State Reduction

Sedion5.7 State Reduct io nand Assignment 221

222 ChapterS SynchronousSequentla llogJc

There are aninfinitenumberof inputsequences thai maybeapplied to the circuit:each re- sultsin a uniqueoutputsequence,Asanexample.considerthe input sequence 01010110100 starting fro m theinitialstatea.Eachinp utof 0 or Iproduces an outputof0 or Iandcauses the circuit togo10the nextstale.From thestare^diagram.we obtain theccrpn^and..tateseqcence

for the given input seque nce asfollows:With the circuitininitialstatea.aninputof0produces anoutpu tof0and thecirc uitremainsinstalea.withpresent^stateaandaninpu tofI. theocr-

put is0and the next state isb.With presentstateband an input ofO.theoutput h. 0and the next stareisc.Conti nuing thisprocess.we find theco mplete sequeoce tobeas follows:

stat e inpul output

a

o o

a I

o

b

o o

c

o

I d

o o

f

I I

f o o

g I I

f o o

g

o o

a

Ineachcolumn.wehavethe presentstate. inputvalue.andoutputvalue.The nextstale is writ- len on la pof thenextcolumn.Itis importa nt torealizethai in thiscircuit thestates themsel ves areofsecondaryimportance.bec ausewe areinterested only inoutputsequences causedbyinput sequences.

Now let usassume that ..lie have foundasequential circuitwhose statediagramhas fewer tha nsevenstates. and..uppose wewish 10 compare this circuitwith the circuitwhosestate di- agramisgivenb)'Fig.5.25 .Ifidenticalinput sequencesare appliedto thetwo circuitsand iden- ticaloutputs occur for allinputsequences.the nthetwo circuits are said 10beequivalent(as farasthe input-outputisconcerned)andonemaybereplacedby theotber.Tbeproblemof stale reductionistofindwaysofreducing the number ofslates ina sequential circuitwithoutaltering the input-outputrelatio nships.

Wenow proceed10 reduce the number of states for thisexample. Firs t. we need the state table:it ismore con ..cnicnt to apply procedures forstate reduction with the use of a table rather than a diagram.The ..tatetable ofthecircuitislisted inTable5.6andisobtained directly from the slatediagram.

Thefollo wingalgorithm forthe statereduction of a completel y speci fiedstate table is given here withou tproof:"Twostates are said tobeequi valentif. foreac h member of the set ofin- puts.they gi..-cexactly thesame outputand sendthecircuit either to the same state or 10 an

Table5.6 Statt Tablt

Next State Output Present State x = 0 x = 1 x = 0 x= 1

a a b 0 0

b c d 0 0

c a d 0 0

d

, _f

0 I

,

a

f

^{0 I}

f s f

^{0 I}

s

^a

f

^{0 I}

SectionS.7 State Reduction and Assignment 223

Table5.7

Rfiiudng theStareTobie

Ne xl Slate Ou t p u t

Present State x = 0 x = 1 K = 0 X = 1

, ,

b 0 0

b c d 0 0

r rr d 0 0

d

, ,

e

f _f

⁰0 ^II

f

^e

f

^{0 I}

equivalentstate."When two states are equivalent.oneofthemcanberemoved without alter- ing the input-o utputrelationships.

Now apply this algorithm toTable 5.6.Goingthroughthe state table.welookfor twopres- entstatesthat gotothesame next state andhavethe sameoutput for both inputcombinations.

Statesgandearetwosuchstares:They bothgotostatesa and

f

and haveoutputsof0and I for.l = 0 and.r = I, respectively.Therefore.statesgandeareequivalent.andone ofthese statescanberemoved.Theprocedure ofremovingastateandreplacingit by its equivalentis demonstratedin Table 5.7.Therow withpresent stategisremoved. andstategisreplacedby stateeeach time itoccurs in the columns headed "Next State,"

Present statefnow has nextstateseandf andoutputs0and I for.r = 0 and x = I,re- spectively.The same nextstatesand outputsappearinthe rowwith present stated.Therefore, statesf anddare equivalent, and statej'canberemovedandreplaced byi/.The finalreduced tableisshownin Table 5.8.Thestate diagramforthereducedtableconsists ofonlyfive states and isshownin Fig.5.26.Thisstatediagramsatisfiestheoriginalinput-outputspecifications andwillproducetherequiredoutputsequencefor any giveninputsequence.Thefollowinglist derived fromthe state diagram ofFig. 5.26is for theinputsequence usedpreviously(note that thesame outputsequence results.althoughthe statesequence is different):

slate a a b c d e d d e d e a

input 0 I 0 I 0 I 0 0 0

output 0 0 0 0 0 1 0 0 0

Ta bl e 5.8

ReducedStote Tobie

Ne xt State Out p ut

Pr e se nt State x

=

^{0 x = 1 x}

=

^{0 x}

=

¹

a a b 0 0

b c d 0 0

c ^a d 0 0

d

,

d 0 I

e a d 0 I

224 ChapterS Synchronou sSequential logic

0,0

•

0.0

I"t)

FIGURE 5.26

Redu ced statediagram

In fact.thissequenceis exactlythesameasthatobtainedfor Fig.5.25ifwereplacegbyf'and f by d.

Checkingeach pair ofstaresforequivalencycanbedonesystematically b)'mean s ofa pro- ced ure thaiemploys animplic ati ontable.which consistsof squares.one foreverysuspected pairofpossibleequivalentstates. By jud icious useofthetable.itis possibletodetermi ne all pairs ofequivalen t Slates inastate table.Theuse ofthe implication table forreducing:thenum- berofstales ina state tableisdemonstratedinSection9.5.

Thesequential circuitofthisexa mplewasreducedfrom seventofive states.Ingeneral. re- ducingthe number of Mates inastaletable may resultina circuitwith less equiprrem,HQ\\"- ever.thefactthat astatetable has been reduced tofewerslatesdoes not guarantee asavin gin the number of flip-flops or the numberof gates.