University of Nigeria Virtual Library

Serial No ISSN 1115-8443

Author 1

OSUAGWU, Charles Chukwudi

Author 2 AGADA, J. O.

Author 3 ANYANWU, C.D.

Title

Fast Minimisation On The Xiao Map Using Row Group Structure Rules

Keywords

Description

Fast Minimisation On The Xiao Map Using Row Group Structure Rules

Category

Engineering

Publisher Large Scale System Research Group

Publication Date July,1989

Signature

.

!- I - . - i I ' ' ! .~ .

FAST MINIMlSATION ON THE XIAO MAP USING ROW GROUP STRUCTURE RULES

C.C. Osuapu, CI9, Anyamwu a d JD. Agada . m t m n t *d klcctrorik Enginocrirg UnivDIsity of Nigcria, Nwkka.

AHS'rRArn

rndipkxcr p;lciUy:~.. uscd in the spthcsis d logic circuits with multiplcxcrs~ 21. LSI dcviccs like pragnmmahlc logic arrays .(PLA) rcquirc rninimis;llum of thc Bwlcan f u n h n if thcy arc to twc.. uwd clli&m~ly. Gale lcvcl minimisation is cx~cnrivcly aapplicd in ~ h c dcaign of IS1 and VLSI circuitry sincc a sibing of onc or m'ac gates could

' trmslatc to suhstanlial saying in silicon real csla~c. Finally, gate lcvcl minimisdon tcchniqucs arc irrcplaccablc . vchiclcs lor teaching .a systematic .,..r\r~\...-I. I* I,.":,. .l..~;"" I* -,,.-I h.. .c , - .N~:~PA. Ad

djsorncics involved in Lhe use af the technique T k rcrhninue ultimatelv uuarmntrrc a minima

2. XIAO MAi'

A W a n fundion amistikg of M midar, wL#t ~ a c b mintcrm bzs a MMblCs is denoted &y

M qx1, --. Xn) = L: Ki

i (1)

Cuoaion is mapped into a Xiao map d b rows and M columns as shown in fw L t=reh column represents a mintera Ki .and

'

mint- are arranged in the ascending a d a d @udc. The map var'ibles (Xi, ..., XI ) m; g!e&kxaul into ' group , d two variables 'XI* X31r:- L i XP stadnp: with CbC most s i n n i r r

T b e n a m b e r o t ~ r c q u ' r r e d i s ~ ' ~ i s 8 i n thiscase.Tbeeigkmwsareanangedastwo partitions AB a d CD. ThE minturns are arranged in ascending c d x d nqphk (6,7,9,11,13,14) as shown in figure 2 The b i i r e p d o n of minterm 6 is 0110. term of the tv& partitions, minterm 6 k AB = 01 and CD = 10. To plot this minterm, dots are p h d at the mtersedion of rows AB = 01 and CD = 10 with the Grst aIumn as ahown in figure 2 In the same way, the .other mint- are plotted on t h e - ~ i a o map.

X i map, . .

r m m m n o N ON mi^ XIAO-MAP

All minimisation, is based od the' cdncepl of. adjacency

- + * = A (3).

Therefore, if two rnintennsare adjacent, a vaciaMe (literal) is eliminated. For two minterms to 'bt adjacent on the Xiao map, the following conditions must be salisiled simultaneously

Fig. 2 : Xiw ...- r

\:!/r:WF;I.:('/ l IWL. I d ,No. I SEPTEMBER 1989 O s u a p . Anyan wu and Agada

,fj)4LeL keypoints muslllie oa-fbe aamc.row within 8 d l i l . b paditk,m.l.4atnt ooe.

(@)IIar&c.pdhnS.arhich tbt.- of& kypoints di&, & .kayptht~ mutit &e .cm llqgically9dwa~aar

S;milafij, the keylines h figure 4@) are adjaamt ;tnd w k n cornbid, eliminaie the variable k Uearly sben, rbe mnb'i-lan of two adjacent

0 -. ctass First class

Fig. 3 : Generation of first closs terms from 0 - class minterms.

, - - " :suiting in tcrm 1' in the first class. The irs1 class >mists of terms in which one variable has been iminated. The Oclass consists of the minrerm list.

Unlike the Karnaugh map technique, mL on a Xiao map is no1 a one step p

a!----. r ---- I- .L, n ,I,-- ,,, ,,L:,.

kevline ioinin~ rows 11 and 10 in the CD oartition re u e l - - - -. _ - - - - _ . . .- -. _. . . .

rimisation rocedhre.

l iu~accn~ Lcr- rn ulr; w-uas ill= w r r w u ~ d to form first cL rust cla CIS (1 US 3u;uuu U- wlwwts ul w r r r w ru nluru

two rariables have been eliminated). In genera!, the procedure k repeated with adjacent terms in the i-th class combined to form terms in the (j + 1)th :lass. Ceneraiha of ckssw terminates when there ire no adjacent terms in the (j + 1)th class.

ass terms. SimilarIy, adjacent terms in the ~ss are combined to form terms in the second a- -*..--A -t....., :&" A :" mA.:ph

kevlines results in the elimination of two variables.

Ir two t e r m nave a rcypom~ ano a ncyuuc u LUZ

same partition, they cannot be adjacent.

Proof:

The presence of a keypoint and a keyliae in one pattition. shows that the two terms have different variables eliminated hence thty cannot bc adjacent.

33, MINIMISATION PROCEDURE

Minimisation of a logical fundion on the Xiao map uses thc followinn three stem:

STEP 2: Eliminate any rcduncf.int tcrms using thq fccqucncy of occurrcncc of rnintcrms as criteria. (Xiao calls this elimination step cmcrgcncc of mintcrms).

!STEP 3: Obtain the rcduced logid cxprwion h m the Xiaa map.

We shall now apply the minimisation proccdurc to E k bUawing minterrn lists:

h(A,B,C,D,KF) =X m(0,2,~6,8,1Q,14j16,17;18 f 1,%24,26S,34J7,38,45,46,49,5OS$4,58,62)

(4) aud

W 4 & C @ B =

h(O,6&lfl,f2,1~,1~~;20,22~~,~,3).

Ia applying the pcedure, we- strall; pro~ose rules rfs;anprl r n ~ n h m n ~ jhp. I\F IPEP n F t h ~ Y;on

tbe adjacmq d e s and. of sections 3.1 d 3 2 respediveIy and tk h l b k g additional

Whcn a row,gxoup structure is cvidcnt imthc most cignifimnt' partition. ofL any class, crhaurlivc. in1 ro-group comhinsrion~ of at1 adjaccnu turn; i n 1 I gcncratc terms i l

Appl~cation, a7 m s rule ra the C)-cJaris, cd+'~&umS- r hth t k C 3-

in the Quincz-McCluskcy, mcthocifykitis ~~ im thc Is1 cubcs.

Note that minfcrms S,, 22, 37 and 9 do) ~mt! h- adj~cent intra-gmup,tcrms.. We- appiy rule 2iir:thii case.

4 3 RULE 2

I f any term c a m be wmhied withim. 3s rrnv eroum them i t shod& bc camfiined w i t h asy

ic r n h ~ m ( s )

byw-in

~errin~tr;bE&me5, t h e ~ s t b t c r m s ~ ~ a a i d an generated from O-cEass terms 5,21 rPd Ti, 53 respectively simce 5 and 37 have no sdjaotnt i~tr;tgroup ~crms. he, r i cias tern PL is na adjacent to any other class t e r n but its cornponen& &terms. an cwercd by s e d ders terms 42 and 52.

& sigmricanr parutma contams rne r w s mosc If a term cannot be combiied within or outside its ~ @ 5 c m t variables Such a row group s h d n r c group and the minterm(s) it covers do not muy also be e v i d e ~ in the first or higher dasses. appear as compents of any term in the next

Fm earmple, in lipre 5 &terms (0,5fi,t@,10,14) class, a search should be made in the preceding form a mw group because they all have a keypoint class to generate new terms that can conlbi . .

m m ; n i m ; r i s a i s c ~ n F ~ ( u o ~ p ) i U i h ruIc is used to gtmate the c&ss term 1 @) which can then k combiid with ~' (6~14) so as to generate the 2nd clam term I* (2', 11') = (6,14,22,30) needed t~ cwtr miaterm 6.

NiIOTECH VOL 13-No. 1 SEPTEMBER 1989 Osuagwu, Anym wu.nndAga& , " ' % .

(1) group structuring - &ses in Xiao map and . cubes in Quine McCIuskey.

(ii)Use of a selection mechanism to identify essential prime implicants - emergence table in the Xiao map and prime implicant table in the

' QGne McCluskey method. '

Infact, the Xiao map can be said to ,be a n approximate graphical representation of the Quine McCIuskey method;

The fundamental difference between the two 'techniques is in the of intermediate product terms.. While the- Quine McCluskey method requires - the formation of all possible product terms (48 l - ~ u b e terms in the &ample of ligwe 3, the Xiao'map k s a reduced set of these terms to cover the, fundion (27 Qt class terms as shown in, figure 5). This. is a significant reductian. This trend continuts in 'subsequent corresponding sets of product terms.

Tbis redliction in. the number of intermediate W u c t .terms makes the Xiao map faster to use thin the Quine McCIuskey method for functioos'of fie or more variables' The Quine McCluskey mothod is algorithmic and .so can be easily

..automated whereas the Xiao ma'p technique is

rpeed of minimisation and ease of use is valid only $ the manual .mode of minimisation. , . .

(2 CONCLUSION

We have pre&nte&an'adjaceky theorem and a set of. rules which exploit the row group structure oE the Xiao map to effect fast minimisation of Boolean Wens of five or more variables using the Xiao map. %e d e s ma$ miuimkation on the Xiao map more systematic. The Xiao, map resembles in many ways, a graphical represeptation of the Quine McCluskey technique. . - . . ...... -.. P l _ Our eMluatlou IS that me +ao qap IS a usqw new .graphical technique :for the synthesis of logic circuits, using gates. 1t should serve as a u+eful appplement to the familiar h a u g h map methd .fsr functions of five or more yubbles. -

REFERENCES

1.. BLAKESLEE, C T . ~ . , .Djgital Design with Standard l4jJ & LSI, John Wdey, New York 1979,2nd edition.

3, WWIN, D., Logical Design of Switching Circuits, Thomas ~ e k o n , h d o i 1974.2 nd edition

4. MANO, MM., Digital Logic and Computer. Design, PrenticeHall, Inc. New Jersey, lX"7..

5. YONG-XIN, XIAO., Xiao map for mhimkation of Boolean expressions, International J. of Electronics, Vol. 63, 1987, pp 353 - 358,

6.Hi4 F J., and PETERSON, G.R., Introductiqn to Switching Theory and Logical Design,, John Wiey, New YO* 1968. ..

4 .

. ..

2 BENNETH, ' L.A.M., . The,. application of mapentered variables to the use of multiplexers in the synthesis of logic functions, international 1. of Elearonics, Vol. 45, 1978, pp 373 - 379.

WIJOTECf f VOL. 13 No. ZSEPTEMRER J989 Osuapu, Anycruwu and. 59