1 Quine-McCluskey Method. 2 Motivation Karnaugh maps are very effective for the minimization of expressions with up to 5 or 6 inputs. However they are.
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1
Quine-McCluskey Method
2
Motivation
Karnaugh maps are very effective for the minimizationof expressions with up to 5 or 6 inputs. However theyare difficult to use and error prone for circuits withmany inputs.
Karnaugh maps depend on our ability to visuallyidentify prime implicants and select a set of primeimplicants that cover all minterms. They do notprovide a direct algorithm to be implemented in acomputer.
For larger systems, we need a programmable method!!
3
Quine-McCluskey
Quine, Willard, “A way to simplify truth functions.” American Mathematical Monthly, vol. 62, 1955.
Quine, Willard, “The problem of simplifying truth functions.” American Mathematical Monthly, vol. 59, 1952.
Willard van Orman Quine 1908-2000, Edgar Pierce Chair of Philosophy at Harvard University.http://members.aol.com/drquine/wv-quine.html
McCluskey Jr., Edward J. “Minimization of Boolean Functions.” Bell Systems Technical Journal, vol. 35, pp. 1417-1444, 1956
Edward J. McCluskey, Professor of ElectricalEngineering and Computer Science at Stanfordhttp://www-crc.stanford.edu/users/ejm/McCluskey_Edward.html
4
Outline of the Quine-McCluskey Method
1. Produce a minterm expansion (standard sum-of-products form) for a function F
2. Eliminate as many literals as possible by systematically applying XY + XY’ = X.
3. Use a prime implicant chart to select a minimum set of prime implicants that when ORed together produce F, and that contains a minimum number of literals.
5
Determination of Prime Implicants
AB’CD’ + AB’CD = AB’C
1 0 1 0 + 1 0 1 1 = 1 0 1 -
(The dash indicates a missing variable)
A’BC’D + A’BCD’
0 1 0 1 + 0 1 1 0
We can combine the minterms above because theydiffer by a single bit. The minterms below won’t combine
6
Quine-McCluskey MethodAn Example
1. Find all the prime implicants
)14,10,9,8,7,6,5,2,1,0(),,,( mdcbaf
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Group the minterms according to the numberof 1s in the minterm.
This way we only have tocompare minterms fromadjacent groups.
7
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
Combininggroup 0 and
group 1:
8
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000-
Combininggroup 0 and
group 1:
9
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0
Combininggroup 0 and
group 1:
10
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000
Does it makesense to nocombine group 0with group 2 or 3?
No, there are atleast two bits thatare different.
11
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000
Does it makesense to nocombine group 0with group 2 or 3?
No, there are atleast two bits thatare different.
Thus, next we combine group 1and group 2.
12
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01
Combine group 1and group 2.
13
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01
Combine group 1and group 2.
14
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001
Combine group 1and group 2.
15
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001
Combine group 1and group 2.
16
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001
Combine group 1and group 2.
17
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001
Combine group 1and group 2.
18
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10
Combine group 1and group 2.
19
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10
Combine group 1and group 2.
20
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010
Combine group 1and group 2.
21
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010
Combine group 1and group 2.
22
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010
Combine group 1and group 2.
23
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100-
Combine group 1and group 2.
24
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0
Again, there isno need to tryto combine group1 with group 2.
25
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0
Again, there isno need to tryto combine group1 with group 3.
Lets try to combinegroup 2 with group 23.
26
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1
Combine group 2and group 3.
27
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1
Combine group 2and group 3.
28
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011-
Combine group 2and group 3.
29
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110
Combine group 2and group 3.
30
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110
Combine group 2and group 3.
31
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110
Combine group 2and group 3.
32
Quine-McCluskey MethodAn Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110
Combine group 2and group 3.
33
Quine-McCluskey MethodAn Example
Column I Column II
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
We have nowcompleted thefirst step. Allminterms in column I wereincluded.
We can dividecolumn II intogroups.
34
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
35
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
36
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
37
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
38
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
39
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
40
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
41
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
42
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
43
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
44
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
45
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
46
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
47
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0
48
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0
49
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00-
50
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00-
51
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0
52
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0
53
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0
54
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0
55
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0
56
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10
57
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10
58
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10
59
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10
60
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10
No more combinationsare possible, thus westop here.
61
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10
We can eliminate repeatedcombinations
62
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d
Now we form f with theterms not checked
63
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d + a’bd
64
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d + a’bd + a’bc
65
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d + a’bd + a’bc + b’c’
66
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d + a’bd + a’bc + b’c’ + b’d’
67
Quine-McCluskey MethodAn Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 0101 6 0110 9 100110 1010
7 011114 1110
0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -11010,14 1-10
Column III
0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
68
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
b
c
d 1
69
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
b
c
d 1
70
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
b
c
d 1
71
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
b
c
d 1
72
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1
b
c
d 1
73
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized. Using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d 1
74
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = a’bd
75
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = a’bd + cd’
76
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = a’bd + cd’ + b’c’
77
Quine-McCluskey MethodAn Example
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
What are the extra terms in the solution obtainedwith the Quine-McCluskey method?
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = a’bd + cd’ + b’c’
Thus, we need a method to eliminate this redundant termsfrom the Quine-McCluskey solution.
78
The Prime Implicant Chart
The prime implicant chart is the second part ofthe Quine-McCluskey procedure.
It is used to select a minimum set of prime implicants.
Similar to the Karnaugh map, we first selectthe essential prime implicants, and then weselect enough prime implicants to cover allthe minterms of the function.
79
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
Question: Given the prime implicant chart above, how can we identify the essential prime implicants of the function?
mintermsP
rime
Impl
ican
ts
80
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
Similar to the Karnaugh map, all we have to do is to look for minterms that are covered by a singleterm.
mintermsP
rime
Impl
ican
ts
81
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
Once a term is included in the solution, all theminterms covered by that term are covered.
Therefore we may now mark the covered mintermsand find terms that are no longer useful.
mintermsP
rime
Impl
ican
ts
82
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
mintermsP
rime
Impl
ican
ts
83
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
As we have not covered all the minterms withessential prime implicants, we must chooseenough non-essential prime implicants to cover the remaining minterms.
mintermsP
rime
Impl
ican
ts
84
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
What strategy should we use to find a minimumcover for the remaining minterms?
mintermsP
rime
Impl
ican
ts
85
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
We choose first prime implicants that cover themost minterms. Should this strategy always work??
mintermsP
rime
Impl
ican
ts
86
Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14 (0,1,8,9) b’c’ X X X X (0,2,8,10) b’d’ X X X X (2,6,10,14) cd’ X X X X (1,5) a’c’d X X (5,7) a’bd X X (6,7) a’bc X X
Therefore our minimum solution is:
f(a,b,c,d) = b’c’ + cd’ + a’bd
mintermsP
rime
Impl
ican
ts
87
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Which ones are the essential prime implicants in this chart?
There is no essential prime implicants, how we proceed?
minterms
Prim
e Im
plic
ants
88
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.
minterms
Prim
e Im
plic
ants
F(a,b,c) = a’b’
89
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.
minterms
Prim
e Im
plic
ants
F(a,b,c) = a’b’ + bc’
90
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Thus, we get the minimization:
F(a,b,c) = a’b’ + bc’ + ac
minterms
Prim
e Im
plic
ants
91
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
92
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = a’c
93
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = a’c + b’c’
94
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = a’c + b’c’+ ab
95
Cyclic Prime Implicant Chart
F(a,b,c) = m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7 (0,1) a’b’ X X (0,2) a’c X X (1,5) b’c X X (2,6) bc’ X X (5,7) ac X X (6,7) ab X X
This time we obtain:
F(a,b,c) = a’c + b’c’+ ab
minterms
Prim
e Im
plic
ants
96
Cyclic Prime Implicant Chart
Which minimal form is better?
F(a,b,c) = a’b’ + bc’ + ac
F(a,b,c) = a’c + b’c’+ ab
Depends on what terms we must form for otherfunctions that we must also implement.
Often we are interested in examining all minimalforms for a given function.
Thus we need an algorithm to do so.
97
Petrick’s Method
S. R. Petrick. A direct determination of the irredundant forms of a boolean function from the set of prime implicants. Technical Report AFCRC-TR-56-110, Air Force Cambridge Research Center, Cambridge, MA, April, 1956.
Goal: Given a prime implicant chart, determine all minimum sum-of-products solutions.
98
Petrick’s MethodAn Example
0 1 2 5 6 7 P1 (0,1) a’b’ X X P2 (0,2) a’c X X P3 (1,5) b’c X X P4 (2,6) bc’ X X P5 (5,7) ac X X P6 (6,7) ab X X
Step 1: Label all the rows in the chart.
Step 2: Form a logic function P with the logic variables P1, P2, P3 that is true when all the minterms in the chart are covered.
minterms
Prim
e Im
plic
ants
99
Petrick’s MethodAn Example 0 1 2 5 6 7 P1 (0,1) a’b’ X X P2 (0,2) a’c X X P3 (1,5) b’c X X P4 (2,6) bc’ X X P5 (5,7) ac X X P6 (6,7) ab X X
The first column has an X in rows P1 and P2. Therefore we must include one of these rowsin order to cover minterm 0. Thus the followingterm must be in P:
(P1 + P2)
minterms
Prim
e Im
plic
ants
100
Petrick’s MethodAn Example 0 1 2 5 6 7 P1 (0,1) a’b’ X X P2 (0,2) a’c X X P3 (1,5) b’c X X P4 (2,6) bc’ X X P5 (5,7) ac X X P6 (6,7) ab X X
Following this technique, we obtain:
P = (P1 + P2) (P1 + P3) (P2 + P4) (P3 + P5) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P5 + P3) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P4 + P6) (P5 + P3) (P5 + P6)
P = (P1 + P2 P3) (P4 + P2 P6) (P5 + P3 P6)
minterms
Prim
e Im
plic
ants
101
Petrick’s MethodAn Example
P = (P1 + P2) (P1 + P3) (P2 + P4) (P3 + P5) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P5 + P3) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P4 + P6) (P5 + P3) (P5 + P6)
P = (P1 + P2 P3) (P4 + P2 P6) (P5 + P3 P6)
P = (P1 P4 + P1 P2 P6 + P2 P3 P4 + P2 P3 P6) (P5 + P3 P6)
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P2 P3 P5 P6
+ P1 P3 P4 P6 + P1 P2 P3 P6 + P2 P3 P4 P6 + P2 P3 P6
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P1 P4 P3 P6 + P2 P3 P6
102
Petrick’s MethodAn Example
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P1 P4 P3 P6 + P2 P3 P6
This expression says that to cover all the mintermswe must include the terms in line P1 and line P4 and line P5, or we must include line P1, and line P2, and line P5, and line P6, or …Considering that all the terms P1, P2, … have the samecost, how many minimal forms the function has?
The two minimal forms are P1 P4 P5 and P2 P3 P6:
F = a’b’ + bc’ + ac F = a’c’ + b’c + ab
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