**Originally published on Computer Jagat in 1994
**

*A.T.M. Shafiqul Khalid (Tuhin)*

This article presents a new technique to simplify Boolean Expressions. The technique is simplier, more generalized and efficient than conventional K-map method and applicable for any number of variables.

**INTRODUCTION**

Karnough map method of simplification of switching function becomes very difficult as the numbers of variables increase. For each combination of n variables or bits there exist exactly n different binary codes with Hamming distance 1. i.e., each binary combination has n adjacent combinations. Representation of such adjacency relation by Karnaugh map is very difficult for large values of n. The proposed KH-map method, however, will be able to overcome such difficulties.

**PROPOSED KH-MAP**

METHODOLOGY & REPRESENTATION OF FUNCTIONS

A KH-map is a modified form of a truth table in which the arrangement of the combinations is particularly convenient. The new maps for functions of three and four variables are shown in fig. 1 and fig. 2 respectively. Proposed map for n variables containing 2″ cells equally spaced within a circle represents all possible combinations of n variables. Line(s) passing through the center that have been used for dividing cells, are MSB line(s). If v is the ith MSB variable then ith MSB lines or simply v-lines will be max( 1, 2*-2) in number. Those v-lines divide all the cells equally as formed previously by other MSB variables

of v. The combination contained in a cell can be calculated using following simple KH-algorithm:

SIMPLE KH-ALGORITHM

1. LetCi,C2, …,C2n represent all cells.

2. Set Ci = 0, m = 1 for i = 1 to n do

for k = 1 to m do

Cm+k = Cm-k+l +m

end – for

m =2*m end – for

Alternative approach is to add 2k with each cell formed by mirror image of previous 2k cells, where 0 <= k < n and initialize first cell with 0. The process is quite similar to the process of gray code generation.

End points of the ith MSB line or v-lines is in such way that if one moves anti-clockwise direction from a cell then he will get at first v provided value of v in that cell is 1, otherwise he will get v’. The mid point of an arc of a cell is marked by • (small circle), x, or – sign depending on function values TRUE, FALSE or DONT CARE for the combination in that cell. The points marked by • are identified as TRUE or 1 vertices and points marked by x are FALSE or 0 vertices. Two vertices’ vl and v2 will be adjacent if line vl-v2, joining vl and v2 intercept even number of Mk MSB-line(s) while a single Mi MSB-line is perpendicular bisector of vl-v2 line where j<k and vl & v2 differ by jth MSB bit. Mi indicates 1st MSB and Mn indicates last MSB bit.

**SIMPLIFICATION AND MINIMIZATION OF FUNCTIONS**

A polygon consists of a collection of 2m vertices each adjacent to m vertices of the collection, is called a KH-polygon, and the KH-polygon is said to cover these vertices, where, 0 <= m <= n. Two consecutive vertices of the KH-polygon must be adjacent. Each KH-plygon of 2m vertices can be expressed by a product containing (n – m) literal, where n is the number of variables in the expression. Function f can be expressed as a sum of those product terms that correspond to the KH-polygon(s) necessary to cover all-of it’s 1 vertices. The number of product terms in the expression for f is equal to the number of KH-polygon. In order to obtain a minimal expression, all 1 vertices must be covered with the smallest possible number of KH-polygons, such that each KH-polygon is as large as possible. A KH-polygon contained in another larger KH-polygon must never be selected. If there is more than one way of covering the map with the minimal number of KH-polygons, then the covering that consists of larger KH-polygon must be selected.

From the foregoing discussions the following steps for obtaining simplified expression for f can be suggested:

1. Draw proposed KH-map for the function f of n variables. Start by covering with KH-polygons those 1 vertices that cannot be combined with any other 1 vertices.

2. Continue step 1 to those vertices that have only a single adjacent vertex for making KH-polygon of two vertices.

3. Next cover these TRUE vertices that yield KH-polygon of 2**- vertices and not part of any larger KH-polygon of 2k+i vertices, where i > 0 and k =

2,3…….n.

4. A minimal expression is one that corresponds to a collection of KH-polygon that are as large and as few in number as possible covering all TRUE vertices in the map of the fuction. A minterm for a KH-polygon of 2k vertices, where 0<=k<=n, will contain n-k literal. Minterm for the KH-polygon will not contain those ith MSB, where ith MSB line bisects any arm of the KH-polygon.

EXAMPLES

Example l.Fig. 3 represents KH-map for the expression f(w,x,y,z) = 1(0,1,2,5,7,8,9,10,13,15)

For abed polygon, ab and ad arm are bisected by w-line and z-line and minterm for the KH-polygon

abed will be x’ y’ that contain only x and y variables. If one start from any vertices and move anti-clockwise then he will get for the first time negative y-line and x-line and hence y’ and x’ points. Minterm x’ y’ can also be found from any vertex of KH-polygon eliminathing appropriate variable(s). In similar way expression for KH-polygon abef is x’ z’ and for ghij xz.

Hence, f(w,x,y,z) = x’y’ + xz + x’z’.

Example 2. Fig 4 represents the expression f(u, w, x, y, z) = (1, 2, 3, 5, 7, 10, 11, 12, 13, 14, 15, 18, 19, 21,23,25, 26,27)

Arms ab, be, and cd of KH-polygon abedefgh have been bisected by u-line, z-line and w-line respectively and minterm is x’ y may come from vertex u’w'x’yx’ (vertex 2) or tracing end point of MSB line. Combining

expressions for all polygon the expression can be evaluated as f(u,v,x,y,z)= x’y (abedefgh) + u’w'z (alkr) + w’xz(ijkl) u’wx (mnop) + uwx’z (eq). Content within parentheses represents corresponding polygon.

**CONCLUSION**

This KH-map provides a regular geometric structure that is more intuitive than other methods having non-geometric pattern. Concatenation of two KH-map is possible by placing one inside the other. By software implementation it can be extended to solve for a huge

number of variables. X-OR realization through this method will also be convenient. The author hopes that this technique will replace Karnaugh map and Quine McCluskey procedure by its inherent simplicity.

**ACKNOWLEDGEMENT**

Author acknowledges encouragements and co-operation of Dr. M. Shahjahan, Vice-Chancellor, Dr. M. A. Choudhury, Associate Professor and Dr. M. Kaykobad, Asst. Professor of Bangladesh Unversity of Engineering and Technology (BUET).

**A.T.M. Shqfiqul Khalid (Tuhin)**

is a final year undergraduate student of the Department of Computer Science and Engineering, Bangladesh University of Engineering & Technology, Dhaka.

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