Discrete Mathematics CS 2610

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Discrete Mathematics CS 2610
February 19, 2009
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Logic Gates the basic elements of circuits

• Electronic circuits consist of so-called gates
  connected by wires

x
x
Inverter (NOT gate)
OR gate
x
xy
AND gate
y
 
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Half Adder
x
y
sum
carry
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Full Adder
sum
Half Adder
Carry in
Half Adder
x
Carry out
y
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Add Three Bits
x0
Half Adder
s0
y0
c0
Full Adder
s1
x1
y1
c1
s2
Full Adder
x2
y2
c2s3
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Circuit Minimization

• We have seen that a function can be represented
  by many different equivalent expressions
• Before building our circuit, we want to find a
  simple expression for the function
• When building circuits, we want to use the
  minimum possible number of gates (why?)
• For example, instead of xyz xyz, we can use xz
• How do we find the minimum expression?

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Adjacent Minterms

• To minimize circuits, start by writing your
  function in DNF (sum of products)
• Two minterms are adjacent if they differ by one
  variable, which is negated in one minterm and not
  negated in another
• The sum of two adjacent minterms is equivalent to
  the single term that results when this variable
  is removed
• E.g., xyz xyz xz

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Why Adjacent?

• We can represent functions by using Karnaugh maps

y
y
x
x y
x y
x
x y
x y
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Example

• xy xy xy x y

y
y
x
1
1
x
1
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Three Variable Karnaugh Maps

• With the three variables x, y, z, we can let x
  and x be on the vertical side as before
• The table will now have 4 columns yz, yz, yz,
  and yz
• Order is important! Columns must be adjacent to
  each other
• We also consider the first and last columns to be
  adjacent
• Picture the table as a flattened cylinder
• A block of 2 cells cancels out 1 variable
• A block of 4 cells cancels out 2 variables
• What if we have a block of 8 cells?

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3-Variable Example

• xyz xyz xyz xyz xyz z xy

yz
yz
yz
yz
x
1
1
1
1
1
x
implicant, prime implicant, essential prime
implicant