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Representation of
Multiple-Valued Logic Functions
Synthesis Lectures on Digital
Circuits and Systems
Editor
MitchellA.Thornton,SouthernMethodistUniversity
TheSynthesisLecturesonDigitalCircuitsandSystemsseriesiscomprisedof50-to100-page
bookstargetedforaudiencememberswithawide-rangingbackground.TheLecturesinclude
topicsthatareofinteresttostudents,professionals,andresearchersintheareaofdesignand
analysisofdigitalcircuitsandsystems.EachLectureisself-containedandfocusesonthe
backgroundinformationrequiredtounderstandthesubjectmatterandpracticalcasestudiesthat
illustrateapplications.TheformatofaLectureisstructuredsuchthateachwillbedevotedtoa
specifictopicindigitalcircuitsandsystemsratherthanalargeroverviewofseveraltopicssuchas
thatfoundinacomprehensivehandbook.TheLecturescoverbothwell-establishedareasaswellas
newlydevelopedoremergingmaterialindigitalcircuitsandsystemsdesignandanalysis.
RepresentationofMultiple-ValuedLogicFunctions
RadomirS.Stankovic´,JaakkoT.Astola,andClaudioMoraga
2012
ArduinoMicrocontroller:ProcessingforEveryone!SecondEdition
StevenF.Barrett
2012
AdvancedCircuitSimulationUsingMultisimWorkbench
DavidBáez-López,FélixE.Guerrero-Castro,andOfeliaDelfinaCervantes-Villagómez
2012
CircuitAnalysiswithMultisim
DavidBáez-LópezandFélixE.Guerrero-Castro
2011
MicrocontrollerProgrammingandInterfacingTexasInstrumentsMSP430,PartI
StevenF.BarrettandDanielJ.Pack
2011
MicrocontrollerProgrammingandInterfacingTexasInstrumentsMSP430,PartII
StevenF.BarrettandDanielJ.Pack
2011
iii
PragmaticElectricalEngineering:SystemsandInstruments
WilliamEccles
2011
PragmaticElectricalEngineering:Fundamentals
WilliamEccles
2011
IntroductiontoEmbeddedSystems:UsingANSICandtheArduinoDevelopment
Environment
DavidJ.Russell
2010
ArduinoMicrocontroller:ProcessingforEveryone!PartII
StevenF.Barrett
2010
ArduinoMicrocontroller:ProcessingforEveryone!PartI
StevenF.Barrett
2010
DigitalSystemVerification:ACombinedFormalMethodsandSimulationFramework
LunLiandMitchellA.Thornton
2010
ProgressinApplicationsofBooleanFunctions
TsutomuSasaoandJonT.Butler
2009
EmbeddedSystemsDesignwiththeAtmelAVRMicrocontroller:PartII
StevenF.Barrett
2009
EmbeddedSystemsDesignwiththeAtmelAVRMicrocontroller:PartI
StevenF.Barrett
2009
EmbeddedSystemsInterfacingforEngineersusingtheFreescaleHCS08MicrocontrollerII:
DigitalandAnalogHardwareInterfacing
DouglasH.Summerville
2009
DesigningAsynchronousCircuitsusingNULLConventionLogic(NCL)
ScottC.SmithandJiaDi
2009
iv
EmbeddedSystemsInterfacingforEngineersusingtheFreescaleHCS08MicrocontrollerI:
AssemblyLanguageProgramming
DouglasH.Summerville
2009
DevelopingEmbeddedSoftwareusingDaVinci&OMAPTechnology
B.I.(Raj)Pawate
2009
MismatchandNoiseinModernICProcesses
AndrewMarshall
2009
AsynchronousSequentialMachineDesignandAnalysis:AComprehensiveDevelopmentof
theDesignandAnalysisofClock-IndependentStateMachinesandSystems
RichardF.Tinder
2009
AnIntroductiontoLogicCircuitTesting
ParagK.Lala
2008
PragmaticPower
WilliamJ.Eccles
2008
MultipleValuedLogic:ConceptsandRepresentations
D.MichaelMillerandMitchellA.Thornton
2007
FiniteStateMachineDatapathDesign,Optimization,andImplementation
JustinDavisandRobertReese
2007
AtmelAVRMicrocontrollerPrimer:ProgrammingandInterfacing
StevenF.BarrettandDanielJ.Pack
2007
PragmaticLogic
WilliamJ.Eccles
2007
PSpiceforFiltersandTransmissionLines
PaulTobin
2007
v
PSpiceforDigitalSignalProcessing
PaulTobin
2007
PSpiceforAnalogCommunicationsEngineering
PaulTobin
2007
PSpiceforDigitalCommunicationsEngineering
PaulTobin
2007
PSpiceforCircuitTheoryandElectronicDevices
PaulTobin
2007
PragmaticCircuits:DCandTimeDomain
WilliamJ.Eccles
2006
PragmaticCircuits:FrequencyDomain
WilliamJ.Eccles
2006
PragmaticCircuits:SignalsandFilters
WilliamJ.Eccles
2006
High-SpeedDigitalSystemDesign
JustinDavis
2006
IntroductiontoLogicSynthesisusingVerilogHDL
RobertB.ReeseandMitchellA.Thornton
2006
MicrocontrollersFundamentalsforEngineersandScientists
StevenF.BarrettandDanielJ.Pack
2006
Copyright© 2012byMorgan&Claypool
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotationsin
printedreviews,withoutthepriorpermissionofthepublisher.
RepresentationofMultiple-ValuedLogicFunctions
RadomirS.Stankovic´,JaakkoT.Astola,andClaudioMoraga
www.morganclaypool.com
ISBN:9781608459421 paperback
ISBN:9781608459438 ebook
DOI10.2200/S00420ED1V01Y201205DCS037
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONDIGITALCIRCUITSANDSYSTEMS
Lecture#37
SeriesEditor:MitchellA.Thornton,SouthernMethodistUniversity
SeriesISSN
SynthesisLecturesonDigitalCircuitsandSystems
Print1932-3166 Electronic1932-3174
Representation of
Multiple-Valued Logic Functions
Radomir S.Stankovic´
UniversityofNiš,Serbia
JaakkoT.Astola
TampereUniversityofTechnology,Finland
Claudio Moraga
EuropeanCentreforSoftComputing,Spain
TechnicalUniversityofDortmund,Germany
SYNTHESISLECTURESONDIGITALCIRCUITSANDSYSTEMS#37
M
&C Morgan &cLaypool publishers
ABSTRACT
Comparedtobinaryswitchingfunctions,themultiple-valuedfunctions(MV)offermorecompact
representationsoftheinformationcontentofsignalsmodeledbylogicfunctionsand,therefore,their
usefitsverywellinthegeneralsettingsofdatacompressionattemptsandapproaches.Thefirsttask
in dealing with such signals is to provide mathematical methods for their representation in a way
thatwillmaketheirapplicationinpracticefeasible.
RepresentationofMultiple-ValuedLogicFunctionsisaimedatprovidinganaccessibleintroduc-
tiontothesemathematicaltechniquesthatarenecessaryforapplicationofrelatedimplementation
methodsandtools.
Thisbookpresentsinauniformwaydifferentrepresentationsofmultiple-valuedlogicfunc-
tions,includingfunctionalexpressions,spectralrepresentationsonfiniteAbeliangroups,andtheir
graphicalcounterparts(variousrelateddecisiondiagrams).
Three-valued,orternaryfunctions,aretraditionallyusedasthefirstextensionfromthebinary
case.Theyhaveagoodfeaturethattheratiobetweenthenumberofbitsandthenumberofdifferent
valuesthatcanbeencodedwiththespecifiednumberofbitsisfavourableforternaryfunctions.
Four-valued functions, also called quaternary functions, are particularly attractive, since in
practical realization within today prevalent binary circuits environment, they may be easy coded
by binary values and realized with two-stable state circuits.At the same time,there is much more
considerableadventindesignoffour-valuedlogiccircuitsthanforotherp-valuedfunctions.
Therefore, this book is written using a hands-on approach such that after introducing the
generalandnecessarilyabstractbackgroundtheory,thepresentationisbasedonalargenumberof
examples for ternary and quaternary functions that should provide an intuitive understanding of
variousrepresentationmethodsandtheinterconnectionsamongthem.
KEYWORDS
multiple-valued logic functions, Reed-Muller expressions, Reed-Muller-Fourier ex-
pressions,Galoisfieldexpressions,spectralexpressions,decisiondiagrams
