ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean...

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ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee Prof. Hsien-Hsin Sean Lee School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Tech Georgia Tech

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Page 1: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

ECE2030 Introduction to Computer Engineering

Lecture 8: Quine-McCluskey Method

Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee

School of Electrical and Computer EngineeringSchool of Electrical and Computer Engineering

Georgia TechGeorgia Tech

Page 2: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Quine-McCluskey Method• A systematic solution to K-Map when

more complex function with more literals is given

• In principle, can be applied to an arbitrary large number of inputs, i.e. works for BBn n

where nn can be arbitrarily large

• One can translate Quine-McCluskey method into a computer program to perform minimization

Page 3: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Quine-McCluskey Method • Two basic steps

– Finding all prime implicants of a given Boolean function

– Select a minimal set of prime implicants that cover this function

Page 4: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method (I)

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Transform the given Boolean function into a canonical SOP function

• Convert each Minterm into binary format• Arrange each binary minterm in groups

– All the minterms in one group contain the same number of “1”

Page 5: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method: Grouping minterms 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

(29) 1 1 1 0 1(30) 1 1 1 1 0

(2) 0 0 0 1 0(4) 0 0 1 0 0(8) 0 1 0 0 0(16) 1 0 0 0 0

A B C D E(0) 0 0 0 0 0

(6) 0 0 1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0(7) 0 0 1 1 1(11) 0 1 0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1

Page 6: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method (II)• Combine terms with Hamming

distance=1 from adjacent groups• Check () the terms being combined

– The checked terms are “covered” by the combined new term

• Keep doing this till no combination is possible between adjacent groups

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Q-M Method: Grouping minterms 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

(29) 1 1 1 0 1(30) 1 1 1 1 0

(2) 0 0 0 1 0(4) 0 0 1 0 0(8) 0 1 0 0 0(16) 1 0 0 0 0

A B C D E(0) 0 0 0 0 0

(6) 0 0 1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0

(7) 0 0 1 1 1(11) 0 1 0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1

A B C D E

(0,2) 0 0 0 – 0

(0,4) 0 0 - 0 0

(0,8) 0 - 0 0 0

(0,16) - 0 0 0 0

(2,6) 0 0 - 1 0

(2,10) 0 - 0 1 0

(2,18) - 0 0 1 0

(4,6) 0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0 - 0(8,12) 0 1 - 0 0(16,18) 1 0 0 - 0

(6,7) 0 0 1 1 -

(6,14) 0 - 1 1 0(10,11) 0 1 0 1 -

(10,14) 0 1 - 1 0(12,13) 0 1 1 0 -

(12,14) 0 1 1 - 0(18,19) 1 0 0 1 -

A B C D E(13,29) - 1 1 0 1

(14,30) - 1 1 1 0

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Q-M Method: Grouping minterms

A B C D E

(0,2) 0 0 0 – 0

(0,4) 0 0 - 0 0(0,8) 0 - 0 0 0

(0,16) - 0 0 0 0

(2,6) 0 0 - 1 0 (2,10) 0 - 0 1 0

(2,18) - 0 0 1 0(4,6) 0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0 - 0(8,12) 0 1 - 0 0(16,18) 1 0 0 - 0

(6,7) 0 0 1 1 -(6,14) 0 - 1 1 0(10,11) 0 1 0 1 -(10,14) 0 1 - 1 0(12,13) 0 1 1 0 -(12,14) 0 1 1 - 0(18,19) 1 0 0 1 -

(13,29) - 1 1 0 1(14,30) - 1 1 1 0

A B C D E(0,2,4,6) 0 0 - – 0

(0,2,8,10) 0 - 0 – 0

(0,2,16,18) - 0 0 – 0

(0,4,8,12) 0 - - 0 0

(2,6,10,14) 0 - - 1 0

(4,6,12,14) 0 - 1 - 0

(8,10,12,14) 0 1 - - 0

A B C D E(0,2,4,6 0 - - - 0 8,10,12,14)

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

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Prime Implicants

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

(6,7) 0 0 1 1 -

(10,11) 0 1 0 1 -

(12,13) 0 1 1 0 -

(18,19) 1 0 0 1 -

(13,29) - 1 1 0 1

(14,30) - 1 1 1 0(0,2,16,18) - 0 0 – 0

(0,2,4,6 0 - - - 0 8,10,12,14)

A B C D E

• Unchecked terms are prime implicants

Page 10: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Prime Implicants

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

CDBADCBA

(6,7) 0 0 1 1 -

(10,11) 0 1 0 1 -

(12,13) 0 1 1 0 -

(18,19) 1 0 0 1 -

(13,29) - 1 1 0 1

(14,30) - 1 1 1 0(0,2,16,18) - 0 0 – 0

(0,2,4,6 0 - - - 0 8,10,12,14)

A B C D E

• Unchecked terms are prime implicants

DBCADCBAEDBCEBCD

ECBEA

Page 11: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method (III)• Form a Prime Implicant Table

– X-axis: the minterm– Y-axis: prime implicants

• An is placed at the intersection of a row and column if the corresponding prime implicant includes the corresponding product (term)

Page 12: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method: Prime Implicant Table

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

Page 13: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method (IV)

• Locate the essential row from the table– These are essential prime implicants– The row consists of minterms covered by a single “”

• Mark all minterms covered by the essential prime implicants

• Find non-essential prime implicants to cover the rest of minterms

• Form the SOP function with the prime implicants selected, which is the minimal representation

Page 14: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

Page 15: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14)

Page 16: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14)

Page 17: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7)

Page 18: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7)

Page 19: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

Page 20: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

Page 21: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

Page 22: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

Page 23: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

Page 24: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

Page 25: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

Page 26: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

Page 27: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30)• Now all the minterms are covered by selected prime implicants !

Page 28: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method

0 2 4 6 7 8 10

11

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30)• Now all the minterms are covered by selected prime implicants !• Note that (12,13), a non-essential prime implicant, is not needed

Page 29: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method Result0 2 4 6 7 8 1

011

12

13

14

16

18

19

29

30

(6,7) XX XX

(10,11) XX XX

(12,13) XX XX

(18,19) XX XX

(13,29) XX XX

(14,30) XX XX

(0,2,16,18) XX XX XX XX

(0,2,4,6,8,10,12,14)

XX XX XX XX XX XX XX XX

EAECBEBCDEDBCDCBADCBACDBA

,10,12,14)(0,2,4,6,8

)(0,2,16,18(14,30)(13,29)(18,19)(10,11)(6,7)

30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,

Page 30: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Q-M Method Example 2• Sometimes,

simplification by K-map method could be less than optimal due to human error

• Quine-McCluskey method can guarantee an optimal answer

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F

00 01 11 10

00 1 1 0 0

01 1 X X 1

11 1 0 0 X

10 1 1 0 1

ABCD

Page 31: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Grouping minterms

(1) 0 0 0 1(4) 0 1 0 0(8) 1 0 0 0

A B C D (0) 0 0 0 0

(5) 0 1 0 1(6) 0 1 1 0(9) 1 0 0 1(10) 1 0 1 0(12) 1 1 0 0

(7) 0 1 1 1(14) 1 1 1 0

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(0,1) 0 0 0 -(0,4) 0 - 0 0(0,8) - 0 0 0

(1,5) 0 - 0 1(1,9) - 0 0 1(4,5) 0 1 0 –(4,6) 0 1 – 0(4,12) – 1 0 0 (8,9) 1 0 0 –(8,10) 1 0 – 0(8,12) 1 – 0 0

(5,7) 0 1 - 1(6,7) 0 1 1 -(6,14) - 1 1 0(10,14) 1 – 1 0(12,14) 1 1 - 0

A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

Page 32: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Page 33: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

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Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Page 34: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

34

Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Page 35: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

35

Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Page 36: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

36

Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Page 37: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

37

Prime Implicants

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D

(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

0 1 4 6 8 9 10 12 5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

Don’t Care

Essential PI

Essential PI

Non-Essential PI

Page 38: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

38

Q-M Method Solution

0 1 4 6 8 9 10

12

5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

A B C D

(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

Don’t Care

BADA CB

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F

Page 39: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

39

Yet Another Q-M Method Solution

0 1 4 6 8 9 10

12

5 7 14

(0,1,4,5) XX XX XX XX

(0,1,8,9) XX XX XX XX

(0,4,8,12) XX XX XX XX

(4,5,6,7) XX XX XX XX

(4,6,12,14) XX XX XX XX

(8,10,12,14) XX XX XX XX

A B C D

(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0

(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0

Don’t Care

DB DA CB

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F

Page 40: ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering.

40

To Get the Same Answer w/ K-Map

BADA CB

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F

00 01 11 10

00 1 1 0 0

01 1 X X 1

11 1 0 0 X

10 1 1 0 1

ABCD

00 01 11 10

00 1 1 0 0

01 1 X X 1

11 1 0 0 X

10 1 1 0 1

ABCD

DB DA CB

d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F


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