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Boolean
Logic
Boolean
Logic
Introduction
to^ Computer
Architecture
Difficult
Textbook
Topics
-^ How^ to
use^ the^ Boolean
Identities
to^ simplify
l^ i^
ti logic^ equations • I^ don't^ fully
understand
the^ implementation
of
the^ brute
force^ method
-^ Determining
which^ identity
laws^ were
used^ in
each stepeach^ step • Better^ examples
of^ sum‐of
‐products
and
products‐
of‐sum
Boolean
Expression
Forms
-^ You^ can
express^ a
Boolean^
equation^
in^ many
ways. • The^ Sum^ of
Products
form^ ORs
together
sub‐
expressions
that^ are^ ANDedF =^ ABC^ +
A’B’C^ +
AB’
-^ The^ Product
of^ Sums^ form
ANDs^ together
sub‐
expressions
that^ are^ ORedF =^ (A+B+C)
*^ (B’^ +^
C’)
Standard
Forms
-^ Using^ the
standard
forms^ can
make^ it^ easier
to
B^ l^
ti use^ a^ Boolean
equation
-^ It^ may^ be
possible^
to^ simplify
an^ equation
if
you^ do^ not
use^ the^ standard
forms.
F^ A * (B
C)
F^ =^ A^ *^ (B
+^ C)
F^ =^ AB^ +
AC
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Converting
Truth^ Table
to^ Expressions
-^ To^ create
a^ Sum^ of
Products
equation
-^ For^ each
row^ where
the^ output
is^ “1”,^ AND
together^ the
input^ variables.
NOT^ the
input
variables^
that^ are^ zero.
-^ OR^ together
all^ of^ the
AND^ statements.
Truth^ Table
to^ Equation
A^ B^ C^
F 0 0 0
1 0 0 0
1 0 0 1
0 0 1 0
0 0 1 1
1 1 0 0
F^ =^ A’B’C’ 0
+^ A’BC^ +
AB’C^ +^ ABC
1 0 0
0 1 0 1
1 1 1 0
0 1 1 1
1
Create^
a^ SoP Expression
A^ B^ C^
F 0 0 0
0 0 0 0
0 0 0 1
1 0 1 0
0 0 1 1
0 1 0 0
1 1 0 0
1 1 0 1
0 1 1 0
0 1 1 1
1
Create^
a^ SoP Expression
A^ B^ C^
F 0 0 0
0 F
=^ A’B’C
+^ AB’C’
+^ ABC
0 0 0
0 0 0 1
1 0 1 0
0 0 1 1
0 1 0 0
1 1 0 0
1 1 0 1
0 1 1 0
0 1 1 1
1
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Inverse
Implementation • Sometimes
the^ equation
has^ a l t^ f t A^ B^ C^
F 0 0 0
1 lot^
of^ terms. • It may^ be^ simpler
to^ solve^
the inverse^ and
then^ NOT
it. F^ =^ A’B’C’
+^ A’BC’^ +
A’BC^ + AB’C’ + AB’C + ABC’ + ABC 0 0 0
1 0 0 1
0 0 1 0
1 0 1 1
1 1 0 0
1 AB’C’
+^ AB’C^ +
ABC’^ +^ ABC F = (A’B’C)’
1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
1
Basic^ Truth
Tables
X^ Y^
X^ AND^ Y 0 0
0
X^ Y^
X^ OR^ Y 0 0
0
0 0
0 0 1
0 1 0
0 1 1
1
0 0
0 0 1
1 1 0
1 1 1
1
Boolean
Identities
Name^
AND^ version
OR^ version
Identity^
X^ *^1 =^ X^
X^ +^0 =^ X
Complement
X^ *
X’^ =^0
X^ +^ X’^ =^1
Commutative
X^ *^ Y
=^ Y^ *^ X^
X^ +^ Y^ =^ Y^ +^ X
Distribution^
X^ ^ (Y +^ Z)^ =^ XY
+^ X*Z^ X^ +
(Y*Z) =^ (X+Y)
*^ (X+Z)
Idempotent^
X^ *^ X^ =^ X^
X^ +^ X^ =^ X
Null^
X^ *^0 =^0
X^ +^1 =^1
Involution^
(X’)’^ =^ X^
‐‐
Absorption^
X^ *^ (X^ +^ Y)^ =^ X
X^ +^ (X^ *^ Y)^ =^ X
Associative^
X *^ (Y^ *^ Z)^ =^ (X
*^ Y)^ *^ Z^
X^ +^ (Y^ +^ Z)^ =^ (X
+^ Y)^ +^ Z
de^ Morgan^
(X^ *^ Y)’^ =^ X’^ +
Y’^
(X^ +^ Y)’^ =^ X’^ *
Y’
Boolean
Simplification
-^ Just^ like
regular^ algebra,
you^ can^ use
Boolean
l^ b^ t^
i^ lif^
ti algebra^ to
simplify^ an
equation.
-^ The^ identities
can^ be^ used
to^ remove
terms
and^ variables
that^ do^ not
make^ a^ difference. A^ *^ (^ A’^ +^
B )^ rule AA’^ +^ AB
Distribution 0 +^ A*B^
Complement A*B^
Identity
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Simplification
Example
(A^ +^ B)^ *^ (A
+^ B’)^ *^ (A’^
+^ B) [A^ (B * B’)] * (A’
B)^ OR Di
ib^ i [A^ + (B^ *^ B’)]
*^ (A’^ +^ B)^
OR^ Distribution (A^ + 0)^ *^ (A’
+^ B)^
AND^ Complement A^ *^ (A’^ +^ B)
OR
Identity A^ *^ A’^ +^ A^ *
B^ AND
Distribution 0 +^ A^ *^ B^
AND^ Complement 0 A^ B^
AND^ Complement A^ *^ B^
OR^ Identity
What^ rule
is^ being
used? 25%^
25%25% 25%
(A^ +^ B)^ *^ (A^ +
B’)A + (B * B’)^ Distribution A^ +^ (B^ ^ B ) 1. Identity 2 Complement Distribution A + 0?* A Identity
Identity^ Complement
Morgan (^) de Distribution
- Complement3. Distribution4. de^ Morgan
Logic^ Circuit
Design^
Process
-^ A^ simple
logic^ design
process^
involves
-^ Problem specification•^ Problem^
specification
-^ Truth^ table
derivation
-^ Derivation
of^ logical^ expression
-^ Simplification
of^ logical^ expression
-^ Implementation
TTL^ Chips GND^ is^ groundVcc is^ power
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Death^ of
Moore’s
Law
-^ Libraries
could^ be^
filled^ with
articles^ written th^ d^ d
di ti
th^
d^ f
over^ the^ d
ecades^ predicting
the^ end^ of
Moore’s^ Law. • Moore’s^ Law
continues
to^ hold
-^ Obviously
Moore’s^
Law^ must
end^ sometime.
Transistors cannot be smaller than an atomTransistors
cannot^ be
smaller^ than
an^ atom.
-^ Some^ have
written^ welcoming
the^ end^ of
Moore’s^ law
so^ hardware
would^ stablize.
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