Quine-McCluskey Algorithm Useful for minimizing equations with more - - PowerPoint PPT Presentation

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Quine-McCluskey Algorithm Useful for minimizing equations with more - - PowerPoint PPT Presentation

Feb 05, 2023 290 likes •536 views

Quine-McCluskey Algorithm Useful for minimizing equations with more than 4 inputs. Like K-map, also uses combining theorem Allows for automation Chapter 2 <1> Edward McCluskey (1929-2016) Pioneer in Electrical Engineering

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SLIDE 1

Chapter 2 <1>

  • Useful for minimizing equations with more

than 4 inputs.

  • Like K-map, also uses combining theorem
  • Allows for automation

Quine-McCluskey Algorithm

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SLIDE 2

Chapter 2 <2>

  • Pioneer in Electrical

Engineering

  • First president of IEEE
  • Professor at Princeton, then

Stanford

  • 1955: Quine-McCluskey

Algorithm

Edward McCluskey (1929-2016)

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SLIDE 3

Chapter 2 <3>

Method:

  • 1. 1’s Table:
  • List each minterm, sorted by the number
  • f 1’s it contains.
  • Combine minterms.
  • 2. Prime Implicant Table: When you can’t

combine anymore, list prime implicants and the minterms they cover.

  • 3. Select Prime Implicants to cover all

minterms.

Quine-McCluskey Algorithm

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SLIDE 4

Chapter 2 <4>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

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SLIDE 5

Chapter 2 <5>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

1’s Table

Number

  • f 1's

Size 1 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6

Order the minterms by the number of 1’s they have.

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SLIDE 6

Chapter 2 <6>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

1’s Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)

01- m(2,3)

  • 10 m(2,6)

0-0 m(0,2)

Combine minterms in adjacent groups (starting with the top group).

  • To combine terms, minterm can differ (be 1 or 0) in only 1 place
  • ‘-’ indicates “don’t care”
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SLIDE 7

Chapter 2 <7>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

1’s Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)

01- m(2,3)

  • 10 m(2,6)

0-0 m(0,2) 0-- m(0,1,2,3)

Combine minterms in adjacent groups (starting with the top group).

  • To combine terms, minterm can differ (be 1 or 0) in only 1 place
  • ‘-’ (don’t care) acts like another variable (0, 1, -)
  • Match up ‘-’ first.
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SLIDE 8

Chapter 2 <8>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

1’s Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)*

01- m(2,3)

  • 10 m(2,6)*

0-0 m(0,2) 0-- m(0,1,2,3)*

List prime implicants:

  • Largest implicants containing a given minterm
  • For example m(0,1) is an implicant but not a prime implicant,

because m(0,1,2,3) is a larger implicant containing those minterms.

  • However, m(1,5) is a prime implicant – no larger implicant exists

that contains minterm 5.

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SLIDE 9

Chapter 2 <9>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

Prime Implicants Minterms ABC m(0,1,2,3) 1 2 3 5 6 m(1,5) m(2,6) X X X X X X X X 0 - -

  • 01
  • 10

1’s Table Prime Implicant Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)*

01- m(2,3)

  • 10 m(2,6)*

0-0 m(0,2) 0-- m(0,1,2,3)* List prime implicants. Show which of the required minterms each includes.

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SLIDE 10

Chapter 2 <10>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

Prime Implicants Minterms ABC m(0,1,2,3) 1 2 3 5 6 m(1,5) m(2,6) X X X X X X X X 0 - -

  • 01
  • 10

1’s Table Prime Implicant Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)*

01- m(2,3)

  • 10 m(2,6)*

0-0 m(0,2) 0-- m(0,1,2,3)* Select columns with only 1 X. Corresponding prime implicants must be included in equation.

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SLIDE 11

Chapter 2 <11>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

Prime Implicants Minterms ABC m(0,1,2,3) 1 2 3 5 6 m(1,5) m(2,6) X X X X X X X X 0 - -

  • 01
  • 10

1’s Table Prime Implicant Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)*

01- m(2,3)

  • 10 m(2,6)*

0-0 m(0,2) 0-- m(0,1,2,3)* Select columns with only 1 X. Corresponding prime implicants must be included in equation.

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SLIDE 12

Chapter 2 <12>

Quine-McCluskey Example 1

ABC Y 000 1 001 1 010 1 011 1 100 0 101 1 110 1 111 0

Prime Implicants Minterms ABC m(0,1,2,3) 1 2 3 5 6 m(1,5) m(2,6) X X X X X X X X 0 - -

  • 01
  • 10

1’s Table Prime Implicant Table

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 000 m0 001 m1 1 010 m2 2 101 m5 011 m3 110 m6 00- m(0,1) 0-1 m(1,3)

  • 01 m(1,5)*

01- m(2,3)

  • 10 m(2,6)*

0-0 m(0,2) 0-- m(0,1,2,3)*

Y = A + BC + BC

Select columns with only 1 X. Corresponding prime implicants must be included in equation.

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SLIDE 13

Chapter 2 <13>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

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SLIDE 14

Chapter 2 <14>

Quine-McCluskey Example 2

1’s Table

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0 Number

  • f 1's

Size 1 Implicants 0001 m1 1 2 3 0010 m2 0100 m4 1000 m8 0011 m3 0110 m6 1001 m9 1010 m10 1110 m14

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SLIDE 15

Chapter 2 <15>

Quine-McCluskey Example 2

1’s Table

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0 Number

  • f 1's

Size 1 Implicants Size 2 Implicants 0001 m1 1 2 3 00-1 m(1,3) 001- m(2,3) 0010 m2 0100 m4 1000 m8 0011 m3 0110 m6 1001 m9 1010 m10 1110 m14 0-10 m(2,6)

  • 010 m(2,10)

01-0 m(4,6) 100- m(8,9)

  • 110 m(6,14)

1-10 m(10,14)

  • 001 m(1,9)

10-0 m(8,10)

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SLIDE 16

Chapter 2 <16>

Quine-McCluskey Example 2

1’s Table

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0 Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 0001 m1 1 2 3 00-1 m(1,3) 001- m(2,3)

  • -10 m(2,6,10,14)

0010 m2 0100 m4 1000 m8 0011 m3 0110 m6 1001 m9 1010 m10 1110 m14 0-10 m(2,6)

  • 010 m(2,10)

01-0 m(4,6) 100- m(8,9)

  • 110 m(6,14)

1-10 m(10,14)

  • 001 m(1,9)

10-0 m(8,10)

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SLIDE 17

Chapter 2 <17>

Quine-McCluskey Example 2

1’s Table

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0 Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 0001 m1 1 2 3 00-1 m(1,3)* 001- m(2,3)*

  • -10 m(2,6,10,14)*

0010 m2 0100 m4 1000 m8 0011 m3 0110 m6 1001 m9 1010 m10 1110 m14 0-10 m(2,6)

  • 010 m(2,10)

01-0 m(4,6)* 100- m(8,9)*

  • 110 m(6,14)

1-10 m(10,14)

  • 001 m(1,9)*

10-0 m(8,10)*

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SLIDE 18

Chapter 2 <18>

Quine-McCluskey Example 2

1’s Table

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Number

  • f 1's

Size 1 Implicants Size 2 Implicants Size 4 Implicants 0001 m1 1 2 3 00-1 m(1,3)* 001- m(2,3)*

  • -10 m(2,6,10,14)*

0010 m2 0100 m4 1000 m8 0011 m3 0110 m6 1001 m9 1010 m10 1110 m14 0-10 m(2,6)

  • 010 m(2,10)

01-0 m(4,6)* 100- m(8,9)*

  • 110 m(6,14)

1-10 m(10,14)

  • 001 m(1,9)*

10-0 m(8,10)*

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

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SLIDE 19

Chapter 2 <19>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

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SLIDE 20

Chapter 2 <20>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

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SLIDE 21

Chapter 2 <21>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

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SLIDE 22

Chapter 2 <22>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

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SLIDE 23

Chapter 2 <23>

Quine-McCluskey Example 2

ABCD Y 0000 0 0001 X 0010 1 0011 X 0100 1 0101 0 0110 X 0111 0 1000 1 1001 1 1010 X 1011 0 1100 0 1101 0 1110 1 1111 0

Prime Implicant Table

Prime Implicants Minterms ABCD m(2,6,10,14) 2 4 8 9 14 m(1,3) m(1,9) X X X X

  • -10

00-1

  • 001

m(2,3) m(4,6)

001- 01-0

X X m(8,9) m(8,10)

100- 10-0

X X

Y = CD + ABD + ABC