lecture six chapter 5: quine-mccluskey method dr. s.v. providence comp 370

Lecture Six Chapter 5: Quine-McCluskey Method

Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370

Computer Minimization Techniques

• Boolean Algebra

• Karnaugh Maps

• Quine-McCluskey Method

Boolean AlgebraBoolean Algebra

Review of Boolean Postulates Review of Boolean Identities Example1 Example2

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

Review Boolean IdentitiesReview Boolean Identities

0 & A = 0, A & 0 = 0Contradiction (always false)

A # 1 = 1, 1 # A = 1Tautology (always true)

A & A = A A # A = A Idempotence

A & (B & C) = (A & B) & C 0 # A = A Associative Laws

!(A & B) = !A # !B orA NAND B = !A OR !B

!(A # B) = !A & !B orA NOR B = !A AND !B

DeMorgan’s Theorem

!!A = A Involution

Example1Example1A # A & B = A

Proof:

1. A # A & B = A & 1 # A & B Identity

2. = A & ( 1 # B ) Distribution

3. = A & 1 Identity

4. = A

(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

Example2Example2

Karnaugh MapsKarnaugh Maps

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

2-Variable K -map2-Variable K -map

F(X,Y) = (0,1,2,3)

00 01 11 10

(0,1,2,3,4,5,6,7)

Example1Example1

F(X,Y,Z) = (1,3,4,5,6,7)

Example1Example1

00 01 11 10

F(X,Y,Z) = (1,3,4,5,6,7)

Example1Example1

00 01 11 10

F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

Example1Example1

00 01 11 10

F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

Example1Example1

00 01 11 10

F(X,Y,Z) = X # Z

Example2Example2

F(X,Y,Z) = (0,2,4,6)

Example2Example2

00 01 11 10

F(X,Y,Z) = (0,2,4,6)

Example2Example2

00 01 11 10

F(X,Y,Z) =

Example2Example2

00 01 11 10

F(X,Y,Z) = !Z

00 01 11 10

10 m8 m9 m11 m10

m12 m13 m15 m14W

ZF(W,X,Y,Z) = (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)

F(W,X,Y,Z) = (5,7,9,11,13,15)

Example1Example1

00 01 11 10

F(W,X,Y,Z) = (5,7,9,11,13,15)

Example1Example1

00 01 11 10

F(W,X,Y,Z) =X & Z # W & Z =(X # W) & Z

Example2Example2

F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)

Example2Example2

00 01 11 10

F(W,X,Y,Z) =(2,3,6,7,8,10,11,12,14,15)

Example2Example2

00 01 11 10

F(W,X,Y,Z) = W & !Z # Y

10 m8m9 m11

m12 m13 m15 m14W

01 11 10

m16m17

m20m21

m19m18

m23m22

10 m24m25 m27

m28 m29 m31 m30W

01 11 10

V=0 V=1

Quine-McCluskey MethodQuine-McCluskey Method

Prime Implicants Table3 or 4 steps

Essential Prime Implicants Table

Finding Prime Implicants (PIs)

F(W,X,Y,Z) = (5,7,9,11,13,15)

Step 1 Step 2 Step 3

List minterms by the number of 1s it contains.

F(W,X,Y,Z) = (5,7,9,11,13,15)

5 0101

9 1001

7 0111

11 1011

13 1101

15 1111

F(W,X,Y,Z) = (5,7,9,11,13,15)

5 0101 5,7

9 1001 5,13

7 0111 9,13

11 1011

13 1101 7,15

15 1111 13,15

Enter combinations of minterms by the number of 1s it contains.

F(W,X,Y,Z) = (5,7,9,11,13,15)

5 0101 5,7 01-1

9 1001 5,13 -101

9,11 10-1

7 0111 9,13 1-01

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

Check off elements used from Step 1.

F(W,X,Y,Z) = (5,7,9,11,13,15)

5 0101 5,7 01-1 5,7,13,15 -1-1

9 1001 5,13 -101 5,13,7,15 -1-1

9,11 10-1 9,11,13,15 1- -1

7 0111 9,13 1-01 9,13,11,15 1- -1

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

Enter combinations of minterms by the number of 1s it contains.

F(W,X,Y,Z) = (5,7,9,11,13,15)

5 0101 5,7 01-1 5,7,13,15 -1-1

9 1001 5,13 -101 5,13,7,15 -1-1

9,11 10-1 9,11,13,15 1- -1

7 0111 9,13 1-01 9,13,11,15 1- -1

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

The entries left unchecked are Prime Implicants.

Finding Essential Prime Implicants (EPIs)

Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15

- 1 - 1 5,7,13,15

1 - - 1 9,13,11,15

Enter the Prime Implicants and their minterms.

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

Enter Xs for the minterms covered.

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

Circle Xs that are in a column singularly.

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

The circled Xs are the Essential Prime Implicants,so we check them off.

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

We check off the minterms covered by each of the EPIs.

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

W X Y Z

- 1 - 1

1 - - 1

EPIs: F = X & Z # W & Z = (X # W) & Z

Finding Prime Implicants (PIs)F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)

Step 1 Step 2 Step 3 Step 4

2 0010

8 1000

3 0011

6 0110

10 1010

12 1100

7 0111

11 1011

14 1110

15 1111

2 0010 2,3 001-

8 1000 2,6 0-10

2,10 -010

3 0011 8,10 10-0

6 0110 8,12 1-00

10 1010

12 1100 3,7 0-11

3,11 -011

7 0111 6,7 011-

11 1011 6,14 -110

14 1110 10,14 1-10

10,11 101-

15 1111 12,14 11-0

7,15 -111

11,15 1-11

14,15 111-

2 0010 2,3 001- 2,3,6,7 0-1-

8 1000 2,6 0-10 2,6,3,7 0-1-

2,10 -010 2,3,10,11 -01-

3 0011 8,10 10-0 2,6,10,14 - - 10

6 0110 8,12 1-00 2,10,3,11 - 01-

10 1010 2,10,6,14 - - 10

12 1100 3,7 0-11 8,10,12,14 1 - - 0

3,11 -011 8,12,10,14 1 - - 0

7 0111 6,7 011-

11 1011 6,14 -110 3,7,11,15 - - 11

14 1110 10,14 1-10 3,11,7,15 - - 11

10,11 101- 6,7,14,15 - 11 -

15 1111 12,14 11-0 6,14,7,15 - 11 -

10,14,11,15 1 - 1 -

7,15 -111 10,11,14,15 1 - 1 -

11,15 1-11

14,15 111-

2 0010 2,3 001- 2,3,6,7 0-1- 2,3,6,7,10,14,11,15 - - 1 -

8 1000 2,6 0-10 2,6,3,7 0-1- 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 -

3 0011 8,10 10-0 2,6,10,14 - - 10 2,6,10,14,3,7,11,15 - - 1 -

6 0110 8,12 1-00 2,10,3,11 - 01- 2,10,3,11,6,7,14,15 - - 1 -

10 1010 2,10,6,14 - - 10 2,10,6,14,3,11,7,15 - - 1 -

12 1100 3,7 0-11 8,10,12,14 1 - - 0

3,11 -011 8,12,10,14 1 - - 0

7 0111 6,7 011-

11 1011 6,14 -110 3,7,11,15 - - 11

14 1110 10,14 1-10 3,11,7,15 - - 11

10,11 101- 6,7,14,15 - 11 -

15 1111 12,14 11-0 6,14,7,15 - 11 -

10,14,11,15 1 - 1 -

7,15 -111 10,11,14,15 1 - 1 -

11,15 1-11

14,15 111-

Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15

1 - - 0 8,12,10,14

- - 1 - 2,3,6,7,10,11,14,15

1 - - 0 8,12,10,14 X X X X