a synchronous logic elements

8/8/2019 A Synchronous Logic Elements

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V.O.VASYUKEVICH

ASYNCHRONOUSLOGICELEMENTS.

VENJUNCTIONANDSEQUENTION

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:)GargantuaandPantagruel

FrancoisRabelais

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TranslatedfromRussian2009

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V.Vasyukevich-Asynchronouslogicelements 3

CONTENTS

PREFACE... 2

1. VENJUNCTION

1.1.Binarysetsandsequences............................................................ 6

(formatofbinaryset,asynchronoussequences,

intersectionofsetsandsequences)

1.2.Logicalswitchings.................................................................................... 9

(momentsofswitchings,backgroundofswitchings,rulesforswitchings) 1.3.Methodsforrepresentationofvariablescollections..............................................10

(sequenceofbinarysets,setofasynchronoussequences,

sequenceoflogicalswitchings)

1.4.Switchingfunctionvenjunction......................................................... 12

(venjunctionoperation,venjunctionasfunction)

1.5.Venjunctionincomparisonwithconjunction........................................................15

1.6.Methodsofvenjunctionformalizeddefinition..................................... 17

(onabasisofbinarysetsandsequences,withinvolvinganundetermined

value,withinvolvingaconjunction,withinvolvingtheadditionalbinary

variable,verbalreadingofvenjunction)

1.7.Truthtablesforvenjunction................................................................. 19

(venjunctivedependences,mastertruthtableforvenjunctions)

1.8.Venjunctivefunctionsandtheirenumeration...................................................23

(venjunctivecompleteform,enumerationoffunctionsoftwovariables)

1.9.Graphsofswitchingsforvenjunctivefunctions....................................... 28

(graphofvenjunction,graphsofvenjunctivefunctions,

graphsandvenjunctivecompleteforms)

1.10.Venjunctionproperties.Basicfunctions........................................................... 30

(relationbetweenoperationsofconjunctionandvenjunction,inversionof

venjunction,operationswithmirrorvenjunctions,commutativityand

associativity,distributivity,idempotency,absorptions,rulesofzeroing,

venjunctionwithlogicalunity)

1.11.Venjunctiverepresentationoflogicalindeterminacy........................... 33

(indeterminacyofvenjunction,criterionforindeterminacy)


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2. APPLICATIONAREA

2.1.Bistablecell.................................................................................................. 35

(functionsofbistablecell,bistablecellswithelementsNANDandNOR)

2.2.Bistableelementsofasynchronouslogic................................................. 39

(logicalelements,venjunctor,doublevenjunctor)

2.3.Venjunctor............................................................................................................. 40

(schemerealizationsofvenjunctor,logicalcircuitofdoublevenjunctor)

2.4.Logicalcircuitsforexoticfunctions.................................................................. 43

(truncatedvenjunctions,logicalindeterminacy)

2.5.Triggers.....................................................................................................45

(SRflip-flop,JKflip-flop,toggleflip-flop,clockedD-latch)

2.6.Triggerfunction................................................................................................ 53

(representationandspecificoftriggerfunction,realizationfeatures,

conditionaltriggerfunctions,memoryformula)

2.7.Triggerapplications...................................................................................... 56

(lambdatrigger,triggeredfunctionofC-element,

triggerfunctionfordoubledbistablecell)

2.8.Examplesoflogicaldevices........................................................................ 61

2.9.Zonemodelforswitchingfunction.......................................................... 64

(1-zone,0-zone,-zone,basictransitionsofzonemodel,

zonedgraphofswitchingfunction)

2.10.Asynchronouslogicoffeedbacks............................................................. 70

(variantsofpositivefeedbacks,negativefeedback,

oscillationanditsinterruption,retentionofshort-pulsedsetting)

3. SEQUENTION

3.1.Sequentionasanorderedset........................................................ 76

3.2.Sequentionfunction....................................................................................... 77

(definition,sequention-functionincomparisonwithsequention-set,

correctsequentions)

3.3.Simpleandcomplicatedsequentions..................................................................... 81

(elementarysequentions,compositesequentions,embeddedsequentions,

embeddinglayers)


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3.4.Binaryrelationsincompositesequentions................................................... 83

(sequentionoftwosequences,sequentionandsequence,combinationoftwo

sequentions,orderrelationsandrelationofsequentions,sequentionswith

commonelements)

3.5.Functionallyimperfectsequentions............................................................... 87

(definition,examplesofimperfectsequentions,criterionsforfunctional

imperfection,compatiblesequentions)

3.6.Splittingandsplicingofsequentions................................................................. 91

3.7.Sequentionallaws........................................................................................ 93

(commutativity,associativity,distributivity,rulesofzeroing,absorption

rules,splicingrule,splittingrule,postulates)

3.8.Methodsfordecompositionofsequentions........................................................... 96 (non-systemizedsplitting,regularsplittingintominisequentions,separation

ofelements)

3.9.Formsforrepresentationofsequentions............................................................... 99

(relationofsequentionwithvenjunction,representativeforms,conjunctive

form,venjunctiveform,findings)

3.10.Algorithmsofsequentionsincreasing.............................................................. 102

3.11.Sequentor............................................................................................................... 103

(logicalcircuitsofsequentor)

3.12.Booleanoperationswithsequentions................................................ 106

(generalprinciplesforconjunctions,generalprinciplesfordisjunctions,

generalprinciplesforvenjunctions)

3.13.Transformationofcomplicatedsequentions......................................................... 110

(conjunctiveexpansion,venjunctiveseparation,unificationofsequentions)

3.14.Graphicsofsequentionsandmemorydepth......................................................... 114

(graphofsequention,memorydepthofcomplicatedsequentions)

AFTERWORD.............................................................................................................. 117


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1.VENJUNCTION

1.1.Binarysetsandsequences

Letusassumethat 1 2 n[ ... ] x x x isanorderedcollectionconsistingofallelementsofaset

1 2 n{ , ,..., } x x x .Theseelementsarebinaryvariableswithlogicalvalues0or1.Byreplacing

thevariableswiththeirvaluesabinarycombinationisformed.Generallythiscombination

canbeconsideredasawordinthealphabet{0,1},andthereforetermedbinaryset.

InthecontextofBooleanalgebra,asetissaidtobebinary,ifitcontainslogicalzero

(0) and unity (1) signs as values of certain Boolean variables. For example, variables

1 2 3 4 5{ , , , , }x x x x x with values 1 1x = , 2 1x = , 3 0x = , 4 1x = , 5 0x = form the binary set

[11010] . Here, the first in order unit determines a value of the variable 1x because it

occupiesthefirstpositioninthecorrespondingset.Similarly,secondunitassociateswith

the second symbol of the set, that is, 2x . And so on, up to zero value of the last, fifth

variable 5x .Thus,thereis positionalcorrespondencebetweenabinarysetanda sequence

ofsymbolslocatedinasetofBooleanvariables.

Formatofbinaryset

For realization of the positional principle established above, the concept of a format of

variables is entered. The format is represented in the form of a set of variables, the

sequenceofwhichpredeterminesvaluesofthesevariablesintheappropriatebinaryset.

Let's explain on an example. In view of the coordination of positions at values 1 1x = ,

2 1x = , 3 0x = , 4 1x = , 5 0x = ,theappropriatingbinaryset[11010] ispresentedinaformat

ofasetofvariables,ormoresimple,intheformat 1 2 3 4 5[ ]x x x x x .Therecordisallowed:

1 2 3 4 5[ ] [11010]x x x x x = . (1-1)

That same binary set, but formalized in a different format, namely 3 2 1 5 4[ ] x x x x x ,

corresponds to values 3 1x = , 2 1x = , 1 0x = , 5 1x = , 4 0x = . As a result of the change of

formatsotherequality,differentfromthepreviousone,isobtained:


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1 2 3 4 5[ ] [01101]x x x x x = . (1-2)

Thus,it ispossibletoconsiderthatbinarysetsarepositionedinaccordancewithaformat

ofvariables.Thankstotheformat,binarysetsareabletorepresentBooleanvariablesinanexplicitform.

Binarysequences

Unlikeset abinary sequence representsvaluesonly forsingle variable,butnot for

wholesetofthem.Sothesequence 11010 meansthatacertainvariablexispresentedby

five values fixed during time, since the moment 1t and finishing 5t . Point in time 1t

corresponds to the unity value 1( ) 1x t = , point 2t to 2( ) 1x t = , and further: 3( ) 0x t = ,

4( ) 1x t = and 5( ) 0x t = . Time depending variable ( )x t forms the sequence

1 2 3 4 5( ) ( ) ( ) ( ) ( )x t x t x t x t x t , which determines a format for the proper construction of

binarysequences,answering x .Inconsideredcaseafollowingequalitytakesplace:

1 2 3 4 5( ) ( ) ( ) ( ) ( ) 11010x t x t x t x t x t = . (1-3)

Thus,abinarysequenceispositionedintime.Eachvalueofacorrespondingvariableis

causedbyamomentoftimewhenthisvariableappearsinanexplicitform:0or1.

Asynchronoussequences

Adistinctivefeatureofasynchronousbinarysequenceisthatitdoesnotimplyanyexternal

controloftimepoints 1 2 3, ,t t t ,andsoon.Thereisnoexternalsynchronizerwhichwould

setthesepoints,placingthecertainmarksontheaxisoftimeduringwhichvariable ( )x t is

traced.Inotherwords,thebinarysequenceofaformat 1 2 3 m( ) ( ) ( ) ... ( )x t x t x t x t iscalled

asynchronousifatemporalsequence 1 2 3 m...t t t t isasynchronous.

Intersectionofsetsandsequences

Followingsaftereachotherbinarysets,aswellasasynchronoussequencescollected

together,in fact,representhowandwhichvariableschangetheirvalueovera fixedtime.

Inaddition,theserepresentationsareorthogonal;theyoverlap.Thereforebinarysetsand

sequences are able to be jointly displayed within the framework of a common table

(Table1.1).


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Table1.1.Binarysetsandasynchronoussequences.

1x 2x 3x 4x

1t 1X 0 0 0 1

2t 2X 0 1 0 1

3t 3X 1 1 0 1

4t 4X 1 1 0 0

5t 5X 1 1 1 0

Thistableisstructuredasfollows.Therearehorizontallylocatedbinarysets:

1 [0001]X = , (1-4.1)

2 [0101]X = , (1-4.2)

3 [1101]X = , (1-4.3)

4 [1100]X = , (1-4.4)

5 [1110]X = , (1-4.5)

whicharefixedinmomentsoftime 1t , 2t , 3t , 4t , 5t respectively.Allsetsarepresentedin

the format 1 2 3 4[ ] x x x x composed of four binary variables. To these variables answer

verticallylocatedsequences:

1 ( ) 0 0111x t = , (1-5.1)

2 ( ) 01111x t = , (1-5.2)

3 ( ) 0 00 01x t = , (1-5.3)

4 ( ) 1110 0x t = , (1-5.4)

which are presented in the format 1 2 3 4 5X X X X X composed of binary sets.

Asynchronouspropertyofbinarysequencesiscausedbyasynchronoustemporalsequence

1 2 3 4 5t t t t t .


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1.2.Logicalswitchings

AttentionisdrawntothatdatapresentedinTable1.1canbeexaminedasaseriesoflogical

switchings. Thus, underswitching is meant changing of valueof one or another binary

variablefromalogicalzerotologicalunityorfromunitytozero:0/1or1/0respectively.

Switchingsarecarriedoutinthefollowingorder.Firstitconcernsavariable 2x .Itchanges

thevaluefromlogicalzerotounity,becauseatthemomentoftime 1t equality 2 1( ) 0x t = is

observed, and at the next moment 2t 2 2( ) 1x t = . A switching on hand is 2 0 /1x = , it

causeschangeofabinaryset 1 [0001]X = to 2 [0101]X = .Furtherswitchings 1 0 /1x = ,

4 1/ 0x = and 3 0 /1x = , which meet the moments of time 3t , 4t and 5t with binary sets

3 [1101]X = , 4 [1100]X = and 5 [1110]X = respectively,takeplace.

Momentsofswitchings

Momentsofswitchingsaretiedtothemomentsoftimewhenthevaluesofvariablesare

fixed. This obvious circumstance admits different interpretations because of binary

variablebehaviordirectlyatthetimeoflogicalswitching.Therefore,eliminatingpossible

collisionsandwithoutlosingagenerality,webelievethatanyasynchronoussequence ( )x t

intheneighborhoodsofthemomentoftime jt obeysthefollowingrules:

j j-1( ) ( )x t t x t < = , (1-6.1)

j j( ) ( )x t t x t = . (1-6.2)

Untilthetime jt variablexdoesnotchangeitsvalue.Atthemoment jt switchingoccurs

andvariabletakesonanothervaluethatmeets j( )x t .Thisnewvalueholdsconstantuntil

the moment of the next switching. Concretely, if j-1( ) 0x t = but j( ) 1x t = , then at the

momentoftime jt variablexisswitchedfromlogicalzerotounity: 0 /1x= .If j-1( ) 1x t =

but j( ) 0x t = ,switching 1 / 0x= takesplace.

Within the framework of the offered concept an imaginary, in sense of "pseudo",

switchings of type 0 / 0x= and 1/1x= areallowedaswell.Asthistakesplace,equality

j-1 j( ) ( ) x t x t = accordingtowhichavariabledoesnotchangeitsvalueatthemoment jt is

obeyed.Realswitchingisalwaysaccompaniedbyaninequality j -1 j( ) ( ) x t x t .


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Backgroundforswitchings

If logical switching is an action, the role of background is rightly assigned to

pseudoswitching.Anexamplemayclarifythis.Whenpassingfrombinaryset 2X toset

3X (Table1.1)thefollowinghappens.Variable 1x changesitsvaluefromlogicalzeroto

unity.Atthesametimeothervariablesdonotchangetheirownvaluesandthuscreatea

peculiarbackgroundforswitching 1 0 /1x = .Backgroundswitchesatthemomentoftime 3t

areformallydisplayedintheformofpseudoswitchings: 2 1/ 1x = , 3 0 / 0x = and 4 1/ 1x = .

A background in itself is meaningful only in a combination with switching, for

exampleswitching 1 0 /1x = onthebackground 2 1/ 1x = .Asimplifiedrecordisallowed

also:switching 1 0 /1x = Xonthebackground 2 1x = .

Rulesforswitchings

Transitionfromanybinarysettothefollowingoccursduetoswitchingsofvariables.

Orderoftheseswitchingsiscausedbythecertainrules.

1.Allswitchingsareasynchronous.Theyareseparatedintimeinsuchamannerthat

duringeachmomentaswitchingofonlyonevariablecanhappen.

2.Eachswitchinggeneratesanewbinaryset.Thissetdiffersfrompreviousbyvalue

ofasingleonevariablewhoseswitching,infact,fixesthemomentoftimewhenbinaryset

changes.

1.3.Methodsforrepresentationofvariablescollections

(Table1.1asanexample)

Judgingfromconfigurationoftable,itscontentcanbeexpressedinanalyticalform.For

thispurposeitisenoughtotakeadvantageofthefollowingmethods.

Method1.Sequenceofbinarysets:

[0 0 01] [0101] [1101] [110 0] [1110] . (1-7)

Here,allvaluesofbinaryvariablesareinagreementwiththefollowingdenotations:

1 1 2 1 3 1 4 1 1[0 0 01] [ ( ) ( ) ( ) ( )]x t x t x t x t X = = , (1-8.1)

1 2 2 2 3 2 4 2 2[0101] [ ( ) ( ) ( ) ( )]x t x t x t x t X = = , (1-8.2)


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1 3 2 3 3 3 4 3 3[1101] [ ( ) ( ) ( ) ( )]x t x t x t x t X = = , (1-8.3)

1 4 2 4 3 4 4 4 4[110 0] [ ( ) ( ) ( ) ( )]x t x t x t x t X = = , (1-8.4)

1 5 2 5 3 5 4 5 5[1110] [ ( ) ( ) ( ) ( )]x t x t x t x t X = = . (1-8.5)

Forageneralcaseasynchronoussequenceofbinarysetsisgivenbythefollowingformal

expression:

1 2 3 m -1 m[X] ...X X X X X = , (1-9)

Heremisalengthofsequence,whichiscausedbynumberofpointsusedforfixingthe

valuesofBooleanvariables.

Method2. Set(collection)ofasynchronoussequences:

[ 00111 01111 00 001 11100 ] . (1-10)

Here,valuesassignedtobinaryvariablescorrespondtothecolumnsoftable:

1 10 0111 ( ) x t x = = , (1-11.1)

2 201111 ( ) x t x = = , (1-11.2)

3 30 0 0 01 ( ) x t x = = , (1-11.3)

4 41110 0 ( ) x t x = = . (1-11.4)

Inthegeneralcaseasetofbinarysequencesispresentedbytheexpression:

1 2 3 n -1 n[ X ] [ ... ] x x x x x = , (1-12)

wherenisthecardinalityofasetofinvolvedBooleanvariables.

Method3.Sequenceoflogicalswitchings.

Switchings of variables occur in accordance with priority, which is defined in the

followingsequence:


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2 1 4 3( 0 /1) ( 0 /1) ( 1/ 0) ( 0 /1) x x x x = = = = , (1-13)

With due account of the variables formatting logical switchings naturally present in

equality:

2 1 4 3 (0/1)(0/1)(1/0)(0/1)x x x x = , (1-14)

whosecomponentssatisfythefollowingrelations:

2 2 1 2 20 /1 ( ) / ( ) x x t x t = = , (1-15.1)

1 1 2 1 30 /1 ( ) / ( ) x x t x t = = , (1-15.2)

4 4 3 4 41/ 0 ( ) / ( ) x x t x t = = , (1-15.3)

3 3 4 3 50 /1 ( ) / ( ) x x t x t = = . (1-15.4)

1.4.Switchingfunctionvenjunction

On the basis of logical switchings a switching function is built. For its realization anoperationcalledvenjunctionisinvolved.

Operationofvenjunction

Verbal expression written as switchingon the background has a formalized

mathematicalrepresentation. Forthis purposea specialoperation named venjunctionand

designatedbysign isintended.Thissignlinksbinaryvariables.Asanexample,for

pairofvariablesxandytheexpressionswitchingxonthebackgroundyispresentedby

formulax y .

Venjunctionisanasymmetricallogicdynamicaloperation.Ittakesintoaccountthe

valuesofvariablesnotonlycurrent,butalsotheirpreviousmomentoftime.Theoperation

isasymmetricalinthesenseofinequality:

x y y x . (1-16)

Switchingxonthebackgroundyandreverseswitchingyonthebackgroundxaredifferent

actions, not compatible in time of their realization. Venjunctions associated with these


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switchingsaremirrorsintherelationtooneanother.

Switchingfunction

As a switching is named a binary function if it essentially depends on switchings of

arguments Booleanvariables. Thecorrespondingchangesareconsidered owing to thevenjunction, logical truth of which is determined by the value of binary variablez in

formula:

z x y= . (1-17)

Thisformulaserves asa basisfor construction ofswitchingfunctionwhich components

aretabulated(Table1.2)inviewoftheformatofTable1.1.Variablesxandy,asbeing

arguments,arefixedatthemomentsoftime j-1t and jt .Valuesofthesevariables j-1( )x t ,

j-1( )y t and j( )x t , j( )y t belongtobinarysets j-1X and jX respectively.

Table1.2.Componentsofswitchingfunction.

x y z

j-1t -1X j-1( )x t j-1( )y t j-1( )z t

jt jX j( )x t j( )y t j( )z t

Switchingfunctionispresentedbyavariablezdeterminedatthemomentoftime jt

withhelpofthefunctionofthefollowingtype:

j j j j -1 j-1 j-1( ) ( ( ), ( ), ( ), ( ), ( ))z t x t y t x t y t z t = . (1-18)

Takingintoaccountswitchingsofvariablesxandyadependenceobtainedisadequately

displayedbythefollowingformula:

j j -1 j j-1 j j-1( ) ( ( ) / ( ), ( ) / ( ), ( ))z t x t x t y t y t z t = . (1-19)


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Venjunctionasfunction

Operationofvenjunctionimpliesthatthe truthofthe respectivefunction,which is

expressedbythelogical unity j( ) 1z t = , is caused by switching j -1 j( ) / ( ) 0 /1x t x t = onthe

background j-1 j( ) / ( ) 1/1y t y t = . Corresponding values of components of the switching

functionarepresentedinTable1.3.

Table1.3.Truthtableforvenjunction.

x y z

j-1t j-1X 0 1 J

jt jX 1 1 1

When a logical switching 0 /1x= on the background 1/1y= takes place, the

operation of venjunction x y ensures the switching J/1z = , where J {0,1} . Symbol J

characterizesindeterminacy,whichinthiscaseresultsfromtwoabilities: 0/1z = or 1/1z = .

Equality j( ) 1z t = indicatesthatswitchingfunctionattainsalogicaltruevalue.Afalse,that

is zero, value j( ) 0z t = , accompanies switchings which answer data of Table1.3. These

casesare 0 / 1x= and/or 1/1y = .

Fromequality j-1( ) Jz t = itfollowsthatavalueassignedtovariable j-1( )z t doesnot

influenceon j( )z t .Therefore,forvenjunctionasforswitchingfunctionitischaracteristic

thefollowingdependence:

j j-1 j-1 j j( ) ( ( ), ( ), ( ), ( ))z z t x t y t x t y t = = . (1-20)

Inthecauseofswitchingofvariablesx,yandz,venjunctivefunctionisabletogenerate

variousbinarycombinations.AllofthemarecollectedintensubtablesofTable1.4.Each

subtableisconstructedinviewoftheformatacceptedforTable1.3.


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Table1.4.Switchingscausedbyvenjunction.

1) 2) 3) 4) 5)

0 1 0 0 0 0 1 1 0 1 1 1 1 0 0

1 1 1 1 0 0 0 1 0 0 1 0 0 0 0

6) 7) 8) 9) 10)

1 0 0 0 0 0 1 1 0 1 1 1 0 1 0

1 1 0 0 1 0 1 0 0 1 0 0 0 0 0

Givenanalyticalcalculationsserveasabasisforanumberofassertions,bymeansof

whichfunctionalpossibilitiesofvenjunctivedependencearecharacterizedinfullmeasure.Intermsoflogicalswitchingstheseassertionsareformulatedasfollows:

o if 0 /1x= onthebackground 1/ 1y= ,then 0 /1x y = ; (1-21.1)o if 0 /1y= onthebackground 1/1x= ,then 0 / 0x y = ; (1-21.2)o if 1/ 0x= onthebackground 1/1y= ,and j-1( ) 1z t = ,then 1/ 0x y = ; (1-21.3)o if 1/ 0x= onthebackground 1/1y= ,and j-1( ) 0z t = ,then 0 / 0x y = ; (1-21.4)o if 1/ 0y= onthebackground 1/ 1x= ,and j-1( ) 1z t = ,then 1/ 0x y = ; (1-21.5)o if 1/ 0y= onthebackground 1/ 1x= ,and j-1( ) 0z t = ,then 0 / 0x y = ; (1-21.6)o ifabackgroundis 0 / 0x= or 0 / 0y= ,then 0 / 0x y = . (1-21.7)

Formally,consideringpseudoswitchingsthefollowingexpressionsshouldbeadded:

o if 1/1x= and 1/1y = ,but j-1( ) 1z t = ,then 1/1x y = ; (1-22.1)o if 1/1x= and 1/1y = ,but j-1( ) 0z t = ,then 0 / 0x y = ; (1-22.2)o if 0 / 0x= and 0 / 0y= ,then 0 / 0x y = . (1-22.3)

1.5.Venjunctionincomparisonwithconjunction

JudgingfromTable1.3function z x y= ,aswellasconjunction x y ,producesunity

value,ifandasaresultofunityvaluesofinputvariables: 1x y= = .Butregarding 1x y =


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thereisonemoresufficientcondition,namely:logicalunitysettingforxmustoccuratthe

momentwhenunityvalue 1y= isalreadyfixed.Thatis,logicalswitching 0 /1x= mustbe

realizedonthebackgroundofasteadyvalue 1y= .Anessenceofoperationisexplainedby

adiagraminFigure.1.1.

Accordingtothediagram,atthemomentsoftimet1andt9,when 1y= ,andvariable

x changes the valuein connection with its switching from logical zero to unity (0/1), a

resultingfunction reducestounity.On acontrary dueto 0y= , switching 0 / 1x= atthe

moment t5 does not produce an effect on the graph of venjunction x y . Unity of the

correspondingfunctionismaintainedbytheunityvaluesofarguments.Whenavariable x

or y reduces to zero, function value becomes zero. In the diagram it is shown by the

moments of time t2 and t10, which are connected with switchings 1/ 0y = and 1 / 0x =

respectively.

x

y

x y

y x

x y

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10

Figure.1.1.Venjunctionandconjunction.Comparativediagram.

Amirrorfunction,asbeingvenjunction y x ,behavessimilarly.Ittakesonaunity

valueatthemomentst3and t6 when switching 0 / 1y= happensonthebackground 1x= .

Functionbecomeszeroatthemomentsoftime t4andt7.Accordingtodiagrammentioned

above mirroring is expressed in a form of the inequality ( ) ( ) y x x y , with the

following relation: if ( ) 1x y = , then ( ) 0y x = and vice versa. By such a manner a

venjunctionfunctiondemonstratesitssensitivitytothevariablesinterchanging.Behaviors of venjunctions x y and y x in Figure1.1 are displayed in


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comparisonwithconjunction x y .Soitisseen,thateachvenjunctioncanbeinterpreted

asatruncatedconjunction.Actually,whileitisrealizedunityswitching 0 /1x= onthe

background 1y= ,acoincidencebetweenvaluesoffunctions x y andx y isobserved.

In the opposite case, when signaly produces a unity value on the background 1x= a

conjunctionx y astoitsvaluedoesnotdifferfromthemirrorvenjunctiony x .

Thus,functionofconjunctionunitesconnectedwithitmirrorvenjunctionsbymeans

oftheformula:

( ) ( )x y x y y x = . (1-23)

The givenexpression formalizes the rule of disjunctiveexpansion of a conjunctioninto

twovenjunctionsascomponents.Orotherwise(reversereading),itisaruleofmerging

ofvenjunctions.

1.6.Methodsofvenjunctionformalizeddefinition

Venjunction,orvenjunctivefunction,isdefinedinaccordancewithdependence z x y= .

Without leaving the framework of Boolean algebra, we believe that operation x y

generatesafunctionwitharangeofvalues{0,1}.Belowtherearepresentedvariouskinds

ofmethods,whichareapplicableforavenjunctiondefining.

Methods1.Onabasisofthebinarysetsandsequences.

o if [ ] [11]x y = withcondition [ ] [01] [11]x y = ,then 1x y = ; (1-24.1)o if [ ] [11]x y = withcondition [ ] [10] [11]x y = ,then 0x y = ; (1-24.2)o if 0x= or 0y= ( 0x y = ),then 0x y = . (1-24.3)

Method2.Withinvolvinganundeterminedvalue.

Uncertain valueof avariabletakes placein the case, whenthis variableis not uniquely

defined.To this variable canbe assignedunity aswellas zerovalue. Uncertaintyin the

form of a symbol J isconvenient for using, if value of a variable is not known, or the

choice of value is free or indifferent (depending on a context). Formally, J is a third,

alongsidewith1and0,valueofabinaryvariable.However,thisvalueisnotself-sufficient


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becauseof J {10} .

Basedontheforegoingthefunctionofvenjunctionisdefinedasfollows:

oif [ ] [[01] [11]x y = ,then 0 /1x y

= ; (1-25.1)o if [ ] [10] [11]x y = ,then 0 / 0x y = ; (1-25.2)o if [ ] [J J] [ J 0]x y = ,then J / 0x y = ; (1-25.3)o if [ ] [J J] [0 J]x y = ,then J / 0x y = . (1-25.4)

Method3.Withinvolvingaconjunction.

o if 1x y = afteramoment 0 /1x= ,then 1x y = ; (1-26.1)o if 0x y = or 1y x = ,then 0x y = . (1-26.2)

Method4.Withinvolvingtheadditionalbinaryvariable.

Ontheabovediagram(Figure1.1)alongsidewithvariableswhichserveasargumentsfor

venjunction function, there is the auxiliary binary variable designated by a symbol .

Valuesofthisvariablearedefinedasfollows:

o 1= afteramoment 0 /1x= when 1y= ; (1-27.1)o 0= afteramoment 0 /1y = when 1x= . (1-27.2)

Momentsofswitching 0/1= arepointsintimet1andt9;switching 1/ 0= occursatthe

momentt3.Byusingtheauxiliaryvariablevenjunctionisdefinedthroughaconjunctionin

accordancewiththefollowingformula:

x y x y = . (1-28)

Similar expression for mirror venjunction differs only by the value of the auxiliary

variable:

y x x y = . (1-29)


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Method5.Notformal.Verbalreadingofvenjunction.

Thereareavarietyofoptions.Forexample,thedetailed(developed)record: thefunction

z x y= takesonaunityvalueinthecaseofswitchingofvariablexfromzerotounityat

theconstantunityvalueofvariabley.Establishedunityvalueoffunctionremainsuntilthemomentwhenxoryreducestozero .Truncatedvariant:thefunction z x y= becomes

unityatswitching 0 /1x= onthebackground 1y= ;itstaysinunityuntilvariablesxandy

retaintheirunityvalues .Exampleofminimizedreadingofvenjunction: 1z= fromthe

moment 0 /1x= onthebackground 1y= ,andupto 0x= or 0y= .

As required theopportunity exists,anditispossibleto introducevenjunctioninto

algorithmicconstructionswithformulation:ifxwheny,then

11.7.Truthtablesforvenjunction

Truthofthefunction z x y= iscausedbyvaluesandswitchingsofvariablesx,y,andit

canbeestablishedwiththehelpofthecorrespondingTables1.51.8.Table1.5represents

thefollowingdependence:

j-1 j j -1 j j-1 j( ) / ( ) ( ( ) / ( ), ( ) / ( ))z t z t x t x t y t y t = , (1-30)

according to which a function (z) and its arguments (x, y) take the form of logical

switchings.

Table1.5.Venjunctivedependence(1-30).

yz

1/1 0/1 1/0 0/0

1/1 J/J 0/0 J/0 0/0

0/1 0/1 0/0

1/0 J/0 0/0x

0/0 0/0 0/0 0/0 0/0

Leftcolumnofthetablecontainsswitchingsofthebinaryvariablex.Switchingsof


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thebinaryvariableyareplacedinupperrow.Inthesquareslocatedattheintersectionof

columnsandrows,switchingsofresultingvariablezarepresented.Forexample,if 0 /1x=

and 1/1y = ,then 0 /1z = . Indeterminationof J / Jz = causedbypseudoswitchings 1/1x=

and 1/1y= means that in the present context two results 0 / 0z = and 1/1z = can be

equallysupposed.

If switching values contradict the rules accepted for the binary variables, the

correspondingsquaresforzremainnotfilled.Forexample,thecombinationofswitchings

0 /1x= and 1/ 0y= is not subject to examination. This combination is considered

inadmissible because at once two variables simultaneously change their values that is

forbiddenapriori(p.1.2).

Table1.5contains all switchings concerned with the operation of venjunction,

including thosewhicharetypicalforavenjunctivefunction. Inview of accepted above

definitions(p.1.6)valuesofthisfunctionarepresentedbythesecondcomponentofthe

switching j-1 j( ) / ( )z t z t , that is j( )z t . The corresponding functional dependence is

representedbyformula:

j j -1 j j-1 j( ) ( ( ) / ( ), ( ) / ( ))z t x t x t y t y t = . (1-31)

Onthebasisofgivendependenceatruthtableofvenjunctionisconstructed(Table1.6).

Table1.6.Venjunctivedependence(1-31).

yz

1 0/1 1/0 0

1 0 0

0/1 1 0

1/0 0 0x

0 0 0

Anotherwayofvenjunctionrepresentationiscausedbyfunctionaldependenceofthe

followingtype:


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[01] [11] [11] [10] [10] [00] [00][10] [10] [11] [11] [01] [01] [00] [00][01] .

(1-33)

Thesequencecanbeadequatelyminimizedbyremovingallrepeatedsets.Then:

[01] [11] [10] [0 0] [10] [11] [01] [0 0] [01] . (1-34)

Inthegivensequenceallswitchingswhicharetypicalfortwovariablesarepresented.This

featureserves asabasisfor constructing yet anothertable (Table1.8).Thismastertable

asidefromabasicvenjunctionx y incorporatesallitsmodificationswhicharegenerated

bypermutationandlogicalinversionofthevariablesxandy.

Table1.8.Mastertruthtableforvenjunctions.

z x y

x y y x x y x y y x x y y x y x

1t 1X 0 1 0 0 0 0 0 J 0 J

2t 2X 1 1 1 0 0 0 0 0 0 0

3t 3X 1 0 0 1 0 0 0 0 0 0

4t 4X 0 0 0 0 1 0 0 0 0 0

5t 5X 1 0 0 0 0 1 0 0 0 0

6t 6X 1 1 0 0 0 0 1 0 0 0

7t 7X 0 1 0 0 0 0 0 1 0 0

8t 8X 0 0 0 0 0 0 0 0 1 0

9

t 1

X 0 1 0 0 0 0 0 0 0 1

Inthetablevenjunctivedependencesarepresentedinfunctionform:

( ) ( [X] )z t = . (1-35)

Theorder,inwhichbinarysets [ ]X x y= arearranged,iscausedbytheformat:


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1 2 3 4 5 6 7 8 1[X] X X X X X X X X X = . (1-36)

The values of each venjunctive function are displayed in the form of asynchronous

sequence:

1 2 3 4 5 6 7 8 9( ) ( ) ( ) . ( ) ( ) ( ) ( ) ( ) ( ) ( )z t z t z t z t z t z t z t z t z t z t = . (1-37)

Forexample,truthvalueofthevenjunction y x ischaracterizedbyasequence:

( ) 0 0 1 0 0 0 0 0 0z t = . (1-38)

1.8.Venjunctivefunctionsandtheirenumeration

Venjunctivefunctionsinessenceareswitchingfunctions;theiranalyticalrepresentationis

inextricably linked with application of the operation named venjunction. Venjunctive

function in the minimal form is an elementary venjunction of two variables. All logic

functionsoftwovariablesaresubjecttotheannouncedenumeration.

Venjunctivecompleteform

Venjunctivecompleteformistheuniversalformalwayavailableforrepresentingthelogic

functions having two variables. Such opportunity is a result of the informative part of

Table1.8, where each eight venjunctions becomes unity exclusively at one set of the

binarysequence.Itisnotablethatunderthissetallothervenjunctionsarereducedtozero.

Therefore, it is not difficult to construct any of possible 28 functions by means of the

Booleandisjunctiveoperation.Commonexpressionforthevenjunctivecompleteformis

givenbythefollowing:

i ii

( V )k , i {1, 2,...,8} . (1-39)

LetterVdenotesvenjunction.Fordefiniteness,itisassumedthat:

1Vx y = , (1-40.1)

2Vy x = , (1-40.2)


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3Vx y = , (1-40.3)

4Vx y = , (1-40.4)

5Vy x = , (1-40.5)

6Vx y = , (1-40.6)

7Vy x = , (1-40.7)

8Vy x = . (1-40.8)

Coefficientkregulatesthesamplingofsuchvenjunctionsthatarechosenforrepresentation

ofoneoranotherfunctioninitscompleteform.Representationtakesplaceatunityvalue

ofk( 1k= ).Intheoppositecase( 0k= )thecorrespondingvenjunctiondoesnottakepart

informationoffunction.

For example, if venjunctions V1, V2, V4, V5, and V6 are chosen, they form the

functionwhichinitscompleteformlooksasfollows:

( ) ( ) ( ) ( ) ( )z x y y x x y y x x y = . (1-41)

In view of the rule established above (p.1.5,1-23) for mirror venjunctions, the given

formulaisminimizedtothecompactexpression:

( ) z x x y= . (1-42)

Itis necessary tonote that acompact form cannot include more than fourvenjunctions.

Wholesetofvenjunctionscontainsfourmirrorpairs.Therefore,anyfifthvenjunctionisa

mirrorinrelationtoexistingvenjunction,andtogethertheygenerateaconjunction.

Note

In an effort to enumerate the functions, another expression is also applicable. It is an

alternativecompleteform,wherelogicaloperationsofconjunctionandnegationareused

insuchamanner:

i ii

( V )k , i {1, 2,...,8} . (1-43)

In this case chosen above venjunctions form a function represented by the following


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expressions:

1 2 4 5 6V V V V V ( ) ( ) ( ) ( ) ( )z x y y x x y y x x y = = , (1-44.1)

3 7 8V V V ( ) ( ) ( )z x y y x y x x y y x = = = . (1-44.2)

It is obvious that venjunctive completeformbased on the disjunctive operation is more

convenient for enumeration in contrast with another form based on the conjunction.

Alternative form is burdened with additional calculations, which are caused by logical

negation.

Enumerationoffunctionsoftwovariables

General aim of the proposed enumeration is to obtain a full collection of binary

functionswithtwovariables.Inthepresentcontextonlyvenjunctivefunctionsareimplied,

thatisfunctionscontainingatleastoneoperationofvenjunction.

All functional expressions listed bellowincorporate the venjunction ( )x y asat

leastoneoperation.Asamatterofconvenience,functionsarepresentedincompactform

and, what is more, they are classified depending on number of combined venjunctions,

conjunctionsandvariables.

Insuchamanner:

Onevenjunction:itis ( ) z x y= .

Venjunctionandavariable:

( ) z x y x= , (1-45.1)

( ) z x y y= . (1-45.2)

Venjunctionandtwovariables:( )z x y x y= . (1-46)

Venjunctionandconjunction:

( ) z x y x y= , (1-47.1)

( )z x y x y= , (1-47.2)

( )z x y x y= . (1-47.3)


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Venjunctionandtwoconjunctions:

( )z x y x y x y= . (1-48)

Twovenjunctions:

( ) ( ) z x y x y = , (1-49.1)

( ) ( ) z x y y x = , (1-49.2)

( ) ( ) z x y x y = , (1-49.3)

( ) ( ) z x y y x = , (1-49.4)

( ) ( ) z x y x y = , (1-49.5)

( ) ( ) z x y y x = . (1-49.6)

Twovenjunctionsandavariable:

( ) ( )z x y x y x = , (1-50.1)

( ) ( )z x y y x x = , (1-50.2)

( ) ( )z x y x y y = , (1-50.3)

( ) ( )z x y y x y = . (1-50.4)

Twovenjunctionsandconjunction:

( ) ( )z x y x y x y = , (1-51.1)

( ) ( )z x y x y x y = , (1-51.2)

( ) ( )z x y x y x y = , (1-51.3)

( ) ( )z x y x y x y = , (1-51.4)

( ) ( )z x y x y x y = , (1-51.5)

( ) ( )z x y x y x y = , (1-51.6)

( ) ( )z x y y x x y = , (1-51.7)

( ) ( )z x y y x x y = , (1-51.8)

( ) ( )z x y y x x y = , (1-51.9)

( ) ( )z x y y x x y = , (1-51.10)

( ) ( )z x y y x x y = , (1-51.11)

( ) ( )z x y y x x y = . (1-51.12)


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01

00

11

10

Fourvenjunctions:

(( ) ( ) ( ) )z x y x y x y x y = , (1-54.1)

(( ) ( ) ( ) )z x y x y x y y x = , (1-54.2)

( ) ( ) ( ) ( )z x y x y y x x y = , (1-54.3)

( ) ( ) ( ) ( )z x y x y y x y x = , (1-54.4)

( ) ( ) ( ) ( )z x y x y y x x y = , (1-54.5)

( ) ( ) ( ) ( )z x y x y y x y x = , (1-54.6)

( ) ( ) ( ) ( )z x y x y y x y x = , (1-54.7)

( ) ( ) ( ) ( )z x y y x y x y x = . (1-54.8)

To continuesearching for newfunctions andin such a manner to completecommenced

enumeration,it issufficienttoperformrearrangementandlogicalnegationoperationsfor

binaryvariablesofthegivenexpressions.

Thus, having 16functions which can be constructed in the framework ofBoolean

algebrausingtwovariables,itispossibletoadd240(!)venjunctivefunctions.

1.9.Graphsofswitchingsforvenjunctivefunctions

Alongsidewithtabularform(p. 1.7),avisualpresentationofvenjunctivefunctionscanbe

madebyusinganadequategraphicalconstructions.Inparticular,dataofTable 1.6arein

agreementwithagraphofvenjunction ( ) z x y= displayedinFigure1.2.

1

0

0 0 0 0

0

0

Figure1.2.Graphofoperationofvenjunction.

Nodesofthegrapharebinarysets[01] ,[11] ,[10] and[0 0] ,presentedintheformat


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Graphsandvenjunctivecompleteforms

Graphsof switchingsofvenjunctivefunctionsinessencearegraphicimagesofcomplete

formsofthesefunctions(p.1.8).Thereisone-to-onecorrespondencebetweenunityarcsof

a graph and venjunctions of complete form. For examples in Figure1.4, mentionedcorrespondenceforunityarcs(unbrokenlines)isvalidatedbythefollowingformulae:

( ) ( ) ( )z x y x y y x = , (1-55.1)

( ) ( ) ( ) ( ) ( )z x y x y x y x y y x = . (1-55.2)

1.10.Venjunctionproperties.Basicformulae

Relationbetweenoperationsofconjunctionandvenjunction

Expansionofconjunctionintotwomirrorvenjunctions:

( ) ( )x y x y y x = . (1-56)

Absorptionofconjunctionbyvenjunction:

( ) ( ) ( )x y x y x y = . (1-57)

Absorptionofvenjunctionbyconjunction:

( ) ( ) ( )x y x y x y = . (1-58)

Inversion(negation)ofvenjunction

LogicalnegationofvenjunctiondoesnotobeyDeMorganslaw:

( ) ( )x y x y y x = , (1-59.1)

( ) ( ) ( ) x y x y y x = . (1-59.2)

Negationofnegation,ordoublenegation:


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( ) ( ) x y x y = . (1-60)

Operationswithmirrorvenjunctions

Disjunctionrules:

( ) ( )x y y x x y = , (1-61.1)

( ) ( ) ( )x y y x y x = , (1-61.2)

( ) ( ) 1 x y y x = . (1-61.3)

Conjunctionrules:

( ) ( ) 0 x y y x = , (1-62.1)

( ) ( )x y y x x y = , (1-62.2)

( ) ( )x y y x x y = . (1-62.3)

Commutativityandassociativity

Commutativityandassociativityrulesarenotinherenttovenjunction:

( ) ( ) x y y x , (1-63.1)

( ) ( )x y z x y z . (1-63.2)

Alongwithinequalities,thefollowingformulaearevalid:

( ) ( ) ( )x y z x y y z = , (1-64.1)

( ) ( )x y z x y z = , (1-64.2)

( )x y y x y = . (1-64.3)

Distributivity

Mainly,distributivityisnotinherenttovenjunction:

( ) ( ) ( )x y z x y x z , (1-65.1)

( ) ( ) ( )x y z x z y z , (1-65.2)


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Rulesofzeroing

Inthegivencontextzeroingrulescoincidewiththesimilarrulesforconjunctionsascanbe

seenfromformulae:

0x x = , (1-71.1)

0x x = , (1-71.2)

0 0x = , (1-71.3)

0 0x = . (1-71.4)

Venjunctionwithlogicalunity

Inthecaseofvenjunctionabsorptions,whicharetypicalforconjunctions,donothappen.Theinequalitiestakeplace:

1x x ; (1-72.1)

1 x x . (1-72.2)

1.11.Venjunctiverepresentationoflogicalindeterminacy

Ifalogicfunctionafterandasaresultofsettingofknownvaluesofthevariablesdoesnot

certainlyreduceeithertozeroorunity,suchafunctionisconsideredtobenotdetermined,

thatisindeterminatefunction.Indeterminacygeneratesanambiguousnessintheformof

not uniquely defined value J {0,1} . This uncertain situation can be equally decided in

favorof J 1= aswellasinfavorof J 0= .

Indeterminacy of venjunction z x y= is caused by the factor of switching of

variables. Inthis context it is believed that the current values of argumentsare known.

Provided that unity values 1x y= = takeplace,thecorrespondingswitchingfunctioncan

notproduce aone-valuedsolution. In contrast, thevalue 0x= or 0y = ensuresacertain

value 0z = .Other value 1z= isproducedattheunityset [ ] [11]x y = ifandonlyifthisset

have beengenerated by the switching 0 /1x= on the background 1y= .Inothercaseof

mirror switching 0 / 1y = onthebackground 1x= , a value 0z = takesplace.Inthecase,

whenaswitchingisnotdeterminedproperly,anappropriativevenjunctiontakesaformof

thefollowingoperation:


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( ) (1 1)x y = . (1-73)

Thus,asconcerningthetermofindeterminacythefollowinglawsarepostulated:

1) if 1z and 0z ,then Jz = ;2) if 1 1z = ,then Jz = .

The above concept is corroborated and can be proved onthe following formal grounds.

Earlier (p.1.6) it was defined that a binary variable in the formula

( ) z x y = changes the value exclusively at the moment of switchings 0 /1x= and

0 /1y = .Therefore,iftheswitchingsthemselvesareonlypartlydeterminedas J /1x= and

J /1y= ,existingindeterminacyisextendedtoauxiliaryvariable,and J= .Then:

( ) J z x y= . (1-74)

As expected in the case of unity values 1x= and 1y= , logical indeterminacy is really

formed: Jz = .

Criterionforindeterminacy

Thereisarule,whichisvalidinlawonlyinreferencetoindeterminatevalues.This

ruleservesasasortofdistinguisherforJincontrastwith1and0.

Rule.Logicalnegationoftheindeterminacyisindeterminacy: J = J .

Assumption.Operation1 1 obeystherule J = J .

Statement.Venjunctionoftwologicalunitiesisindeterminacy:1 1 J = .

Proof:

J (1 1 1 1 (1 1) 0 0 (1 1) 1 1) J = = = = = . (1-75)


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defined.Correspondinglogicalrelationshipsaredisplayedintheformofswitchingtables.

Tables2.1showwhatkindofbinaryswitchingsappearontheoutputsofbistablecell

in response to the switchings of input signals. For example, appearance X J/1Z = in

response to 1/0X= and 1/1Y= means that owing to switching 1/0X= on the

background 1Y= bistable cell is transferred to unity state of the output XZ . Sign J

symbolizesdependencyoftheoutputsignalfromitsinitial,andthereforeunknown,value.

According to p.1.4, venjunctive function is defined at such values that complete

switchingsofoutputsignalinTables2.1.Inviewofthegivencircumstance,theexamined

bistablecellrespondstoinputswitchingsincompliancewiththedataofTables 2.2.These

tablesrepresentthefollowingfunctionaldependences:

X X ( , ) Z X Y = , (2-1.1)

Y Y ( , ) Z X Y = , (2-1.2)

where {1, 0/1,1/0, 0}X , {1, 0/1, 1/ 0, 0}Y , {1, 0}Z .

Tables2.2.Bistablecell(Figure2.1)functioning.

Y YZX

1 0/1 1/0 0 ZY

1 0/1 1/0 0

1 0 0 1 1 1

0/1 1 0 0/1 0 1

1/0 1 1 1/0 0 1X

0 1 1

X

0 0 1

Transforming tabulated switchings into venjunctions, and applying these

venjunctions to the given above dependences, the following logical expressions are

formed:

X ( ) ( ) ( ) ( ) ( )Z X Y X Y Y X Y X X Y = , (2-2.1)


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Y ( ) ( ) ( ) ( ) ( )Z Y X Y X X Y X Y Y X = . (2-2.2)

Theseexpressionsareoutputfunctionspresentedintheirvenjunctivecompleteform.They

combine all input venjunctions, unity values of which cause the outputs of X 1Z = and

Y 1Z = .Afterminimizations(p.1.10)onthebasisofequalities:

( ) ( )X Y Y X X Y = , (2-3.1)

( ) ( )X Y Y X X Y = , (2-3.2)

X Y X Y X = , (2-3.3)

outputfunctionsarereducedtothecompactformsasfollows:

X ( ) Z X X Y = , (2-4.1)

Y ( ) Z Y Y X = . (2-4.2)

Thus, unity value of outputZX (ZY) takes place at zero signal of inputX (Y), or at the

switching 0 /1X= ( 0 /1Y= ) on the background 1Y= ( 1X= ). Specifics of formulae

allowtheirotherwisereading,forexamplereferringtoZX:ifsignalX=0occurs,function

takesunityvalueandretainsit,evenwhenthevalueofXswitches,butonlyuntil asignal

Y=1isunchangeable,thatisuntilthemomentofswitchingY=1/0.

For X 0Z = (Tables2.2)acompactformulaanditscompleteformarepresentedas

follows:

X ( ) Z X Y Y X = , (2-5.1)

X ( ) ( ) ( )Z X Y Y X Y X = . (2-5.2)

Similarexpressionsforzerovalue Y 0Z = aregivenbytheformulae:

Y ( ) Z X Y X Y = , (2-6.1)

Y ( ) ( ) ( )Z X Y Y X X Y = . (2-6.2)


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BistablecellwithNOR-elements

BistablecellbasedonthelogicalelementsofNORispresentedbyitsstructuralschemein

Figure2.2.Output signalsZXandZY depending ontheinput switchings are displayedin

Table2.3.

ZX ZY

NOR

NOR

X Y

Figure.2.2.LogicalcircuitofbistablecellbuiltwithNOR-elements.

Table2.3.Bistablecell(Figure2.2)functioning.

Y YZX

1 0/1 1/0 0 ZY

1 0/1 1/0 0

1 0 0 1 0 1

0/1 0 0 0/1 0 1

1/0 1 0 1/0 0 1X

0 1 1

X

0 0 0

From comparing the data in Tables2.3 with the corresponding data in Tables2.2

followsthatafterreplacementoflogicalNAND-elements(Figure2.1)withNOR-elements

(Figure2.2)functioningofbistablecellchangesinacertainway.Formerlogicoperations

andstructureofformulaeremainvalid,butinputandoutputsignalsaresubjecttonegation.

Thus,functionsassumetheformofdependences:

X X ( , ) Z X Y = , (2-7.1)

Y Y ( , ) Z X Y = . (2-7.2)

Correspondingchangeshave aneffectonvenjunctiveexpressions.Logicofbistablecell(Figure2.2)functioningisdefinedbyformulae:


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X ( ) Z X X Y = , (2-8.1)

Y ( ) Z Y Y X = , (2-8.2)

X ( ) Z X Y Y X = , (2-8.3)

Y ( ) Z X Y X Y = . (2-8.4)

2.2.Bistableelementsofasynchronouslogic

Bistableelementsof asynchronouslogic aredisplayed inFigure2.3.Theseelementsare

chosen for consideration because of theirunique functionality.There are fourelements.

Bistablecellispresentedbytworealizations:BCbasedonNAND-elements(Figure2.1)

and BC based on NOR-elements (Figure2.2). Other asynchronous elements realize

venjunctivefunctionsofspecialkind.Thecorrespondingdevicesarevenjunctormarkedby

characterVanddoublevenjunctorDV.

ZX ZY ZX ZY Z ZX ZY

1) BC 2) BC

3) V

4) DV

X Y X Y X Y X Y

Figure2.3.Asynchronouslogicelements:

1)BCbistablecell;2)BC invertedbistablecell;

3)VVenjunctor;4)DVDoublevenjunctor.

Venjunctorisa two-inputlogicaldeviceintendedforimplementingthevenjunctive

function:

Z X Y = . (2-9)

Bistable V-element is displayed taking into account that venjunction is asymmetrical

operation. ThereforeZ output is displaced in the direction ofX input. In this way it is

denotedthattheinputsofvenjunctorarenotgraphicallyequalintheirrelationswiththe

output.Figure2.3-3presentsV-element,whichisintendedtorealizeaswitching 0 /1X=

on the background 1Y= . Accordingly,X input is assumed to be active in contrast to


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passiveinputY.

Double venjunctor unitestwovenjunctors so thattwo mirror venjunctivefunctions

areperformed.Theyare:

X Z X Y = , (2-10.1)

Y Z Y X = . (2-10.2)

2.3.Venjunctor

Venjunctorasalogicalelementisbuiltonabaseofbistablecell.Functionalpropertiesof

thiscellallowconstructingavenjunctorusingdifferentmethods.Ifabistablecell(Figure

2.3-1,Figure2.1)isapplied,itissuitabletodeduceavenjunction X Y bythefollowing

transformations:

X( ( ))Z X Z X X X Y X Y = = = , (2-11.1)

Y( ( ))Z X Z X X Y X Y X Y = = = , (2-11.2)

X( )Z X Y Z X Y = = , (2-11.3)

Y( ) Z X Y Z X Y = = . (2-11.4)

Thegivenfourformulaeserveas areferencefor constructinglogicalcircuits(Figure2.4,

Figure2.5, Figure2.6, Figure2.7). Each of these circuits performs the operation of

venjunction: Z X Y = .

Fromastructuralredundancyandsignalsracestandpoint,itcanbedeterminedthat

the logical circuit shown in Figure2.5 realizes a functionality of a venjunctor most

efficiently.

Z

AND

NAND NAND

X Y

Figure2.4.Logicalcircuitofvenjunctor(2-11.1).


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involved:

X ( ( ))X Z X X X Y X Y = = , (2-12.1)

Y ( ( ))X Z X X Y X Y X Y = = . (2-12.2)

Doublevenjunctor

Double venjunctor (Figure2.3-4) is presented by its logical circuit in Figure2.8.

Logicschemecontainstwobistablecellsconnectedconsecutivelyinsuchamannerthat

outputsofthefirstcellarelinkedwiththeinputsofanothercell.Inputsignalsareapplied

tobothcells.Onthecircuitoutputs,twoinventors(NOT-elements)areplaced.

Accordingtothelogicalcircuit,adependenceofoutputsignalZXfrominputsignals

XandYisrepresentedbythefollowingformula:

X ( ( )) (( ( )) ( ( )))Z X X X Y X X X Y Y Y Y X = . (2-13)

Taking into account equalities ( ( ))X X X Y X Y = and ( ( ))Y Y Y X Y X = ,

thegivenexpressionisminimizedasfollows:

X ( ) (( ) ( ))Z X Y X Y Y X = . (2-14)

ZX ZY

NOT NOT

NAND NAND

NAND NAND

X Y

Figure2.8.Logicalcircuitofdoublevenjunctor.


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Further logical actions, including zeroing ( ) ( ) 0 X Y Y X = , produce a

venjunctionasaresultofthefollowingoperation:

X

( ) Z X Y X Y = = . (2-15)

SimilartransformationsrelatedtooutputsignalZYensurededucibilityofanother(mirror)

venjunction:

Y ( ) Z Y X Y X = = . (2-16)

Thus,astodoublevenjunctor,itsfunctionalvalidityiscertainlyconfirmedbythelogical

circuitinFigure2.8.

2.4.Logicalcircuitsforexoticfunctions

Asexoticarenamedfunctions,whicharepracticallyuseless.Theironlyjustificationis

thattheyexistandcanbeexpressedanalyticallyaswellasdisplayedinformsoftablesand

logicalcircuits.Correspondingfunctionstakeplaceinthecaseofvenjunctiveoperations

withaconstantvaluelogicalunity.

Example1.Truncatedvenjunctionwithfunction 1Z X= .

ThelogicofvenjunctionfunctioningispresentedinTables 2.4.Correspondingrealization

ismadeonthebasisofvenjunctor(Figures2.42.7)keepinginviewthatthefunctionality

is restricted (truncated) because of the constant value 1Y = . In these conditions the

resultinglogicalcircuitisdisplayedinFigure2.9.

Tables2.4.Venjunction 1Z X= :

1)signalsswitchings; 2)switchingfunction.

1) Y 2) Y

Z

1/1 Z

1

1/1 J/J 1 J

0/1 0/1 0/1 1

1/0 J/0 1/0 0

X

0/0 0/0

X

0 0


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Z

AND NAND NOT

X

Figure2.9.Logicalcircuitofvenjunction 1Z X= .

Example2.Truncatedvenjunctionwithfunction 1Z Y= .

ThelogicofvenjunctionfunctioningispresentedinTables 2.5.Correspondingrealization

ismadeonthebasisofvenjunctor(Figures2.42.7)keepinginviewthatthefunctionality

istruncatedbecauseoftheconstantvalue 1X = .Intheseconditionstheresultinglogical

circuitisdisplayedinFigure2.10.

Tables2.5.Venjunction 1Z Y= :

1)signalsswitchings; 2)switchingfunction.

1) Y 2) Y

Z

1/1 0/1 1/0 0/0

Z

1 0/1 1/0 0/0

X 1/1 J/J 0/0 J/0 0/0 X 1 J 0 0 0

Z

AND

Y

Figure2.10.Logicalcircuitoffunction 1Z Y= .

Example3.Indefinitevenjunctionwithdegenerativefunction 1 1Z = .

Thelogic of venjunction functioning is presentedin Table2.6, whereit is seen that the

functiondegeneratesintoindeterminacy.Correspondingrealizationismadeonthebasisof

venjunctor(Figures2.42.7)keepinginviewthatthefunctionalityisrestrictedbecauseof


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the constant values 1X = and 1Y = .Intheseconditionstheresultinglogicalschemeis

displayedinFigure2.11.

Table2.6.Indeterminacy.

YZ

1

X 1 J

Figure2.11.Imageoflogicalindeterminacy1 1 .

(Elementwithoutinputdeadloop,orpointelementoutputfromnowhere).

Remark

Above mentioned examples are remarkable because of their clarity. It seems to be the

simplestwaytodemonstratethosenottrivialpossibilitiesthatbecomeavailableowingto

mathematicsofvenjunctiveformulae.

2.5.Triggers

Atriggerisassumedtobeacommonnameforallkindsofbinarydevices(includingflip-

flops,latches,bistablemultivibrators,etc.)withtwostableoutputstates.Eachstate,being

set,remainsstableuntilswitchingsignalappears.Logicalcircuitsfortriggersofvarious

typesareconstructedonthebasisofbistablecell.

SRflip-flop

ThelogicoftriggerfunctioningisgiveninTable2.7.Symbols SandRdenotethe

input signals. Output signal of the device is marked as Q. The S input is intended for

settingthetriggerinitsunitystate,thatmeanslogicalunityappearanceatthe Qoutput:

or


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setting.Furthermore,SinputshouldbereplacedwithJ,andRinputwithK.Asaresult,

thefunctiondisplayedinTable2.8,aswellastheJKtrigger,isformed.

Table2.8.JKflip-flopfunctioning.

KQ

1 0/1 1/0 0

1 0 1

0/1 1 1

1/0 0 1J

0 0 0

Accordingtothetable,logicofJKflip-flopfunctioningisdefinedbythefollowing

venjunctiveexpressions:

( ) ( ) ( ) ( )Q J K K J J K J K = , (2-21.1)

( ) ( )Q J K J K J K = . (2-21.2)

Anotheranalyticalrepresentationofthetriggerisgivenbytheformula:

( ) ( ) ( )Q J K J K J K J K J K K J = , (2-22)

containinglogicalexpressions ( )J K J K and ( )J K K J ,whichareassignedfor

outputunityandzerosettingsrespectively.

Classical implementation of JK flip-flop is based on the feedback usage with the

output signal participation. As a result, the function takes the form of functional

dependence ( , , )Q J K Q= ,whichisspecifiedbytheformula:

( ) ( ) ( )Q J Q J Q K Q= . (2-23)

Fromanasynchronouslogicstandpoint,theobtainedfunctionisnotwhollycorrect,


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becausethereareinputswitchingswithindefiniteoutputafter-effects.While 1J= setting

at the time of 1K = and 0Q = takes place, the output signal changes because of the

switching 0 /1Q = .Underthisswitchingthefollowingchangesoccur: ( ) 1/ 0J Q = and

( ) 0 /1K Q = . I f applied t o v enjunction ( ) ( ) J Q K Q , the occurred changes donot

ensure an expected switching ( ) 0 /1J Q = on t he background ( ) 1K Q = , while itis

necessaryforoutputsignaltobekeptinitsunitystate.Inthesecircumstancesareverse

switching 1/ 0Q = isadmissibleaswell.So,itispossibleforoutputsignaltoreturntothe

initialzerostate,thatsymbolizeanunsuccessfulefforttoperformtheunityvaluesetting:

1Q = .

Inactualpractice,aproblemoffunctionalindeterminacyisconsideredandsolvedasa problem of race hazard dangerous race of signals. To avoid these hazards various

schematic andtechnical methods areused. Usually, suchtime delays andsuch waysfor

signalspassingaredesigned,thattherightsignalalwaysturnsouttobeawinner.

In principle, existing race problem can be solved even at a function level. It is

sufficienttoblockundesirablesignals,andinthiswaytoexcludeindeterminacy.AstoJK

trigger,apossibleindeterminationsarecausedbyswitchings 0 /1Q = and 1/ 0Q = onthe

background[ ] [11]J K = .Thisproblemcanbesolvedanalytically,forexample,bymeansofthefollowingfunction:

( ) ( ) ( )Q J Q J Q K Q = , (2-24)

where venjunctions J Q and K Q are involved for performing unity and zero state

settingsrespectively.Correspondingswitchingsremovetheproblemofhazards.Duetothe

givenupdatesaJKflip-flopisfreefromhazardousfunctionalrace.

Remark

Generally JK flip-flop is realized as a synchronous device with additional clock signal.

Therefore,itwillbemorecorrecttorepresentupdatedtriggerasanasynchronousmodule

ofJKtrigger.

Tflip-flop(toggletrigger)

TherearetwotypesofTflip-flops:staticanddynamic.Theydifferfromeachother

bythemannerofactingoftheclockingprocedure.Statictriggeristoggledrespondingto


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theclocksignalidentifiedbyitsvalue,anddynamictriggerrespondstotheswitchingof

thesignal.

InstaticmodeTflip-floprealizesthefollowingfunction:

( ) ( )Q T Q T Q T Q= . (2-25)

Forthisfunction(aswellasinthecaseofJKflip-flop),araceproblemischaracteristic.

Switching 0 /1T = on the background 0Q = producesoutputswitching 0 /1Q = ,which

provokesraceprocessprolongeduptothemomentof 1/ 0T = .Atthistimeastableoutput

stateisformed,anditssignalcanbesettozeroaswellastounityvalue.Inthefirstevent

after output setting 1Q = invalid switching 1 / 0Q = takes place. In other case trigger

maintainstheunitystate,andsoperformsitsowntogglefunction.

InanefforttoprovideavalidfunctioningfortheTflip-flop,racehazardsmustbe

blocked,forexample,bytheclockpulsereduction.Combiningmentionedaboveandother

schematicandtechnicalmethods,itispossibletoavoidthehazardousswitchings 0 /1Q =

and 1/ 0Q = on the background 1T= completely. The logic of static toggle trigger

functioningispresentedinTable2.9.

Table2.9.StaticTflip-flopfunctioning.

QQ

1 0/1 1/0 0

1 F F

0/1 0 1

1/0 1 0

T

0 1 0

Dynamic T flip-flop in contrast to static trigger is free from dangerous problem

causedbyracedsignals.Allsolutionsrelatedwithblockingofhazardousswitchingsare

avoided not during the process of construction or realization, but beforehand when

logical function is produced. Considering that input signal always overtakes returnedoutput signal (in practice such assumption is quite justified), dynamic trigger can be


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implementedonthebaseofthefollowingvenjunctivefunction:

( ) ( )Q T Q T Q T Q = . (2-26)

Obtained toggle trigger is dynamical because of its output settings, that are

performed not during the time of unity signal 1T= , but at the moment of switching

0 /1T= .Allpossibleinputswitchingstherewithareallowedandeachofthemcausesa

validhazard-freetransitiontothenextstate.Thesestatesarepresentedbytriggeroutput

valuesinTable2.10.

Table2.10.DynamicTflip-flopfunctioning.

QQ

1 0/1 1/0 0

1 1 0

0/1 0 1

1/0 1 0T

0 1 0

Remark

Dynamic toggle trigger is able to function without any restrictions in relation to input

signals,andthereforedynamicTflip-flopbelongstoasynchronouslogic.

ClockedD-latch

In analogy with a toggle trigger, clocked D-latches are also subdivided into two

types.Staticanddynamictriggersdifferfromoneanotherbytheirfunctionalpossibilities.

CorrespondingTables2.11clearlydemonstratehowmentioneddifferencesarepresented

ontheoutputsofcomparedtypesoftriggers.

D-latch is clocked by T signal. Symbol J with apostrophe characterizes so called

conditional (partial) indeterminacy, when a signal is not certainly defined, because it

depends on another signal or on the previous value of the same signal. Partial

independency is secondary in relation to unconditional independency marked J. In this


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case,updatedsymboldenotes,thatapreviousoutputsignalisrepeated,previousvalueis

storedaccordingtopseudoswitchings 0 /0Q= or 1/1Q= .

Tables2.11.ClockedD-latchfunctioning:

1)statictype; 2)dynamictype.

D DQ

1 0/1 1/0 0 Q

1 0/1 1/0 0

1 1 0 1 J J

0/1 1 0 0/1 1 0

1/0 1 0 1/0 J

J

T

0 J J

T

0 J J

Analytically,functionsofstaticanddynamictriggersareexpressedbythefollowing

formulae:

( ) ( )Q T D T D T D= , (2-27.1)

(( ) ( ))Q T D T D T D = . (2-27.2)

Static D-latch is able to change its output signal repeatedly during a clock period

when 1T= . This circumstance produces independency as a factor of trigger behavior,

because it is not known, what a state will beset at the moment ofclocksignal reverse

switching 1/ 0T= .Toavoidconfusion,itisusuallyrecommendedtoshortenclockpulse.

However, even a short clock signal does not eliminate but only reduces the hazard of

triggernotproperfunctioning.Tosolvethisproblemcompletely,itisnecessarytoblock

dangerousswitchings 0 /1D = and 1/ 0D = onthebackground 1T= ,ortoassignoutput

valuesfortheseswitchings.Thelastsolutionisbasicallyusedfordynamictypetriggers

constructingpurposes.

Setting actions for dynamic D-latch are performed by venjunctions instead of

conjunctionsrelatedtostaticlatch.Asaresultofthisreplacement,triggerbecomesfree

from invalid consequences caused by repeated switchings. Changes of output state are


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exclusively controlled by switching 0 /1T= , under impact of which trigger reaches a

stable condition. Signal ofD input remains blocked until the next switching 0/1T=

appears.

When clock pulse is regulated in time, static D-latch functions as a synchronizedlogicaldevice.Astodynamictrigger,itsbelongingtoasynchronouslogicisundoubted.

Note

ResultsofD-latchswitchings(Tables2.11)arepartlynotdetermined.Thereareswitchings

inresponsetowhich,function Qreactsambiguously(J symbol). Itmeans thatmemory

depth of trigger device exceedsthe levelwhichcan be provided and maintained bythe

elementary venjunctions.Due to thisreason thefunctionof D-latchcannotbeexpanded

intocomponentsofvenjunctivecompleteform(p.1.8).

2.6.Triggerfunction

Allkindsofpresentedabovetriggersintheirimplementationsexploit,asa rule,a logical

constructionsbasedonthefollowinggeneralformulae:

Z X X Y = , (2-28.1)

Z Y Y X = . (2-28.2)

Theseformalexpressionsdefineatriggerfunction.Toidentifythisfunctionitisnecessary

toobeythefollowingconditions:

0X Y = ,( 0X , 0Y ); (2-29.1)

X Y , (2-29.2)

0X Y , (2-29.3)

0Y X . (2-29.4)

In the case when the conditional requirements are wholly satisfied, any function is

recognizedasthetriggerfunction,andaprocedureofitsrealizationbylogicaldeviceof

triggertypebecomesmucheasier.

Triggerfunctionis controlled bysubfunctionsXand Yaswellasbyvenjunctions


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X Y and Y X .Unitysignal 1X= formsthecorrespondingoutputstateof 1Z= .Zero

stateissetbyasignal 1Y= .

Aroleofkeeperforfixedstablestatesisassignedtomirrorvenjunctions.Switching

1/ 0X= on the background 0Y= maintainsthefunctioninitsunitystate,andswitching

1/ 0Y= onthebackground 0X= holdszerostate.

Trigger function in essence is a specific representation form for venjunctive

functionsofasynchronouslogic.Thisspecificityisexpressedbymeansofthefollowing

features.

Negationwithoutcalculations

Aresultoflogicalnegationorinversionprocedurefortriggerfunctionisatriggerfunction.

Structureof the functiondoesnot change.Negation asa logical operation is reduced to

interchangingthearguments:subfunctionXisreplacedwithsubfunctionY,andviceversa.

Conflict-freesettings

Conflictsofsettingsareusuallyprovokedbysimultaneousactionsofsignalswithopposite

purposes: 1X= and 1Y= . Trigger function is protected from suchlike conflicts by

definition, owing to zero equality 0X Y = .Conflictingsituationbetweenoutputvalues

1Z= and 0Z= isexpelled already atfunction levelas it isseenfromthe conjunction,

whichisreducedtozeroasfollows:

(( ) ( ) ) 0Z Z X X Y Y Y X X Y = = = . (2-30)

Butifinequality 0X Y isallowed,then 0Z Z ;thatisnotlogicallycorrectatleast

intheframeworkofBooleanalgebra.

Stabilizationofsettings

Stabilizationofsetstatesisensuredintriggerfunctionowingtoinequalities 0X Y and

0Y X .Correspondingvenjunctionsarenotzero,andsoeachofthemisabletokeep

its own state. Conflicts between the states are expelled, because of zero conjunction:

( ) ( ) 0 X Y Y X = .

Realizationfeatures

Triggerfunctionisperformedbyrealizingthe correspondingoperationsat theoutputsZX

andZY,logicalunityvaluesofwhicharesetbysignalsXandYrespectively.Obeyingtherule:


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Forexample,amemoryofSRflip-flopisexpressedbythefollowingformula:

S R S R= , (2-33)

accordingtowhichandtakingintoaccountthelockedbinaryset [ ] [11]S R ,outputstate

isstoredduringthetimeof[ ] [00]S R = .AsforstaticDlatch,itsmemoryismaintainedby

zerovalueofclocksignal,asfollows:

( ) ( )T D T D T = = . (2-34)

Memory formulaisrelated with triggerfunctionby meansofthe rule:if 0 /1 = , then

1/1Z= or 0 /0Z= .

Note

Initially,triggerfunctionwasconsideredtobeananalyticalrepresentationfortriggertype

devices.Howeveralongwiththis,triggerfunctionpossessesfundamentalopportunityto

represent wide variety of venjunctive expressions on the basis of this specific form. In

particular,itcanbeusedasanoff-beattemplateforgeneratingvariouslogicaldevices.As

well trigger function can represent sequential devices with memory that could not be

displayedusingvenjunctivecompleteform.

2.7.Triggerapplications

Triggeredlambda-function

Introduced above (p.1.6)auxiliaryvariable(lambda) is controlled byswitchings

0/ 1x= on the background 1y= and 0/1y= on the background 1x= .Inthecontextof

triggerfunctionstheseswitchingsserveasabasisfordefiningsettingsubfunctionsbythe

following venjunctions: X x y= and Y y x= . Availability of settings permits to

characterizethelogicoflambda-variablebehaviorbymeansoftriggertypefunction:

( ) ( )Z x y x y y x = . (2-35)

It is real trigger function because all required conditions are satisfied by the following


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01

00

11

10

expressions:

(( ) ( )) 0 X Y x y y x = = , (2-36.1)

( ) ( ( ) ( ))X x y Y y x x y x y = = = , (2-36.2)

0X Y (forexample,at 1/ 0x= onthebackground 1y= ); (2-36.3)

0Y X (forexample,at 1/ 0y= onthebackground 1x= ). (2-36.4)

Memoryformulaisformedasaresultofthefollowingcalculations:

(( ) ( ) ( ) ( ))x y y x x y y x x y x y x y = = = . (2-37)

Logicoflambda-triggerfunctioningispresentedinTable2.12.Thecorrespondinggraphis

showninFigure2.12.

Table2.12.Lambda-triggerfunctioning.

YZ

1 0/1 1/0 0

1 0 J

0/1 1 J

1/0 J J

X

0 J J

1

J

J J 0J

J

J

Figure2.12.Graphoflambda-function.


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In Figure2.13, a structural logic scheme for lambda-trigger is displayed. It is

constructedonthebasisofasynchronouselements:DVand BC .Twooutputsofdoubled

venjunctorarelinkedtotheinputsofnegatedbistablecell.

Z Z

BC

DV

x y

Figure2.13.Structuralschemeoflambda-trigger.

TriggerfunctionofC-element(MullerC-gate)

C-element is asynchronous logical gate. Regardless of realization, the logic of its

functioningiscertainlypresentedbyTable2.13.

Table2.13.C-elementfunctioning.

B

1 0/1 1/0 0

1 1 1

0/1 1 0

1/0 1 0

A

0 0 0

Tabulateddataareadequatelyconvertedintothefollowingvenjunctiveforms:

( ) ( )C A B A B B A = , (2-38.1)

( ) ( )C A B A B B A = . (2-38.2)


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Outputunityvalueissetbythebinarycombination[ ] [11]A B = ,andzerovalueby

[ ] [0 0]A B = .Otherswitchingsensureastoragemodeforoutputstates.

Setting signals are formed by conjunctions X A B= and Y A B= , which are

wholly applicable forconstructing a trigger function,becauseof their satisfaction to the

followingrules:

(( ) ( )) 0 X Y A B A B = = , (2-39.1)

( ) ( ( ) ))X A B Y A B A B= = = , (2-39.2)

0X Y (forexample,at 1/ 0A= onthebackground 1B= ); (2-39.3)

0Y X (forexample,at 0/1A= onthebackground 0B= ). (2-39.4)

Thus,C-elementrealizestriggerfunctionasfollows:

( ) ( )C A B A B A B= . (2-40)

MemoryformulacombinesinputsignalsbymeansofXORoperationinaccordancewith

theexpression:

( )X Y A B A B A B = = = . (2-41)

Triggerfunctionfordoubledbistablecell

In Figure2.14 a logical scheme is displayed. It consists of two bistable cells

connected in such a manner that outputs of one bistable cell are attached to inputs of

anothercell.Functionrealizedbythegivenschemeisdefinedbythefollowingformulae:

( ) ( ) ( )Z x x y x x y y y x = , (2-42.1)

( ) ( ) ( )Z y y x y y x x x y = . (2-42.2)

Requiredsettingsarecausedbysubfunctions:

( ) X x x y= , (2-43.1)


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Conditions, required for the function of doubled bistable cell to be considered as

triggerfunction,aresatisfiedinviewoftheexpressions:

(( ) ( ) ( ) ( )) 0X Y x x y y y x x y y x y x x y = = = , (2-44.1)

( ) ( )X x x y x y y x Y y y x = = = , (2-44.2)

0X Y (forexample,at 1/ 0x= onthebackground 0y= ); (2-44.3)

0Y X (forexample,at 1/ 0y= onthebackground 0x= ). (2-44.4)

Memoryofthedoubledbistablecellisgivenbytheformula:

x y= . (2-45)

Remark

AspresentedinTable2.14,thelogicofdoubledcellfunctioningcopiesthesimilarlogicof

C-elementwithanaccuracytotheinputassociations:A x= ,B y= .

2.8.Examplesoflogicaldevices

Inthecontext of trigger functions, logical devices withatypical, thatmeans venjunctive

settings,areofspecialinterest;aswellasdevices,storagemodeofwhichinsteadoftypical

binarysetsismaintainedbyinputswitchings.

Example1.

Trigger,outputsettingsofwhicharecausedbysignalswitchingsformedonthebasisof

venjunctionsofinitialvariables.

Unitysetting: X x y= .

Zerosetting:Y y x= .

Function: (( ) )Z x y x y y x = .

Memoryformula: ( ) ( )x x y x y x x y x y x y y x = = .

Tabulatedswitchings:


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Table2.15.

yZ

1 0/1 1/0 0

1 J J

0/1 1 J

1/0 J J

x

0 J 0

Example2.

Triggerwithtwopairsofmirrorvenjunctionsintendedforsettingactions.Unitysetting: ( ) ( ) X x y x y = .

Zerosetting: ( ) ( )Y y x y x = .

Function: ( ) ( ) ( )Z x y x y x y x y y x y x = .

Memoryformula: x y x y x y = = .


Table2.16.

yZ

1 0/1 1/0 0

1 0 J

0/1 1 J

1/0 J 1x

0 J 0

Example3.

Triggerwithsettingscausedbythreepairsofmirrorvenjunctions.

Unitysetting: ( ) ( ) ( )X x y x y y x = .

Zerosetting: ( ) ( ) ( )Y y x y x x y = .

Function: ( ) ( ) ( )Z x y x y y x x y x y y x y x y x x y = .

Memoryformula: x y = .


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Table2.17.

yZ

1 0/1 1/0 0

1 0 J

0/1 1 J

1/0 0 1x

0 1 0

Example4.

Triggerwithtwovenjunctionsintendedformemorystate.

Unitysetting: X x y y x y x = .

Zerosetting:Y y x x y y x = .

Function: ( ) ( ) ( )Z x y y x y x x y y x y x y x x y y x = .

Memoryformula: ( ) ( ) x y x y = .


Table2.18.

yZ

1 0/1 1/0 0

1 0 0

0/1 J 1

1/0 0 J

x

0 1 1

Example5.

Triggerwiththreevenjunctionsintendedformemory.

Unitysetting: X x y x y = .

Zerosetting:Y y x x y y x = .

Function: ( ) ( ) ( )Z x y x y x y x y y x x y y x = .


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the expansion is limited, becauseof the limits of venjunctive complete form itself.Any

compactlogicalexpressionusesnotmorethanfourvenjunctions(p.1.8).Thereforetwo-

input triggers, their setting functions, and memory formulae are restricted by four

components.

2.9.Zonemodelforswitchingfunction

Trigger-typeswitchingfunctionalwaysstaysinoneofthreestates.Theyare:thestateof

unity setting, the state of zero setting, and the state of memory. In accordance with

mentionedstates,threezonesareformed(Figure2.16).

Figure2.16.Zonemodelforswitchingfunction.

1-zoneisintendedforlogicalunitysettings.Itunitesallinputsignals(binarysetsand

theirsequences)requiredforoutputunityvaluesetting.Withinthiszone,settingsignalsof

X subfunction are active. While and because signal 1X= acts, trigger device holds the

unitystate 1Q= .

0-zoneisintendedforlogicalzerosettings.Itunitesallinputsignals(binarysetsand

theirsequences)requiredforoutputzerovaluesetting.Withinthiszone,settingsignalsof

Ysubfunctionareactive.Whileandbecausesignal 1Y= acts,triggerdeviceholdsthezero

state 0Q= .

M-zoneisintendedforstoringtheoutputstatethathasbeensetpreviously.Itunites

allinputsignals(binarysetsandtheirsequences)requiredforkeepingoutputstateinits

1-zone

( 1X= )

0-zone

( 1Y = )

-zone

( 1 = )


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unityorzerovalue.Withinthiszone,memoryformulaisactive.Whileandbecausesignal

1= acts, trigger device holds the constant output value, invariability of which is

maintainedduetopseudoswitchings 1/1Q= and 0 / 0Q= .

Allthreezonesareconnectedtogetherbytransitionchannels.Transitionfrom0-zoneinto 1-zone is initiated by switching 0 / 1X= on the background 0 = , and is

accompaniedbyzeroing 1/ 0Y= andsetting 0 /1Q= .Reversetransitionfrom1-zoneinto

0-zoneisinitiatedbyswitching 0 /1Y= onthebackground 0 = ,andisaccompaniedby

zeroing 1/ 0X= withthesetting 1 / 0Q= .

Transition from 1-zone into M-zone is initiated by switching 1 / 0X= on the

background 0Y= , and is accompanied by switching 0 / 1 = and keeping 1/1Q= .

Transitionfrom0-zoneintoM-zoneisinitiatedbyswitching 1/ 0Y= onthebackground

0X= ,andisaccompaniedbyswitching 0 /1 = andkeeping 0 / 0Q= .

Transition from M-zone into 1-zone is initiated by switching 0 / 1X= on the

background 0Y= ,andisaccompaniedbyzeroing 1/ 0 = andsetting 1Q= .Transition

fromM-zoneinto0-zoneisinitiatedbyswitching 0 /1Y= on the background 0X= ,and

isaccompaniedbyzeroing 1/ 0 = andsetting 0Q= .

Therearetransitions,whichdonotpresumemovingtootherzone.They arecalledintrazone(ascontrastwithinterzone)transitions.Intrazonetransitionsarecausedbysignal

actions,thatoccurredinsideofzoneanddonotinitiateexitfromit.Asaresult,zonemodel

retainsitsstate.Switchingfunction,subfunctionsXandY,andmemoryformulakeeptheir

states.

In accordance with the represented model, every switching of any input signal

activates one of transitions, thus initiating the movement inside or outside the

corresponding zone. Transition from one zone to another zone is performed without

participation of the third zone, which holds zero state. Every transition process is

completedbystablestatesetting.

Dependingonthemodeledfunction,someswitchingscanprovetobenotrealized.

Forthisreasonitispossiblethatsometransitions,aswellastheirchannels,disappear.

Thiscircumstanceis reflectedin theobjectsof asynchronouslogic. Inthespecificzone

configurationsonlyreal(active)transitionsaredisplayed.

InTables2.21typicalzonemodelsarepresented.Thesemodelsareconstructedon

thebasisoftheabovementionedtriggerfunctions(Examples16,p.2.8),whichareused


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source function (Example1, p.2.8), zone model presented in Table2.21-1 is also

characteristicofdynamicD-latch(p.2.5)andlambda-trigger(p.2.7).

AccordingtoTable2.21-2,onlyfourinterzonetransitionscanbeused.Theyconnect

both of the setting zones with the memory zone. Aside from the source function(Example2, p.2.8), the corresponding zone model is also characteristic of SR flip-flop

(p.2.5)andC-element(p.2.7).

Zone models presented in Table2.21-3 and Table2.21-4 are equally configured

without transitions inside of the memory zone. Such a model is characteristic of the

doubledbistablecell(p.2.7).

In zone model presented in Table2.21-5, transition within the setting 1-zone is

absent.ZonemodelinTable2.21-6doesnotuseintrazonetransitionsofthesettingzones.

AnalogousconfigurationischaracteristicofstaticD-latch(p.2.5).

Modeledtriggerdevisesareabletofunctionunderconditionswhenthenumberof

transitionsisrestricted.Accordingtothetriggerfunctiondefinition(p.2.6),thefollowing

transitionsmustbeobligatoryused:

bothoftransitionsfromsettingzonesintothememoryzone; atleastonetransitionintosetting1-zoneaswellasinto0-zone.

Four transitions is a minimal number of interzone transitions for any model of trigger

function.Correspondingbasic configurationsarepresentedin Figure2.17. Basic models

donotcontainoptionalintrazonetransitions.

(1) (2) (3)

Figure2.17.Basictransitionsofzonemodel.

Basicmodel(1)exactlyconformstotransitionsthatareshowninTable 2.21-2.As

forTable2.21-1,itsdatadiffersfromthebasicmodel(1)byoneintrazonetransition.In

otherexemplarytransitiontables,incontrasttofourbasictransitions,allsixofthemare

used.Similarconfigurationisformedasaresultofmergingthebasicmodels(2)and(3),

1 0 1 0 1 0


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that means(2plus3). Compositions, which combinebasic model(1) with model(2) as

wellaswithmodel(3)containfiveinterzonetransitions.

Zoneconfigurationsconformedtobasicmodel(1),aswellascompositionsformed

by merging basicmodels (2) and (3), are symmetric from the memory zone standpoint.

This symmetry is adequately revealed in Tables2.21. Interzone transitions are located

symmetrically about the diagonal thatcrosses squares assign

a synchronous logic elements

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