Simplification of Boolean functions

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Lecture 4 EGR 270 Fundamentals of
Computer Engineering
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)

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Lecture 4 EGR 270 Fundamentals of
Computer Engineering

• 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 1s 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.

Example Use a K-map to minimize the function
f(x,y) xy xy xy
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Lecture 4 EGR 270 Fundamentals of
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• 3-variable K-maps
• The K-map has 23 8 cells.

Example Use a K-map to minimize each function
below. 1. 2. f(x, y, z) ?(0, 1, 5,
7) 3. f(x, y, z) xyz xyz xyz
yz 4. f(a, b, c) ab ac ab b
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Lecture 4 EGR 270 Fundamentals of
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• 4-variable K-maps The K-map has 24 16 cells.

Example Use a K-map to minimize each function
below.
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Lecture 4 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

• Example Use a K-map to minimize each function
  below.
• 1. f(A, B, C, D) ABC ACD BCD ABD
  ABD ABCD
• 2. f(A, B, C, D) B CD BD AD BC

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Lecture 4 EGR 270 Fundamentals of
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Example Use a K-map to minimize each function
below. Identify the essential prime implicants
in each problem.
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Lecture 4 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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Lecture 4 EGR 270 Fundamentals of
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5-variable K-maps The K-map has 25 32
cells. Note the odd arrangement of ABCDE. This
arrangement results in the left half of the K-map
having the same minterm ordering as a 4-variable
map. Other arrangements of ABCDE will result in
different minterm ordering.
Example Use a K-map to minimize the function
below. F(A, B, C, D, E) ?(1, 2, 5, 6, 8, 10,
12, 14, 17, 18, 21, 22, 24, 26, 28, 30)
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Lecture 4 EGR 270 Fundamentals of
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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
• Group 0s in the K-map instead of 1s (these
  groupings correspond to F).
• Find a minimal SOP expression for F.
• Find F from F by applying DeMorgans theorem.
  This yields a minimal POS expression.

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.
Example Determine SOP and POS expressions for
f(a,b,c,d) ?(0, 2, 4-9, 13-15). Draw the
corresponding circuits.
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Lecture 4 EGR 270 Fundamentals of
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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.
Example Determine SOP and POS expressions for
f(a,b,c,d) ?(0-2, 4-5, 8-10,14). Draw the
corresponding circuits.
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Lecture 4 EGR 270 Fundamentals of
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• Dont Care conditions
• There are often unused input combinations or
  input combinations that are illegal and should
  never occur. In such cases, we often dont
  care what the output is since the input should
  never occur anyway. We typically represent
  dont cares in a K-map using either d or X.

Rule Include a dont care in a grouping only
if it helps to make for larger groupings. It is
not necessary to include all dont cares in
groupings.
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).
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Lecture 4 EGR 270 Fundamentals of
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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

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 XORs to the
number that would be required in SOP form.
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Lecture 4 EGR 270 Fundamentals of
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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 XORs to the
number that would be required in SOP form.