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Simplifying Boolean Expressions Using K-Map Method

- CS 3402-- Digital Logic Design

Simplifying Switching Functions

- SOP and POS expressions
- gt 2-level circuits
- Minimum SOP/POS expression
- Minimize the number of literals
- Minimum number of terms
- How?
- Algebraically I.e. using the axioms and theorems

of Boolean algebra. - Karnaugh Map
- McCluskey Method

Simplifying Algebraically

- Simplify F(X,Y,Z) X YZ XYZ XYZ
- Simplify F(A,B,C) ABC ABC ABC

Simplifying Switching Functions K-Map

- Simplifying Theorem
- XY XY Y
- Definition Logical Adjacency
- Two terms are logically adjacent iff they differ

in only one literal the literal is complemented

in one term and non-complemented in the other. - Two Logically adjacent terms can be combined into

one term consisting of only the common literals

Simplifying Boolean Expressions

- Example XY XY
- ABC ABC
- ABCD ABC
- (AB CD).(ABCD)
- AB AB
- ABC ABC
- (XY) (XY)

Karnaugh-Map (K-Map)

- 2-dimensional representation of a truth table.
- Logically adjacent terms are physically adjacent

in the map. - 2-Variable Functions F(X,Y) XY XY

X Y

F(X,Y)

0 0 0 1 1 0 1 1

m0 m1 m2 m3

2-Variable K-Map

XY

XY

XY

XY

- Note
- Logically adjacent cells are physically adjacent

in the k-map - Each cells has two adjacent cells

m0

m2

m1

m3

Function Minimization Using K-Maps

- 1. Each square (minterm) in a k-map of 2

variables has 2 logically adjacent squares, each

square in a 3-variable k-map has 3 adjacent

squares, etc. - 2. Combine only the minterms for which the

function is 1. - 3. When combining terms on a k-map, group

adjacent squares in groups of powers of 2 (I.e.

2, 4, 8, etc.). Grouping two squares eliminates

one variables, grouping 4 squares eliminates 2

variables, etc. - Can't combine a group of 3 minterms

Function Minimization Using K-Maps

- 4. Group as many squares together as possible

the larger the group is the fewer the number of

literals in the resulting product term - 5. Select as few groups as possible to cover all

the minterms of the functions. A minterm is

covered if it is included in at least one group.

Each minterm may be covered as many times as it

is needed however, it must be covered at least

once. - 6. In combining squares on the map, always begin

with those squares for which there are the fewest

number of adjacent squares (the loneliest"

squares on the map).

Definitions

- Implicant a product term that could be used to

cover one or more minterms - Prime Implicant A product term obtained by

combining the maximum number of adjacent squares

in the map. - Essential Prime Implicant A prime implicant

that covers at least one minterm that is not

covered by any other prime implicant. - All essential prime implicants must be included

in the final minimal expression.

Definitions (Cont.)

- Cover of function is a set of prime implicants

for which each minterm of the function is covered

by at least one prime implicant. - All essential prime implicants must be included

in the cover of a function.

Algorithm for Deriving the Minimal SOP

- 1. Circle all prime implicants on the k-map
- 2. Identify and select all essential prime

implicants - 3. Select a minimum subset of the remaining prime

implicants to cover those minterms not covered by

the essential prime implicants.

Four Variable K-map

Five Variable K-map

More Examples

- Find min. SOP and POS expression for each of the

following functions - F(A,B,C,D) ? m(2,3,5,7,10,11,13,14,15)
- G(W,X,Y,Z) ? M(1,3,4,5,7)
- H(A,B,C,D) ?m(1,3,4,7,11)d(5,12,13,14,15)
- F2(A,B,C,D) ? m(1,2,7,12,15)d(5,9,10,11,13)
- F3(A,B,C,D,E) ? m(0,1,2,4,5,6,13,15,16,18,22,24,

26,29)

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