Lecture 6:
Boolean Algebra and
minimization techniques
Quine-McCluskey Minimization Method
• Advantages over K-maps
– Can be computerized
– Can handle functions of more than six variables
• 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
Quine-McCluskey Method: Outline
1.
Express each minterm in n variables by a bit string of length n
with the i-th position filled by 1 if x_i occurs in the expression
and 0 if ~x_i occurs.
2. Group the bit strings according to the number of 1's in them.
3. Use the Resolution Rule to combine as many n-variable terms
into terms containing n-1 variables as possible.
4. Continue applying the Resolution Rule to combine Boolean
products into fewer variables as long as possible.
5. Find the smallest set of the resulting combined Boolean
products, together with any of the original terms that were
never combined with anything, so that the sum of these
products represents the Boolean function.
Quine-McCluskey Method: Representation
Represent each product in the function as a Boolean sequence,
according to whether the variables or their negations occur in the
product.
For the example on the previous slide, the seven products would be
represented as:
w
w
w
~w
~w
~w
~w
x y ~z
~x y z
~x y ~z
x y z
x ~y z
~x y z
~x ~y z
1110
1011
1010
0111
0101
0011
0001
Quine-McCluskey Method: Combining
Order the terms by the number of 1's in them; then form all the 3-term
sequences you can by looking for 4-term sequences that differ in only one
position; note that we only need to compare each sequence to ones that
have exactly one more 1 in them.
Label each term with its numeric value.
term F
-----------1 0001 1
-----------3 0011 1
5
10
0101
1010
1
1
-----------7 0111 1
11 1011 1
14 1110 1
------------
--> 1,3
--> 1,5
00-1
0-01
-->
-->
-->
-->
-->
0-11
-011
01-1
1011-10
3,7
3,11
5,7
10,11
10,14
none
none
none
Quine-McCluskey Method: Combining
Repeat the process with the 3-term sequences, forming 2-term sequences:
Second phase:
-----------1,3
00-1
1,5
0-01
3,7
0-11
3,11
-011
5,7
01-1
10,11
10110,14
1-10
--> 1,3,5,7
--> 1,3,5,7
none
none
none
none
none
0--1
0--1 (duplicate)
Q-M Method: Selecting Prime Implicants
---------------------------------------Prime
Implicants
Coverage Map
1
3
5
7 10 11 14
---------------------------------------1,3,5,7
0--1
x
x
x
x
1,3
00-1
x
x
1,5
0-01
x
3,7
0-11
x
3,11
-011
x
5,7
01-1
10,11
101-
x
10,14
1-10
x
x
x
x
or
x
x
x
----------------------------------------
Essential for 14
Quine-McCluskey Method: Formation
Finally, translate the selected Boolean sequences back into the appropriate product
terms:
0--1
1-10
101-
wz  wy z  wxy
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)
A
AB
CD
00
0
01
4
00
11
12
1
1
5
13
1
9
1
3
7
15
11
1
D
11
1
2
10
8
1
01
C
10
6
1
14
1
10
1
B
Step 1 -- List Prime Implicants in Groups
Step 2 -- Generate Prime Implicants
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
Definition of terms for two-level simplification
• Implicant
• Single product term of the ONset (terms that create a logic 1)
• Prime implicant
• Implicant that can't be
combined with another to form
an implicant with fewer literals.
Systematic Simplification
•A Prime Implicant is a product term obtained by
combining the maximum possible number of
adjacent squares in the map.
•A prime implicant is called an Essential Prime
Implicant if it is the only prime implicant that
covers (includes) one or more minterms.
•Prime Implicants and Essential Prime Implicants
can be determined by inspection of the K-Map.
•A set of prime implicants that "covers all
minterms" means that, for each minterm of the
function, there is at least one prime implicant in
the selected set of prime implicants that includes
the minterm.
Karnaugh maps: Don’t cares
•
•
•
In some cases, outputs are undefined
We “don’t care” if the logic produces a 0 or a 1
This knowledge can be used to simplify
functions.
AB
00
CD
C
A
01 11
10
00
0
0
X
0
01
1
1
X
1
11
1
1
0
0
10
0
X
0
0
B
D
- Treat X’s like either 1’s or 0’s
- Very useful
- OK to leave some X’s uncovered
Karnaugh maps: Don’t cares
•
f(A,B,C,D) =  m(1,3,5,7,9) + d(6,12,13)
without don't cares
f=
A’D + C’D
A
AB
00 01 11 10
CD
C
00
0
0
X
0
01
1
1
X
1
11
1
1
0
0
10
0
X
0
0
B
D
A
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
B C
0 0
0 0
0 1
0 1
1 0
1 0
1 1
1 1
0 0
0 0
0 1
0 1
1 0
1 0
1 1
1 1
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
f
0
1
0
1
0
1
X
1
0
1
0
0
X
X
0
0
Don’t Care Conditions
• In
some situations, we don’t care
about the value of a function for
certain combinations of the
variables.
•these combinations may be
impossible in certain contexts
•or the value of the function
may not matter in when the
combinations occur
Don’t Care Conditions
• In
such situations we say the function is
incompletely specified and there are multiple
(completely specified) logic functions that can
be used in the design.
• so
we can select a function that gives the
simplest circuit
• When
constructing the terms in the
simplification procedure, we can choose to
either cover or not cover the don’t care
conditions.
Quine-McCluskey (tabular) method
1. Arrange all minterms in group such that all terms
in the same group have the same # of 1’s in their
binary representation.
2. Compare every term of the lowest-index group
with each term in the successive group. Whenever
possible, combine two terms being compared by
means of
gxi+gxi´=g(xi+xi´)=g.
Two terms from adjacent groups are combinable if
their binary representation differ by just a single
digit in the same position.
3. The process continues until no further
combinations are possible. The remaining
unchecked terms constitute the set of PI.
Ex) f(x1,x2,x3,x4) = (0,1,2,5,6,7,8,9,10,13,15)
#
0
1
2
8
5
6
9
10
7
13
15
x1,x2,x3,x
x1,x2,x3,x4
4
0
0
0
0
10
10
0
1
1
1
0
0
11
1
1
1
0
0
0
01
0
0
1
00
1
0
0
1
1
1
1
1
1 
0 
0 
1
0

0
1







0
0
-
0
0
0
0
0
0
0



(0,1)
(0,2)
(0,8)
0
0
1
1
0
0
0
0
0
0
1
1
0
-
1
1
0
0
0


(1,5)
(1,9)
(2,6)
(2,10)
(8,9)
(8,10)
0
0
1
1
1
1
-
0
1
0
1
1
1


1
1
1
1
-




1 
1 
(5,7)
(5,13)
(6,7)
(9,13)
(7,15)
(13,15)
x1,x2,x3,x
x1,x2,x3,x
4
4
-
0
0
1
0
0
-
0
1
1
(0,1,8,9)
(0,2,8,10)
(1,5,9,13)
(5,7,13,15)
Using prime implicant chart, we can find
essential PI
0 1
15
(2,6)
(6,7)
(0,1,8,9)

(0,2,8,10) 
(1,5,9,13)
(5,7,13,15)
2
5

6
7
9
10
13

 

 




8







The reduced PI chart
1
(2,6)
(6,7)
(0,1,8,9)
(1,5,9,13)
6
The essential PI’s are (0,2,8,10) and (5,7,13,15) . So,
f(x1,x2,x3,x4) = (0,2,7,8) + (5,7,13,15) + PI’s
9


Here are 4 different choices (2,6) + (0,1,8,9), (2,6) +
(1,5,9,13)
(6,7) + (0,1,8,9), or (6,7)
+ (1,5,9,13)
A PI pj dominates PI pk iff every minterm covered by pk is also covered by pj.




m1 m2 m3 m4
pj
pk


Branching
method m
p1
p2
p3
p4
p5
1


 

(can
remove)
m2 m3 m4 m5

 





If we choose p1 first, then p3, p5 are next.
p1
p3
p5
p4
p2
p3
Quine – McCluskey method (no limitation of the # of variables)