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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