedon'tcarehasdroppedaterm(thegreenrectangle);simplifiedanother(theredone);andremovedtheracehazard(removingtheyellowtermasshowninthefollowingsectiononracehazards). Theinversecaseissimplifiedasfollows: f ( A , B , C , D ) ¯ = A ¯ B ¯ + A ¯ C ¯ + A ¯ D {\displaystyle{\overline{f(A,B,C,D)}}={\overline{A}}\,{\overline{B}}+{\overline{A}}\,{\overline{C}}+{\overline{A}}D} ThroughtheuseofDeMorgan'slaws,theproductofsumscanbedetermined: f ( A , B , C , D ) = f ( A , B , C , D ) ¯ ¯ = A ¯ B ¯ + A ¯ C ¯ + A ¯ D ¯ = ( A ¯ B ¯ ¯ ) ( A ¯ C ¯ ¯ ) ( A ¯ D ¯ ) = ( A + B ) ( A + C ) ( A + D ¯ ) {\displaystyle{\begin{aligned}f(A,B,C,D)&={\overline{\overline{f(A,B,C,D)}}}\\&={\overline{{\overline{A}}\,{\overline{B}}+{\overline{A}}\,{\overline{C}}+{\overline{A}}\,D}}\\&=\left({\overline{{\overline{A}}\,{\overline{B}}}}\right)\left({\overline{{\overline{A}}\,{\overline{C}}}}\right)\left({\overline{{\overline{A}}\,D}}\right)\\&=\left(A+B\right)\left(A+C\right)\left(A+{\overline{D}}\right)\end{aligned}}} Racehazards[edit] Elimination[edit] Karnaughmapsareusefulfordetectingandeliminatingraceconditions.RacehazardsareveryeasytospotusingaKarnaughmap,becausearaceconditionmayexistwhenmovingbetweenanypairofadjacent,butdisjoint,regionscircumscribedonthemap.However,becauseofthenatureofGraycoding,adjacenthasaspecialdefinitionexplainedabove–we'reinfactmovingonatorus,ratherthanarectangle,wrappingaroundthetop,bottom,andthesides. Intheexampleabove,apotentialraceconditionexistswhenCis1andDis0,Ais1,andBchangesfrom1to0(movingfromthebluestatetothegreenstate).Forthiscase,theoutputisdefinedtoremainunchangedat1,butbecausethistransitionisnotcoveredbyaspecifictermintheequation,apotentialforaglitch(amomentarytransitionoftheoutputto0)exists. Thereisasecondpotentialglitchinthesameexamplethatismoredifficulttospot:whenDis0andAandBareboth1,withCchangingfrom1to0(movingfromthebluestatetotheredstate).Inthiscasetheglitchwrapsaroundfromthetopofthemaptothebottom. Racehazardsarepresentinthisdiagram. Abovediagramwithconsensustermsaddedtoavoidracehazards. Whetherglitcheswillactuallyoccurdependsonthephysicalnatureoftheimplementation,andwhetherweneedtoworryaboutitdependsontheapplication.Inclockedlogic,itisenoughthatthelogicsettlesonthedesiredvalueintimetomeetthetimingdeadline.Inourexample,wearenotconsideringclockedlogic. Inourcase,anadditionaltermof A D ¯ {\displaystyleA{\overline{D}}} wouldeliminatethepotentialracehazard,bridgingbetweenthegreenandblueoutputstatesorblueandredoutputstates:thisisshownastheyellowregion(whichwrapsaroundfromthebottomtothetopoftherighthalf)intheadjacentdiagram. Thetermisredundantintermsofthestaticlogicofthesystem,butsuchredundant,orconsensusterms,areoftenneededtoassurerace-freedynamicperformance. Similarly,anadditionaltermof A ¯ D {\displaystyle{\overline{A}}D} mustbeaddedtotheinversetoeliminateanotherpotentialracehazard.ApplyingDeMorgan'slawscreatesanotherproductofsumsexpressionforf,butwithanewfactorof ( A + D ¯ ) {\displaystyle\left(A+{\overline{D}}\right)} . 2-variablemapexamples[edit] Thefollowingareallthepossible2-variable,2 × 2Karnaughmaps.Listedwitheachisthemintermsasafunctionof ∑ m ( ) {\textstyle\summ()} andtheracehazardfree(seeprevioussection)minimumequation.Amintermisdefinedasanexpressionthatgivesthemostminimalformofexpressionofthemappedvariables.Allpossiblehorizontalandverticalinterconnectedblockscanbeformed.Theseblocksmustbeofthesizeofthepowersof2(1,2,4,8,16,32,...).Theseexpressionscreateaminimallogicalmappingoftheminimallogicvariableexpressionsforthebinaryexpressionstobemapped.Herearealltheblockswithonefield. Ablockcanbecontinuedacrossthebottom,top,left,orrightofthechart.Thatcanevenwrapbeyondtheedgeofthechartforvariableminimization.Thisisbecauseeachlogicvariablecorrespondstoeachverticalcolumnandhorizontalrow.Avisualizationofthek-mapcanbeconsideredcylindrical.Thefieldsatedgesontheleftandrightareadjacent,andthetopandbottomareadjacent.K-Mapsforfourvariablesmustbedepictedasadonutortorusshape.Thefourcornersofthesquaredrawnbythek-mapareadjacent.Stillmorecomplexmapsareneededfor5variablesandmore. Σm(0);K=0 Σm(1);K=A′B′ Σm(2);K=AB′ Σm(3);K=A′B Σm(4);K=AB Σm(1,2);K=B′ Σm(1,3);K=A′ Σm(1,4);K=A′B′+AB Σm(2,3);K=AB′+A′B Σm(2,4);K=A Σm(3,4);K=B Σm(1,2,3);K=A'+B′ Σm(1,2,4);K=A+B′ Σm(1,3,4);K=A′+B Σm(2,3,4);K=A+B Σm(1,2,3,4);K=1 Relatedgraphicalmethods[edit] Furtherinformation:Logicoptimization§ Graphicalmethods Relatedgraphicalminimizationmethodsinclude: Marquanddiagram(1881)byAllanMarquand(1853–1924)[5][4] Veitchchart(1952)byEdwardW.Veitch(1924–2013)[3][4] Mahoneymap(M-map,designationnumbers,1963)byMatthewV.Mahoney(areflection-symmetricalextensionofKarnaughmapsforlargernumbersofinputs) ReducedKarnaughmap(RKM)techniques(from1969)likeinfrequentvariables,map-enteredvariables(MEV),variable-enteredmap(VEM)orvariable-enteredKarnaughmap(VEKM)byG.W.Schultz,ThomasE.Osborne,ChristopherR.Clare,J.RobertBurgoon,LarryL.Dornhoff,WilliamI.Fletcher,AliM.Rushdiandothers(severalsuccessiveKarnaughmapextensionsbasedonvariableinputsforalargernumbersofinputs) Minterm-ringmap(MRM,1990)byThomasR.McCalla(athree-dimensionalextensionofKarnaughmapsforlargernumbersofinputs) Seealso[edit] Algebraicnormalform(ANF) Binarydecisiondiagram(BDD),adatastructurethatisacompressedrepresentationofaBooleanfunction Espressoheuristiclogicminimizer ListofBooleanalgebratopics Logicoptimization Punnettsquare(1905),asimilardiagraminbiology Quine–McCluskeyalgorithm Reed–Mullerexpansion Venndiagram(1880) Zhegalkinpolynomial Notes[edit] ^Thisshouldnotbeconfusedwiththenegationoftheresultofthepreviouslyfoundfunction. References[edit] ^abKarnaugh,Maurice(November1953)[1953-04-23,1953-03-17]."TheMapMethodforSynthesisofCombinationalLogicCircuits"(PDF).TransactionsoftheAmericanInstituteofElectricalEngineers,PartI:CommunicationandElectronics.72(5):593–599.doi:10.1109/TCE.1953.6371932.Paper53-217.Archivedfromtheoriginal(PDF)on2017-04-16.Retrieved2017-04-16.(NB.AlsocontainsashortreviewbySamuelH.Caldwell.) ^Curtis,HerbertAllen(1962).Anewapproachtothedesignofswitchingcircuits.TheBellLaboratoriesSeries(1 ed.).Princeton,NewJersey,USA:D.vanNostrandCompany,Inc.ISBN 0-44201794-4.OCLC 1036797958.S2CID 57068910.ISBN 978-0-44201794-1.ark:/13960/t56d6st0q.(viii+635pages)(NB.ThisbookwasreprintedbyChinJihin1969.) ^abVeitch,EdwardWestbrook(1952-05-03)[1952-05-02]."AChartMethodforSimplifyingTruthFunctions".Transactionsofthe1952ACMAnnualMeeting.ACMAnnualConference/AnnualMeeting:Proceedingsofthe1952ACMAnnualMeeting(Pittsburgh,Pennsylvania,USA).NewYork,USA:AssociationforComputingMachinery(ACM):127–133.doi:10.1145/609784.609801. ^abcdefgBrown,FrankMarkham(2012)[2003,1990].BooleanReasoning-TheLogicofBooleanEquations(reissueof2nd ed.).Mineola,NewYork:DoverPublications,Inc.ISBN 978-0-486-42785-0.[1] ^abMarquand,Allan(1881)."XXXIII:OnLogicalDiagramsfornterms".TheLondon,Edinburgh,andDublinPhilosophicalMagazineandJournalofScience.5.12(75):266–270.doi:10.1080/14786448108627104.Retrieved2017-05-15.(NB.Quitemanysecondarysourceserroneouslycitethisworkas"Alogicaldiagramfornterms"or"Onalogicaldiagramfornterms".) ^Wakerly,JohnF.(1994).DigitalDesign:Principles&Practices.NewJersey,USA:PrenticeHall.pp. 48–49,222.ISBN 0-13-211459-3.(NB.ThetwopagesectionstakentogethersaythatK-mapsarelabeledwithGraycode.ThefirstsectionsaysthattheyarelabeledwithacodethatchangesonlyonebitbetweenentriesandthesecondsectionsaysthatsuchacodeiscalledGraycode.) ^Belton,David(April1998)."KarnaughMaps–RulesofSimplification".Archivedfromtheoriginalon2017-04-18.Retrieved2009-05-30. ^Dodge,NathanB.(September2015)."SimplifyingLogicCircuitswithKarnaughMaps"(PDF).TheUniversityofTexasatDallas,ErikJonssonSchoolofEngineeringandComputerScience.Archived(PDF)fromtheoriginalon2017-04-18.Retrieved2017-04-18. ^Cook,Aaron."UsingKarnaughMapstoSimplifyCode".QuantumRarity.Archivedfromtheoriginalon2017-04-18.Retrieved2012-10-07. Furtherreading[edit] Katz,RandyHoward(1998)[1994].ContemporaryLogicDesign.TheBenjamin/CummingsPublishingCompany.pp. 70–85.doi:10.1016/0026-2692(95)90052-7.ISBN 0-8053-2703-7. Vingron,ShimonPeter(2004)[2003-11-05]."KarnaughMaps".SwitchingTheory:InsightThroughPredicateLogic.Berlin,Heidelberg,NewYork:Springer-Verlag.pp. 57–76.ISBN 3-540-40343-4. Wickes,WilliamE.(1968)."3.5.VeitchDiagrams".LogicDesignwithIntegratedCircuits.NewYork,USA:JohnWiley&Sons.pp. 36–49.LCCN 68-21185.p. 36:[…]arefinementoftheVenndiagraminthatcirclesarereplacedbysquaresandarrangedinaformofmatrix.TheVeitchdiagramlabelsthesquareswiththeminterms.Karnaughassigned1sand0stothesquaresandtheirlabelsanddeducedthenumberingschemeincommonuse. Maxfield,Clive"Max"(2006-11-29)."Reed-MullerLogic".Logic101.EETimes.Part3.Archivedfromtheoriginalon2017-04-19.Retrieved2017-04-19. Lind,LarryFrederick;Nelson,JohnChristopherCunliffe(1977)."Section2.3".AnalysisandDesignofSequentialDigitalSystems.MacmillanPress.ISBN 0-33319266-4.(146pages) Holder,MichelElizabeth(March2005)[2005-02-14]."AmodifiedKarnaughmaptechnique".IEEETransactionsonEducation.IEEE.48(1):206–207.doi:10.1109/TE.2004.832879.eISSN 1557-9638.ISSN 0018-9359.S2CID 25576523.[2] Cavanagh,Joseph(2008).ComputerArithmeticandVerilogHDLFundamentals(1 ed.).CRCPress. Kohavi,Zvi;Jha,NirajK.(2009).SwitchingandFiniteAutomataTheory(3 ed.).CambridgeUniversityPress.ISBN 978-0-521-85748-2. Externallinks[edit] KarnaughmapatWikipedia'ssisterprojects DefinitionsfromWiktionaryMediafromCommonsNewsfromWikinewsQuotationsfromWikiquoteTextsfromWikisourceTextbooksfromWikibooksResourcesfromWikiversity DetectOverlappingRectangles,byHerbertGlarner. UsingKarnaughmapsinpracticalapplications,Circuitdesignprojecttocontroltrafficlights. K-MapTutorialfor2,3,4and5variables KarnaughMapExample POCKET–PCBOOLEANFUNCTIONSIMPLIFICATION,LedionBitincka—GeorgeE.Antoniou K-Maptroubleshoot Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Karnaugh_map&oldid=1081411777" Categories:BooleanalgebraDiagramsElectronicsoptimizationLogicincomputerscienceHiddencategories:ArticlesneedingtranslationfromGermanWikipediaArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataUsedmydatesfromApril2019Uselist-definedreferencesfromJanuary2022PagesusingSisterprojectlinkswithdefaultsearch Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages العربيةБългарскиCatalàČeštinaDeutschEestiEspañolEuskaraفارسیFrançaisGalego한국어BahasaIndonesiaItalianoעבריתMagyarNederlands日本語PolskiPortuguêsRomânăРусскийSlovenčinaСрпски/srpskiSuomiSvenskaதமிழ்TürkçeУкраїнськаTiếngViệt吴语中文 Editlinks