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Lecture Six Chapter 5: Quine-McCluskey Method Dr. S.V. Providence COMP 370.

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1 Lecture Six Chapter 5: Quine-McCluskey Method Dr. S.V. Providence COMP 370

2 Computer Minimization Techniques Boolean Algebra Karnaugh Maps Quine-McCluskey Method Dr. S.V. Providence COMP 370

3 Boolean Algebra  Review of Boolean Postulates  Review of Boolean Identities  Example1  Example2 Dr. S.V. Providence COMP 370

4 Review of Boolean Postulates A & B = B & AA # B = B # ACommutative Laws A & (B # C) = (A & B) # (A & C)A # (B & C) = (A # B) & (A # C)Distributive Laws (not like ordinary algebra) 1 & A = A0 # A = AIdentity Elements A &!A = 0A # !A = 1Inverse Elements A #A & B = AA & ( A # B ) = AAbsorption Dr. S.V. Providence COMP 370

5 Review Boolean Identities 0 & A = 0, A & 0 = 0 Contradiction (always false) A # 1 = 1, 1 # A = 1 Tautology (always true) A & A = AA # A = AIdempotence A & (B & C) = (A & B) & C0 # A = AAssociative Laws !(A & B) = !A # !B or A NAND B = !A OR !B !(A # B) = !A & !B or A NOR B = !A AND !B DeMorgan’s Theorem !!A = AInvolution Dr. S.V. Providence COMP 370

6 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

7 (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)] DeMorgan’s 3. = ![(!X # X) & !Y] Distribution 4. = ![1 & !Y] Identity 5. = ![!Y] = Y Involution Example2 Dr. S.V. Providence COMP 370

8 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 Dr. S.V. Providence COMP 370

9 2-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m2m2 m3m3 0 1 0 1 X Y F(X,Y) =  (0,1,2,3) X Y

10 3-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m6 00 01 11 10 0 1 X YZ  (0,1,2,3,4,5,6,7) X Y Z

11 Example1 Dr. S.V. Providence COMP 370 F(X,Y,Z) =  (1,3,4,5,6,7)

12 Example1 Dr. S.V. Providence COMP 370 11 00 01 11 10 0 1 X YZ 11 11 F(X,Y,Z) =  (1,3,4,5,6,7)

13 Example1 Dr. S.V. Providence COMP 370 11 00 01 11 10 0 1 X YZ 11 11 F(X,Y,Z) =  (1,3,4,5,6,7) = m 1 # m 3 # m 4 # m 5 # m 6 # m 7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

14 Example1 Dr. S.V. Providence COMP 370 11 00 01 11 10 0 1 X YZ 11 11 F(X,Y,Z) =  (1,3,4,5,6,7) = m 1 # m 3 # m 4 # m 5 # m 6 # m 7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

15 Example1 Dr. S.V. Providence COMP 370 11 00 01 11 10 0 1 X YZ 11 11 F(X,Y,Z) = X # Z

16 Example2 Dr. S.V. Providence COMP 370 F(X,Y,Z) =  (0,2,4,6)

17 Example2 Dr. S.V. Providence COMP 370 1 00 01 11 10 0 1 X YZ 1 11 F(X,Y,Z) =  (0,2,4,6)

18 Example2 Dr. S.V. Providence COMP 370 1 00 01 11 10 0 1 X YZ 1 11 F(X,Y,Z) =

19 Example2 Dr. S.V. Providence COMP 370 1 00 01 11 10 0 1 X YZ 1 11 F(X,Y,Z) = !Z

20 4-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m6 00 01 11 10 00 WX YZ 01 11 10m8m8 m9m9 m 11 m 10 m 12 m 13 m 15 m 14 W Y X Z F(W,X,Y,Z) =  (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)

21 Dr. S.V. Providence COMP 370 F(W,X,Y,Z) =  (5,7,9,11,13,15) Example1

22 Dr. S.V. Providence COMP 370 00 01 11 10 00 WX YZ 01 11 10 W Y X Z 11 11 11 F(W,X,Y,Z) =  (5,7,9,11,13,15)

23 Example1 Dr. S.V. Providence COMP 370 00 01 11 10 00 WX YZ 01 11 10 W Y X Z 11 11 11 F(W,X,Y,Z) = X & Z # W & Z = (X # W) & Z

24 Example2 Dr. S.V. Providence COMP 370 F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15)

25 Example2 Dr. S.V. Providence COMP 370 00 01 11 10 00 WX YZ 01 11 10 W Y X Z 1 1 11 11 1 1 1 1 F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15)

26 Example2 Dr. S.V. Providence COMP 370 00 01 11 10 00 WX YZ 01 11 10 W Y X Z 1 1 11 11 1 1 1 1 F(W,X,Y,Z) = W & !Z # Y

27 5-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m6 00 WX YZ 01 11 10m8m8 m9m9 m 11 m 10 m 12 m 13 m 15 m 14 W Y X Z 011110 m 16 m 17 m 20 m 21 m 19 m 18 m 23 m 22 00 WX YZ 01 11 10m 24 m 25 m 27 m 26 m 28 m 29 m 31 m 30 W Y X Z 011110 V=0V=1

28 Quine-McCluskey Method  Prime Implicants Table 3 or 4 steps  Essential Prime Implicants Table Dr. S.V. Providence COMP 370

29 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3 5 9 7 11 13 15 List minterms by the number of 1s it contains. 2 3 4 Dr. S.V. Providence COMP 370

30 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3 50101 91001 70111 111011 131101 151111 Dr. S.V. Providence COMP 370

31 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3 501015,7 910015,13 9,11 701119,13 111011 1311017,15 11,15 15111113,15 Enter combinations of minterms by the number of 1s it contains. 2 3 Dr. S.V. Providence COMP 370

32 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3  501015,701-1  910015,13-101 9,1110-1  701119,131-01  111011  1311017,15-111 11,151-11  15111113,1511-1 Check off elements used from Step 1. Dr. S.V. Providence COMP 370

33 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3  501015,701-15,7,13,15-1-1  910015,13-1015,13,7,15-1-1 9,1110-19,11,13,151- -1  701119,131-019,13,11,151- -1  111011  1311017,15-111 11,151-11  15111113,1511-1 Enter combinations of minterms by the number of 1s it contains. Dr. S.V. Providence COMP 370

34 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3  50101  5,701-15,7,13,15-1-1  91001  5,13-1015,13,7,15-1-1  9,1110-19,11,13,151- -1  70111  9,131-019,13,11,151- -1  111011  131101  7,15-111  11,151-11  151111  13,1511-1 The entries left unchecked are Prime Implicants. Dr. S.V. Providence COMP 370

35 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15 - 1 5,7,13,15 1 - - 19,13,11,15 Enter the Prime Implicants and their minterms. Dr. S.V. Providence COMP 370

36 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15 - 1 5,7,13,15XXXX 1 - - 19,13,11,15XXXX Enter Xs for the minterms covered. Dr. S.V. Providence COMP 370

37 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15 - 1 5,7,13,15XXXX 1 - - 19,13,11,15XXXX Circle Xs that are in a column singularly. Dr. S.V. Providence COMP 370

38 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15  - 1 5,7,13,15XXXX  1 - - 19,13,11,15XXXX The circled Xs are the Essential Prime Implicants, so we check them off. Dr. S.V. Providence COMP 370

39 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15  - 1 5,7,13,15XXXX  1 - - 19,13,11,15XXXX  We check off the minterms covered by each of the EPIs. Dr. S.V. Providence COMP 370

40 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 5 7 9 11 13 15  - 1 5,7,13,15XXXX  1 - - 19,13,11,15XXXX  WXYZ -1-1 1--1 EPIs: F = X & Z # W & Z = (X # W) & Z Dr. S.V. Providence COMP 370

41 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4 20010 81000 30011 60110 101010 121100 70111 111011 141110 151111 Dr. S.V. Providence COMP 370

42 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4  200102,3001-  810002,60-10 2,10-010  300118,1010-0  601108,121-00  101010  1211003,70-11 3,11-011  701116,7011-  1110116,14-110  14111010,141-10 10,11101-  15111112,1411-0 7,15-111 11,151-11 14,15111- Dr. S.V. Providence COMP 370

43 Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4  20010  2,3001-2,3,6,70-1-  81000  2,60-102,6,3,70-1-  2,10-0102,3,10,11-01-  30011  8,1010-02,6,10,14- - 10  60110  8,121-002,10,3,11- 01-  1010102,10,6,14- - 10  121100  3,70-118,10,12,141 - - 0  3,11-0118,12,10,141 - - 0  70111  6,7011-  111011  6,14-1103,7,11,15- - 11  141110  10,141-103,11,7,15- - 11  10,11101-6,7,14,15- 11 -  151111  12,1411-06,14,7,15- 11 - 10,14,11,151 -  7,15-11110,11,14,151 -  11,151-11  14,15111- 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) Step 1Step 2Step 3Step 4  20010  2,3001-  2,3,6,70-1-2,3,6,7,10,14,11,15- - 1 -  81000  2,60-10  2,6,3,70-1-2,3,10,11,6,14,7,15- - 1 -  2,10-010  2,3,10,11-01-2,6,3,7,10,11,14,15- - 1 -  30011  8,1010-0  2,6,10,14- - 102,6,10,14,3,7,11,15- - 1 -  60110  8,121-00  2,10,3,11- 01-2,10,3,11,6,7,14,15- - 1 -  101010  2,10,6,14- - 102,10,6,14,3,11,7,15- - 1 -  121100  3,70-118,10,12,141 - - 0  3,11-0118,12,10,141 - - 0  70111  6,7011-  111011  6,14-110  3,7,11,15- - 11  141110  10,141-10  3,11,7,15- - 11  10,11101-  6,7,14,15- 11 -  151111  12,1411-0  6,14,7,15- 11 -  10,14,11,151 -  7,15-111  10,11,14,151 -  11,151-11  14,15111- Dr. S.V. Providence COMP 370

45 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15 1 - - 08,12,10,14 - - 1 -2,3,6,7,10,11,14,15 Dr. S.V. Providence COMP 370

46 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15 1 - - 08,12,10,14XXXX - - 1 -2,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

47 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15 1 - - 08,12,10,14XXXX - - 1 -2,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

48 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15  1 - - 08,12,10,14XXXX  - - 1 -2,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

49 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15  1 - - 08,12,10,14XXXX  - - 1 -2,3,6,7,10,11,14,15XXXXXXXX  Dr. S.V. Providence COMP 370

50 Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms 2 3 6 7 8 10 11 12 14 15  1 - - 08,12,10,14XXXX  - - 1 -2,3,6,7,10,11,14,15XXXXXXXX  WXYZ 1--0 --1- EPIs: F = (W & !Z) # Y Dr. S.V. Providence COMP 370

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