Two-Level Simplification

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Two-Level Simplification

• All Boolean expressions can be represented in
  two-level forms
• Sum-of-products
• Product-of-sums

Canonical S.O.P. form

• Canonical forms are very easy to produce
• Just read them off of a truth table
• But, theyre not the most efficient representation

Reduced S.O.P. form

• Reduced two-level forms are more efficient

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Venn Diagrams
Consider a Venn Diagram for 2 sets, A and B
AB
AB
AB
AB
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Karnaugh maps
2-variable K-map
0
1
Space for AB
1
0
Space for AB
Space for AB
Space for AB
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Karnaugh maps
K-maps can represent up to four variables easily
f(A,B,C)
f(A,B,C,D)
3-variable K-map
4-variable K-map
Numbering Scheme 00, 01, 11, 10 Gray Code only
a single bit changes from one number to the next
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Filling in a K-map
F(A,B,C,D) ABCD ABCD ABCD ABCD
ABCD
F (A,B,C) ABC ABC ABC ABC
1
1
1
1
1
0
0
1
1
1
0
0
1
4-variable K-map
3-variable K-map
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Finding Combinations with K-maps
We can combine AB and AB
F AB AB B
We can combine AB and AB
G AB AB A
With Karnaugh maps, adjacent 1s mean we can
combine them
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Adjacencies in the K-map
Neighbors
Wrap from top to bottom
Wrap from left to right
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3-variable K-map examples
F(C,B,A) ABC ABC AB
In the K-map, adjacency wraps from left to
right and from top to bottom
F(C,B,A) AC BC
Same function, alternative circling Note
Larger circles are better
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3-variable K-map examples
We can use the combining theorem on larger units
as well.
G(A,B,C) ABC ABC ABC ABC
AB(C C) AB(C C) AB
AB B(A A)
B

• What can we circle?
• Any rectangle that contains all ones
• As long as its size is a power of two
• 1, 2, 4, 8, 16, ...
• No rectangles of 3, 5, 6, ...

Find the smallest number of the largest possible
rectangles that cover all the 1s at least once
(overlapping circles are allowed)
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4-variable K-map example
F(A,B,C,D) åm(0, 1, 2, 3, 4, 5, 6, 7, 8, 10,
11, 13, 14)
Find the smallest number of the largest possible
rectangles that cover all the 1s

• Start at upper left corner and search for 1s
• Circled? Go to next 1
• Not circled? Circle largest term that contains
  this 1 and go to next 1
• Tie? Skip this square for now and come back to
  it later...

F(A,B,C,D)
A
BCD
CD
BD
BC
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K-maps for XORs and XNORs
F AB AB A Å B
G ABC ABC ABC ABC A Å B Å
C
Q A Å B Å C Å D
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Product-of-Sums
We can circle 0s to find a sum-of-products for
the complement
F(A,B,C,D) Sm(0,1,5,8,10,12,14)
F
AC
ABD
AD
DeMorgans Law
F (AC)(ABD)(AD)
Product-of-Sums!

• Circling 1s gives S.O.P. for F
• Complementing S.O.P. of F gives P.O.S. for F
• Circling 0s gives S.O.P. for F
• Complementing S.O.P. for F gives P.O.S. for F

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K-maps and Dont Cares
Invalid Inputs (Dont Cares) can be treated as
1's or 0's if it is advantageous to do so
F(A,B,C,D) Sm(1,3,5,7,9) Sd(6,12,13)
F assuming xs are zero
AD
BCD
Tie! - Skip and come back
F using dont cares
AD
CD
By treating this X as a "1", a largerrectangle
can be formed
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Example 2-bit Comparator
Will need a 4-variable K-map for each of the 3
output functions
DC and BA are two-bitbinary numbers
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K-maps for 2-bit comparator
F1 ABCD
F2 ABltCD
F3 ABgtCD
F1

F2
ABCD
ABCD
ABCD
ABCD
ABD
AC
BCD
-OR-
F3
BCD
AC
ABD
F1

BD
AC
BD
AC
F1 (BD)(AC)(BD)(AC)
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BCD Decrement by 1

• BCD Binary Coded Decimal
• Represents 0 through 9 in four bits
• Binary patterns for 10-15 are invalid inputs

• Decrement by 1 function
• N2 N1 1
• 0 1 9 (rolls over)

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BCD Decrement by One
W
ABCD
AD
X
BD
BC
AD
Y
CD
BCD
AD
Z
D