### Chapter 2B – Karnaugh Maps

```Chapter 2
EGR 270 – Fundamentals of Computer Engineering
1
Reading Assignment: Chapter 2 in Logic and Computer Design
Fundamentals, 4th Edition by Mano
Simplification of Boolean functions
There are 3 common methods:
1. Boolean algebra
– The basis for all methods
– Difficult to see the best path to take and to know when you are finished
2. Karnaugh maps
– Fast and easy for 2 – 6 variables
– Pictorial approach based on recognition of patterns
– Difficult for large numbers of variables
3. Tabulation methods
– Tedious to perform by hand, but well suited to computer implementation
– No limit on the number of variables
– Examples: Quine-McCluskey, Expresso-II by UC Berkeley
– Discussed in online supplement (not covered in this course)
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
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Karnaugh Maps (K-maps)
• Essentially a 2D truth table arranged in a table or map so that adjacent cells in
the map differ in only one bit position.
• Each cell in the map corresponds to a minterm.
• Combining 1’s in the Kmap into groups of 1, 2, 4, 8, etc. can yield a quick
minimization of a Boolean function.
• Kmaps are most commonly used for functions involving 2-6 variables.
2-variable K-maps
The K-map has 22 = 4 cells.
y
MSB
x
0
1
0
m
m
0
2
1
m
m
1
3
Example: Use a K-map to minimize the function f(x,y) = x’y + xy’ + xy
EGR 270 – Fundamentals of Computer Engineering
Chapter 2
3-variable K-maps
The K-map has 23 = 8 cells.
yz
MSB
x
0
1
00
m
0
m
4
01
m
1
m
5
11
m
3
m
7
10
m
2
m
6
Note the strange ordering
Example: Use a K-map to minimize each function below.
1.
yz
00
01
11
10
0
0
0
1
1
1
1
0
0
1
x
Answer: f(x,y,z) = x’y + xz’
Group is x’y
(x = 0 and y = 1)
“Wraparound”
group is xz’
(x = 1 and z = 0)
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Chapter 2
EGR 270 – Fundamentals of Computer Engineering
2. f(x, y, z) = (0, 1, 5, 7)
3. f(x, y, z) = xyz + x’yz + x’y’z’ + y’z
4. f(a, b, c) = ab’ + ac + ab + b
4
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
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4-variable K-maps The K-map has 24 = 16 cells.
CD
A = MSB
AB
00
01
11
10
00
m
01
m
11
m
10
m
m
m
m
m
0
m
m
4
12
8
1
m
m
5
13
9
3
m
m
Note the strange ordering
2
7
m
15
m
11
6
14
10
Example: Use a K-map to minimize each function below.
1.
CD
2.
CD
00
01
11
10
0
1
1
0
AB
00
01
0
0
0
1
11
0
0
0
10
1
1
1
AB
00
00
01
11
10
1
1
0
1
01
1
1
0
0
0
11
0
0
0
1
1
10
1
1
0
1
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Group size vs. number of literals:
Note how the number of literals in a term is
reduced by one each time the size of grouping is
doubled. For the 4-variable K-map:
Size of
Grouping
1
2
4
8
6
Number of
Literals
4
3
2
1
Example: Use a K-map to minimize each function below.
1. f(A, B, C, D) = A’B’C’ + A’CD + B’CD’ + ABD + AB’D + AB’C’D’
2. f(A, B, C, D) = B’D’ + C’D + BD’ + A’D + BC
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Prime Implicants
Prime Implicant – a product term which cannot be combined with other terms
to yield a term with fewer literals. Hint: A prime implicant corresponds to a
maximal grouping in a Kmap.
Essential Prime Implicant – a prime implicant that covers at least one
minterm of the function that is not covered by any other prime implicant.
A good way to approach finding minimal expressions in Karnaugh maps is to
find essential prime implicants first. Once those groups have been drawn on
the map, it is easier to see what other groupings would most efficiently cover
the rest of the minterms.
In general, a function can be expressed as:
F = sum of essential prime implicants + (some combination of non-essential
prime implicants)
Example: Identify all prime implicants and all essential prime implicants in
f(A, B, C, D) = (0, 1, 3, 4, 5, 7, 11, 15)
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EGR 270 – Fundamentals of Computer Engineering
Chapter 2
Example: Use a K-map to minimize each function below. Identify the
essential prime implicants in each problem.
1.
CD
00
01
11
10
00
0
1
1
0
01
1
1
0
11
1
0
10
1
0
AB
2.
CD
00
01
11
10
00
1
1
0
1
1
01
1
1
1
0
1
0
11
0
1
1
1
1
1
10
1
1
0
1
AB
8
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
9
5-variable K-maps
The K-map has 25 = 32 cells.
Note odd arrangement of the variables ABCDE. This arrangement results in the
left half of the K-map having the same minterm ordering as a 4-variable map.
Note that groups combine as if the left map was over the top of the right map
(like 3D chess).
These two groups
combine to form B’DE’
BC
000
001
011
010
1 00
101
111
110
00
m0 m1 m3 m2
m16 m17 m19 m18
01
m4 m5 m7 m6
m20 m21 m23 m22
11
m12 m13 m15 m14
m28 m29 m31 m30
10
m8 m9 m11 m10
m24 m25 m27 m26
These two groups do
NOT combine!
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
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5-variable K-maps (alternate minterm ordering)
Shown below is an alternate arrangement for the variables A, B, C, D and E.
Note that changing the arrangement of the variables changes the ordering of the
minterms.
CDE
AB 000 001 011 010
00 m0 m1 m3 m2
01
m8 m9 m11 m10
1 00 101 111 110
m4 m5 m7 m6
m12 m13 m15 m14
11 m24 m25 m27 m26
m28 m29 m31 m30
m16 m17 m19 m18
m20 m21 m23 m21
10
These 8 minterms
combine to form AE’
(A = 1 and E = 0)
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Example: Use a K-map to minimize the function below. Carefully consider
how to label A, B, C, D, and E on the K-map.
F(A, B, C, D, E) = (1, 2, 5, 6, 8, 10, 12, 14, 17, 18, 21, 22, 24, 26, 28, 30)
000 001 011 010
00
01
11
10
1 00 101 111 110
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EGR 270 – Fundamentals of Computer Engineering
Chapter 2
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SOP and POS Expressions from K-maps
The examples so far have all generated minimal SOP expressions.
POS expressions can be formed as follows:
1. Group 0’s in the K-map instead of 1’s (these groupings correspond to F’).
2. Find a minimal SOP expression for F’.
3. Find F from F’ by applying DeMorgan’s theorem. This yields a minimal
POS expression.
Example: Determine a POS expression for F(A, B, C, D) = (1 - 3, 6, 9 - 11).
Draw the corresponding circuit.
CD
Solution
:
00
01
11
10
0
1
1
1
F  AB  C D  BD
01
0
0
0
1
F  AB  C D  BD
11
0
0
0
0
F  AB  C D  BD
10
0
1
1
1
AB
00



F  A  B C  D  B  D

With practice it
is easy to go
straight from
the first to the
last step.
EGR 270 – Fundamentals of Computer Engineering
Chapter 2
Example: Determine SOP and POS expressions for
f(A, B, C, D) = (0, 2, 4-9, 13-15). Draw the corresponding circuits.
CD
AB
00
00
01
11
10
CD
AB
00
01
01
11
11
10
10
00
01
11
10
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EGR 270 – Fundamentals of Computer Engineering
Chapter 2
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Example: Determine SOP and POS expressions for
f(A, B, C, D) = (0-2, 4-5, 8-10,14). Draw the corresponding circuits.
CD
AB
00
00
01
11
10
CD
AB
00
01
01
11
11
10
10
00
01
11
10
Note: We cannot in general predict whether the minimal SOP expression
or minimal POS expression will result in the fewest gates. It is often useful
to check both expressions to see which gives the best result.
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
“Don’t Care” conditions
There are often unused input combinations or input combinations that are
illegal and should never occur. In such cases, we often “don’t care” what the
output is since the input should never occur anyway. We typically represent
“don’t cares” in a K-map using either “d” or “X”.
Rule: Include a “don’t care” in a grouping only if it helps to make for larger
groupings. It is not necessary to include all “don’t cares” in groupings.
Example: Find a minimal SOP expression for
f(A, B, C, D) = (1, 8, 9 ) + d(0, 6, 10, 11).
CD
00
01
11
10
X
1
0
0
01
0
0
0
X
11
0
0
0
0
10
1
1
X
X
AB
00
Solution: F = B’C’
(Note that AB’ and A’BCD’
are both unnecessary)
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EGR 270 – Fundamentals of Computer Engineering
Chapter 2
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Example: BCD inputs are used to represent the digits 0 – 9, so the input codes
corresponding to 10 – 15 are illegal inputs. Find both minimal SOP and POS
expressions for f(A, B, C, D) = (0, 2-5, 7) + d(10-15).
CD
AB
00
01
11
10
00
01
11
10
EGR 270 – Fundamentals of Computer Engineering
Chapter 2
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Recognizing XOR and XNOR functions from K-maps
First a review of the various ways to express the outputs of XOR and XNOR
gates: x
F  x  y  x y  xy
y
x
y
F  x y  x  y  x  y  x  y  x  y  x  y
Note: Diagonal or staggered groupings in K-maps often indicate that using
XOR gates will yield a simpler solution than either SOP or POS form.
Example: Minimize each function shown below. Make use of XOR gates as
much as possible. Compare the number of gates using XOR’s to the number
that would be required in SOP form.
1.
yz
x
2.
00 01 11 10
yz
x
00 01 11 10
0
0
1
0
1
0
0
0
0
0
1
1
0
1
0
1
0
1
0
1
EGR 270 – Fundamentals of Computer Engineering
Chapter 2
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Example: Minimize each function shown below. Make use of XOR gates as
much as possible. Compare the number of gates using XOR’s to the number
that would be required in SOP form.
3.
CD
AB 00 01 11
10
4. CD
AB 00 01 11
10
5. CD
AB 00 01 11
10
00 1
0
1
0
00
0
0
1
1
00
0
0
0
1
01 0
1
0
1
01
1
1
0
0
01
0
0
1
0
11
1
0
1
0
11
0
0
1
1
11
0
1
0
0
10 0
1
0
1
10
1
1
0
0
10
1
0
0
0
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Cost Criteria
We need some sort of measure that will indicate how efficiently logic
expressions can be implemented. Different methods include:
1) Literal cost
2) Number of logic gates
3) Gate input cost
4) Number of gate delays
Literal cost – defined earlier as the number of literal appearances in a logic
expression. A literal is any complemented or uncomplemented variable.
Example: F = AB + A’B’ + AC
has 6 literals
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Chapter 2
EGR 270 – Fundamentals of Computer Engineering
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Propagation delay – the amount of time for the output of a gate to respond to a
change in the input. Example: tp = 10 ns
Number of gate delays – The number of gate delays for the longest path
through a logic circuit. For example, in the logic diagram below:
• If D changes, there is only 1 gate delay before F responds
• If C changes, there are 2 gate delays before F responds
• If A or B change, there are 3 gate delays before F responds (the path for
signal A is highlighted in green)
• So the number of gate delays (longest path) = 3
A
B
C
F
D
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
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Gate Input Cost
Gate input cost is simply the number of inputs to all gates if the
implementation corresponds exactly to the expression given. Note that input
inverters are typically not counted as inputs are often available in
complemented or uncomplemented form.
Example: Find the gate input cost for the circuit below. It is easy! Just count
the inputs (numbered 1-6 below)
A
1
B
2
Gate input cost = 6
3
C
4
5
F
D
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Chapter 2 EGR 270 – Fundamentals of Computer Engineering
Example: Evaluate each expression using various cost criteria.
F = ABCD + A’B’C’D’
Literal cost: __________
Number of gates: _______
Gate input cost: __________
Number of gate delays: __________
F = (A’+B)(B’+C)(C’+D)(D’+A)
Literal cost: __________
Number of gates: _______
Gate input cost: __________
Number of gate delays: __________
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Which method is best? Gate input cost is considered to be a better measure than literal
cost of the number of logic gates. Gate input cost is a good measure for contemporary
logic implementations since it is proportional to the number of transistors and wires
used in implementing a logic circuit. However, the number of gate delays is also
important as it relates to the speed of the circuit. There is often a tradeoff between these
two.
Gate input cost
(area or size of the circuit)
Number of gate delays
(speed of the circuit)
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Multiple-Level Circuit Optimization
Recall that SOP and POS expressions are both two-level circuit
implementations. Although these implementations may result in the shortest
propagation delay, other implementations may have lower gate input cost.
Example: (Refer to Figure 2-26 from the text shown below)
G = ABC + ABD + E + ACF + ADF  see Fig a below. Gate input cost =
______
G = AB(C+D) + E + AF(C+D)  see Fig b below. Gate input cost = ______
G = (AB+AF)(C+D) + E
 see Fig c below. Gate input cost = ______
G = A(B+F)(C+D) + E
 see Fig d below. Gate input cost = ______
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Chapter 2
EGR 270 – Fundamentals of Computer Engineering
24
Transformations for Multiple-Level Circuit Optimization
As seen in the last example, the circuit cost can sometimes be reduced through
algebraic manipulation (or transformations). As with Boolean algebra, there are
no specific rules for these transformations, but five types of transformations can
be defined as follows:
1) Factoring – finding a factored form of a SOP or POS expression
2) Decomposition – expression of a function as a set of new functions
3) Extraction – expression of multiple functions as a set of new functions
(decomposing multiple functions to extract common subexpressions)
4) Substitution – substituting function G into function F is when F is
expressed as a function of G and some or all of the original variables of F
5) Elimination – the inverse of substitution in which function G in an
expression for function F is replaced by the expression for G. Elimination is
also called flattening or collapsing.
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Example - Factoring
Reduce circuit cost through factoring for the function G below:
Original expression
Gate input cost: ________
Number of gate delays: ________
Factored expression
Gate input cost: ________
Number of gate delays: ________
25
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Chapter 2 EGR 270 – Fundamentals of Computer Engineering
Example - Decomposition
Function G was factored in the previous example as follows:
G = A(C’+D’)(E+F) + BCDE’F’
Now introduce two new functions, X1 and X2 (note their complements):
X1 = CD
X2 = E + F
Next rewrite G in terms of X1 and X2 (i.e., decompose G in terms of X1 and X2). Draw
the circuit also.
G=
Original expression
Gate input cost: ________
Number of gate delays: ________
Factored expression
Gate input cost: ________
Number of gate delays: ________
Note: Decomposition is more useful
Decomposed expression
when there are two or more outputs
Gate input cost: ________
that may share gates or functions (as
Number of gate delays: ________
in the next example)
27
Chapter 2 EGR 270 – Fundamentals of Computer Engineering
Example - Extraction
Extraction is the process of decomposing multiple expressions to extract common subexpressions.
First consider the original expressions for functions G and H:
H = A’BCD + ABE + ABF + BCE + BCF (expression in the text is incorrect)
Next factor the functions as follows:
G = A(C’+D’)(E+F) + BCDE’F’
H = B(A’CD + (A+C)(E+F))
Now introduce three new functions, X1, X2 and X3 as follows:
X1 = CD
X2 = E + F
X3 = A + C
Next extract X1, X2 and X3 from G and H: (see circuits on the following two pages)
G=
H=
Original expressions for G and H
Gate input cost: ________
Expressions for G and H with X1, X2 and X3 extracted
Gate input cost: ________
Chapter 2
EGR 270 – Fundamentals of Computer Engineering
Original expressions for functions G and H (label the output of each gate):
H = A’BCD + ABE + ABF + BCE + BCF
Original expressions for G and H
Gate input cost: ________
Number of gate delays: ________
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Chapter 2 EGR 270 – Fundamentals of Computer Engineering
Reduced expressions for G and H using extraction (label the output of each gate):
X1 = CD
X2 = E + F
X3 = A + C
G = AX1’X2 + BX1X2’
H = B(A’X1 + X3X2) = BA’X1 + BX3X2
Reduced expressions for G and H
Gate input cost: ________
Number of gate delays: ________
End of Test #1 material
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