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Quine-McCluskey Method (Class 6.1 – 10/2/12)
CSE 2441 – Introduction to Digital Logic
Fall 2012
Instructor – Bill Carroll, Professor of CSE
Today’s Topics
• Return and discuss exam 1
• Quine-McCluskey simplification method
Quine-McCluskey Minimization Method
• Advantages over K-maps – Can be computerized
– Can handle functions of more than six variables
– Can be adapted to minimize multiple functions
• Overview of the method – Given the minterms of a function
– Find all prime implicants (steps 1 and 2) • Partition minterms into groups according to the number of 1’s
• Exhaustively search for prime implicants
– Find a minimum prime implicant cover (steps 3 and 4) • Construct a prime implicant chart
• Select the minimum number of prime implicants
– Note – the method can also be described for maxterms and implicates
Example 3.24 -- Use the Q-M method to find the MSOP of the function
f(A,B,C,D) = ∑m(2,4,6,8,9,10,12,13,15)
CD
AB
1
00 01 11 10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
00
01
11
10
B
D
1 1
1
1 1 1
A
C
1 1
Figure 3.32 K-map for example 3.30.
Step 1 -- List Prime Implicants in Groups (Example 3.24)
Step 2 -- Generate Prime Implicants (Example 3.24)
Step 3 -- Prime Implicant Chart (Example 3.24)
642 8 10 12 13 15
´
Ä
Ö
PI2
PI3
PI4
PI5
PI6
* * PI7
* * PI1
´
´
´ ´
´
´
´
´
´
´
´ ´ ´
9
Ö Ö Ö Ö
Ä
Step 4 -- Reduced Prime Implicant Chart (Example 3.24)
642 10
´
Ö
PI2
*PI3
*PI4
PI5
PI6
´
´
´
ÖÖ Ö
´ ´
´
´
The Resulting Minimal Realization of f
f(A,B,C,D) = PI1 + PI3 + PI4 + PI7
= 1-0- + -010 + 01-0 + 11-1
= AC + B CD + A BD + ABD
How the Q-M Results Look on a K-map
CD
AB
00 01 11 10
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
00
01
11
10
B
1
A
C
1 1
1
D
1
1
1
1 1
Figure 3.33 Grouping of terms.
Covering Procedure
Step 1 -- Identify any minterms covered by only one PI. Select these PIs for the cover.
Step 2 -- Remove rows covered by the PIs identified in step 1. Remove minterms covered by the removed rows.
Step 3 -- If a cyclic chart results from step 2, go to step 5. Otherwise, apply the reduction procedure of steps 1 and 2.
Step 4 -- If a cyclic chart results from step 3, go to step 5. Otherwise return to step 1.
Step 5 -- Apply the cyclic chart procedure. Repeat step 5 until a void chart or noncyclic chart chart is produced. In the latter case, return to step 1.
Coverage Example f(A,B,C,D) = m(0,1,5,6,7,8,9,10,11,13,14,15)
510 6 8 9 10 11
´
Ä
Ö
* * PI1
PI2
PI3
PI4
PI5
PI6
* * PI7
´
´
´
´
´
´ ´ ´
7
Ö Ö Ö Ö Ö
13 14 15
Ö Ö
Ä
´ ´
´
´
´
´
´ ´ ´
´ ´
´
´
´
´
´
´
Reduced PI Charts
11105 13
´
PI2
PI3
PI4
PI5
PI6
´
´ ´
´
´
´ ´
´ ´´
Ö Ö
´
105
* PI2
*PI4
Cyclic PI Charts
1. No essential PIs.
2. No row or column coverage.
321 4
´
*PI1
PI2
PI3
PI4
PI5
PI6
´ ´
ÖÖ
´ ´
´
5 6
´´
´ ´
´´
542 6
´
PI2
PI3
PI4
PI5
PI6
´
´
´
´
´
´
´
542 6
´
Ö
*PI3
PI4
*PI5
´
ÖÖ Ö
´ ´
´
´
Using the Q-M Method with Incompletely Specified Functions
1. Use minterms and don’t cares when generating prime implicants
2. Use only minterms when finding a minimal cover
Example 3.25 -- Find a minimal sum of products of the following function
using the Quine-McCluskey procedure.
Minimizing Table for Example 3.25
PI Chart for Example 3.25
732 10 15 27
Ä
Ö
PI1
PI2
PI3
´
´
12
Ö Ö Ö Ö Ö
Ä
´ ´
* * PI4
* * PI5
* * PI6
* * PI7
Ä
Ä
´ ´
´
Results of Minimization for Example 3.25
f(A,B,C,D,E) = PI1 + PI4 + PI5 + PI6 + PI7 OR
= PI2 + PI4 + PI5 + PI6 + PI7
Minimizing Circuits with Multiple Outputs
(12,15)(0,2,7,10)=),,,( dmDCBAf
(6,7,8,10)(2,4,5)=),,,( dmDCBAf
(0,5,13)(2,7,8)=),,,( dmDCBAf
Minimizing Table for Example 3.26
Prime Implicant Chart for Example 3.26
720 10 4 5 2 7
Ä
Ö
* * PI2
PI3
PI4
* * PI5
PI6
PI7
PI8
PI9
´
´
2
Ö Ö Ö Ö Ö
8
´
´ ´
´
Ö
f f f
* * PI1
PI10
PI11
PI12
PI13
Ä ´
Ä
´
´ ´ ´
´
´´
´
´
Reduced Prime Implicant Chart for Example 3.26
877
´
* PI3
PI7
PI9 ´
ÖÖ Ö
´
f f
PI11
* PI13
´´
´
Minimum Realizations for Example 3.26
1352= PIPIPIf
51= PIPIf
1332= PIPIPIf
BCDADCBDBAf =
DCBBAf =
BCDADCBDBAf =
Figure 3.34 Reduced multiple-output circuit.
A CB D
f
f
PI1
PI2
PI3
PI5
PI13
A CB D
fa
