IntroductiontoDigitalLogic

CS64:ComputerOrganizationandDesignLogicLecture#11

ZiadMatni

Dept.ofComputerScience,UCSB

Administrative

•  Re:MidtermExam#1

•  Re:MidtermExam#2– NextThursday!– Everythingfromlectures7thru12

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Administrative

•  Lab#6(nextweek)areexercisesthatdonotrequireacomputer–  Youare*excused*fromgoingtoMondaylabnextweek,butT.Aswillbetheretohelp,ifyouneedit

– MaterialwecovertodayandonTuesdaywillbenecessarytofinishLab#6

•  Lab#6willstillbedueendofdayFriday

•  Lab#7willbedifferent…!

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Administrative

•  Mid-quarterevaluationsforT.AsandforProf.– LinksonPiazza– Optionaltodo,butveryappreciatedbyusall!– DeadlineisMonday

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LectureOutline

•  IntrotoBinary(Digital)LogicGates

•  TruthTableConstruction

•  LogicFunctionsandtheirSimplifications

•  TheLawsofBinaryLogic

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Digitali.e.BinaryLogic

•  Electroniccircuitswhenusedincomputersareaseriesofswitches

•  2possiblestates:eitherON(1)andOFF(0)

•  Perfectforbinarylogicrepresentation!

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BasicBuildingBlocksofDigitalLogic

•  Sameasthebitwiseoperators:ANDORXORNOTetc...

•  Weoftenrefertotheseas“logicgates”indigitaldesign

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ElectronicCircuitLogicEquivalents

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GraphicalSymbolsandTruthTablesNOT

A Aor!A0 11 0

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GraphicalSymbolsandTruthTablesANDandNAND

A B A.B0 0 00 1 01 0 01 1 1

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A B A.Bor!(A.B)

0 0 10 1 11 0 11 1 0

≡

GraphicalSymbolsandTruthTablesORandNOR

A B A+B0 0 00 1 11 0 11 1 1

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A B A+Bor!(A+B)

0 0 10 1 01 0 01 1 0

≡

GraphicalSymbolsandTruthTablesXOR

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A B A+B0 0 00 1 11 0 11 1 0

ConstructingTruthTables

•  T.TscanbeappliedtoANYdigitalcircuit

•  TheyshowALLpossibleinputswithALLpossibleoutputs

•  NumberofentriesintheT.T.

=2N,whereNisthenumberofinputs

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Example:ConstructingtheT.Tofa1-bitAdder

•  Recallthe1-bitadder:

•  3inputs:I1andI2andCI–  Input1,Input2,andCarry-In–  HowmanyentriesintheT.T.isthat?

•  2outputs:RandCO–  Result,andCarry-Out–  Youcanhavemultipleoutputs:eachwillstilldependonsomecombinationoftheinputs

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EXAMPLE:

1 1

0

0

1

Example:ConstructingtheT.Tofa1-bitAdder

T.TConstructionTime!

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Example:ConstructingtheT.Tofa1-bitAdder

# I1 I2 CI CO R0 0 0 0 0 01 0 0 1 0 12 0 1 0 0 13 0 1 1 1 04 1 0 0 0 15 1 0 1 1 06 1 1 0 1 07 1 1 1 1 1

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INPUTS OUTPUTS

LogicFunctions•  AnoutputfunctionFcanbeseenasacombinationof1or

moreinputs•  Example:

F=A.B+C (allsinglebits)EquivalentinC/C++:

booleanf(booleana,booleanb,booleanc) { return((a&b)|c) }

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ORandANDasSumandProduct•  Logicfunctionsareoftenexpressedwithbasiclogicbuilding

blocks,likeORsandANDsandNOTs,etc…

•  ORissometimesreferredtoas“logicalsum”or“logicalunion”–  Partlywhyit’ssymbolizedas“+”–  BUTIT’SNOTTHESAMEASNUMERICALADDITION!!!!!!

•  ANDas“logicalproduct”or“logicaldisjunction”–  Partlywhyit’ssymbolizedas“.”–  BUTIT’SNOTTHESAMEASNUMERICALMULTIPLICATION!!!!!!

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Example

•  AXORBtakesthevalue“1”(i.e.isTRUE)ifandonlyif–  A=0,B=1 i.e.!A.BisTRUE,or–  A=1,B=0 i.e.A.!BisTRUE

•  Inotherwords, AXORB isTRUEiff(ifandonlyif) A!B+!ABisTRUE

A+B=!A.B+A.!BWhichcanalsobewrittenas:A.B+A.B

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A B A+B0 0 00 1 11 0 11 1 0

RepresentingtheCircuitGraphically

A+B=!A.B+A.!B

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A B A+B0 0 00 1 11 0 11 1 0

Q:Doesittakeanytimeforaelectronicsignaltogothru3“layers”oflogicgates?

A:Ideally,NO,itallhappenssimultaneously.Inreality,OFCOURSEittakestime(it’scalledlatency)

A

A

B

B

A+B

WhatisTheLogicalFunctionforTheHalfAdder?

# I1 I2 CO R0 0 0 0 01 0 1 0 12 1 0 0 13 1 1 1 0

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INPUTS OUTPUTS

CO=I1.I2R=I1+I2

HalfAdder1-bitadderthatdoesnothaveaCarry-In(Ci)bit.Thislogicblockhasonly21-bitinputsand21-bitoutputs

Ourattempttodescribetheoutputsasfunctions

oftheinputs:

WhatisTheLogicalFunctionforAFull1-bitadder?

# I1 I2 CI CO R0 0 0 0 0 01 0 0 1 0 12 0 1 0 0 13 0 1 1 1 04 1 0 0 0 15 1 0 1 1 06 1 1 0 1 07 1 1 1 1 1

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INPUTS OUTPUTS

Ans.: CO=!I1.I2.CI+I1.!I2.CI+I1.I2.!CI+I1.I2.CI R=!I1.!I2.CI+!I1.I2.!CI+I1.!I2.!CI+I1.I2.CI

MinimizationofBinaryLogic•  Why?

–  It’sMUCHeasiertoreadandunderstand…–  Savesmemory(software)and/orphysicalspace(hardware)–  Runsfaster/performsbetter

•  Why?...rememberlatency?

•  Forexample,whenwedotheT.T.for(seedemoonboard): X=A.B+A.!B+B.!A,wefindthatitisthesameas

(savedourselvesabunchoflogicgates!)

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A+B

UsingT.Tsvs.UsingLogicRules•  Inanefforttosimplifyalogicfunction,wedon’talways

havetouseT.Ts–wecanuselogicrulesinstead

Example:Whatarethefollowinglogicoutcomes?A.AA+A

A.1A+1

A.0A+0

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AAA10A

UsingT.Tsvs.UsingLogicRules•  BinaryLogicworksinAssociativeways–  (A.B).C isthesameas A.(B.C)–  (A+B)+C isthesameas A+(B+C)

•  ItalsoworksinDistributiveways–  (A+B).C isthesameas:A.C+B.C–  (A+B).(A+C) isthesameas:

A.A+A.C+B.A+B.C=A+A.C+A.B+B.C=A+B.C

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MoreExamplesofMinimizationa.k.aSimplification

•  Simplify: R =A.B+!A.B =(A+!A).B =B

•  Simplify: R =!ABCD+ABCD+!AB!CD+AB!CD

=BCD(A+!A)+!AB!CD+AB!CD =BCD+B!CD(!A+A) =BCD+B!CD =BD(C+!C) =BD

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Let’sverifyitwithatruth-table

Let’sverifyitwithatruth-table

Note:often,theANDdotsymbol(.)isomitted,butunderstoodtobethere(likewithmultiplicationdotsymbol)

MoreSimplificationExercises

•  Simplify: R =!A!BC+!A!B!C+!ABC+!AB!C+A!BC =!A(C+!C)+!AB(C+!C)+A!BC =!A+!AB+A!BC =!A+A!BC

•  ReformulateusingonlyANDandNOTlogic:

R =!AC+!BC =C(!A+!B) =C.!(A.B) ßDeMorgan’sLaw

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Let’sverifyitwithatruth-table

Important:LawsofBinaryLogic

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DigitalCircuitDesignProcess

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CANTHISPROCESSBEREVERSED?

MoreSimplificationExamplesSimplifytheBooleanexpression:•  (A+B+C)(D+E)'+(A+B+C)(D+E)SimplifytheBooleanexpressionandwriteitoutonatruthtableasproof•  XZ+Z(X'+XY)

UseDeMorgan’sTheormtore-writetheexpressionbelowusingatleastoneORoperation•  NOT(X+YZ)

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ScalingUpSimplification

•  Whenwegettomorethan3variables,itbecomeschallengingtousetruthtables

•  WecaninsteaduseKarnaughMapstomakeitimmediatelyapparentastowhatcanbesimplified

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ExampleofaK-MapA B f(A,B)0 0 a0 1 b1 0 c1 1 d

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A B f(A,B)0 0 00 1 11 0 11 1 1

0123

MoreonK-MapsNextWeek!

YOURTO-DOs

•  FinishLab#5bytomorrow!

•  Optional:Dotheonlinemid-termevaluations–  SeeannouncementonPiazzaforlinks

•  NextTime:MoreK-MapsandCombinatorialLogic

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