sổ tay cdt chuong 36-logic so va tke to hop

8/6/2019 s tay cdt Chuong 36-Logic So Va Tke to Hop

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36Cc khi nim v logic s v thit k

logic t hp

George I. CohnCalifonia State University, Fullerton

36.1 Gii thiu ........................................................... .1

36.2 Biu din thng tin dng s ........................... .....1

36.3 Cc h s .............................................................2

36.4 Biu din s ................................................. .......2

36.5 S hc .......................................................... .......2

36.6 Chuyn i c s ........................................ ........336.7 Phn b ........................................................ .......4

36.8 M ......................................................................6

36.9 i s Boolean ....................................................7

36.10 Cc hm Boolean ..............................................9

36.11 Cc mch cng ..................................................9

36.12 Cc dng khai trin ........................................ .10

36.13 Nhn din ........................................................10

36.14 Cc lc thi gian ......................................11

36.15 Cc nguy c ................................................ ....12

36.16 Cc dng gin K .......................... ......... .....1236.17 Gin K v ti thiu ho ..............................14

36.18 Ti thiu ho vi gin K ............................15

36.19 Ti gin ho theo bng Quine-McCluskey .... .16

36.1 Gii thiu

Logic s gii quyt cc vn v biu din, truyn, thao tc, lu tr thng tin dng s. Mt i lng dng s ch c ccgi tr ri rc nht nh ngc li vi dng tng t c th c gi tr bt k nm trong di cho php. u im ni tri cadng s so vi dng tng t l n lm gim c cc nh hng ca nhiu nu nhiu khng vt qu mt ngng cho php.

36.2 Biu din thng tin dng s

Thng tin c th c m t v mt s lng hoc cht lng. Thng tin s lng cn mt h c s biu din cho n.Thng tin cht lng th khng nh vy. Tuy nhin trong c hai trng hp, thng tin c s ho u c biu din bimt tp hp hu hn cc k t khc nhau. Mi k t s l mt nh lng ri rc ca thng tin. Tp hp cc k t ny cs dng to thnh ch ci.

BNG 36.1 K hiu cho cc s

Dng lin k Dng a thc

S nguyn 010

21 NNnNnNN =1

0

nk

kk

N N R

=

=

S thp phn 1 2 1o

m mF F F F F + =1

kk

k m

F F R

=

=

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S thc 1 2 1 0 1 1 1o

n n m mX X X X X X X X X + = 1n

kk

k m

X X R

=

=

36.3 Cc h s

Thng tin s lng c biu din bi mt h c s. Mt k t biu din cho thng tin v s lng c gi l mt con

s. S cc gi tr khc nhau m mt con s c th c c gi l c s, c k hiu l R. Nhng k hiu biu din nhng gitr khc nhau ca mt con s c th c c gi l cc k t s. Cc k t s thng c s dng nhiu nht l 0, 1, 2, trong 0 biu din cho gi tr nh nht. Gi tr ln nht c th c trong mt h c s l c s c rt gn, r = R 1. Cc cs c gi tr khc nhau th i din cho cc h s khc nhau. V d nh R = 2 th h c s gi l h nh phn (binary), viR = 3 th ta gi l h tam phn (ternary), vi R = 8 ta gi l h bt phn (octal), vi R = 10 th ta gi l h thp phn(decimal), v vi R = 16 th ta gi l h c s 16 (hexadeximal).

Bt k mt gi tr no c biu din n l bng cc con s th gi l s nguyn. Mt s nguyn m l mt s nguyn cc bng cch ly mt s nguyn tr i s nguyn c gi tr ln hn v ta k hiu du tr ng trc s nguyn . Ly mts nguyn bt k chia cho s nguyn ln hn n ta s c mt phn s. Mt s c c phn nguyn v phn phn s gi lmt s thc.

Tt c cc s trong mt h s c cng mt c s, c s l c s cho mt h s. C l con ngi c mi ngn tay nnngi ta dng h c s 10 cho d s dng. Mi k t i din cho mt s khc nhau bao gm 0, 1, 2, 3, 4, 5, 6, 7, 8, v 9. H

c s 2 th hin thc cht nht cho cc h in t s bi v cc trng thi ri rc ng tin trong mt h thng c th l hiuqu nht v n c thc hin vi hai trng thi n nh ca cc phn t. Bnh thng hai k t c dng biu din ccs trong h 2 l 0 v 1. H 16 bao gm cc k t ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, v F vi c s R = 16), n rt quantrng bi v n c th chia ra thnh mt cm bao gm mt chui 4 s mang thng tin nh phn ci m c lu tr v x ltrong cc my tnh s.

36.4 Biu din s

Cc ch s cn mt hoc nhiu k t c th biu din theo cch khc nh l trong bng 36.1. Cc dng khc nhau sthun tin hn khi x l bng cc th tc khc nhau. Cc thut ton d dng nht vi cc s vit theo dng lin k. Cn dnga thc th d dng hn trong nghin cu l thuyt.

36.5 S hc

Cc thut ton x l thng dng nht: cng, tr, nhn, v chia c thc hin thun tin vi vic dng cc k hiu. Vicdng cc a thc rt c ch khi pht trin cc cng thc mang tnh h thng. K t khi p dng cc s vo trong k thut s,logic c s dng tnh ton vi cc s th gi l logic s. Logic s thng c thc hin bi phn cng. Tuy nhin nkhc vi logic ca i s boolean. Bn php tnh c bn l hnh thc to ra phng trnh hoc mt dng x l thut ton,nh trong hnh 36.2.

Vic x l thut ton trong h c s 2 c da vo hai php tnh l php cng v php nhn nh phn nh trong bng36.3.

Bng 36.4 a ra v d cho mi php tnh c bn.

BNG 36.2 Cc php ton s hc

BNG 36.3 Bng s nh phn n nguyn

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Cc khi nim v logic s v thit k logic t hp

BNG 36.4 Cc v d v ton t vi s nh phn

36.6 Chuyn i c s

Vic chuyn i mt s t h ny sang h khc c thc hin bng vic s dng cc khai trin a thc chui, minh hotrong bng 36.5.

Vic phn tch cc a thc c thc hin d dng hn vi dng ghp vo nhau ca chng. Dng ghp thu c t dngchui khi phn tch cc bin thnh tha s lin tip t cc gii hn, nh trong bng 36.6. Cc php nhn c phn tch didng ghp lm tng bc ca cc a thc mt cch tuyn tnh, trong khi cc php nhn dng chui lm tng bnh phngcc h s.

Ta c th dng phng php gim bc ca a thc chuyn i cc s nguyn t c s ny sang c s khc, nh trongbng 36.6(b). K t c t ngha nht ca s trong h c s mi l phn nh thu c sau khi chia s cn i cho c s mi.K t t ngha tip theo l phn nh thu c sau khi chia a thc lm gim u tin cho c s mi. Quy trnh ny c

lp li cho n khi thu c k t quan trng nht trong h c s mi l phn nh, khi m c s mi khng hp vi a thc lm gim cui cng na. V d trong bng 36.7 cng vi li ch dn ca thut ton minh ho v th hin c th phngphp chui a thc.

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chuyn i mt s thp phn t c s ny sang c s khc ta nhn lin tip s thp phn vi c s ca h s m taang nh chuyn sang. Mi php nhn vi c s s cho mt kt qu, n c cc k t c chuyn qua pha tri mt v tr.Php nhn chuyn k t quan trng nht ca s thp phn ti v tr bn tri du chm c s, du phn chia phn nguyn vphn thp phn, do vy ta tch n ra khi phn s. Quy trnh ny c minh ho trong dng i s ct bn tri ca bng36.8 v trong dng i s ct k tip. Hai v d minh ho cho phng php chuyn h c s th hin trong bng 36.8.

Bng 36.8 ch lm vi phn s hu hn, ci m phn thp phn s bin mt sau mt s bc nht nh. Trong trng hpphn s v hn ngi ta ch tnh sau mt s bc ln thu c mt s, vi kt qu gn chnh xc. Trong bng 36.9minh ho trng hp v hn. Mt dy cc s lp i lp li v hn v kt thc bng ng gch.

Vic chuyn sang h c s 2 t mt c s l bi s ca 2 c th c thc hin n gin nht bi vic chuyn i c lpca mi k t lin tip, nh minh ho trong bng 36.10(a). Vic chuyn ngc li t c s 2 sang c s 2 k c th thc hinn gin bng cch nhm k (bit) trong dy s thnh tng nhm, mi nhm bt u vi bt c ngha t nht cho phn nguynv vi bt c ngha nht cho phn thp phn, nh minh ho trong bng 36.10(b).

36.7 Phn bMi h c s u c hai phn b c s dng thun tin:

B c s ca N = NRC = Rn - N4


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Phn b h c s gim ca N = NrC = NRC - 1

R l c s v n l s cc k t trong s h N. Nhng phng trnh trn a ra cc phn b ca nhng s c ln lN.

Mt s dng c biu din bi mt k t trong hai k t ca ngn ng my l 0 v 1, trong h c s 2, th n ch nggi tr ca n. Mt s m th cn mt bit m ho du trong h c s hai. Ta c th thc hin vic ny bng cch m horing bit du v cc bit ln hoc bng cch m ho mt s m nh mt thc th n. Bng 36.11 minh ho bn dng mho khc nhau cho mt s m. Cc s m c th c biu din bng cch s dng bit u tin th hin bit du (0 cho du

+ v 1 cho du -) v cc k t cn li miu t ln ca s. Cc phn b v php dch rt c ngha trong vic m ho cc sm nh l mt thc th n thay cho vic phi dng cc k t ri rc m ho bit du ca chng. Vic s dng cc phn bcung cp cc dy gi tr bng nhau cn thit cho c s m v s dng. Cc s c chuyn cng c th cung cp cc dybng nhau cn thit cho c s m v s dng bng vic chn la cc gi tr dch chuyn tr thnh mt na cn thit cas nh phn ln nht v n c th c biu din bng cc k t sn c. M dch chuyn thu c bi vic tr m c xtcho gi tr dch chuyn, n c th hin ct ngoi cng trong bng 36.11.

Cc phn b lm cho vic tr cc s c thc hin qua vic cng vi phn b ca n. Nu kt qu thu c va trongs lng k t ti a cho php th s l hp l. Ta phi d on khi kt qu khng ng trong trng hp hin tng trns xy ra, s c nhiu k t hn so vi mc cho php. Bng 36.12 minh ho cc php tnh dng v khng dng phnb. Hai ct tn cng bn phi minh ho cc trng hp m cc kt qu trn so vi kch c cho php ca ln. iu kintrn c th c biu din trong cc gii hn ca hai thng s nh di y:

C0, tn hiu ra c nh t k t v tr bn tri nht

C1, tn hiu ra c nh t k t v tr th hai tnh t bn tri sang (tn hiu ra c ly t cm k t nu c sdng du)

Nu c hai thng s nh trn l ng nht (v d nh cng gi tr) th kt qu va vn vi s k t cho php v ta thu ckt qu ng. Cn ngc li, nu hai thng s l khng ng nht th kt qu l sai.

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36.8 M

Ngi ta pht trin nhiu dng m phc v cho nhiu mc ch khc nhau. C cc m ly t cc k t trong bngch ci c biu din c lp trong gii hn ca m trong mt bng ch ci nh hn. V d nh ta c th thm cc chci cng vi ch s biu din h BCD (binary- coded decimal) hoc m 8421 trong bng 36.13. Bc ca m 8421 thhin mc c a ti mi k t nh phn trong qu trnh m ho.

Mt s m c dng cho qu trnh gii ton c thun tin hn. M 2421 c th c dng trong vic biu din hthp phn. M 2421 c u im l m cho phn b hai cng ging nh phn b hai cho m, v iu ny khng ng trong mBCD. iu ny lm m 2421 thun tin hn trong vic nhn cc nhm k t c tch ring thnh tng cm. Ngi tacn a ra cc m nhm pht hin li xy ra trong lu tr cng nh trong khi chuyn i.

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HNH 36.1 Cm bin v tr 8-phn t vi li tip xc khng ng k: (a) cu hnh vt l m nh phn, (b) cu hnh vt l mGray

V d nh cc bit trng thi chn v l v 2 trong s 5 m c biu din trong bng 36.13. M pht hin li 2 trong s 5nh th l mi gi tr thp phn u c chnh xc hai gi tr trng s. Bit trng thi gn vi mt bit thm v n c gi tr l khitng s cc trng s l l (nu bit trng thi l c s dng), c gi tr chn khi tng s cc trng s c gi tr chn (nu bittrng thi chn c s dng). Mt bit trng thi n khng c kh nng pht hin mt s chn cc bit li. T , cc mtrng thi n ch cho cc mc bit li thp. Khi cn thm vo mt s bit cho cho cc bit trng thi kh nng pht hin v sa cha tt c cc li.

Ngi ta a ra mt s bit dng ngn chn vic biu din nhm cc kch thc da vo nhng li nh trong s linkt cc cm bin. Cc m Gray c s dng cho mc ch ny. M Gray, mt trong s nhng m dng cho cc v tr lin kc v mt vt l v logic v chng ch khc nhau mt k t nh phn. M Gray c th c to thnh cho bt k s no cacc k t bi s phn nh m Gray cho cc trng hp vi 1 k t t hn, nh trong hnh 36.13, cho cc trng hp 1, 2, v 3

bit m. u im ca vic m ho cm bin v tr tuyn tnh bi m Gray c minh ho trong hnh 36.1 cho trng hp ctm on m.

36.9 i s Boolean

i s Boolean cung cp phng tin phn tch v thit k cc h nh phn v da trn c s 7 nh ton hc trongbng 36.14. Tt c cc quan h khc trong i s Boolean c ly ra t 7 nh ny. Cc nh ny c biu din theodng hnh hc gi l s Venn v chng xut hin nhiu hn trong t nhin v trong logic. ch li ny thu c t vic thhin hnh nh hai chiu, gii phng vic th hin khi s bt buc phi th hin theo mt chiu bi nh dng ngn ng tuyntnh.

Cc ton t OR v AND bnh thng c ch nh bi cc k hiu ton t thut ton + v v m ch nh l tng vtch s cc ton t trong ti liu logic s c bn. Tuy nhin, trong cc h thng s thc hin cc php tnh thut ton th k

hiu ny khng r rng v cc k hiu cho OR v L cho AND lm hn ch s m h gia thut ton v cc ton tboolean. Vic a ra mt tp cc nguyn l l cch tt nht hiu ngha ca cc ton t boolean ny da trn khi nim, tpny s dng ton t hp U cho OR v ton t giao cho AND. Mt phn t trong mt tp ci m l giao ca cc tp lmt thnh phn ca mt tp AND mt thnh phn ca tp khc trong php giao.

Mt tp cc nh l c ly ra t cc nh d dng cho cc pht trin xa hn. Cc nh l c tng hp trong bng36.15. Vic s dng cc nh c minh ho bi s kim chng mt nh l trong hnh 36.2.

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HNH 36.2 Chng minh cho nh l 8: Khng thay i gi tr (a): x + x = x

HNH 36.3 V d v biu din cho nh ngha cc hm Boolean: (a) nh ngha biu thc Boolean, (b) phn tch biu thcBoolean, (c) nh ngha bng chn

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36.10 Cc hm Boolean

Cc hm Boolean c th c nh ngha v gii thiu trong cc s hng ca cc biu thc boolean v trong cc bngchn l nh minh ho trong hnh 36.3(a,c). Ta c th chuyn t dng ny sang dng khc. Cc gi tr hm cn cho vic xydng mt bng chn l c th c ly ra t vic phn tch cc hm nh minh ho trong hnh 36.3(b). S chuyn i ngcli s c minh ha sau .

Vi mt s c a ra ca cc bin th c mt s gii hn ca cc hm boolean. T mi bin boolean c th c hai

gi tr, 0 hoc 1, mt tp n bin c 2n

gi tr khc nhau. Mt hm boolean c mt gi tr c bit cho mi gi tr c th m ccbin c lp c th c. T c hai gi tr c th cho gi tr ca cc bin c lp, c 22

n hm boolean khc nhau ca n bin.

S lng ca hm tng ln rt nhanh vi s bin c lp, c ch ra trong bng 36.16.

Mi su hm boolean khc nhau ca hai bin c lp c nh ngha trong i s trong bng 36.17 v trong dng bng

chn l 36.18.

36.11 Cc mch cng

C th to thnh cc hm boolean t cc mch s. Cc mch thc hin cc hm boolean phc tp c th c chia nhhn vo cc mch n gin hn, thc hin cc hm boolean n gin hn. Cc mch thc hin cc hm boolean n ginnht gi l cc phn t c bn gi l cc cng v c biu din

bng cc k t c bit. Cc cng thc hin cc hm vi hai bin c lp c biu din trong bng 36.17. Cc cng cxc nh da vo tnh cht kt qu php ton m chng thc hin. Cc cng c bn nht l AND, OR, NAND, NOR, XOR,v COIN. Cc cng n u vo c tm quan trng l cng o hoc cng NOT. Cc cng l cc phn t c bn gip chngta xy dng cc mch logic s phc tp hn.

Mt mch logic c u ra n nh v ch ph thuc vo u vo n inh( v khng ph thuc vo cc tn hiu vo trc

) c gi l mch logic t hp. c th ph thuc vo cc tn hiu u vo trc th cn phi c b nh, nhng ycc mch logic t hp li khng c cc phn t nh.

i s Boolean cho php xy dng bt k mt mch logic t hp no ch da trn cc phn t cng c bn l AND, OR,v NOT. Ta cng c th xy dng bt k mt mch logic t hp no m ch da trn phn t NAND cng nh phn t NOR.

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36.12 Cc dng khai trin

Tng ca cc tch (SP) l mt dng c bn, n c th cha tt c cc hm boolean. Tch ca cc tng (PS) li l mt dngkhc m n cng c th cha tt c cc hm boolean. Cc v d minh ho c a ra trong hnh 36.4(b,c) l khai trin ca vd trong hnh 36.4(a)

HNH 36.4 Cc v d v chuyn i cc hm Boolean gia cc dng: (a) v d, (b) chuyn sang dng SP, (c) chuyn sangdng PS, (d) chuyn sang dng SP quy chun, (e) chuyn sang dng PS quy chun, (f) dng SP nh nht quy chun/k hiu,

(g) dng PS cc i quy chun/k hiu

Cc dng ti gin l mt tp c bit cc hm v n ch l duy nht. Mi dng ti gin c mi bin dng b hoc khngb kt hp vi nhau bi phn t AND. Mt khai trin SP ch c cc dng ti gin trong , l mt s khai trin SP theo quytc. Hnh 36.4(d) ch ra vic khai trin ca mt SP theo quy tc cho nhng v d trc . Ta c th biu din vic khai trinSP theo quy tc mt cch n gin bi vic lit k cc phn t ti gin nh trong hnh 36.4(f). Vic so snh bng chn lvi dng khai trin cc hm ti gin ch ra rng mi gi tr hm ca 1 biu din mt dng ti gin ca hm v ngc li. Ttc cc gi tr hm khc l 0.

Cc dng cci c cc tnh cht ging nh cc dng ti gin nhng chng c hnh thnh bi cc phn t OR. Mtkhai trin PS vi cc cc i trong cng l mt khai trin PS theo quy tc nh hnh 36.4(e) v chng cng c th cbiu din n gin bng cch lit k cc phn t cc i nh trong hnh 36.4(g). Vic so snh bng chn l vi dng khaitrin cc hm cc i ch ra rng mi gi tr hm ca 1 biu din mt dng cc i ca hm v ngc li. Tt c cc gi trhm khc l 1.

36.13 Nhn din

Cc dng khc nhau ca vic khai trin boolean em n cc mch khc nhau cho vic pht sinh cc hm mi. Mt hmbiu din theo cc SP c nhn ra trc tip bi vic thc hin phn t AND- OR nh trong hnh 36.5(a). Mt hm biu dintheo dng PS c nhn ra trc tip bi vic thc hin phn t OR AND nh trong hnh 36.5(b). Qua vic s dng s bini v nh l deMorgan, ta c th khai trin dng SP theo cc phn t NAND NAND v dng PS theo cc phn t NOR NOR nh trong hnh 36.5(c,d). Vic o ngc gi tr c bit u vo c th c cung cp t c phn t NAND hocNOR, nh hnh 36.5(g,h), sau chng a n cc mch NAND NAND NAND v NOR NOR NOR nh trong hnh36.5(i,j).

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HNH 36.5 Cc v d v nhn dng da trn cc dng khai trin bin i: (a) nhn dng AND-OR ca hmadbcdcadcbaf ++=),,,( ,(b) nhn dng OR-AND ca hm ))()((),,,( dcbcabadcbaf ++++= ,(c) chuyn i

AND-OR thnh NAND-NAND,(d) chuyn i OR-AND thnh NOR-NOR, (e) nhn dng NAND-NAND ca hmadbcdcadcbaf ++=),,,( ,(f) nhn dng NOR-NOR ca hm ))()((),,,( dcbcabadcbaf ++++= , (g) nhn dng

cng NAND ca cng NOT, (h) nhn dng cng NOR ca cng NOT,(i) nhn dng NAND-NAND-NAND ca hm

adbcdcadcbaf ++=),,,( ,(j) nhn dng NOR-NOR-NOR ca hm = + + + +( , , , ) ( )( )( )f a b c d a b a c b c d

HNH 36.6 Cc lc thi gian cho mch cng NAND: (a) lc vi thi gian, (b) lc vi thi gian

36.14 Cc lc thi gianCc lc thi gian bao gm hai dng chnh l lc thi gian nh v lc thi gian ln. Lc thi gian nh c

phm vi thi gian m rng trong khong trng hin th r rng tr ca cng nh trong hnh 36.6(a) cho mt cngAND. Lc thi gian ln c phm vi thi gian co li nh trong khong trng n ni tr ca cng l khng ng k,nh trong hnh 36.6(b) cho mt cng AND.

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HNH 36.7 V d v mt hazard (bin thin u ra) gy nn bi cc hng i tr khng phng: (a) mch cho minh ho mthazard, (b) trng hp l tng, khng tr, 0321 === , khng c hazard c a ra, (c) cc tn hiu vi tr khc

nhau, 3321 >++ , c hazard

u im ca lc thi gian ln l nhng khong thi gian ln hn c th c biu din trong mt kch c thi giancho php v chng c th c pht trin nhanh chng hn na. Nhng nhc im ca n l chng khng th hin cthng tin yu cu cn cho nhng s xem xt v gii hn tc .

36.15 Cc nguy c

S bin thin trong tr cc tn hiu khi chng qua cc phn t mch khc nhau trong cc ng dn khc nhau c thkhin cc tn hiu ra thay i bt thng so vi d on bi bng chn l khng ph thuc vo thi gian cho cc phn t. Sthay i bt thng c th gy ra mt kt qu khng ri rc v chnh l mt ri ro. iu ny c minh ho trong hnh36.7.

36.16 Cc dng gin K

bng chn l cc gi tr ca mt hm boolean c biu din theo mt dy mt chiu. Mt gin K cha ng thngtin tng t c sp t theo nhiu chiu nh hng nh l cc bin c lp trong mt hm. Vic biu din qua mt dng

c bit em n mt th tc n gin trong vic ti gin cc biu thc v t s lng cc b phn cn thit cho vic thchin mt hm theo mt dng cho. Hm biu din trong khng gian ca biu Venn c gi l mt tp vn nng. Gin K l mt dng c bit ca biu Venn. S phn chia khng gian lm hai phn l khc nhau cho mi bin c lp. Mimt bin c lp khng gian c chia lm hai vng c kch thc ng nht khc nhau, mi ci i din cho mt dng tigin ca hm. Vi n bin c lp ta s c 2n vng c kch thc ng nht khc nhau, mi vng i din cho mt dng tigin ca 2n dng ti gin ca hm. iu ny c minh ho trong mt dy cc hnh t 36.8 36.15.

Hnh 36.8 a ra cc dng gin K mt bin. Hnh 36.8(a) a ra vic chia khng gian thnh hai vng bng nhau, mivng dnh cho mt trong hai dng ti gin c th ca mi bin c lp. Ta cng c th xc nh cc din tch bi cc binc t ngoi khong trng nh r vng thuc phm vi ca bin, cng vi khong trng cha nh tn thuc phm viphn b ca bin, nh trong hnh 36.8(b). Cch khc trong vic xc nh cc khu vc l dng mt s vng cho dng tigin, nh hnh 36.8(c). Mt cch khc na l t cc gi tr ca bin theo mt t l c th dc theo khong trng nh tronghnh 36.8(d). Vic dn nhn a hp, hnh 36.8(e), xut hin ra cc d tha nhng chng cng thng c ch bi v cc dngkhc nhau c th s dng vi cc loi nhn khc nhau tu theo tng li suy ngh. Vic t cc biu thc ti gin hin thi vo

trong mi hnh vung l qu ln xn v him khi c s dng ngoi tr mc ch ging dy v l thuyt v gin K. Mcd c s dng rng ri nhng vic s dng cc nhm dng ti gin thng lm ln xn lc v phng php lun ca ra y s tr nn tha khi m cc khi nim c hiu.

HNH 36.8 Cc dng gin K mt bin: (a) cc nhn ti thiu bn trong, (b) nhn vng ngoi, (c) cc nhn s ti thiubn trong, (d) nhn t l bn ngoi, (e) dn nhn tng hp

HNH 36.9 Dng cu trc ca th K hai bin: (a) cc vng cho a, (b) cc vng cho b, tng hp

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HNH 36.10 Cc dng gin K hai bin: (a) cc nhn ti thiu bn trong, (b) nhn vng ngoi, (c) cc nhn s ti thiubn trong, (d) nhn t l bn ngoi, (e) dn nhn tng hp

HNH 36.11 Dng cu trc thay i ca gin K hai bin: (a) cc vng cho a, (b) cc vng cho b, (c) tng hp

HNH 36.12 Dng th K thay i 2 bin: (a) cc nhn ti thiu, (b) dn nhn tng hp

HNH 36.13 Cc dng gin K-3 bin: (a) ba chiu, (b) hai na tri, phi 3 chiu, (c) hai chiu

HNH 36.14 Cc dng gin K- 2 chiu ba bin: (a) cc nhn ti thiu, (b) cc nhn t l tng hp

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S tay C in t

HNH 36.15 Dng cc gin K cho cc hm t 2-6 bin c lp theo t l ph hp: (a) trng hp 2 bin f(a,b), trng hpba bin f(a,b,c), (c) trng hp 4 bin f(a,b,c,d) ; (d) trng hp 5 bin f(a,b,c,d,e); trng hp 6 bin f(a,b,c,d,e,f)

Vic sp xp gin K dng hai bin c minh ho trong hnh 36.9. Khong trng c chia nh theo ct thnh haivng cho bin a v phn b ca n, nh hnh 36.9(a,b). iu cn ch y l s so snh gia s nhn ca dng ti gintrong h nh phn v h thp phn. S dng ti thiu nh phn l n gin v vic ni ghp gia s t l ng v ngang chocc v tr. Vic s dng cch xc nh ny lm cc nhn trong cc hnh vung tr nn hon ton d tha.

Mt cch phn chia khong trng mt cch lun phin cho trng hp hai bin c minh ho trong hnh 36.11 v ct tn mt cch ln lt nh trong hnh 36.12. Cu hnh s dng trong hnh 36.9 s dng hai chiu cho hai bin, trong khi cu hnh s dng trong hnh 36.11 s dng mt chiu cho hai bin. Cu hnh hai chiu tr nn logic hn v thun tin hnvic s dng cu hnh mt chiu cho trng hp hai bin. Trong trng hp c nhiu bin hn th ta phi dng cu hnh cbit khc.

Vic sp xp cho dng gin K ba bin minh ha trong hnh 36.13. L hp l khi a thm vo mt khong trng chomi bin c lp c thm vo nh m t trong hnh 36.13(a). Tuy nhin vic khng thun tin qu mc khi lm vic vidng ny lm cho chng tr nn khng thc t. to ra s quy trnh thc hin th chng phi c t ra ngoi dng haichiu. Ta c th lm iu ny theo hai cch. Cch th nht l a ra cc lt ct ring l ca cu hnh 3D v t chng lin kvi ci khc nh minh ho trong hnh 36.13(b). Cch th hai l s dng cu hnh mt chiu cho hai bin nh trong hnh 36.12v 36.13(c). Trong trng hp c 3 n 4 bin c lp th dng a ra nh trong hnh 36.12 c nhiu tin ch hn v n cngc s dng thun tin trong trng hp c nhiu bin hn na( hnh 36.13. b). Hnh 36.15 minh ho cc dng trn. Hnh36.14 minh ho vic t tn cho 3 bin c lp.

Cc bin boolean c lp trong cc mc sp xp ng th t c chnh xc theo cng th t ging nh trong danh schi s ca hm boolean, nh m t trong hnh 36.15. S sp xp ng th t ca cc bin c lp vi mc sp xp tronggin K to nn mt chui cc v tr kt hp cho mt dng ti thiu( hoc mt dng cc i) ng nht vi s dng ti

thiu( hoc dng cc i). Vic s dng ng nht thc ny kh i cc yu cu sp t ca s nhn dng ti thiu xc nhtrong mi vung hoc cho mt bng nhn dng v tr ring bit. c im ny gia tng thi gian cn thit xy dng mtgin K v trnh cc li d xy ra trong khi xy dng. S dng ti thiu, c a ra bi s sp xp thnh chui ca cc skt hp theo dng ng hoc ngang, l r rng nu s dng cc h nh phn hay h thp phn.

36.17 Gin K v ti thiu ho

Mt hm c sp xp theo dng gin K thng qua vic nhp s liu cho gi tr mi dng ti thiu trong khong trng.Ta c th lu tr cc gi tr theo nhiu cch nh l t bng chn l cho hm, t biu thc cho hm, hoc t cc cch nhngha khc cho hm. Mt v d l bng chn l trong hnh 36.4(a) c lp li trong hnh 36.16(a) v gin K ca n cbiu din theo nhiu dng t 36.16(b-d).

Cc hm c th c sp xp vo trong gin K theo mc c th t trc tip t quy tc khai trin, iu ny tr nn

cn thit theo mt quy trnh nhp cc dng ti thiu( hoc dng ti a) t bng chn l. Cc hm cng c th c sp xptrc tip t bt k mt dng khai trin PS hay SP no. ngha khc ca vic lu tr theo gin K l xc nh n nh lmt bng hm, nh minh ho trong php nhn mt s c mt ch s vi s c hai ch s trong hnh 36.17.

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HNH 36.16 Cc v d gin K: (a) bng chn l, (b) gin K vi tt c cc gi tr biu din, (c) gin K dng cc tiu,(d) gin K dng cc i

HNH 36.17 Cc v d v gin K c cng thc ho nh l cc bng hm: (a) Gin K cho tch ca hai s dng 1-b P= x * y; (b) Gin K cho tch ca hai s 2-a, 3 2 1 0 1 0 1 0P P PP x x y y= ; (c) Gin K cho cc k t P3 ca tch hai s 2-b; (d)

gin K cho k t P2 ca tch hai s 2-b; (e) Gin K cho k t P1 ca tch hai s 2-b; (f) Gin K cho k t P0 ca tchhai s 2-b

36.18 Ti thiu ho vi gin K

c trng quan trng ca gin K l n lm cho chng thun tin hn trong qu trnh ti thiu ho vi cc dng tithiu trong cc khng gian lin k theo cc phng thng ng v phng ngang theo th t hp l. Dng lin k hp lca cc dng ti thiu l ng nht trong tt c cc bin tr 1 bin. iu ny cho php kt hp hai dng ti thiu thnh 1 vimt bin t hn( minh ho trong hnh 36.18). Hai dng ti thiu lin k kt hp vo trong mt lin kt bc mt. Mt lin ktbc mt l s kt hp ca tt c cc bin c lp tr 1. Trong v d ny, lin kt bc mt biu din trong cc s hng ca dngti thiu ci m cha 8 ch nhng biu thc ti gin th ch cha c 3 ch. Vic thc hin mch cho s t hp OR ca haidng ti thiu c hai cng AND v mt cng OR, trong khi s thc hin mc tng ng ch cn mt cng ANDn.

S t hp cc dng ti thiu trong lin kt theo th t u tin c th c thc hin cht ch hn na bi vic s dngcc k hiu biu din dng ti thiu n vi cc ch s xc nh dng ti thiu c bit biu din trong h nh phn, minh

ho trong hnh 36.18(d).C th kt hp hai lin kt theo th t u tin lin k thnh mt nh hnh 36.19. Mt lin kt theo th t u tin cha

ng tt c cc bin c lp tr hai. Nhn chung mt lin kt theo th t th n cha ng tt c cc bin tr n v cn mt tpgm 2n dng ti thiu c xp x theo nhm.

Cc dng ti thiu ti cc gc i din cng mt ct hoc mt hng l lin k hp l khi m chng hn km nhau 1bin. Nu khng gian mt phng c cun trn vo trong mt hnh tr vi cc gc i din lin quan vi nhau, th cc cpgc lin k hp l tr thnh lin k vt l. Khi s lng bin s l ln, ta s dng gin K vi cc mng song song, cc vtr tng ng trn cc mng khc nhau l lin k hp l vi nhau. Nu cc mng c xem xt che ph nhau th cc vung tng ng l lin k vt l. Ta c th thu c biu thc ti gin vi vic che tt c cc dng ti gin vi s lng tnht ca cc lin kt kh nng nht. Mt dng ti gin l mt lin kt theo th t 0. Hnh 36.20 minh ho nhiu loi v d.iu khng cn quan tm y l mt gi tr khng bao gi xy ra hoc nu n c xy ra th n cng khng c s dng vt s khng c vn g d gi tr l bao nhiu. Mt iu khng cn quan tm na l vic minh ho ci m chng s

dng ti gin ho cc hm qua vic trao i gi tr ca chng vi gi tr cc i trong cc th t lin kt.C th s dng cc gin K dng cc i thu c cc biu thc ti gin qua vic kt hp cc dng cc i vo trong

cc th t lin kt cao hn, minh ho trong v d hnh 36.21.

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S tay C in t

HNH 36.18 V d ca ti gin ho vi 1 gin K: (a) v d gin K, (b) biu din theo dng ti thiu ca vic nh nghahm trong (a), (c) n gin ho biu thc trong (b), (d) n gin ho biu thc trong (b) s dng dng ti gin k hiu n v

k hiu hm

Hnh 36.19 V d ca ti gin ho vi K-map

Hnh 36.20 V d ti gin vi Gin K su bin

Hnh 36.21 V d gin K dng cc i 3 bin

36.19 Ti gin ho theo bng Quine-McCluskey

Ti gin ho theo gin K l qu phc tp khi c t 6 bin tr ln v rt kh khn lp trnh cho n. Phng phpbng, c th thc hin vi bt k s lng bin no v vic lp trnh cho n cng d dng hn, bao gm cc bc sau:

Lit k tt c cc dng ti gin trong hm boolean (vi m nh phn) theo cc nhm c cha cng mt ch s ca1s. Cc nhm phi c lit k theo th t lin tip ca ch s ca 1s.

Xy dng danh sch theo cc lin kt theo th t th nht. S dng cc c xc nh cc dng ti gin, khngquan tm, hoc cc lin kt cng vi cc hm ca n( ch cc dng ti gin trong cc nhm lin k c kh nng trthnh lin k v t phng php th t ny gim mt cch ng k cng sc trong vic bin dch cc lin kt).

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Xy dng danh sch theo cc lin kt theo th t th hai v lit k tt c cc lin kt th t cao hn, cho n khikhng cn th t lin kt cao hn no c to.

Xy dng bng lin kt chnh. Bng lin kt chnh biu din lin kt chnh no bao ph cc dng ti gin no can.

La chn ch s nh nht ca cc lin kt chnh ln nht cho cc dng ti thiu.

Th tc trn c minh ho bng di vi hai hm boolean.

HNH 36.22 Minh ho cho phng php Quine-McClusky ca ti gin ng thi

Cc thut ng

C s: s cc gi tr khc nhau m mt con s n c th c. S m mt con s cn c nhn bng cch chuyn n i mtcon s sang bn tri.

Thp phn m ho theo nh phn (BCD): cc con s theo h thp phn c biu din dng nh phn.

Chui ghp ni: cc k t ni vi nhau to thnh mt chui ln hn, nh l cc k t trong mt t v cc con s trongmt s.

M: chuyn i biu din t h thng bng ch ci ny sang bng ch ci khc.

Phn b mt s: i lng t c bng cch ly i lng ln nht tr i s trong mt h thng s.

Con s: mt k t biu din thng tin v s lng.

M gray: l mt tp hp cc m c thuc tnh k lin logc

Lin k theo quan h logic: hai m no c cng mt s cc con s m gi tr ca chng ch khc nhau mt con s.

Biu Macrotiming: mt biu din ho th hin s bin i ca cc dng sng theo thi gian, nhng vi mt thang othi gian m khng phn gii th hin tr c to ra bi cc phn t c s ring ca mch s.

Dng cc i: l mt hm ca mt tp hp cc bin Boolean m s ch cho mt gi tr thp vi mt t hp cc gi tr bin vcho gi tr cao i vi tt c cc t hp khc ca cc gi tr bin.

Biu Microtiming: mt biu din ho th hin s bin i ca cc dng sng theo thi gian, nhng vi mt thang othi gian phn gii th hin mt cch r rng tr c to ra bi cc phn t c s ring ca mch s.

Dng ti gin: l mt hm ca mt tp hp cc bin Boolean m s ch cho mt gi tr cao vi mt t hp cc bin v c gitr thp vi tt c cc t hp khc ca cc gi tr bin.

Trn s: l mt phn ca mt php ton s tnh ton ra kt qu ra khng nm trong trng c ch nh.

Bt th hin tnh chn l: mt bt thm c to thnh mt chui ca mt m v mt gi tr a ra lm sao m s tng cacc bt cao l chn i vi trng thi chn v l l i vi trng thi l.

Tch ca cc tng (PS): Php t hp AND cc ai lng m c t hp theo lut OR t cc bin Boolean.S thc: mt s c c phn phn s v phn s nguyn.

Tng ca cc tch (SP): Php t hp OR ca cc i lng m c t hp theo lut AND t cc bin Boolean.

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S tay C in t

Bng chn l: l mt bng cc gi tr m mt hm Boolean c th c vi cc bin c lp c xem xt nh l mt s nhiuc s v c sp xp theo th t lin tip.

Ti liu tham kho

[1] Hayes, J.P. 1993.Introduction of Digital Logic Design. Addison-Wesley, Reading, MA.

[2] Humphrey, W.S., Jr. 1958. Switching Circuits with Computer Applications. McGraw-Hill, New York.

[3] Hill and Peterson. 1974.Introduction to Switching Theory and Logical Design, 2nd ed. Wiley, New York.[4] Johnson and Karim. 1987.Digital Design a Pragmatic Approach. Prindle, Weber and Schmidt, Boston.

[5] Karnaugh, M. 1953. The map method for synthesis of combinational logic circuits. AIEE Trans. Comm.

[6] Elec. 72 (Nov.): 593599.

[7] Mano, M.M. 1991.Digital Design. Prentice-Hall, Englewood Cliffs, NJ.

[8] McClusky, E.J. 1986.Logic Design Principles. Prentice-Hall, Englewood Cliffs, NJ.

[9] Mowle, F.J. 1976.A Systematic Approach to Digital Logic Design. Addison-Wesley, Reading, MA.

[10] Nagle, Carrol, and Irwin. 1975.An Introduction to Computer Logic, 2nd ed. Prentice-Hall, Englewood

[11] Cliffs, NJ.

[12] Pappas, N.L. 1994.Digital Design West, St. Paul, MN.

[13] Roth, C.H., Jr. 1985.Fundamentals of Logic Design, 3rd ed. West, St. Paul, MN.

[14] Sandige, R.S. 1990. Modern Digital Design. McGraw-Hill, New York.

[15] Shaw, A.W. 1991.Logic Circuit Design. Saunders, Fort Worth, TX.

[16] Wakerly, 1990.Digital Design Principles and Practices . Prentice-Hall, Englewood Cliffs, NJ.

Thng tin b sung

Thng tin hn na v cc khi nim logic c bn v thit k logic t hp c th tm thy trong cc bi bo v cc tp chsau:

Lecture Notes in Computer Science (annual)

International Journal of Electronics (monthly)

IEEE Transactions on Education (quarterly)

IEEE Transactions on Computers (monthly)IEEE Transactions on Software Engineering(monthly)

IEEE Transactions on Circuits and Systems 1. Fundamental Theory and Applications (monthly)

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sổ tay cdt chuong 36-logic so va tke to hop

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