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CHAPTER 5
KARNAUGH MAPS
Karnaugh map (K-map) is a graphical method for obtaining the simplest sum-of- products and simplest product-of-sums expressions for a Boolean function. A simplest sum-of-products expression has a minimum number of product terms and the total number of literals in all the product terms combined is also a minimum. Similarly, a simplest product-of-sums expression for a function also has a minimum number of sum terms as well as a minimum number of literals in all the sums combined. A function may have more than one simplest sum-of-products expression and/or more than one simplest product-of-sums expression.
5.1 Function Representation by Karnaugh Map
A Karnaugh map for an n-variable Boolean function is partitioned into 2n^ parts. Each of the 2n^ parts is a box or cell designated to one of the 2n^ combinations of input values or input states. Figure 5.1(a) is a 2-variable Karnaugh map with four cells. The left half of the map is assigned to A = 0 and the right half is for A = 1. The upper half is for B = 0 and the lower half for B = 1. The vertical partition for A and the horizontal partition for B create four cells for the four combinations of input values. The values of A and B are labeled respectively on the top and the left side of the map. The corresponding values of AB for each cell are also written at the lower right corner of each cell. Figure 5.1(b) shows the Karnaugh map for the function AB. AB = 1 when A = B = 1. Therefore the lower right cell is filled with a value of 1. Each of the other three cells has a value of 0 because AB = 0 for the other three combinations of input values. The value of either 0 or 1 in each cell is actually the function value.
(a) (b) Figure 5.1 (a) Two-variable Karnaugh map. (b) Karnaugh map for AB.
The Karnaugh map for a 3-variable function F(A,B,C) is shown in Figure 5.2(a). The map is partitioned into four columns and two rows, which consists of eight cells for
A
B
A
B
the eight combinations of input values. The values of A and B assigned to the four columns are in the order of 00, 01, 11, 10, not 00, 01, 10, 11. The upper half of the map is for C = 0 and the lower half for C = 1. When the values of A and B for the columns are assigned in the order of 00, 01, 11, 10, their values between two neighboring columns, as well as the values between the right and the left columns, differ in only one bit. This characteristic of Karnaugh map is referred to as logical adjacency. Logical adjacency of the two columns at the right and the left could be considered as a piece of paper wrapped around and those two columns were actually next to each other. The paper had to be cut along a line between two columns so that it can be unfolded. Therefore the values of the four columns can be in other orders, such as 11, 10, 00, 01, as they as they exhibit the characteristics of logical adjacency. When comparing the values of ABC for two logically adjacent cells vertically or horizontally, only one variable has different values. Each of the other two variables has the same value in those two cells. Thus these two cells are said to be logically adjacent to each other. In Figure 5.2(a), the corresponding values of A, B, C for each cell are labeled at the lower right corner in decimal. Cell are numbered starting from the upper left cell. Since C is the least significant variable, the cells are numbered from top to bottom in the same direction the values of C changes. Because of logical adjacency, after the second column from the left, the cells in the last (rightmost) column should be numbered before the cells in the third column.
(a) (b) Figure 5.2 Three-variable Karnaugh maps.
Figure 5.3 Two different 3-variable Karnaugh maps.
The labeling of variables on a Karnaugh map is arbitrary. Figure 5.3 shows two different ways of labeling. Note that the position for a cell may change if the labeling is different. For example, the cells for ABC = 001 and 110 are in different positions, while the cells for ABC = 111 remain in the same position. Figures 5.4 (a) and (b) show the representation of the truth table in Table 4.3 using the two different 3-variable Karnaugh
AB
C
AB
C
AB
C
BC
A
cells from the two sub-maps. Two cells in the same location of the two sub-maps are logically adjacent to each other. An example is cell 21 and cell 5.
Figure 5.5 Four-variable Karnaugh map. Figure 5.6 Karnaugh map for F = m( 0, 2, 3, 5, 7, 10, 13, 14, 15)
Figure 5.7 Five-variable Karnaugh map.
5.2 Prime Implicants
When a function is expressed in terms of a minterm list, a truth table, or a Boolean expression, it is not easy to see how two canonical products can be combined to one using the combination theorem x'y + xy = y. On a Karnaugh map, these two canonical products can be spotted easily because they are logical adjacent to each other.
As shown on the Karnaugh map in Figure 5.8, m 5 and m 13 are two logically adjacent 1-cells. Their corresponding canonical products are A’BC’D and ABC’D
CD
AB
0
1 5
4
13
12 8
9
3
2 6
7
14
15 11
10
AB
CD
DE
BC
0
1 5
4
13
12 8
9
3
2 6
7
14
15 11
10
A = 0
00 DE
01 11 10 BC
16
17 21
20
29
28 24
25
19
18 22
23
30
31 27
26
A = 1
respectively. They can be combined to become BC’D. The combination not only reduces the number of products from two to one, but also eliminates one variable. The resulting product can be read off directly from the map without any algebraic manipulation. To find this product, examine the corresponding values of the variables on the top and at the left side of the K-map for these two 1-cells. When a variable has the same value for the two cells, the variable remains in the product. The variable is in true form if it has a value of 1, and in complemented form if its value is 0. The only variable that has different values for the two adjacent cells is discarded. A grouping of two logically adjacent 1-cells is called a 1-cube. Three other examples of 1-cube are shown in Figure 5.9.
Figure 5.8 Two logically adjacent minterms.
Figure 5.9 Examples of 1-cubes.
AB
CD^01 11
A B C D
m 5 0 1 0 1
m 13 1 1 0 1
B C’ D
A’BC’D + ABC’D
= (A’ + A) BC’D
5 13
AB
CD^01 11
A’BD’
1 AC’D’
B’C’D
B’C’
When two 1-cubes are logically adjacent to each other, the four 1-cells in the two 1- cubes should be in a square or rectangle. The latter appears to be either a column or a row. The product resulted from the combination of four logically adjacent 1-cells can also be read off directly from the map. A variable with different values from the four 1-cells should be discarded. A grouping of four logically adjacent 1-cells is called a 2-cube. Some more examples are given in Figure 5.11. When the four 1-cells are from one column, the product can be read off just from the values on the top of this column. Similarly, when they are from one row, the values to the left of this row provide the values for the product.
The grouping of eight 1-cells from two logical adjacent 2-cube creates a 3-cube. The 1-cells in a 3-cube occupy either two logically adjacent columns or rows. In other words, a 3-cube occupies half of the 4-variable K-map. Some examples of 3-cubes are given in Figure 5.12. The product from the two end columns is B’. D’ is the product from the combination of the top and the bottom rows. From the development of combining 1- cells, it is seen that every time two lower order logically adjacent cubes are combined to form a high order cube, one variable is eliminated. The number of 1-cells in a grouping can be extended to sixteen, thirty-two, etc.
Figure 5.12 Examples of 3-cubes.
Definition 5.1: A j-cube is a grouping of 2j^ logically adjacent 1-cells on a K-map for an n- variable function, which can be combined to form a product of (n-j) literals. j is a positive integer, 0 j n.
Note that a minterm is a 0-cube.
Definition 5.2: An implicant is a cube of any order.
Definition 5.3: A j-cube is called a prime implicant if it cannot combine with another j- cube to form a (j+1)-cube.
D’
AB
CD^01 11
C’
B
AB
CD^01 11
A
A prime implicant can also be defined as an implicant that cannot be included in or be part of another implicant. In other words, it is a grouping that cannot be enlarged.
Some examples of prime implicants are shown on the four K-maps in Figure 5.13. Some of the 1-cells are highlighted, which will be explained later. In Figure 5.13(a), there is an isolated 1-cell not adjacent to any other 1-cells. Therefore the isolated 1-cell is a prime implicant. The prime implicants for the four K-maps in Figure 5.13 are listed below.
(a) A’BCD’, A’B’D, ABD, C’D, AC’ (b) A’C’D, A’BC, ACD, ABC’, BD (c) A’BC, ABC’, AC’D’, A’D, BD, CD (d) A’CD’, A’BC, A’BD, B’D’, B’C’, C’D, AC’
An example of prime implicants on a 5-variable Karnaugh map is shown in Figure 5.14. A prime implicant on a 5-variable K-map may occur in either one of the two 4- variable sub-maps, such as ACD’E. The 1-cells of a prime implicant can also be from both sub-maps, such as B’ and CDE’. When a prime implicant comprises of 1-cells from both sub-maps, the 1-cells should fall equally on the two sub-maps and in identical locations. The variable A is always discarded in the product for such a prime implicant.
In Figures 5.13 and 5.14, it is seen that a 1-cell may be included in one or more than one prime implicant. For example, in Figure 5.13(c), m 8 is included in the prime implicant AC’D’ only. m 12 exists in AC’D’ and ABC’. m 7 occurs in A’D, BD, CD, and A’BC.
Definition 5.4: If a 1-cell can exist in one and only one prime implicant, it is called a distinguished 1-cell.
Definition 5.5: A prime implicant is called an essential prime implicant if it includes at least one distinguished 1-cell.
The 1-cells that are highlighted on the K-maps in Figures 5.13 and 5.14 are distinguished 1-cells. The essential prime implicants in Figure 5.13 are as follows:
(a) A’BCD’, A’B’D, ABD, C’D, AC’ (b) A’C’D, A’BC, ACD, ABC’ (c) A’BC, AC’D’, A’D, CD (d) B’D’, AC’
There are five prime implicants in Figure 5.14, B’C, CDE’, AD’E’, A’B’DE, and ACE’. All are essential except ACE’.
5.3 Simplest Sum-of-Products Expression
In finding the simplest sum-of-products expression for a function from Karnaugh map, each minterm must be covered or grouped at least once. When a minterm is a distinguished 1-cell, there is no other option except to select the prime implicant including this 1-cell, which is an essential prime implicant by Definition 5.5. When a prime implicant is selected, except an isolated 1-cell, the prime implicant will also covers some other minterms. After the selection of all the essential prime implicants, if there are still some uncovered minterms, it is necessary to select some other prime implicants for the coverage of these remaining minterms. To ensure that the sum-of-products expression is minimal, they must be covered by a minimum number of prime implicants called secondary essential prime implicants. Also the number of literals in the secondary essential prime implicants should be a minimum. The selection of essential prime implicants is unique for a function, whereas there may be more than one option in the selection of secondary essential prime implicants. The procedure is summarized below in two steps.
(i) Select all the essential prime implicants. (ii) Select a minimum number of secondary essential prime implicants with a minimum number of literals for all the 1-cells not covered by the essential prime implicants.
The minimization algorithm just developed is used to find the simplest sum-of- products expressions for the Karnaugh maps in Figures 5.13 and 5.14. Since all the five prime implicants in Figure 5.13(a) are essential, the simplest sum-of-products expression is
F(A,B,C,D) = A’BCD’ + A’B’D + ABD + C’D + AC’
Similarly, the simplest expression for Figure 5.14 is
F(A,B,C,D,E) = B’C + CDE’ + ACD’E +AD’E’
For the Karnaugh map in Figure 5.13(b), all minterms are covered after the selection of the essential prime implicants.
F(A,B,C,D) = A’C’D + A’BC + ACD + ABC’
The prime implicant BD is not required for the simplest sum-of-products. This example suggests that selection must begin with the essential prime implicants, not the largest groupings.
The Karnaugh maps in Figures 5.13 (c) and (d) are re-drawn in Figures 5.15(a) and 5.15(b) to show the selection of secondary essential prime implicants. The highlighted minterms are those not covered by the essential prime implicants. After the selection of four essential prime implicants, only one 1-cell, m 13 , is not covered in Figure 5.15(a). It requires just one secondary essential prime implicant. m 13 can be covered by
either BD or ABC’. The choice is obviously BD because it has two literals, one less than ABC’. Thus
F(A,B,C,D) = A’BC + AC’D’ + A’D + CD + BD
For the Karnaugh map in Figure 5.15(b), there are four 1-cells not covered by the essential prime implicants B’D’ and AC’. They can be covered by two prime implicants, which are C’D and A’BC.
F(A,B,C,D) = B’D’ + AC’ + C’D + A’BC
(a) (b) Figure 5.15 Minimization of the Karnaugh maps in Figures 5.13(c) and 5.13(d).
Example 5.
This is another example for the minimization of a 4-variable K-map. There is a 3- cube that could have been overlooked sometimes. The simplest sum-of-products expression is
F(A,B,C,D) = B’ + C’D + A’CD’ + AD
All four products are essential prime implicants.
Example 5.
As shown in Figure 5.17, there are five uncovered minterms after the selection of the only essential prime implicant A’D’. These five minterms can be covered by three secondary essential prime implicants. Each of them is a 1-cube. The selections of secondary essential prime implicants are shown in Figure 5.18.
AB
CD^01 11
AB
CD^01 11
Thus the simplest sum-of-products expressions corresponding to the four different selections of secondary essential prime implicants are listed below. Secondary essential prime implicants in the expressions are underlined.
(a) F(A,B,C,D) = A’D’ + BCD + AC’D + AB’C’ (b) F(A,B,C,D) = A’D’ + BCD + AC’D + B’C’D’ (c) F(A,B,C,D) = A’D’ + BCD + ABD + AB’C’ (d) F(A,B,C,D) = A’D’ + A’BC + ABD + AB’C’
Example 5.
The prime implicants of a 5-variable function may occur on either one of two 4- variable sub-maps or from both sub-maps. As shown on the 5-variable K-map in Figure 5.19, there are four (highlighted) distinguished 1-cells. They are m 1 , m 15 , m 27 , and m 28. The essential prime implicants that include these four 1-cells are A’B’D’E, A’CE, ACDE, and B’CE’. The first two essential prime implicants occur on the sub- map for A = 0. The third one is found on the sub-map for A = 1. The minterms of the last essential prime implicant come from both sub-maps.
After selecting the essential prime implicants, there are two uncovered minterms, m 21 and m 23. They can be covered a secondary essential prime implicant B’CE. Thus the simplest sum-of-products expression is
F(A,B,C,D,E) = A’B’D’E + A’CE + AC’DE + B’CE’ + B’CE
Figure 5.19 K-map for Example 5.3.
DE
BC
A = 0
00 DE
BC
A = 1
AC’DE
A’CE
CD’E’
A’B’D’E B’CE
5.4 Simplest Product-of-Sums Expressions
Simplest product-of-sums expressions can be obtained from maxterms as simplest sum-of-products expressions are obtained from minterms. In Figure 5.20, it shows that two logically maxterms or 0-cells can be combined to form a sum. The sum can be read off directly from the Karnaugh map. A variable with different values in two logically adjacent 0-cells is discarded from the resulting sum. A variable with a value of 0 appears in true form in the sum. It appears in complemented form for a value of 1. A j-cube comprising of 2j^ logically adjacent 0-cells is called an implicate. The grouping of 0-cells is exactly the same as how 1-cells are grouped, except that the grouping of 0-cells produces a sum, not a product. The definitions of distinguished 1-cell, prime implicant, and essential prime implicants, and secondary essential prime implicants for 1-cells are now called, respectively, distinguished 0-cell, prime implicate, essential prime implicate, and secondary essential prime implicate for 0-cells. The two-step minimization algorithm in Section 5.3 is also applicable to 0-cells. The product of all the prime implicates selected in the two-step minimization algorithm is a simplest or minimal product-of-sums.
Figure 5.20 Logically adjacent maxterms.
Example 5.
The simplest product-of-sums expressions for the two Karnaugh maps in Figure 5. are given below. Secondary essential prime implicates are underlined.
(a) F(A,B,C,D) = (A’ + B’) (B’ + D’) (C’ + D)
(b) F(A,B,C,D) = (B + C) (C + D’) (A + B’ + C’) (A’ + C’ + D)
AB
CD
A B C D
M 5 0 1 0 1
M 13 1 1 0 1
B’+C+D’
(A+B’+C+D’)(A’+B’+C+D’)
= AA’ + B’ + C + D’
5 13
F(A,B,C,D,E) = (B + C’) (B’ + D’ + E) (A + C + D’ + E’) (A’ + B’ + D’)
5.5 Minimization of Incompletely Specified Functions
In minimizing incompletely specified functions, the minimization algorithm in Section 5.3 is still applicable. The question is when a don’t-care term needs to be included in a grouping. The following rules can be followed to determine when a don’t- care term should be included in the formation of a grouping.
(1) Do not consider a don't-care term to be a distinguished cell. A don’t care term is a distinguished cell if it can be included in only one prime implicant (or implicate).
(2) Always include don't-care terms with 1-cells or 0-cells to form a higher order cube or larger grouping.
For the incompletely specified function in Table 4.8, a sum-of-products expression (w’z + x’y) is obtained using Boolean algebra. This result can be easily verified using K-map. As shown in Figure 5.23, m 1 and m 2 are distinguished 1-cells. The essential prime implicants that covers m 1 is a 2-cube w’z. To cover m 2 , the two don’t-care terms d 10 and d 11 are combined with m 2 and m 3 to form a 2-cube x’y, which satisfied Rule (2).
Figure 5.23 Karnaugh map for the verification of the function in Table 4.8.
A don't-care term assumes a value of 1 when combining with 1-cells; otherwise it is equal to 0. Likewise, a don't-care term combined with 0-cells takes on a value of 0. The value of a don't-care term becomes fixed once it is grouped. After minimization, the don't-care terms that are left out in grouping will take on a different value.
wx yz^01 11
00
d
d
d
d d
d
Example 5.
In Figure 5.24(a), the simplest sum-of-product expression is
F(A,B,C,D) = A’B + AD’
AD’ is essential because m 8 is a distinguished 1-cell. A’B is secondary essential.
In Figure 5.24(b), (A’+D) and (B+C’) are essential prime implicates. (B’+D’) is a secondary essential prime implicate. The simplest product-of-sums expression is
F(A,B,C,D) = (A’ + D) (B + C’) (B’ + D’)
(a) (b) Figure 5.24 Karnaugh maps for Example 5.6.
5.6 Exclusive-OR and Exclusive-NOR Patterns
The propagation delay of exclusive-OR and exclusive-NOR gate is greater than that of AND, OR, NAND, and NOR. Fabrication of exclusive-OR and exclusive-NOR gates also requires more components than other gate types. However, the implementation of digital circuits using exclusive-OR and exclusive-NOR gate may still be more economical for certain Boolean functions.
XOR and XNOR patterns are shown in Figures 5.25 to 5.28. (Both patterns will be referred to as “XOR pattern” from this point.) An XOR pattern consists of two cubes of the same orders and appears in one of the following three patterns: (1) in diagonal, (2) in the same row(s) but there is one cell between them, and (3) in the same column(s) but there is also one cell between them..
AB
CD
10 d
d
d
d d
AB
CD
d d
d
d
(A’ + C + D) (A’ + C’ + D’)
= A’ + (C + D) (C’ + D’) = A’ + (C D) (Pattern 3)
(A + D’) (A’ + D) = (A D)’ (Pattern 1)
Figure 5.27 Exclusive-OR patterns for 2-cubes.
Figure 5.28 Exclusive-OR patterns from 0-cells.
AB
CD
AB
CD^01 11
(A’+C+D) (A’+C’+D’) (A+D’) (A’+D)
AB
CD^01 11
BD
B’D’
AB
CD^01
A’B
AB’
Example 5.
The 3-variable function y 0 in Table 1.2 is used as an example to show the advantage of implementing digital circuits with XOR gates.
y 0 = F(x 2 , x 1 , x 0 ) = m(1, 2, 4,7)
From the K-map in Figure 5.29, it is seen that all the minterms are isolated 1-cells. None of them can be combined with other 1-cells. The simplest sum-of-products expression given below needs four 3-input AND gates and one 4-input OR gate. Three additional inverters are needed to complement the inputs if they are available only in true form. F(x 2 , x 1 , x 0 ) = x 2 ’x 1 ’x 0 + x 2 ’x 1 x 0 ’ + x 2 x 1 ’x 0 ’ + x 2 x 1 x 0
From the K-map, it is obvious that some XOR patterns exist. The following derivation shows that y 0 can be implemented just by two XOR gates.
F(x 2 , x 1 , x 0 ) = (m 1 + m 2 ) + (m 4 + m 7 ) = (x 2 ’x 1 ’x 0 + x 2 ’x 1 x 0 ’) + (x 2 x 1 ’x 0 ’ + x 2 x 1 x 0 ) = x 2 ’(x 1 ’x 0 + x 1 x 0 ’) + x 2 (x 1 ’x 0 ’ + x 1 x 0 ) = x 2 ’(x 1 x 0 ) + x 2 (x 1 x 0 )’ = x 2 x 1 x 0
Figure 5.29 K-map for y 0 in Table 1.2.
5.7 Sub-Functions
As shown in Chapter 4, a function can be expanded to or partitioned into two sub- functions using Shannon’s expansion theorem. The following function is used to illustrate the partition of a K-map into sub-functions.
F(A,B,C,D) = m( 0, 2, 3, 6, 8, 9, 12, 13, 15) + d(5, 7, 10) (5.1)
The K-map of F is shown in Figure 5.30 (a). To partition the map with A as the expansion variable, the two left columns of the 4-variable K-map (A = 0) is separated from the two right columns (A = 1). The partition is shown in Figure 5.30 (b) as two 3-
x 2 x 1 01 11 10
x 0
