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XOR realization using KH-mapA.T.M. Shafiqul Khalid and A. 4 . S. Awwal

Computer Science &- Engineering Department,Wright S tat e University, Dayton,Ohio 45435, USA

akhalidQvalhal1a.cs. Wright .edu

AbstractLogic design using XOR gates is becoming morepopular due to testability and cost effectiveness.For designing logic circuit, K-map is still an un-parallel tool to a logic designer. Due to geometricconstraint of I(-map, derivation of XOR relation isnot as simple as those incorporating A N D - O R rela-tion. However, for large problem one total ly relieson mechanical approach such as computer simula-tion where intu ition is less effective. KH-map isa newly proposed technique of mapping large logicfunctions in limited physical system and offers bet-ter way of realizing minimized logic functions. Thispaper gives soiiie rules that can be effectively usedin realizing XOR-relations from KH-map.

1 IntroductionPhysical construction of a I

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the set of these lines is called X ; line set.starting from X h in a clockwise direction.2n-i+l cells.End-for

Mark the end points of all chords as X i , X j//This step will generate a KH-map of

The minterms (combination of all binary liter-als) associated with each cell can be obtainedusing algorithm I.Identify the cells that contain the combinationsfor which function value F is TRUE. Mark midpoint of the arc of each of those cells by a smallcircle or node. This node will be called a ver-tex.

In all the figures and examples w, IC , y and z havebeen used instead of X q t X 3 , X 2 and X-1 . KX-ma pof three and four variables are shown in fig. 1.aand 1.c respectively. In the figure binary numberin each cell represents variables combinatiotn asso-ciated with that cell.

3 Minimization Using KH-mapDefinition 3.1 Adjacent nodes: Two vertices V kand 14 are called adjacent t o X , line if only a nd onlyif the address associated with the vertices differ byX , variable. For instance, if the address of V k and

are BI, and -B l respectively then BI ,@ Bi =X,.B, indicates binary combinations associated withvertex V, in ith cell. Analysis shows th at the fol-lowing conditions are sufficient for the vertices Vkand I$ to be adjacent to the X , line: (,a) the line( L ) joining the vertices Nk and Nl never intersectsX, ine where j > i ; and (b) X, is a perpendicularbisector of L.Definition 3. 2 KH-polygon: A polygon consist-ing of a collection of 2 vertices each aldjacent tom vertices of the collection, is called a 1I;H-polygon.The KH-polygon is said to cover all 2 vertices,where, 0 5 m 5 n. Two consecutive vertices of theKH-polygon must be adjacent. Th e KH-polygonwill be represented by a product term ( p ) whichcovers all minterms associated with the vertices ofth e collection. This polygon is called a IiH-polygonof dimension m.

Algorithm I:3.1 AND-OR minimization:

1. Start from the node of any cell in an anti-clockwise direction.

2. Collect the n end point markers of all X , linesfor i = 1 , 2 , ...,n on a first-come first--servedbasis. Th e end point markers will correspondto t he node address in cell Ci.

For example, if one moves in an anti-clockwise di-rection from the cell containing 1000 in Fig. 2(b),one gets the literals zyzxzyzw which after includ-ing only the literals that appear first yields zydw.Therefore, the combination associated with the cellis wzyz i.e., 1000. If for any cell the end pointsare wxyz, then the address of the nodes in thatcell is 0101. Algorithm I1 implies that the addressof any node can be easily calculated using the endpoint markers of chords in a KH-map. Therefore,the explicit address placement required in conven-tiona l mapping is no longer required in the proposedrepresent ation .

In order to obtain a minimal expression for aboolean function, all vertices must be covered withthe smallest possible number of KH-polygons, suchth at each KH-polygon is as large a s possible. A KH-polygon contained in another larger KH-polygonmust never be selected. Function F can be ex-pressed as a sum of those product terms t hat corre-spond to the KH-polygon(s) necessary t o cover allits vertices.

Example 1: Figure 2.a and 2.b ma p the func-tion F(x,y,z)= C(O,3,4,6) using KH-map andK-map respectively. In fig. 2.a vertex CL has no ad-jacent vertex(i.e., no mirror image). Therefore. thisis marked as a KH-polygon of zero dimension. lrer-tex b has only a single adjacent vertex d that formsa KH-polygon of one dimension. Uncovered vertexc makes cd KH-polygon. Now, the KH-polygons CL ,bd and cd cover all the vertices and prolduct termsfor them a re dy z, yz and xz respectively. Productterm of bd excludes 5 variable as z variable line(zz)

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perpendicularly bisects bd . Thus the expression forthe function is F ( z ,y, z )=zyz+yz +zz .

Example 2: Figure 2.c and 2.d represent thefunction F(w, x, , ~) C(3,5,8,10,11,12,14,15)using KH-map and K-map respectively. In fig.2.c vertex a form a KH-polygon of zero dimen-sion. Vertices b and g form a IiH-polygon of onedimension. Vertex c has two adjacent vertices dand f . But. froin the figure we observe that d isadjacent to e and e is adjacent to f resulting alarger KH-polygon cdef of two dimensions. TwoIiH-polygons cf and d e produce the similar re-sult. Uncovered vertex g form a one dimensionalKH-polygon with the vertex h, which eventuallyform a larger KH-polygon of two dimensions withtlic d e KH-polygon. Now, the IiH-polygons a , b y .c d e f and g d e h cover all the vertices and productterm for them are wxyz. xyz,wz and wy respec-tively. Product term for c d e f excludes z and yvariables as x variable line zx and y variable lineyy perpendicularly bisect the arms d e ( cf also)and cd . Thus the expression for the function isF ( w , z ,y, z)=wzy z+zyz t wz t wy. In fig.2.d equivalent subcubes of the KH-polygon bg andc d e f are formed by folding the Ii-map.

4 EX-OR minimization andproperties

Definition 4.1: Two vertices VI, nd V, will becalled EX-OR adjacent about lines X , and X , ifthere exist a vertex V such that V is adjacent to I,iand abou t line X , and X , respectively. This im-plies that combination associated with the verticesmust differ by only two variable.

Theorem 4.1: Two vertices V k and 15 will bealways EX-OR adjacent about lines X, and X,-1 ifline joining the vertices passes through the center.

Proof: From the construction of KH-map weknow that the single lines in an n variable map isX, nd X,-1. Al l othe r lines ar e 2(k >0 ) in num-ber. Line passing thro ugh the center intersect allthe Xi lines. Therefore, combination associated with the vertices must differ by X, and X,-1.

Theorem 4.2: If vertices V k and are EX-O R adjacent about lines X ; and X , and L is theline joining the vertices thena. L intersect only one X , line where i j .

Definition 4.2 : EX-OR polygon A poly-gon consisting of 2 vertices is called EX-OR poly-gon if two consecutive vertices are EX-OR adja-cent about any two variable line. EX-OR poly-gon will be uniquely determined by PQ where P =B1 @ B 2@ , ..,$B, where B1 ,B2, ...,B, representsthe address associated with vertices of the polygon.Q is EX- ORof all the variable not present in P.Q is called EX-OR product. For example in fig.3.b. EX- ORpolygon abcd will be repr esented asY(W 8 $ 2).

Polarity determination: If vertices Vk and14 are E X-OR adjacent about lines X , and X ,then EX-0R polygon containing only the verticesVL-nd I$ may be represented by P ( X , @ X , ) orP ( X , @ X , ) . Let the line L joining the vertices in-tersect only one X , line. Consider all the end pointsof X , line intersected by L in on a fixed side. If thereare even number of X , points then take X : other-wise take X,. ow, i f the points are X,,XJ or Xi,lYithen take P ( X ,@ X,) otherwise take P (X ,@ X j ) .

In order to obtain a minimal expression in theform EX-OR A N D O R, all vertices must be coveredwith the smallest possible number of KH-polygonsand EX-OR polygon, such th at each IiH-polygon isas large as possible.

Example 3: Figure 3.a. represents the func-tion F(w, , , 2 )= ( 1, 2, 7, 11, 13, 14) .n the fig-ure vertex a and b are EX-OR adjacent about z andw line. End points of z and wlines on t he s ame sideof ab i s either wx or wx. Therefore EX-OR poly-gon ab will be represented by yz(w@.>.roductterm yz can easily be found applying technique inthe previous section. For the EX- OR polygon c d e f ,E X-OR product would be 20 @ z @ z . Therefore ex-pression is y ( w CE z @ .) +yz(w @ .).

Example 4: Figure 3.b. represents the func-(1,2,4,7,8,11,13)and afterion F ( w , C , y, z ) =

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minimization we get F= y (w GI 5 e? )+wxyz +Z W ( Z ) .

5 ConclusionFrom the foregoing discussion it is clear that thetechnique provides an intuitive and geomietric wayto minimize logic function. It is also cleax that al-most the same approach used in realizing AND-ORrelation using KH-map can be used in XOR-AND-O R for other minimization. Th e only difference isthe way of defining KH-polygon and XOR,-.polygon.Examples used in this paper show how the EX-ORrelation can be realized simply. It is hoped tha tKH-map will be an important tool to analyze anddesign complicated EX-OR based circuits and EX-O R production rules.References1. S. B. Akers, ( Binary Dec is ion Diagram!

IE EE Trans Comp ut. , vol C-27, pp. 509-516,June 1978.

2. M. Karnaugh, The Map Method fofar Synthe-sis of Combinational Logic Circuits, Trans.AIEE. pt. I, vol. 72, no. 9, pp.593-599, 1953.

3. C . Y. Lee., (Representation of switching cir-cuits by binary decision programs, Bell Syst.Tech. J., vo l 38, pp. 985-999, July 1950.

4. E. J. McCluskey, Min imiza t ion of Boolean,func t ions , Bell System Tech. J . , vol. 35,no 6, pp. 1417-1444, Nov. 1956.

5 . S. Muroga , Logic Design and S witching The- ,ory, New York:Wiley, 1979.

6. W. E. Veitch, A Chart Method for S impl i fy -ing Trut h Func t ions , Proc. ACM, Pittsburgh.,May 2-3, 1952, pp. 127-133.

7. A. T. M . Shafiqul Khalid, A New MappingMethod for Boolean Func t ion Manipulat ion,Journal of Institute of Engineers, Bangladesh.,vol. 22 , no. 2, pp . 99-104, 1994.

Sim pl i jc at ion of Mul t i-var iable Boolean Func-tion, IE EE NAECON95 proc. Dayton, Ohio,pp . 256-261, 1995.

8. A. T. M . Shafiqul Khalid, Farid Ahnied, M. A,,Karim A Comp osite M apping Tec,hnique fo r

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J

Figure 1.(a ) ICH-map for 3 variables( b ) IC-map for 3 variables( c ) KH-map for 4variablesj d ) I

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X

c

x

d

Y'

wxyz

XY *

Z

w'Cz

VZ,

Figure 2. (a ) . ( b ) I iH -map and I(-map for the functionF ( 5 . y, Z ) =c(O,3.1.)(c), ( d ) I iH- map and I(-map for the functionF ( E . n',y. z )=~(3 ,5 ,8 ,10 ,11 ,12 ,14 ,13)

Figure 3 . (a)KH-map for the function F ( w , , , ) =( b ) K H - m a p for the functionF (w,z,y ,z)=~ ~ ( 1 , 2 , 4 . 7 , 8 , 1 1 , 1 3 )

(1,2,7,11,13,14).

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