chap_02_p2b quine mc cluskey

7/29/2019 Chap_02_P2b Quine Mc Cluskey

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Quine-McCluskey (Tabular) Minimization

Two step process utilizing tabular listings to: Identify prime implicants (implicant tables)

Identify minimal PI set (cover tables) All work is done in tabular form

Number of variables is not a limitation

Basis for many computer implementations Dont cares are easily handled Proper organization and term identification are

key factors for correct results


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DifficultyNote: Can be 2n minterms ~ 3n/n primes

Thus O(2n) rows and O(3n/n ) columns AND minimum covering problemis NP-complete. Hence can probably be double exponential in size ofinput, i.e. difficulty is O(23n)

0

1

0

0

10

3n/n

minterms

primes

2n


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Example

Primes: y + w +xz

Covering Table

Solution: {1,2} y + w is minimum prime cover.

(also w +xz)

F x y z w x y zw x y zw x yzw

D yz xyw x y zw x y w x y z w

dd

ddd

dd

dd

00

1

11

01

xy xy xy xy

zw

z w

z w

zw

xz

wKarnaugh map

010

011

110

101

y w xz

xyz w

x y z w

x yz w

xyzw

y


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Covering Table

Definition: An essential prime is any prime that uniquelycovers a minterm of f.

010

011

110

101

y w xz

xyz w

x y z w

x yz w

xyzw

Primes of f+d

Minterms of f

Essential prime

Row singleton(essential minterm)


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Quine-McCluskey Minimization (cont.)

Terms are initially listed one per line in groups Each group contains terms with the same number of true

and complemented variables Terms are listed in numerical order within group

Terms and implicants are identified using one ofthree common notations

full variable form cellular form 1,0,- form


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Example of Different Notations

F(A, B, C, D) = m(4,5,6,8,10,13)

1 ABCD 4 0100

ABCD 8 1000

2 ABCD 5 0101

ABCD 6 0110

ABCD 10 1010

3 ABCD 13 1101

Full variable Cellular 1,0,-


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Implication Table (1,0,-)

Quine-McCluskey Method Tabular method to

systematically find all primeimplicants

(A,B,C,D) = m(4,5,6,8,9,10,13) + d(0,7,15)

Part 1: Find all prime implicants Step 1: Fill Column 1 with

active-set and DC-set mintermindices. Group by number oftrue variables (# of 1s).

Implication Table

Column I

0000

01001 000

0101011010011010

0 1 1 11101

1111NOTE: DCs are included in this step!


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Minimization - First Pass (1,0,-)

Quine-McCluskey Method Tabular method to systematically find

all prime implicants

(A,B,C,D) = m(4,5,6,8,9,10,13) + d(0,7,15)

Part 1: Find all prime implicants

Step 2: Apply Adjacency - Compareelements of group with N 1's againstthose with N+1 1's. One bit differenceimplies adjacent. Eliminate variableand place in next column.

E.g., 0000 vs. 0100 yields 0-00

0000 vs. 1000 yields -000

When used in a combination, mark witha check. If cannot be combined, markwith a star. These are the primeimplicants.

Repeat until nothing left.

Implication Table

Column I0000

0100 1000

0101

0110 1001 1010

0111

1101

1111

Column II0-00-000

010-01-0100-

10-0

01-1-101011-

1-01

-11111-1


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Stage 2: find smallest set of prime implicants that cover theactive-setrecall that essential prime implicants must be in

final expression

Prime Implicants (cont.)

Prime Implicants:

DCA00-0

CBA-100

DCB000-

DBA0-10

DCA01-1 DB1-1-

BA--01

X 1 0 1

0 1 1 1

0 X X 0

0 1 0 1

AB

00 01 11

01

11

10C

CD

A

D

B

00

10


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rows = prime implicantscolumns = ON-set elements

place an "X" if ON-set element is covered by the prime implicant

Coverage Chart

0,4 0-00

0,8 -000

8,9 100-

8,10 10-0

9,13 1-01

4,5,6,7 01--

5,7,13,15 -1-1

4X

X

5

X

X

6

X

8

X

X

X

9

X

X

10

X

13

X

X

Coverage Table

Note: Dont include DCs in coveragetable; they dont have covered by thefinal logic expression!


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Row and Column Dominance Definition: A row i2 whose set of primes is contained in the set of

primes of row i1 is said to be dominated.

Example:i1 x x x x x xi2 x x x x

i1 dominates i2

We can remove row i2, because covering i1, i2 is automaticallycovered.


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Coverage Table (cont.)

rows = prime implicantscolumns = ON-set elementsplace an "X" if ON-set element is

covered by the prime implicant

If column has a single X, than theimplicant associated with the rowis essential. It must appear inminimum cover

0,4 0-00

0,8 -000

8,9 100-

8,10 10-0

9,13 1-01

4,5,6,7 01--

5,7,13,15 -1-1

4X

X

5

X

X

6

X

8

X

X

X

9

X

X

10

X

13

X

X

0,4 0-00

0,8 -000

8,9 100-

8,10 10-0

9,13 1-01

4,5,6,7 01--

5,7,13,15 -1-1

4X

X

5

X

X

6

X

8

X

X

X

9

X

X

10

X

13

X

X

Coverage Chart


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Cyclic Core

110

011

101

456

CyclicCore


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chap_02_p2b quine mc cluskey

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