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1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.

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1 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology

2 H.-H. S. Lee 2 Quine-McCluskey Method A systematic solution to K-Map when more complex function with more literals is given B n n In principle, can be applied to an arbitrary large number of inputs, i.e. works for B n where n can be arbitrarily large One can translate Quine-McCluskey method into a computer program to perform minimization

3 H.-H. S. Lee 3 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

4 H.-H. S. Lee 4 Q-M Method (I) 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”

5 H.-H. S. Lee 5 Q-M Method: Grouping minterms (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

6 H.-H. S. Lee 6 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

7 H.-H. S. Lee 7 Q-M Method: Grouping minterms (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 

8 H.-H. S. Lee 8 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)      

9 H.-H. S. Lee 9 Prime Implicants (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

10 H.-H. S. Lee 10 Prime Implicants (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

11 H.-H. S. Lee 11 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)

12 H.-H. S. Lee 12 Q-M Method: Prime Implicant Table 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

13 H.-H. S. Lee 13 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

14 H.-H. S. Lee 14 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

15 H.-H. S. Lee 15 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14)

16 H.-H. S. Lee 16 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14)

17 H.-H. S. Lee 17 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7)

18 H.-H. S. Lee 18 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7)

19 H.-H. S. Lee 19 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

20 H.-H. S. Lee 20 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

21 H.-H. S. Lee 21 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

22 H.-H. S. Lee 22 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

23 H.-H. S. Lee 23 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

24 H.-H. S. Lee 24 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

25 H.-H. S. Lee 25 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

26 H.-H. S. Lee 26 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

27 H.-H. S. Lee 27 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX 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 !

28 H.-H. S. Lee 28 Q-M Method 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX 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

29 H.-H. S. Lee 29 Q-M Method Result 02467810111213141618192930 (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

30 H.-H. S. Lee 30 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 00011110 00 1100 01 1XX1 11 100X 10 1101 AB CD

31 H.-H. S. Lee 31 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 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         

32 H.-H. S. Lee 32 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

33 H.-H. S. Lee 33 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

34 H.-H. S. Lee 34 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

35 H.-H. S. Lee 35 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

36 H.-H. S. Lee 36 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

37 H.-H. S. Lee 37 Prime Implicants 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 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care Essential PI Non-Essential PI

38 H.-H. S. Lee 38 Q-M Method Solution 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX 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 Don’t Care

39 H.-H. S. Lee 39 Yet Another Q-M Method Solution 01468910125714 (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX 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 Don’t Care

40 H.-H. S. Lee 40 To Get the Same Answer w/ K-Map 00011110 00 1100 01 1XX1 11 100X 10 1101 AB CD 00011110 00 1100 01 1XX1 11 100X 10 1101 AB CD

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