Lecture Six Chapter 5: QuineMcCluskey Method

1
Lecture Six Chapter 5 Quine-McCluskey Method
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Computer Minimization Techniques

• Boolean Algebra
• Karnaugh Maps
• Quine-McCluskey Method

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Boolean Algebra

• Review of Boolean Postulates
• Review of Boolean Identities
• Example1
• Example2

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Review of Boolean Postulates
A B B A
A B B A
Commutative Laws
A (B C) (A B) (A C)
A (B C) (A B) (A C)
Distributive Laws (not like ordinary algebra)
1 A A
0 A A
Identity Elements
A
!A
0
A !A
1
Inverse Elements
A
A B A
A ( A B ) A
Absorption
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5
Review Boolean Identities
!!A A
Involution
0 A 0, A 0 0
A 1 1, 1 A 1
Contradiction (always false)
Tautology (always true)
A A A
A A A
Idempotence
A (B C) (A B) C
0 A A
Associative Laws
!(A
B)
!A
!B
!(A
B)
!A
!B
DeMorgans Theorem
or
or
A NAND B !A
OR !B
A NOR B !A
AND !B
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Example1
A
A B A
Proof
1.
A A B A 1 A B Identity
2.
A ( 1 B ) Distribution
3.
A 1 Identity
4.
A
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Example2
(X Y) (!X Y) (X !X) (!X Y) (X
Y) (Y Y)
0 (!X Y) (X Y) Y
(!X X) Y
Y 1
Y Y
Proof
1.
(X Y) (!X Y) !!(X Y) (!X Y)
2.
!(!X !Y) (X !Y)
DeMorgans
3.
!(!X X) !Y
Distribution
4.
!1 !Y Identity
5.
!!Y Y Involution
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Karnaugh Maps

• A 2 Variable K - map
• Review 3 Variable K - maps
• Example1
• Example2
• Review 4 Variable K - maps
• Example1
• Example2
• A 5 Variable K - map

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2-Variable K -map
Y
0 1
X
m1
m0
0
m2
m3
1
X
Y
F(X,Y) ?(0,1,2,3)
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3-Variable K -map
Y
YZ
00 01 11 10
X
m1
m2
m0
m3
0
m5
m6
m4
m7
1
X
Z
?(0,1,2,3,4,5,6,7)
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Example1
F(X,Y,Z) ?(1,3,4,5,6,7)
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Example1
YZ
00 01 11 10
X
1
1
0
1
1
1
1
1
F(X,Y,Z) ?(1,3,4,5,6,7)
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Example1
YZ
00 01 11 10
X
1
1
0
1
1
1
1
1
F(X,Y,Z) ?(1,3,4,5,6,7) m1 m3 m4 m5
m6 m7 !X!YZ !XYZ X!Y!Z X!YZ
XY!Z XYZ
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Example1
YZ
00 01 11 10
X
1
1
0
1
1
1
1
1
F(X,Y,Z) ?(1,3,4,5,6,7) m1 m3 m4 m5
m6 m7 !X!YZ !XYZ X!Y!Z X!YZ
XY!Z XYZ
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Example1
YZ
00 01 11 10
X
1
1
0
1
1
1
1
1
F(X,Y,Z) X Z
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Example2
F(X,Y,Z) ?(0,2,4,6)
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Example2
YZ
00 01 11 10
X
0
1
1
1
1
1
F(X,Y,Z) ?(0,2,4,6)
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Example2
YZ
00 01 11 10
X
0
1
1
1
1
1
F(X,Y,Z)
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Example2
YZ
00 01 11 10
X
0
1
1
1
1
1
F(X,Y,Z) !Z
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4-Variable K -map
Y
YZ
00 01 11 10
WX
m0
m1
m3
m2
00
m4
m5
m7
m6
01
X
m12
m13
m15
m14
11
W
10
m8
m9
m11
m10
Z
F(W,X,Y,Z) ?(0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
15)
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Example1
F(W,X,Y,Z) ?(5,7,9,11,13,15)
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Example1
Y
YZ
00 01 11 10
F(W,X,Y,Z) ?(5,7,9,11,13,15)
WX
00
01
1
1
X
1
1
11
W
10
1
1
Z
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Example1
Y
YZ
00 01 11 10
F(W,X,Y,Z) X Z W Z (X W) Z
WX
00
01
1
1
X
1
1
11
W
10
1
1
Z
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Example2
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
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Example2
Y
YZ
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
00 01 11 10
WX
1
1
00
01
1
1
X
1
1
1
11
W
10
1
1
1
Z
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Example2
Y
YZ
F(W,X,Y,Z) W !Z Y
00 01 11 10
WX
1
1
00
01
1
1
X
1
1
1
11
W
10
1
1
1
Z
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COMP 370
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5-Variable K -map
V0
V1
Y
Y
YZ
YZ
00
01
11
10
00
01
11
10
WX
WX
m17
m18
m1
m2
m16
m19
m0
m3
00
00
m21
m22
m20
m23
m4
m5
m7
m6
01
01
X
X
m28
m29
m31
m30
m12
m13
m15
m14
11
11
W
W
m25
m26
m9
m10
10
m24
m27
10
m8
m11
Z
Z
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Quine-McCluskey Method

• Prime Implicants Table 3 or 4 steps
• Essential Prime Implicants Table

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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
2
3
4
List minterms by the number of 1s it contains.
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
2
3
Enter combinations of minterms by the number of
1s it contains.
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COMP 370
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
Check off elements used from Step 1.
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
Enter combinations of minterms by the number of
1s it contains.
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COMP 370
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(5,7,9,11,13,15)
The entries left unchecked are Prime Implicants.
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Finding Essential Prime Implicants (EPIs)
Enter the Prime Implicants and their minterms.
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Finding Essential Prime Implicants (EPIs)
Enter Xs for the minterms covered.
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Finding Essential Prime Implicants (EPIs)
Circle Xs that are in a column singularly.
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Finding Essential Prime Implicants (EPIs)
The circled Xs are the Essential Prime
Implicants, so we check them off.
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Finding Essential Prime Implicants (EPIs)
We check off the minterms covered by each of the
EPIs.
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Finding Essential Prime Implicants (EPIs)
EPIs
F X Z W Z (X W) Z
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
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COMP 370
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Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
Dr. S.V. Providence
COMP 370
44
Finding Prime Implicants (PIs)
F(W,X,Y,Z) ?(2,3,6,7,8,10,11,12,14,15)
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COMP 370
45
Finding Essential Prime Implicants (EPIs)
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46
Finding Essential Prime Implicants (EPIs)
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Finding Essential Prime Implicants (EPIs)
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COMP 370
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Finding Essential Prime Implicants (EPIs)
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Finding Essential Prime Implicants (EPIs)
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Finding Essential Prime Implicants (EPIs)
EPIs
F (W !Z) Y
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COMP 370