Download presentation
Presentation is loading. Please wait.
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.