Logic gate - Wikipedia

Universal logic gates Logicgate FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Computationalequipment,physicalortheoretical,thatperformsabooleanlogicfunction "Discretelogic"redirectshere.Fordiscretecircuitry,seeDiscretecircuit.FordiscreteTTLlogic,seeTransistor–transistorlogic. AlogicgateisanidealizedorphysicaldeviceimplementingaBooleanfunction,alogicaloperationperformedononeormorebinaryinputsthatproducesasinglebinaryoutput.Dependingonthecontext,thetermmayrefertoanideallogicgate,onethathasforinstancezerorisetimeandunlimitedfan-out,oritmayrefertoanon-idealphysicaldevice[1](seeIdealandrealop-ampsforcomparison). Logicgatesareprimarilyimplementedusingdiodesortransistorsactingaselectronicswitches,butcanalsobeconstructedusingvacuumtubes,electromagneticrelays(relaylogic),fluidiclogic,pneumaticlogic,optics,molecules,orevenmechanicalelements.Now,mostlogicgatesaremadefromMOSFETs(metal–oxide–semiconductorfield-effecttransistors). Withamplification,logicgatescanbecascadedinthesamewaythatBooleanfunctionscanbecomposed,allowingtheconstructionofaphysicalmodelofallofBooleanlogic,andtherefore,allofthealgorithmsandmathematicsthatcanbedescribedwithBooleanlogic. Logiccircuitsincludesuchdevicesasmultiplexers,registers,arithmeticlogicunits(ALUs),andcomputermemory,allthewayupthroughcompletemicroprocessors,whichmaycontainmorethan100millionlogicgates. CompoundlogicgatesAND-OR-Invert(AOI)andOR-AND-Invert(OAI)areoftenemployedincircuitdesignbecausetheirconstructionusingMOSFETsissimplerandmoreefficientthanthesumoftheindividualgates.[2] Inreversiblelogic,Toffoligatesareused. Contents 1Electronicgates 2Historyanddevelopment 3Symbols 4Truthtables 5Universallogicgates 6DeMorganequivalentsymbols 7Datastorage 8Three-statelogicgates 9Manufacturing 9.1Earlytechniques 9.2Logicfamilies 10Seealso 11References 12Furtherreading 13Externallinks Electronicgates[edit] Mainarticle:Logicfamily Afunctionallycompletelogicsystemmaybecomposedofrelays,valves(vacuumtubes),ortransistors.Thesimplestfamilyoflogicgatesusesbipolartransistors,andiscalledresistor–transistorlogic(RTL).Unlikesimplediodelogicgates(whichdonothaveagainelement),RTLgatescanbecascadedindefinitelytoproducemorecomplexlogicfunctions.RTLgateswereusedinearlyintegratedcircuits.Forhigherspeedandbetterdensity,theresistorsusedinRTLwerereplacedbydiodesresultingindiode–transistorlogic(DTL).Transistor–transistorlogic(TTL)thensupplantedDTL.Asintegratedcircuitsbecamemorecomplex,bipolartransistorswerereplacedwithsmallerfield-effecttransistors(MOSFETs);seePMOSandNMOS.Toreducepowerconsumptionstillfurther,mostcontemporarychipimplementationsofdigitalsystemsnowuseCMOSlogic.CMOSusescomplementary(bothn-channelandp-channel)MOSFETdevicestoachieveahighspeedwithlowpowerdissipation. Forsmall-scalelogic,designersnowuseprefabricatedlogicgatesfromfamiliesofdevicessuchastheTTL7400seriesbyTexasInstruments,theCMOS4000seriesbyRCA,andtheirmorerecentdescendants.Increasingly,thesefixed-functionlogicgatesarebeingreplacedbyprogrammablelogicdevices,whichallowdesignerstopackmanymixedlogicgatesintoasingleintegratedcircuit.Thefield-programmablenatureofprogrammablelogicdevicessuchasFPGAshasreducedthe'hard'propertyofhardware;itisnowpossibletochangethelogicdesignofahardwaresystembyreprogrammingsomeofitscomponents,thusallowingthefeaturesorfunctionofahardwareimplementationofalogicsystemtobechanged.Othertypesoflogicgatesinclude,butarenotlimitedto:[3] Logicfamily Abbreviation Description Diodelogic DL Tunneldiodelogic TDL Exactlythesameasdiodelogicbutcanperformatahigherspeed.[failedverification] Neonlogic NL Usesneonbulbsor3elementneontriggertubestoperformlogic. Corediodelogic CDL Performedbysemiconductordiodesandsmallferritetoroidalcoresformoderatespeedandmoderatepowerlevel. 4LayerDeviceLogic 4LDL UsesthyristorsandSCRstoperformlogicoperationswherehighcurrentandorhighvoltagesarerequired. Direct-coupledtransistorlogic DCTL Usestransistorsswitchingbetweensaturatedandcutoffstatestoperformlogic.Thetransistorsrequirecarefullycontrolledparameters.Economicalbecausefewothercomponentsareneeded,buttendstobesusceptibletonoisebecauseofthelowervoltagelevelsemployed.OftenconsideredtobethefathertomodernTTLlogic. Metal-oxide-semiconductorlogic MOS UsesMOSFETs(metal-oxide-semiconductorfield-effecttransistors),thebasisformostmodernlogicgates.TheMOSlogicfamilyincludesPMOSlogic,NMOSlogic,complementaryMOS(CMOS),andBiCMOS(bipolarCMOS). Current-modelogic CML Usestransistorstoperformlogicbutbiasingisfromconstantcurrentsourcestopreventsaturationandallowextremelyfastswitching.Hashighnoiseimmunitydespitefairlylowlogiclevels. Quantum-dotcellularautomata QCA Usestunnelableq-bitsforsynthesizingthebinarylogicbits.Theelectrostaticrepulsiveforceinbetweentwoelectronsinthequantumdotsassignstheelectronconfigurations(thatdefineshigh-levellogicstate1orlow-levellogicstate0)underthesuitablydrivenpolarizations.Thisisatransistorless,currentless,junctionlessbinarylogicsynthesistechniqueallowingittohaveveryfastoperationspeeds. Electroniclogicgatesdiffersignificantlyfromtheirrelay-and-switchequivalents.Theyaremuchfaster,consumemuchlesspower,andaremuchsmaller(allbyafactorofamillionormoreinmostcases).Also,thereisafundamentalstructuraldifference.Theswitchcircuitcreatesacontinuousmetallicpathforcurrenttoflow(ineitherdirection)betweenitsinputanditsoutput.Thesemiconductorlogicgate,ontheotherhand,actsasahigh-gainvoltageamplifier,whichsinksatinycurrentatitsinputandproducesalow-impedancevoltageatitsoutput.Itisnotpossibleforcurrenttoflowbetweentheoutputandtheinputofasemiconductorlogicgate. Anotherimportantadvantageofstandardizedintegratedcircuitlogicfamilies,suchasthe7400and4000families,isthattheycanbecascaded.Thismeansthattheoutputofonegatecanbewiredtotheinputsofoneorseveralothergates,andsoon.Systemswithvaryingdegreesofcomplexitycanbebuiltwithoutgreatconcernofthedesignerfortheinternalworkingsofthegates,providedthelimitationsofeachintegratedcircuitareconsidered. Theoutputofonegatecanonlydriveafinitenumberofinputstoothergates,anumbercalledthe'fan-outlimit'.Also,thereisalwaysadelay,calledthe'propagationdelay',fromachangeininputofagatetothecorrespondingchangeinitsoutput.Whengatesarecascaded,thetotalpropagationdelayisapproximatelythesumoftheindividualdelays,aneffectwhichcanbecomeaprobleminhigh-speedsynchronouscircuits.Additionaldelaycanbecausedwhenmanyinputsareconnectedtoanoutput,duetothedistributedcapacitanceofalltheinputsandwiringandthefiniteamountofcurrentthateachoutputcanprovide. Historyanddevelopment[edit] ThebinarynumbersystemwasrefinedbyGottfriedWilhelmLeibniz(publishedin1705),influencedbytheancientIChing'sbinarysystem.[4][5]Leibnizestablishedthatusingthebinarysystemcombinedtheprinciplesofarithmeticandlogic. Inan1886letter,CharlesSandersPeircedescribedhowlogicaloperationscouldbecarriedoutbyelectricalswitchingcircuits.[6]Eventually,vacuumtubesreplacedrelaysforlogicoperations.LeeDeForest'smodification,in1907,oftheFlemingvalvecanbeusedasalogicgate.LudwigWittgensteinintroducedaversionofthe16-rowtruthtableasproposition5.101ofTractatusLogico-Philosophicus(1921).WaltherBothe,inventorofthecoincidencecircuit,gotpartofthe1954NobelPrizeinphysics,forthefirstmodernelectronicANDgatein1924.KonradZusedesignedandbuiltelectromechanicallogicgatesforhiscomputerZ1(from1935to1938). From1934to1936,NECengineerAkiraNakashima,ClaudeShannonandViktorShetakovintroducedswitchingcircuittheoryinaseriesofpapersshowingthattwo-valuedBooleanalgebra,whichtheydiscoveredindependently,candescribetheoperationofswitchingcircuits.[7][8][9][10]Usingthispropertyofelectricalswitchestoimplementlogicisthefundamentalconceptthatunderliesallelectronicdigitalcomputers.Switchingcircuittheorybecamethefoundationofdigitalcircuitdesign,asitbecamewidelyknownintheelectricalengineeringcommunityduringandafterWorldWarII,withtheoreticalrigorsupersedingtheadhocmethodsthathadprevailedpreviously.[10] Metal-oxide-semiconductor(MOS)devicesintheformsofPMOSandNMOSweredemonstratedbyBellLabsengineersMohamedM.AtallaandDawonKahngin1960.[11]BothtypeswerelatercombinedandadaptedintocomplementaryMOS(CMOS)logicbyChih-TangSahandFrankWanlassatFairchildSemiconductorin1963.[12] Activeresearchistakingplaceinmolecularlogicgates. Symbols[edit] Asynchronous4-bitup/downdecadecountersymbol(74LS192)inaccordancewithANSI/IEEEStd.91-1984andIECPublication60617-12. Therearetwosetsofsymbolsforelementarylogicgatesincommonuse,bothdefinedinANSI/IEEEStd91-1984anditssupplementANSI/IEEEStd91a-1991.The"distinctiveshape"set,basedontraditionalschematics,isusedforsimpledrawingsandderivesfromUnitedStatesMilitaryStandardMIL-STD-806ofthe1950sand1960s.[13]Itissometimesunofficiallydescribedas"military",reflectingitsorigin.The"rectangularshape"set,basedonANSIY32.14andotherearlyindustrystandardsaslaterrefinedbyIEEEandIEC,hasrectangularoutlinesforalltypesofgateandallowsrepresentationofamuchwiderrangeofdevicesthanispossiblewiththetraditionalsymbols.[14]TheIECstandard,IEC60617-12,hasbeenadoptedbyotherstandards,suchasEN60617-12:1999inEurope,BSEN60617-12:1999intheUnitedKingdom,andDINEN60617-12:1998inGermany. ThemutualgoalofIEEEStd91-1984andIEC60617-12wastoprovideauniformmethodofdescribingthecomplexlogicfunctionsofdigitalcircuitswithschematicsymbols.ThesefunctionsweremorecomplexthansimpleANDandORgates.Theycouldbemediumscalecircuitssuchasa4-bitcountertoalargescalecircuitsuchasamicroprocessor. IEC617-12anditssuccessorIEC60617-12donotexplicitlyshowthe"distinctiveshape"symbols,butdonotprohibitthem.[14]Theseare,however,showninANSI/IEEE91(and91a)withthisnote:"Thedistinctive-shapesymbolis,accordingtoIECPublication617,Part12,notpreferred,butisnotconsideredtobeincontradictiontothatstandard."IEC60617-12correspondinglycontainsthenote(Section2.1)"Althoughnon-preferred,theuseofothersymbolsrecognizedbyofficialnationalstandards,thatisdistinctiveshapesinplaceofsymbols[listofbasicgates],shallnotbeconsideredtobeincontradictionwiththisstandard.Usageoftheseothersymbolsincombinationtoformcomplexsymbols(forexample,useasembeddedsymbols)isdiscouraged."ThiscompromisewasreachedbetweentherespectiveIEEEandIECworkinggroupstopermittheIEEEandIECstandardstobeinmutualcompliancewithoneanother. Athirdstyleofsymbols,DIN40700(1976),wasinuseinEuropeandisstillwidelyusedinEuropeanacademia,seethelogictableinGermanWikipedia. Inthe1980s,schematicswerethepredominantmethodtodesignbothcircuitboardsandcustomICsknownasgatearrays.TodaycustomICsandthefield-programmablegatearrayaretypicallydesignedwithHardwareDescriptionLanguages(HDL)suchasVerilogorVHDL. Type Distinctiveshape(IEEEStd91/91a-1991) Rectangularshape(IEEEStd91/91a-1991)(IEC60617-12:1997) BooleanalgebrabetweenA&B Truthtable 1-Inputgates Buffer A {\displaystyle{A}} INPUT OUTPUT A Q 0 0 1 1 NOT(inverter) A ¯ {\displaystyle{\overline{A}}} or ¬ A {\displaystyle\negA} INPUT OUTPUT A Q 0 1 1 0 InelectronicsaNOTgateismorecommonlycalledaninverter.Thecircleonthesymboliscalledabubbleandisusedinlogicdiagramstoindicatealogicnegationbetweentheexternallogicstateandtheinternallogicstate(1to0orviceversa).Onacircuitdiagramitmustbeaccompaniedbyastatementassertingthatthepositivelogicconventionornegativelogicconventionisbeingused(highvoltagelevel=1orlowvoltagelevel=1,respectively).Thewedgeisusedincircuitdiagramstodirectlyindicateanactive-low(lowvoltagelevel=1)inputoroutputwithoutrequiringauniformconventionthroughoutthecircuitdiagram.ThisiscalledDirectPolarityIndication.SeeIEEEStd91/91AandIEC60617-12.Boththebubbleandthewedgecanbeusedondistinctive-shapeandrectangular-shapesymbolsoncircuitdiagrams,dependingonthelogicconventionused.Onpurelogicdiagrams,onlythebubbleismeaningful. ConjunctionandDisjunction AND A ⋅ B {\displaystyleA\cdotB} or A ∧ B {\displaystyleA\landB} INPUT OUTPUT A B Q 0 0 0 0 1 0 1 0 0 1 1 1 OR A + B {\displaystyleA+B} or A ∨ B {\displaystyleA\lorB} INPUT OUTPUT A B Q 0 0 0 0 1 1 1 0 1 1 1 1 AlternativedenialandJointdenial NAND A ⋅ B ¯ {\displaystyle{\overline{A\cdotB}}} or A ↑ B {\displaystyleA\uparrowB} INPUT OUTPUT A B Q 0 0 1 0 1 1 1 0 1 1 1 0 NOR A + B ¯ {\displaystyle{\overline{A+B}}} or A ↓ B {\displaystyleA\downarrowB} INPUT OUTPUT A B Q 0 0 1 0 1 0 1 0 0 1 1 0 ExclusiveorandBiconditional XOR A ⊕ B {\displaystyleA\oplusB} or A ⊻ B {\displaystyleA\veebarB} INPUT OUTPUT A B Q 0 0 0 0 1 1 1 0 1 1 1 0 Theoutputofatwoinputexclusive-ORistrueonlywhenthetwoinputvaluesaredifferent,andfalseiftheyareequal,regardlessofthevalue.Iftherearemorethantwoinputs,theoutputofthedistinctive-shapesymbolisundefined.Theoutputoftherectangular-shapedsymbolistrueifthenumberoftrueinputsisexactlyoneorexactlythenumberfollowingthe"="inthequalifyingsymbol. XNOR A ⊕ B ¯ {\displaystyle{\overline{A\oplusB}}} or A ⊙ B {\displaystyle{A\odotB}} INPUT OUTPUT A B Q 0 0 1 0 1 0 1 0 0 1 1 1 Truthtables[edit] Outputcomparisonof1-inputlogicgates. INPUT OUTPUT A Buffer Inverter 0 0 1 1 1 0 Outputcomparisonof2-inputlogicgates. INPUT OUTPUT A B AND NAND OR NOR XOR XNOR 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 Universallogicgates[edit] Furtherinformationonthetheoreticalbasis:Functionalcompleteness The7400chip,containingfourNANDs.Thetwoadditionalpinssupplypower(+5V)andconnecttheground. CharlesSandersPeirce(during1880–81)showedthatNORgatesalone(oralternativelyNANDgatesalone)canbeusedtoreproducethefunctionsofalltheotherlogicgates,buthisworkonitwasunpublisheduntil1933.[15]ThefirstpublishedproofwasbyHenryM.Shefferin1913,sotheNANDlogicaloperationissometimescalledShefferstroke;thelogicalNORissometimescalledPeirce'sarrow.[16]Consequently,thesegatesaresometimescalleduniversallogicgates.[17] type NANDconstruction NORconstruction NOT AND NAND OR NOR XOR XNOR DeMorganequivalentsymbols[edit] ByuseofDeMorgan'slaws,anANDfunctionisidenticaltoanORfunctionwithnegatedinputsandoutputs.Likewise,anORfunctionisidenticaltoanANDfunctionwithnegatedinputsandoutputs.ANANDgateisequivalenttoanORgatewithnegatedinputs,andaNORgateisequivalenttoanANDgatewithnegatedinputs. Thisleadstoanalternativesetofsymbolsforbasicgatesthatusetheoppositecoresymbol(ANDorOR)butwiththeinputsandoutputsnegated.Useofthesealternativesymbolscanmakelogiccircuitdiagramsmuchclearerandhelptoshowaccidentalconnectionofanactivehighoutputtoanactivelowinputorviceversa.Anyconnectionthathaslogicnegationsatbothendscanbereplacedbyanegationlessconnectionandasuitablechangeofgateorviceversa.AnyconnectionthathasanegationatoneendandnonegationattheothercanbemadeeasiertointerpretbyinsteadusingtheDeMorganequivalentsymbolateitherofthetwoends.Whennegationorpolarityindicatorsonbothendsofaconnectionmatch,thereisnologicnegationinthatpath(effectively,bubbles"cancel"),makingiteasiertofollowlogicstatesfromonesymboltothenext.Thisiscommonlyseeninreallogicdiagrams–thusthereadermustnotgetintothehabitofassociatingtheshapesexclusivelyasORorANDshapes,butalsotakeintoaccountthebubblesatbothinputsandoutputsinordertodeterminethe"true"logicfunctionindicated. ADeMorgansymbolcanshowmoreclearlyagate'sprimarylogicalpurposeandthepolarityofitsnodesthatareconsideredinthe"signaled"(active,on)state.Considerthesimplifiedcasewhereatwo-inputNANDgateisusedtodriveamotorwheneitherofitsinputsarebroughtlowbyaswitch.The"signaled"state(motoron)occurswheneitheroneORtheotherswitchison.UnlikearegularNANDsymbol,whichsuggestsANDlogic,theDeMorganversion,atwonegative-inputORgate,correctlyshowsthatORisofinterest.TheregularNANDsymbolhasabubbleattheoutputandnoneattheinputs(theoppositeofthestatesthatwillturnthemotoron),buttheDeMorgansymbolshowsbothinputsandoutputinthepolaritythatwilldrivethemotor. DeMorgan'stheoremismostcommonlyusedtoimplementlogicgatesascombinationsofonlyNANDgates,orascombinationsofonlyNORgates,foreconomicreasons. Datastorage[edit] Mainarticle:Sequentiallogic Logicgatescanalsobeusedtostoredata.Astorageelementcanbeconstructedbyconnectingseveralgatesina"latch"circuit.Morecomplicateddesignsthatuseclocksignalsandthatchangeonlyonarisingorfallingedgeoftheclockarecallededge-triggered"flip-flops".Formally,aflip-flopiscalledabistablecircuit,becauseithastwostablestateswhichitcanmaintainindefinitely.Thecombinationofmultipleflip-flopsinparallel,tostoreamultiple-bitvalue,isknownasaregister.Whenusinganyofthesegatesetupstheoverallsystemhasmemory;itisthencalledasequentiallogicsystemsinceitsoutputcanbeinfluencedbyitspreviousstate(s),i.e.bythesequenceofinputstates.Incontrast,theoutputfromcombinationallogicispurelyacombinationofitspresentinputs,unaffectedbythepreviousinputandoutputstates. Theselogiccircuitsareknownascomputermemory.Theyvaryinperformance,basedonfactorsofspeed,complexity,andreliabilityofstorage,andmanydifferenttypesofdesignsareusedbasedontheapplication. Three-statelogicgates[edit] Atristatebuffercanbethoughtofasaswitch.IfBison,theswitchisclosed.IfBisoff,theswitchisopen. Mainarticle:Tri-statebuffer Athree-statelogicgateisatypeoflogicgatethatcanhavethreedifferentoutputs:high(H),low(L)andhigh-impedance(Z).Thehigh-impedancestateplaysnoroleinthelogic,whichisstrictlybinary.ThesedevicesareusedonbusesoftheCPUtoallowmultiplechipstosenddata.Agroupofthree-statesdrivingalinewithasuitablecontrolcircuitisbasicallyequivalenttoamultiplexer,whichmaybephysicallydistributedoverseparatedevicesorplug-incards. Inelectronics,ahighoutputwouldmeantheoutputissourcingcurrentfromthepositivepowerterminal(positivevoltage).Alowoutputwouldmeantheoutputissinkingcurrenttothenegativepowerterminal(zerovoltage).Highimpedancewouldmeanthattheoutputiseffectivelydisconnectedfromthecircuit. Manufacturing[edit] Seealso:UnconventionalcomputingandSemiconductordevicefabrication Sincethe1990s,mostlogicgatesaremadeinCMOS(complementarymetaloxidesemiconductor)technologythatusesbothNMOSandPMOStransistors.Oftenmillionsoflogicgatesarepackagedinasingleintegratedcircuit. Earlytechniques[edit] Non-electronicimplementationsarevaried,thoughfewofthemareusedinpracticalapplications.Manyearlyelectromechanicaldigitalcomputers,suchastheHarvardMarkI,werebuiltfromrelaylogicgates,usingelectro-mechanicalrelays.Logicgatescanbemadeusingpneumaticdevices,suchastheSortebergrelayormechanicallogicgates,includingonamolecularscale.[18]LogicgateshavebeenmadeoutofDNA(seeDNAnanotechnology)[19]andusedtocreateacomputercalledMAYA(seeMAYA-II).Logicgatescanbemadefromquantummechanicaleffects,seequantumlogicgate.Photoniclogicgatesusenonlinearopticaleffects. Inprincipleanymethodthatleadstoagatethatisfunctionallycomplete(forexample,eitheraNORoraNANDgate)canbeusedtomakeanykindofdigitallogiccircuit.Notethattheuseof3-statelogicforbussystemsisnotneeded,andcanbereplacedbydigitalmultiplexers,whichcanbebuiltusingonlysimplelogicgates(suchasNANDgates,NORgates,orANDandORgates). Logicfamilies[edit] Thereareseverallogicfamilieswithdifferentcharacteristics(powerconsumption,speed,cost,size)suchas:RDL(resistor–diodelogic),RTL(resistor-transistorlogic),DTL(diode–transistorlogic),TTL(transistor–transistorlogic)andCMOS.Therearealsosub-variants,e.g.standardCMOSlogicvs.advancedtypesusingstillCMOStechnology,butwithsomeoptimizationsforavoidinglossofspeedduetoslowerPMOStransistors. Seealso[edit] And-invertergraph Booleanalgebratopics Booleanfunction Digitalcircuit Espressoheuristiclogicminimizer Fan-out Field-programmablegatearray(FPGA) Flip-flop(electronics) Functionalcompleteness Karnaughmap Combinationallogic Listof4000seriesintegratedcircuits Listof7400seriesintegratedcircuits Logicfamily Logiclevel Logicalgraph NMOSlogic ProgrammableLogicController(PLC) ProgrammableLogicDevice(PLD) Propositionalcalculus Quantumlogicgate Racehazard Reversiblecomputing Truthtable References[edit] ^Jaeger,MicroelectronicCircuitDesign,McGraw-Hill1997,ISBN 0-07-032482-4,pp.226–233 ^Tinder,RichardF.(2000).Engineeringdigitaldesign:RevisedSecondEdition.pp. 317–319.ISBN 0-12-691295-5.Retrieved2008-07-04. ^Rowe,Jim."CircuitLogic–WhyandHow".No. December1966.ElectronicsAustralia. ^Nylan,Michael(2001).TheFive"Confucian"Classics.YaleUniversityPress.pp. 204–206.ISBN 978-0-300-08185-5.Retrieved8June2010. ^Perkins,Franklin.LeibnizandChina:ACommerceofLight.Cambridge:CambridgeUniversityPress,2004.p117.Print. ^Peirce,C.S.,"Letter,PeircetoA.Marquand",dated1886,WritingsofCharlesS.Peirce,v.5,1993,pp.421–23.SeeBurks,ArthurW.,"Review:CharlesS.Peirce,Thenewelementsofmathematics",BulletinoftheAmericanMathematicalSocietyv.84,n.5(1978),pp.913–18,see917.PDFEprint. ^HistoryofResearchonSwitchingTheoryinJapan,IEEJTransactionsonFundamentalsandMaterials,Vol.124(2004)No.8,pp.720–726,InstituteofElectricalEngineersofJapan ^SwitchingTheory/RelayCircuitNetworkTheory/TheoryofLogicalMathematics,IPSJComputerMuseum,InformationProcessingSocietyofJapan ^RadomirS.Stanković(UniversityofNiš),JaakkoT.Astola(TampereUniversityofTechnology),MarkG.Karpovsky(BostonUniversity),SomeHistoricalRemarksonSwitchingTheory,2007,DOI10.1.1.66.1248 ^abRadomirS.Stanković,JaakkoAstola(2008),ReprintsfromtheEarlyDaysofInformationSciences:TICSPSeriesOntheContributionsofAkiraNakashimatoSwitchingTheory,TICSPSeries#40,TampereInternationalCenterforSignalProcessing,TampereUniversityofTechnology ^Lojek,Bo(2007).HistoryofSemiconductorEngineering.SpringerScience&BusinessMedia.pp. 321–3.ISBN 9783540342588. ^"1963:ComplementaryMOSCircuitConfigurationisInvented".ComputerHistoryMuseum.Retrieved6July2019. ^"MIL-STD-806 :GraphicalSymbolsforLogicDiagrams".ASSISTQuickSearch.DefenseLogisticsAgency.RetrievedAugust27,2021. ^abOverviewofIEEEStandard91-1984ExplanationofLogicSymbols,Doc.No.SDYZ001A,TexasInstrumentsSemiconductorGroup,1996 ^Peirce,C.S.(manuscriptwinterof1880–81),"ABoolianAlgebrawithOneConstant",published1933inCollectedPapersv.4,paragraphs12–20.Reprinted1989inWritingsofCharlesS.Peircev.4,pp.218–21,Google[1].SeeRoberts,DonD.(2009),TheExistentialGraphsofCharlesS.Peirce,p.131. ^HansKleineBüning;TheodorLettmann(1999).Propositionallogic:deductionandalgorithms.CambridgeUniversityPress.p. 2.ISBN 978-0-521-63017-7. ^JohnBird(2007).Engineeringmathematics.Newnes.p. 532.ISBN 978-0-7506-8555-9. ^MechanicalLogicgates(focusedonmolecularscale) ^DNALogicgatesArchived2010-06-18attheWaybackMachine Furtherreading[edit] Awschalom,D.D.;Loss,D.;Samarth,N.(5August2002).SemiconductorSpintronicsandQuantumComputation.Berlin,Germany:Springer-Verlag.ISBN 978-3-540-42176-4.Retrieved28November2012. Bostock,Geoff(1988).Programmablelogicdevices:technologyandapplications.NewYork:McGraw-Hill.ISBN 978-0-07-006611-3.Retrieved28November2012. Brown,StephenD.;Francis,RobertJ.;Rose,Jonathan;Vranesic,ZvonkoG.(1992).FieldProgrammableGateArrays.Boston,MA:KluwerAcademicPublishers.ISBN 978-0-7923-9248-4.Retrieved28November2012. 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