Representation of Multiple-Valued Logic Functions

170 Pages·2012·0.936 MB·English

by  Stankovich R.S., Astola J.T., Moraga C.

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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
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illustrateapplications.TheformatofaLectureisstructuredsuchthateachwillbedevotedtoa
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thatfoundinacomprehensivehandbook.TheLecturescoverbothwell-establishedareasaswellas
newlydevelopedoremergingmaterialindigitalcircuitsandsystemsdesignandanalysis.
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MicrocontrollerProgrammingandInterfacingTexasInstrumentsMSP430,PartII
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PragmaticElectricalEngineering:Fundamentals
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Environment
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DigitalandAnalogHardwareInterfacing
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theDesignandAnalysisofClock-IndependentStateMachinesandSystems
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AnIntroductiontoLogicCircuitTesting
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MultipleValuedLogic:ConceptsandRepresentations
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AtmelAVRMicrocontrollerPrimer:ProgrammingandInterfacing
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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
ix
Contents
Acknowledgments.......................................................xiii
1 Multiple-ValuedLogicFunctions ...........................................1
1.1 TabularRepresentations ................................................. 2
1.2 Cubes................................................................. 4
1.3 EncodingofVariables ................................................... 7
1.4 OtherRepresentations................................................... 7
1.5 AlgebraicStructuresforMultiple-ValuedFunctions.......................... 8
1.6 FunctionswithVariousProperties........................................ 12
2 FunctionalExpressionsforMultiple-ValuedFunctions ...................... 15
2.1 FunctionalExpressions ................................................. 15
2.2 GeneralizationstoMultiple-ValuedFunctions ............................. 17
2.3 Sum-of-ProductExpressions ............................................ 19
2.4 GaloisFieldExpressions................................................ 21
2.4.1 GaloisFieldExpressionsforTernaryFunctions ...................... 23
2.4.2 GaloisFieldExpressionsforQuaternaryFunctions................... 24
2.5 Fixed-PolarityGF-expressions........................................... 26
2.5.1 Reed-Muller-FourierTransform ................................... 28
2.5.2 RMFexpressionsforp =3........................................ 30
2.5.3 RMFexpressionsforp =4........................................ 32
2.6 EfficiencyofRepresentations............................................ 35
2.6.1 ComputationalEfficiency......................................... 35
2.6.2 RealizationEfficiency ............................................ 37
2.7 ArithmeticExpressionsforMultiple-ValuedFunctions...................... 40
2.7.1 ArithmeticExpressionsforMultiple-ValuedFunctionsderivedfrom
theGF-expressions ..............................................42
2.7.2 ArithmeticExpressionsderivedfromtheRMF-expressions............ 44
2.7.3 StructureoftheArithmeticRMF-transformmatrices................. 46
2.8 Haar-likeExpressionsforMultiple-ValuedFunctions ....................... 49
2.9 SparseRepresentationsfromCoveringCodes.............................. 53