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Quine-McCluskey Algorithm
Gautam Singh
Outline: step 1Find Prime Implicants:
1) Find all the minterms and sort by number of 1 bits (these are implicants)
2) Compare every implicant with n 1-bits to all implicants with n+1 1-bits.
3) Combine implicants that vary by a single bit. Replace the bit with an X.
4) Check off implicants that have been 'used'
5) Cross off duplicate implicants.
6) Repeat steps 2 through 5 until no more minterms can be combined
7) The prime implicants are those that were not checked off.
Outline: step 2Find Essential Prime Implicants:
Make a prime implicant chart with minterms as columns and prime implicants as rows
Look for columns that are 'covered' by only a single prime implicant. These are called
“Essential Prime Implicants.” Put a check next to them and also check off all the columns
they cover.
Among the remaining columns, select the fewest amount of remaining prime implicants
that cover them.
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
1 1 1
1 1 1
BC CBCB
A
A
CB
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
#1 bits implicants
1 001
1 010
2 011
2 101
2 110
3 111
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
1 1 1
1 1 1
BCB C B CB C
A
A
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
#1 bits implicants
1 001
1 010
2 011
2 101
2 110
3 111
implicants
0X1
X01
01X
X10
X11
1X1
11X
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
1 1 1
1 1 1
BCB C B CB C
A
A
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
#1 bits implicants
1 001
1 010
2 011
2 101
2 110
3 111
implicants
0X1
X01
01X
X10
X11
1X1
11X
implicants
XX1
XX1
X1X
X1X
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
1 1 1
1 1 1
BCB C B CB C
A
A
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
#1 bits implicants
1 001
1 010
2 011
2 101
2 110
3 111
implicants
0X1
X01
01X
X10
X11
1X1
11X
implicants
XX1
XX1
X1X
X1X
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
1 1 1
1 1 1
BCB C B CB C
A
A
XX1
X1X
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
001 010 011 101 110 111
XX1
X1X
1 1 1
1 1 1
BCB C B CB C
A
A
A B C Y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
C +ABC¿
B C+AB ¿A B CA BC+A ¿
A B C+ ¿Y= ¿
¿
001 010 011 101 110 111
XX1
X1X
Y= C + B
1 1 1
1 1 1
BCB C B CB C
A
A
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
1 bits implicants
1 0010
1 1000
2 0011
2 0110
2 1001
2 1100
3 0111
3 1101
3 1110
4 1111
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
1 bits implicants
1 0010
1 1000
2 0011
2 0110
2 1001
2 1100
3 0111
3 1101
3 1110
4 1111
001X
0X10
100X
1X00
0X11
011X
X110
1X01
110X
11X0
X111
11X1
111X
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
1 bits implicants
1 0010
1 1000
2 0011
2 0110
2 1001
2 1100
3 0111
3 1101
3 1110
4 1111
001X
0X10
100X
1X00
0X11
011X
X110
1X01
110X
11X0
X111
11X1
111X
0X1X
0X1X
1X0X
1X0X
X11X
X11X
11XX
11XX
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
1 bits implicants
1 0010
1 1000
2 0011
2 0110
2 1001
2 1100
3 0111
3 1101
3 1110
4 1111
001X
0X10
100X
1X00
0X11
011X
X110
1X01
110X
11X0
X111
11X1
111X
0X1X
0X1X
1X0X
1X0X
X11X
X11X
11XX
11XX
0X1X
1X0X
X11X
11XX
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
0010 0011 0110 0111 1110
11XX
X11X
1X0X
0X1X
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
0010 0011 0110 0111 1110
11XX
X11X
1X0X
0X1X
A B C D Y
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 X
1 0 0 1 X
1 0 1 0 0
1 0 1 1 0
1 1 0 0 X
1 1 0 1 X
1 1 1 0 1
1 1 1 1 X
0010 0011 0110 0111 1110
11XX
X11X
1X0X
0X1X
BCCAY
ABCAY
1 1
1 1
X X X 1
X X
BA
BA
AB
BA
DC DCDC CD
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
00000
00001
00100
10000
00101
10001
10100
01110
10101
01111
11011
11111
00000
00001
00100
10000
00101
10001
10100
01110
10101
01111
11011
11111
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
0000X
00X00
X0000
00X01
X0001
0010X
X0100
1000X
10X00
X0101
10X01
1010X
0111X
X1111
11X11
00000
00001
00100
10000
00101
10001
10100
01110
10101
01111
11011
11111
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
0000X
00X00
X0000
00X01
X0001
0010X
X0100
1000X
10X00
X0101
10X01
1010X
0111X
X1111
11X11
00X0X
X000X
00X0X
X0X00
X000X
X0X00
X0X01
X0X01
X010X
X010X
10X0X
10X0X
00000
00001
00100
10000
00101
10001
10100
01110
10101
01111
11011
11111
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
0000X
00X00
X0000
00X01
X0001
0010X
X0100
1000X
10X00
X0101
10X01
1010X
0111X
X1111
11X11
00X0X
X000X
00X0X
X0X00
X000X
X0X00
X0X01
X0X01
X010X
X010X
10X0X
10X0X
00X0X
X000X
X0X00
X0X01
X010X
10X0X
00000
00001
00100
10000
00101
10001
10100
01110
10101
01111
11011
11111
A B C D E Y
0 0 0 0 0 1
0 0 0 0 1 1
0 0 0 1 0 0
0 0 0 1 1 0
0 0 1 0 0 1
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 1 0
0 1 0 0 0 0
0 1 0 0 1 0
0 1 0 1 0 0
0 1 0 1 1 0
0 1 1 0 0 0
0 1 1 0 1 0
0 1 1 1 0 1
0 1 1 1 1 1
1 0 0 0 0 1
1 0 0 0 1 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 1 0 0 1
1 0 1 0 1 1
1 0 1 1 0 0
1 0 1 1 1 0
1 1 0 0 0 0
1 1 0 0 1 0
1 1 0 1 0 0
1 1 0 1 1 1
1 1 1 0 0 0
1 1 1 0 1 0
1 1 1 1 0 0
1 1 1 1 1 1
0000X
00X00
X0000
00X01
X0001
0010X
X0100
1000X
10X00
X0101
10X01
1010X
0111X
X1111
11X11
00X0X
X000X
00X0X
X0X00
X000X
X0X00
X0X01
X0X01
X010X
X010X
10X0X
10X0X
00X0X
X000X
X0X00
X0X01
X010X
10X0X
X0X0X
X0X0X
X0X0X
00000 00001 00100 00101 01110 01111 10000 10001 10100 10101 11011 11111
X0X0X
11X11
X1111
0111X
00000 00001 00100 00101 01110 01111 10000 10001 10100 10101 11011 11111
X0X0X
11X11
X1111
0111X
00000 00001 00100 00101 01110 01111 10000 10001 10100 10101 11011 11111
X0X0X
11X11
X1111
0111X
ABDEBCDADBY
A B C D Y
0 0 0 0 1
0 0 0 1 1
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 1
0 1 1 0 X
0 1 1 1 1
1 0 0 0 0
1 0 0 1 0
1 0 1 0 1
1 0 1 1 0
1 1 0 0 0
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1
A B C D Y
0 0 0 0 1
0 0 0 1 1
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 1
0 1 1 0 X
0 1 1 1 1
1 0 0 0 0
1 0 0 1 0
1 0 1 0 1
1 0 1 1 0
1 1 0 0 0
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1