Gate Level Minimization1
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Gate-level Minimization
The procedure of simplifying Boolean expressions (in 2-4) isdifficult since it lacks specific rules to predict the successive steps
in the simplification process.
Alternative: Karnaugh Map (K-map) Method.
K-map method can be seen as a pictorial form of the truth table.
m0 m1
m2 m3
xy
''yx yx'
'xy xy
0 1
1
0
y
x
Two-variable map
xy
''yx yx'
'xy xy
0 1
1
0
y
x
Two-variable K-MAP
xy
xyF =1
0 1
1
0
y
x
xy 0 1
1
0
y
x1 1 1
1
xyxyyx
mmmF
++=
=++=
''
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xy 0 1
1
0
y
x 1 1
1
yxF +=2
The three squares can be determined from the intersection
of variable x in the second row and variable y in the secondcolumn.
xy
''yx yx'
'xy xy
0 1
1
0
y
x
Two-variable K-MAP
Any two adjacent squares differ by only one variable.
From the postulates of Boolean algebra, the sum of two minterms
in adjacent squares can be simplified to a simple AND term.
How is this map useful?
Three-Variable K-Map
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2 3 4 5( , , , ) ' ' ' ' ' '
' ( ') '( ') ' '
F m m m m x yz x yz xy z xy z
x y z z xy z z x y xy
= = + + +
= + + + = +
Example 1
Three-Variable K-Map
Example 2
Three-Variable K-Map
)7,6,4,3(),,( =zyxFSimplify:
m6m7m5m4
m2m3m1m0
'xz 'xzyz
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Three-Variable K-Map
)6,5,4,2,0(),,( =zyxF
Example 3
Simplify:
m6m7m5m4
m2m3m1m0
Example 4
Three-Variable K-Map
Given: BCCABBACACBAF +++= '''),,(
(a) Express F in sum of minterms.
(b) Find the minimal sum of products using K-Map
BCAABCAABCCAB
BCABCACCBA
CBABCABBCA
')'('
''')'('
''')'('
+=+
+=+
+=+
)7,5,3,2,1(
''''''),,(
=
++++= ABCCABBCABCACBACBAF(a)
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Three-Variable K-Map
Example 4 (continued) )7,5,3,2,1(),,( =CBAF
m6m7m5m4
m2m3m1m0
Three-variable K-Map: Observations
Encircling one square represents one minterm
A term of how many literals? 3
Two adjacent squares? 2 literals
Four adjacent squares? 1 literal
Eight adjacent squares
function equals to 1
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Four-Variable K-Map
Four-Variable K-Map
Example 5 Simplify F(w,x,y,z) = (0,1,2,4,5,6,8,9,12,13,14)
'''' xzzwyF ++=
1
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Four-Variable K-Map
Example 6 Simplify F(A,B,C,D) = ''''''''' CABBCDACDBCBA +++
'''''' CDACBDBF ++=
Four-variable K-Map: Observations
One square represents one minterm a term of 4 literals
Two adjacent squares a term of 3 literals
Four adjacent squares a term of 2 literal
Eight adjacent squares a term of 1 literal
sixteen adjacent squares the function equals to 1
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' 'F AB CD BD= + +
Simplify the following Boolean function in:
(a) sum of products (b) product of sums
( , , , ) (0,1,2,5,8,9,10)F A B C D =
Combining the zeros:
'AB CD BD+ +
SUM of PRODUCT and PRODUCT OF SUM
Combining the ones:
' ' ' ' ' 'F B D B C A C D= + +(a)
Taking the the complement:
( ')'
( ' ')( ' ')( ' )
F F
A B C D B D
= =
= + + +
(b)
SUM OF PRODUCT (SOP) PRODUCT OF SUM (POS)
SOP and POS gate implementation
