Karnaugh maps

Karnaugh Maps

 Introduction
 Venn Diagrams
 2-variable K-maps
 5-variable and larger K-maps
 Simplification using K-maps

Karnaugh Maps

 Converting to Minterms Form
 Simplest SOP Expressions
 Getting POS Expressions
 Don’t-care Conditions
 Review
 Examples

Introduction
 Systematic method to obtain simplified sum-of-
products (SOPs) Boolean expressions.
 Objective: Fewest possible terms/literals.
 Diagrammatic technique based on a special form of
Venn diagram.
 Advantage: Easy with visual aid.
 Disadvantage: Limited to 5 or 6 variables.

Venn Diagrams
 Venn diagram to represent the space of minterms.
 Example of 2 variables (4 minterms):

a'b'

ab' ab a'b
a
b

Venn Diagrams
 Each set of minterms represents a Boolean function.
Examples:

{ ab, ab' }  ab + ab' = a(b+b') = a
{ a'b, ab }  a'b + ab = (a'+a)b = b
{ ab }  ab
a'b'
{ ab, ab', a'b }  ab + ab' + a'b = a + b
{} 0 ab' ab a'b
{ a'b',ab,ab',a'b }  1 a b

2-variable K-maps
 Karnaugh-map (K-map) is an abstract form of Venn
diagram, organised as a matrix of squares, where
 each square represents a minterm
 adjacent squares always differ by just one literal (so
that the unifying theorem may apply: a + a' = 1)
 For 2-variable case (e.g.: variables a,b), the map can
be drawn as:

2-variable K-maps
 Alternative layouts of a 2-variable (a, b) K-map
Alternative 1: Alternative 2:
a
b b a
OR OR
a a'b m0 m1 a ab' m0 m2
'b' 'b'
a ab a m3 b ab b m1 m3
ab' m2 a'b

Alternative 3:
a a
OR
b ab a'b b m3 m1 and others…
ab' a m2 m0
'b'

2-variable K-maps
 Equivalent labeling:
b b
a 0 1
0
equivalent to:
a
1

a a
b 1 0

equivalent to: 0
b
1

2-variable K-maps
 The K-map for a function is specified by putting
 a ‘1’ in the square corresponding to a minterm
 a ‘0’ otherwise

 For example: Carry and Sum of a half adder.
b b

0 0 0 1

a 0 1 a 1 0

C = ab S = ab' + a'b

3-variable K-maps
 There are 8 minterms for 3 variables (a, b, c).
Therefore, there are 8 cells in a 3-variable K-map.
b
b
bc
bc
a a 00 01 11 10
00 01 11 10
0 m0 m1 m3 m2
0 a'b'c a'b'c a'bc a'bc'
' OR
a m4 m5 m7 m6
a
1 ab'c' ab'c abc abc' 1

c
c

Above arrangement ensures that minterms Note Gray code sequence
of adjacent cells differ by only ONE literal.
(Other arrangements which satisfy this
criterion may also be used.)

3-variable K-maps
 There is wrap-around in the K-map:
 a'b'c' (m0) is adjacent to a'bc' (m2)
 ab'c' (m4) is adjacent to abc' (m6)

bc
a
00 01 11 10
0 m0 m1 m3 m2

m4 m5 m7 m6
1

Each cell in a 3-variable K-map has 3 adjacent neighbours.
In general, each cell in an n-variable K-map has n adjacent
neighbours. For example, m0 has 3 adjacent neighbours:
m1, m2 and m4.

Solve it yourself (Exercise 6.1)
1. The K-map of a 3-variable function F is shown
below. What is the sum-of-minterms expression of
F? b
bc
a
00 01 11 10
0 1 0 0 1

a 0 1 0 0
1

c

2. Draw the K-map for this function A:
A(x, y, z) = x.y + y.z’ + x’.y’.z

4-variable K-maps
 There are 16 cells in a 4-variable (w, x, y, z) K-map.

y
yz
wx 00 01 11 10
00 m0 m1 m3 m2

m4 m5 m7 m6
01
x
m1 m1 m1 m1
11 2 3 5 4
w
m8 m9 m1 m1
10 1 0

z

4-variable K-maps
 There are 2 wrap-arounds: a horizontal wrap-around
and a vertical wrap-around.
 Every cell thus has 4 neighbours. For example, the
cell corresponding to minterm m0 has neighbours
m1, m2, m4 and m8.
yz y
wx
m0 m1 m3 m2

m4 m5 m7 m6
x
m1 m1 m1 m1
w 2 3 5 4
m8 m9 m1 m1
1 0

z

5-variable K-maps
 Maps of more than 4 variables are more difficult to
use because the geometry (hyper-cube
configurations) for combining adjacent squares
becomes more involved.
 For 5 variables, e.g. vwxyz, need 25 = 32 squares.

5-variable K-maps
 Organised as two 4-variable K-maps:
v' v
y y
yz yz
wx 00 01 11 10 wx 00 01 11 10
00 m0 m1 m3 m2 00 m1 m1 m1 m1
6 7 9 8
m4 m5 m7 m6 m2 m2 m2 m2
01 01
x 0 1 3 2 x
m1 m1 m1 m1 m2 m2 m3 m3
11 2 3 5 4 11 8 9 1 0
w w
m8 m9 m1 m1 m2 m2 m2 m2
10 1 0 10 4 5 7 6

z z

Corresponding squares of each map are adjacent.
Can visualise this as being one 4-variable map on TOP of the
other 4-variable map.

Larger K-maps
 6-variable K-map is pushing the limit of human
“pattern-recognition” capability.
 K-maps larger than 6 variables are practically
unheard of!
 Normally, a 6-variable K-map is organised as four
4-variable K-maps, which are mirrored along two
axes.

Larger K-maps w
b

ef a'b' a'b ef
cd 00 01 11 10 10 11 01 00 cd

00 m0 m1 m3 m2 m1 m1 m1 m1 00
8 9 7 6
m4 m5 m7 m6 m2 m2 m2 m2
2 3 1 0
01 m1 m1 m1 m1 m3 m3 m2 m2 01
2 3 5 4 0 1 9 8
m8 m9 m1 m1 m2 m2 m2 m2 11
11 1 0 6 7 5 4

10 10
10 m4 m4 m4 m4 m5 m5 m5 m5 10
0 1 3 2 8 9 7 6
m4 m4 m4 m4 m6 m6 m6 m6
a 4 5 7 6 2 3 1 0
m3 m3 m3 m3 m5 m5 m5 m5
11 6 7 9 8 4 5 3 2
11
m3 m3 m3 m3 m5 m5 m4 m4
01 2 3 5 4 0 1 9 8 01
cd cd
ef
00 01 11 10 10 11 01 00 ef
00 ab' ab 00

Try stretch your recognition capability by finding simpliest
sum-of-products expression for Σ m(6,8,14,18,23,25,27,29,41,45,57,61).

Simplification Using K-maps
 Based on the Unifying Theorem:
A + A' = 1
 In a K-map, each cell containing a ‘1’ corresponds to
a minterm of a given function F.
 Each group of adjacent cells containing ‘1’ (group
must have size in powers of twos: 1, 2, 4, 8, …) then
corresponds to a simpler product term of F.
 Grouping 2 adjacent squares eliminates 1 variable, grouping
4 squares eliminates 2 variables, grouping 8 squares
eliminates 3 variables, and so on. In general, grouping 2n
squares eliminates n variables.

 Group as many squares as possible.
 The larger the group is, the fewer the number of literals in
the resulting product term.

 Select as few groups as possible to cover all the
squares (minterms) of the function.
 The fewer the groups, the fewer the number of product
terms in the minimized function.

 Example:
F (w,x,y,z) = w'xy'z' + w'xy'z + wx'yz'
+ wx'yz + wxyz' + wxyz
= Σ m(4, 5, 10, 11, 14, 15)
y
yz
wx 00 01 11 10
00

01 1 1
x (cells with ‘0’ are not
1 1
w
11 shown for clarity)
1 1
10
z

 Each group of adjacent minterms (group size in
powers of twos) corresponds to a possible product
term of the given function.
y
yz
wx 00 01 11 10
00
A
01 1 1
x
11 1 1
w
10 1 1 B

z

 There are 2 groups of minterms: A and B, where:
A = w'xy'z' + w'xy'z
= w'xy'(z' + z)
= w'xy'

B = wx'yz' + wx'yz + wxyz' + wxyz y
yz
= wx'y(z' + z) + wxy(z' + z) 00 01 11 10
wx
= wx'y + wxy
00
= w(x'+x)y A
01 1 1
= wy x
11 1 1
w
10 1 1 B

z

 Each product term of a group, w'xy' and wy,
represents the sum of minterms in that group.
 Boolean function is therefore the sum of product
terms (SOP) which represent all groups of the
minterms of the function.
F(w,x,y,z) = A + B = w'xy' + wy

 Larger groups correspond to product terms of fewer
literals. In the case of a 4-variable K-map:
1 cell = 4 literals, e.g.: wxyz, w'xy'z
2 cells = 3 literals, e.g.: wxy, wy'z'
4 cells = 2 literals, e.g.: wx, x'y
8 cells = 1 literal, e.g.: w, y', z
16 cells = no literal, e.g.: 1

 Other possible valid groupings of a 4-variable K-map
include:

1 1 1 1
1 1 1 1

1 1 1 1

1

1 1 1 1 1 1
1

  

 Groups of minterms must be
(1) rectangular, and
(2) have size in powers of 2’s.
Otherwise they are invalid groups. Some examples
of invalid groups:

1 1 1 1

1 1 1 1

1 1 1

1 1 1 1 1

 

Converting to Minterms Form
 The K-map of a function is easily drawn when the
function is given in canonical sum-of-products, or
sum-of-minterms form.
 What if the function is not in sum-of-minterms?
 Convert it to sum-of-products (SOP) form.
 Expand the SOP expression into sum-of-minterms
expression, or fill in the K-map directly based on the
SOP expression.

Converting to Minterms Form
 Example:
f(A,B,C,D) = A(C+D)'(B'+D') + C(B+C'+A'D)
= A(C'D')(B'+D') + BC + CC' + A'CD
= AB'C'D' + AC'D' + BC + A'CD
A
AB'C'D' + AC'D' + BC + A'CD AB
CD 00 01 11 10
= AB'C'D' + AC'D'(B+B') + BC + A'CD
00 1 1
= AB'C'D' + ABC'D' + AB'C'D' +
BC(A+A') + A'CD 01
D
= AB'C'D' + ABC'D' + ABC + A'BC + 11 1 1 1
A'CD C
10 1 1
= AB'C'D' + ABC'D' + ABC(D+D') +
A'BC(D+D') + A'CD(B+B') B

= AB'C'D' + ABC'D' + ABCD + ABCD' +
A'BCD + A'BCD' + A'B'CD

Simplest SOP Expressions
 To find the simplest possible sum of products (SOP)
expression from a K-map, you need to obtain:
 minimum number of literals per product term; and
 minimum number of product terms

 This is achieved in K-map using
 bigger groupings of minterms (prime implicants) where
possible; and
 no redundant groupings (look for essential prime implicants)

Implicant: a product term that could be used
to cover minterms of the function.

 A prime implicant is a product term obtained by
combining the maximum possible number of
minterms from adjacent squares in the map.
 Use bigger groupings (prime implicants) where
possible.

1 1 1 1 1 1

1 1 1 1 1 1
 

 No redundant groups:
1 1 1 1

1 1 1 1

1 1
 1 1

1 1 1 1

Essential prime implicants

 An essential prime implicant is a prime implicant that
includes at least one minterm that is not covered by
any other prime implicant.

Q. Identify the prime implicants and the essential prime
implicants of the two K-maps below.

b
A
bc
a AB
00 01 11 10 CD 00 01 11 10

0 1 1 0 1 00 1 1 1

a 0 1 0 0 01 1 1
1 D
11 1 1 1
c C
10 1 1 1

B

 Algorithm 1 (non optimal):
1. Count the number of adjacencies for each minterm on the
K-map.
2. Select an uncovered minterm with the fewest number of
adjacencies. Make an arbitrary choice if more than one
choice is possible.
3. Generate a prime implicant for this minterm and put it in the
cover. If this minterm is covered by more than one prime
implicant, select the one that covers the most uncovered
minterms.
4. Repeat steps 2 and 3 until all the minterms have been
covered.

 Algorithm 2 (non optimal):
1. Circle all prime implicants on the K-map.
2. Identify and select all essential prime implicants for the
cover.
3. Select a minimum subset of the remaining prime implicants
to complete the cover, that is, to cover those minterms not
covered by the essential prime implicants.

 Example:
f(A,B,C,D) = ∑ m(2,3,4,5,7,8,10,13,15)
A
AB
CD 00 01 11 10

00 1 1
All prime implicants
01 1 1
D
11 1 1 1
C
10 1 1

B

A
AB A
CD 00 01 11 10 AB
00 CD 00 01 11 10
1 1
01
1 1 D
00 1 1 Essential prime
C
11 1 1 1 01 1 1 implicants
10 1 1 D
11 1 1 1
B C
10 1 1

B
A
AB
CD 00 01 11 10

00 1 1 Minimum cover
01 1 1
D
11 1 1 1
C
10 1 1

B


A
AB
CD 00 01 11 10
A'BC'
00 1 1 AB'D'
01 1 1
D
11 1 1 1
C
10 1 1 BD
B

A'B'C

f(A,B,C,D) = BD + A'B'C + AB'D' + A'BC'

Q. Find the simplified expression for G(A,B,C,D).
A
AB
CD 00 01 11 10

00 1

01 1 1 1
D
11 1 1 1
C
10 1

B

Getting POS Expressions
 Simplified POS expression can be obtained by
grouping the maxterms (i.e. 0s) of given function.
 Example:
Given F=∑m(0,1,2,3,5,7,8,9,10,11), we first draw
the K-map, then group the maxterms together:
A
AB
CD 00 01 11 10

00 1 0 0 1

01 1 1 0 1
D
11 1 1 0 1
C
10 1 0 0 1

B

Getting POS Expressions
A A
AB AB
CD 00 01 11 10 CD 00 01 11 10

K-map 00 1 0 0 1 00 0 1 1 0 K-map
of F 01 1 1 0 1 01 0 0 1 0 of F'
D D
11 1 1 0 1 11 0 0 1 0
C C
10 1 0 0 1 10 0 1 1 0

B B

 This gives the SOP of F' to be:
F' = BD' + AB
 To get POS of F, we have:
F = (BD' + AB)'
= (BD')'(AB)' DeMorgan
= (B'+D)(A'+B') DeMorgan

Don’t-care Conditions
 In certain problems, some No.
0
A
0
B
0
C
0
D
0
P
1
outputs are not specified. 1 0 0 0 1 0
2 0 0 1 0 0
 These outputs can be either ‘1’ 3
4
0
0
0
1
1
0
1
0
1
0
or ‘0’. 5 0 1 0 1 1
6 0 1 1 0 1
 They are called don’t-care 7
8
0
1
1
0
1
0
1
0
0
0
conditions, denoted by X (or 9 1 0 0 1 1
sometimes, d). 10
11
1
1
0
0
1
1
0
1
X
X
12 1 1 0 0 X
 Example: An odd parity 13 1 1 0 1 X
generator for BCD code which 14
15
1
1
1
1
1
1
0
1
X
X
has 6 unused combinations.

 Don’t-care conditions can be used to help simplify
Boolean expression further in K-maps.
 They could be chosen to be either ‘1’ or ‘0’,
depending on which gives the simpler expression.

C
 For comparison: AB
CD
00 01 11 10
 WITHOUT Don’t-cares: 00
1 1
P = A'B'C'D’ + A'B'CD + A'BC'D 01
1 1
B
+ A'BCD' + AB'C'D 11
A
10 1

D

 WITH Don’t-cares: CD
C

AB 00 01 11 10
P = A'B'C'D' + B'CD + BC'D 00 1 1
+ BCD' + AD 01 1 1
B
11 X X X X
A
10 1 X X

D

Review – The Techniques
 Algebraic Simplification.
 requires skill but extremely open-ended.

 Karnaugh Maps.
 can obtain simplified standard forms.
 easy for humans (pattern-matching skills).
 limited to not more than 6 variables.

 Other computer-aided techniques such as Quine-
McCluskey method (not covered in this course).

Review – K-maps
 Characteristics of K-map layouts:
(i) each minterm in one square/cell
(ii) adjacent/neighbouring minterms differ by only 1 literal
(iii) n-literal minterm has n neighbours/adjacent cells

 Valid 2-, 3-, 4-variable K-maps
b b

a a'b m0 m1
'b' OR
a ab' ab a m2 m3

Review – K-maps
b b
bc bc
a a
00 01 11 10 00 01 11 10
0 a'b'c a'b'c a'bc a'bc' 0 m0 m1 m3 m2
'
a ab'c' ab'c abc abc' a m4 m5 m7 m6
1 1

c c
y
yz
wx 00 01 11 10
00 m0 m1 m3 m2

m4 m5 m7 m6
x
01 m1 m1 m1 m1
w 2 3 5 4

11 m8 m9 m1 m1
1 0
10 z

Review – K-maps
 Groupings to select product-terms must be:
 (i) rectangular in shape
 (ii) in powers of twos (1, 2, 4, 8, etc.)
 (iii) always select largest possible groupings of minterms
(i.e. prime implicants)
 (iv) eliminate redundant groupings

 Sum-of-products (SOP) form obtained by selecting
groupings of minterms (corresponding to product
terms).

Review – K-maps
 Product-of-sums (POS) form obtained by selecting
groupings of maxterms (corresponding to sum terms)
and by applying DeMorgan’s theorem.
 Don’t cares, marked by X (or d), can denote either 1
or 0. They could therefore be selected as 1 or 0 to
further simplify expressions.

Examples
 Example #1:
f(A,B,C,D) = ∑ m(2,3,4,5,7,8,10,13,15)
A
AB
CD 00 01 11 10

00 1 1
Fill in the 1’s.
01 1 1
D
11 1 1 1
C
10 1 1

B

Examples
 Example #1:
f(A,B,C,D) = ∑ m(2,3,4,5,7,8,10,13,15)
A
AB
CD 00 01 11 10
These are all the
00 1 1 prime implicants; but
01 1 1 do we need them
D
11 1 1 1
all?
C
10 1 1

B

Examples
 Example #1:
f(A,B,C,D) = ∑ m(2,3,4,5,7,8,10,13,15)
A
AB
CD 00 01 11 10

00 1 1
Essential prime implicants:
01 1 1
D
11 1 1 1
B.D
C
10 1 1 A'.B.C'
B A.B'.D'

Examples
 Example #1:
f(A,B,C,D) = ∑ m(2,3,4,5,7,8,10,13,15)
A
AB
CD 00 01 11 10

00 1 1
Minimum cover.
01 1 1
D EPIs: B.D, A'.B.C', A.B'.D'
11 1 1 1
C
1 1
+
10

B A'.B'.C

f(A,B,C,D) = B.D + A'.B.C' + A.B'.D' + A'.B'.C

Examples
A
AB A
CD 00 01 11 10 AB
00 CD 00 01 11 10
1 1
01
1 1 D
00 1 1 Essential prime
C
11 1 1 1 01 1 1 implicants
10 1 1 D
11 1 1 1
B C
10 1 1

B
SUMMARY
A
AB
CD 00 01 11 10

00 1 1
Minimum cover
01 1 1
D
11 1 1 1
C
10 1 1
f(A,B,C,D) = BD + A'B'C + AB'D' + A'B.C'
B

Examples
 Example #2:
f(A,B,C,D) = A.B.C + B'.C.D' + A.D + B'.C'.D'
A
AB
CD 00 01 11 10

00 1 1
Fill in the 1’s.
01 1 1
D
11 1 1
C
10 1 1 1

B

Examples
 Example #2:
A
AB
CD 00 01 11 10

00 1 1
Find all PIs:
01 1 1
D
11 1 1 A.D
C
10 1 1 1 A.C
B B'.D'

Are all ‘1’s covered by the PIs? Yes, so the
answer is: f(A,B,C,D) = A.D + A.C + B'.D'

Examples
 Example #3 (with don’t cares):
f(A,B,C,D) = ∑ m(2,8,10,15) + ∑ d(0,1,3,7)
A
AB
CD 00 01 11 10

00 X 1
Fill in the 1’s and X’s.
01 X
D
11 X X 1
C
10 1 1

B

Examples
 Example #3 (with don’t cares):
f(A,B,C,D) = ∑ m(2,8,10,15) + ∑ d(0,1,3,7)
A
AB
CD 00 01 11 10 Do we need to have an
00 X 1 additional term A'.B' to
01 X cover the 2 remaining x’s?
D
11 X X 1 No, because all the 1’s
C
1 1 (minterms) have been
10
covered.
B

f(A,B,C,D) = B'.D' + B.C.D

Examples
 To find simplest POS expression for example #2:
 Draw the K-map of the complement of f, f '.
A From K-map,
AB
CD 00 01 11 10
f ' = A'.B + A'.D + B.C'.D'
00 1 1
Using DeMorgan’s theorem,
01 1 1
D
11 1 1 f = (A'.B + A'.D + B.C'.D')'
C
10 1 = (A+B').(A+D').(B'+C+D)
B

Examples
s To find simplest POS expression for example #3:
f(A,B,C,D) = ∑ m(2,8,10,15) + ∑ d(0,1,3,7)
s Draw the K-map of the complement of f, f '.
f '(A,B,C,D) = ∑ m(4,5,6,9,11,12,13,14) + ∑
d(0,1,3,7) A

CD
AB
00 01 11 10
From K-map,
00 X 1 1 f ' = B.C' + B.D' + B'.D
01 X 1 1 1
D
Using DeMorgan’s theorem,
11 X X 1
C f = (B.C' + B.D' + B'.D)'
10 1 1
= (B'+C).(B'+D).(B+D')
B

Karnaugh maps

More Related Content

Similar to Karnaugh maps (20)

Karnaugh maps