Overview

▪ Part 1 – Gate Circuits and Boolean Equations

• Binary Logic and Gates
• Boolean Algebra
• Standard Forms
▪ Part 2 – Circuit Optimization
• Two-Level Optimization
• Map Manipulation
• Multi-Level Circuit Optimization
▪ Part 3 – Additional Gates and Circuits
• Other Gate Types
• Exclusive-OR Operator and Gates
• High-Impedance Outputs
Binary Logic and Gates
▪ Binary variables take on one of two values.
▪ Logical operators operate on binary values and
binary variables.
▪ Basic logical operators are the logic functions
AND, OR and NOT.
▪ Logic gates implement logic functions.
▪ Boolean Algebra: a useful mathematical system
for specifying and transforming logic functions.
▪ We study Boolean algebra as foundation for
designing and analyzing digital systems!
Binary Variables
▪ Recall that the two binary values have
different names:
• True/False
• On/Off
• Yes/No
• 1/0
▪ We use 1 and 0 to denote the two values.
▪ Variable identifier examples:
• A, B, y, z, or X1 for now
• RESET, START_IT, or ADD1 later
Logical Operations
▪ The three basic logical operations are:
• AND
• OR
• NOT
▪ AND is denoted by a dot (·).
▪ OR is denoted by a plus (+).
▪ NOT is denoted by an overbar ( ¯ ), a
single quote mark (') after, or (~) before
the variable.
Notation Examples
▪ Examples:
• Y = A  B is read “Y is equal to A AND B.”
• z = x + y is read “z is equal to x OR y.”
• X = A is read “X is equal to NOT A.”
▪ Note: The statement:
1 + 1 = 2 (read “one plus one equals two”)
is not the same as
1 + 1 = 1 (read “1 or 1 equals 1”).
 Operator Definitions
▪ Operations are defined on the values
"0" and "1" for each operator:

AND OR NOT
0·0=0 0+0=0 0=1
0·1=0 0+1=1 1=0
1·0=0 1+0=1
1·1=1 1+1=1
 Truth Tables
▪ Truth table - a tabular listing of the values of a
function for all possible combinations of values on its
arguments
▪ Example: Truth tables for the basic logic operations:
AND OR NOT
X Y Z = X·Y X Y Z = X+Y X Z=X
0 0 0 0 0 0 0 1
0 1 0 0 1 1 1 0
1 0 0 1 0 1
1 1 1 1 1 1
Logic Function Implementation
▪ Using Switches Switches in parallel => OR
• For inputs:
▪ logic 1 is switch closed
▪ logic 0 is switch open
• For outputs: Switches in series => AND
▪ logic 1 is light on
▪ logic 0 is light off.
• NOT uses a switch such
that: Normally-closed switch => NOT
▪ logic 1 is switch open C
▪ logic 0 is switch closed
Logic Function Implementation (Continued)

▪ Example: Logic Using Switches

B C
A

▪ Light is on (L = 1) for
L(A, B, C, D) =
and off (L = 0), otherwise.
▪ Useful model for relay circuits and for CMOS
gate circuits, the foundation of current digital
logic technology
Logic Function Implementation
Logic Gates
▪ In the earliest computers, switches were opened
and closed by magnetic fields produced by
energizing coils in relays. The switches in turn
opened and closed the current paths.
▪ Later, vacuum tubes that open and close
current paths electronically replaced relays.
▪ Today, transistors are used as electronic
switches that open and close current paths.
 Logic Gates (continued)
▪ Implementation of logic gates with transistors
+V
+V
•
•
• • +V

X .Y
•• •
F • X • •
X

G = X +Y
• • X •
X •
Y •
Y •
•

(a) NOR (b) NAND (c) NOT

▪ Transistor or tube implementations of logic functions are
called logic gates or just gates
▪ Transistor gate circuits can be modeled by switch circuits
Logic Gates
 Logic Gate Symbols and Behavior
▪ Logic gates have special symbols:
X X
Z 5 X ·Y Z5 X1 Y X Z5 X
Y Y
AND gate OR gate NOT gate or
inverter
(a) Graphic symbols
▪ And waveform behavior in time as follows:
X 0 0 1 1

Y 0 1 0 1

(AND) X ·Y 0 0 0 1

(OR) X1 Y 0 1 1 1

(NOT) X 1 1 0 0
(b) Timing diagram
Breadboard
Breadboard
 Logic Diagrams and Expressions
Truth Table Equation
XYZ F = X + Y Z
000 0 F = X +Y Z
001 1
010 0 Logic Diagram
011 0 X
100 1
Y F
101 1
110 1 Z
111 1
▪ Boolean equations, truth tables and logic diagrams describe
the same function!
▪ Truth tables are unique; expressions and logic diagrams are
not. This gives flexibility in implementing functions.
 Boolean Algebra
▪ An algebraic structure defined on a set of at least two elements, B,
together with three binary operators (denoted +, · and ) that
satisfies the following basic identities:

1. X+0= X 2. X .1 =X
3. X+1 =1 4. X .0 =0
5. X+X =X 6. X .X = X
7. X+X =1 8. X .X = 0
9. X=X

1-4 Existence of 0 and 1 5-6 Idempotence

7-8 Existence of complement 9 Involution

 Boolean Algebra
▪ An algebraic structure defined on a set of at least two elements, B,
together with three binary operators (denoted +, · and ) that
satisfies the following basic identities:

10. X + Y = Y + X 11. XY = YX
12. (X + Y) + Z = X + (Y + Z) 13. (XY) Z = X(YZ)
14. X(Y + Z) = XY + XZ 15. X + YZ = (X + Y) (X + Z)
16. X + Y = X . Y 17. X . Y = X + Y

10-11 Commutative Laws 12-13 Associative Laws

14-15 Distributive Laws 16-17 DeMorgan’s Laws
 Boolean Algebra
▪ An algebraic structure defined on a set of at least two elements, B,
together with three binary operators (denoted +, · and ) that
satisfies the following basic identities:

1. X+0= X 2. X .1 =X
3. X+1 =1 4. X .0 =0
5. X+X =X 6. X .X = X
7. X+X =1 8. X .X = 0
9. X=X
10. X + Y = Y + X 11. XY = YX Commutative
12. (X + Y) + Z = X + (Y + Z) 13. (XY) Z = X(YZ) Associative
14. X(Y + Z) = XY + XZ 15. X + YZ = (X + Y) (X + Z) Distributive
16. X + Y = X . Y 17. X . Y = X + Y DeMorgan’s
Some Properties of Identities & the Algebra

▪ If the meaning is unambiguous, we leave out the symbol “·”

▪ The identities above are organized into pairs. These pairs
have names as follows:
1-4 Existence of 0 and 1 5-6 Idempotence
7-8 Existence of complement 9 Involution
10-11 Commutative Laws 12-13 Associative Laws
14-15 Distributive Laws 16-17 DeMorgan’s Laws
▪ The dual of an algebraic expression is obtained by
interchanging + and · and interchanging 0’s and 1’s.
▪ The identities appear in dual pairs. When there is only
one identity on a line the identity is self-dual, i. e., the
dual expression = the original expression.
Some Properties of Identities & the Algebra
(Continued)

▪ Unless it happens to be self-dual, the dual of an

expression does not equal the expression itself.
▪ Example: F = (A + C) · B + 0
dual F = (A · C + B) · 1 = A · C + B
▪ Example: G = X · Y + (W + Z)
dual G =
▪ Example: H = A · B + A · C + B · C
dual H =
▪ Are any of these functions self-dual?
Some Properties of Identities & the Algebra
(Continued)
▪ There can be more that 2 elements in B, i. e.,
elements other than 1 and 0. What are some
common useful Boolean algebras with more
than 2 elements?
1. Algebra of Sets
2. Algebra of n-bit binary vectors
▪ If B contains only 1 and 0, then B is called the
switching algebra which is the algebra we use
most often.
Boolean Operator Precedence
▪ The order of evaluation in a Boolean
expression is:
1. Parentheses
2. NOT
3. AND
4. OR
▪ Consequence: Parentheses appear
around OR expressions
▪ Example: F = A(B + C)(C + D)
Example 1: Boolean Algebraic Proof
▪ A + A·B = A (Absorption Theorem)
Proof Steps Justification (identity or theorem)
A + A·B
= A·1+A·B X=X·1
= A · ( 1 + B) X · Y + X · Z = X ·(Y + Z)(Distributive Law)
=A· 1 1+X=1
=A X·1=X

▪ Our primary reason for doing proofs is to learn:

• Careful and efficient use of the identities and theorems of
Boolean algebra, and
• How to choose the appropriate identity or theorem to apply
to make forward progress, irrespective of the application.
Example 2: Boolean Algebraic Proofs
▪ AB + AC + BC = AB + AC (Consensus Theorem)
Proof Steps Justification (identity or theorem)
AB + AC + BC
= AB + AC + 1 · BC ?
= AB +AC + (A + A) · BC ?
=
Example 3: Boolean Algebraic Proofs
▪ ( X + Y ) Z + X Y = Y( X + Z )
Proof Steps Justification (identity or theorem)
( X + Y )Z + X Y
=
Useful Theorems
▪ x  y + x  y = y (x + y )(x + y )= y Minimization
▪ x + xy = x x  (x + y ) = x Absorption
▪ x + x  y = x + y x  (x + y )= x  y Simplification
▪ xy + xz + yz = xy + xz Consensus
(x + y ) (x + z ) (y + z ) = (x + y ) (x + z )
▪ x + y = xy xy = x + y DeMorgan' s Laws
Proof of Simplification
xy +xy = y (x + y )(x + y ) = y
Proof of DeMorgan’s Laws
x + y = xy xy = x + y
 Boolean Function Evaluation
F1 = xy z x y z F1 F2 F3 F4
F2 = x + yz 0 0 0 0 0
F3 = x y z + x y z + x y 0 0 1 0 1
F4 = x y + x z 0 1 0 0 0
0 1 1 0 0
1 0 0 0 1
1 0 1 0 1
1 1 0 1 1
1 1 1 0 1
 Boolean Function Evaluation
F1 = xy z x y z F1 F2 F3 F4
F2 = x + yz 0 0 0 0 0 1 0
F3 = x y z + x y z + x y 0 0 1 0 1 0 1
F4 = x y + x z 0 1 0 0 0 0 0
0 1 1 0 0 1 1
1 0 0 0 1 1 1
1 0 1 0 1 1 1
1 1 0 1 1 0 0
1 1 1 0 1 0 0
Expression Simplification
▪ An application of Boolean algebra
▪ Simplify to contain the smallest number
of literals (complemented and
uncomplemented variables):
A B + ACD + A BD + AC D + A BCD
= AB + ABCD + A C D + A C D + A B D
= AB + AB(CD) + A C (D + D) + A B D
= AB + A C + A B D = B(A + AD) +AC
= B (A + D) + A C 5 literals
 Complementing Functions
▪ Use DeMorgan's Theorem to
complement a function:
1. Interchange AND and OR operators
2. Complement each constant value and
literal
▪ Example: Complement F = xy z + x y z
F = (x + y + z)(x + y + z)
▪ Example: Complement G = (a + bc)d + e
G=
Overview – Canonical Forms
▪ What are Canonical Forms?
▪ Minterms and Maxterms
▪ Index Representation of Minterms and
Maxterms
▪ Sum-of-Minterm (SOM) Representations
▪ Product-of-Maxterm (POM) Representations
▪ Representation of Complements of Functions
▪ Conversions between Representations
Canonical Forms
▪ It is useful to specify Boolean functions in
a form that:
• Allows comparison for equality.
• Has a correspondence to the truth tables
▪ Canonical Forms in common usage:
• Sum of Minterms (SOM)
• Product of Maxterms (POM)
Canonical Forms
Canonical Forms
Minterms
▪ Minterms are AND terms with every variable
present in either true or complemented form.
▪ Given that each binary variable may appear
normal (e.g., x) or complemented (e.g., x), there
are 2n minterms for n variables.
▪ Example: Two variables (X and Y)produce
2 x 2 = 4 combinations:
XY (both normal)
X Y (X normal, Y complemented)
XY (X complemented, Y normal)
X Y (both complemented)
▪ Thus there are four minterms of two variables.
Maxterms
▪ Maxterms are OR terms with every variable in
true or complemented form.
▪ Given that each binary variable may appear
normal (e.g., x) or complemented (e.g., x), there
are 2n maxterms for n variables.
▪ Example: Two variables (X and Y) produce
2 x 2 = 4 combinations:
X + Y (both normal)
X + Y (x normal, y complemented)
X + Y (x complemented, y normal)
X + Y (both complemented)
Maxterms and Minterms
▪ Examples: Two variable minterms and
maxterms.
Index Minterm Maxterm
0 xy x+y
1 xy x+y
2 xy x+y
3 xy x+y
▪ The index above is important for describing
which variables in the terms are true and which
are complemented.
 Standard Order
▪ Minterms and maxterms are designated with a subscript
▪ The subscript is a number, corresponding to a binary
pattern
▪ The bits in the pattern represent the complemented or
normal state of each variable listed in a standard order.
▪ All variables will be present in a minterm or maxterm and
will be listed in the same order (usually alphabetically)
▪ Example: For variables a, b, c:
• Maxterms: (a + b + c), (a + b + c)
• Terms: (b + a + c), a c b, and (c + b + a) are NOT in
standard order.
• Minterms: a b c, a b c, a b c
• Terms: (a + c), b c, and (a + b) do not contain all
variables
Purpose of the Index
▪ The index for the minterm or maxterm,
expressed as a binary number, is used to
determine whether the variable is shown in the
true form or complemented form.
▪ For Minterms:
• “1” means the variable is “Not Complemented” and
• “0” means the variable is “Complemented”.
▪ For Maxterms:
• “0” means the variable is “Not Complemented” and
• “1” means the variable is “Complemented”.
Index Example in Three Variables
▪ Example: (for three variables)
▪ Assume the variables are called X, Y, and Z.
▪ The standard order is X, then Y, then Z.
▪ The Index 0 (base 10) = 000 (base 2) for three
variables). All three variables are complemented
for minterm 0 ( X , Y, Z) and no variables are
complemented for Maxterm 0 (X,Y,Z).
• Minterm 0, called m0 is X Y Z .
• Maxterm 0, called M0 is (X + Y + Z).
• Minterm 6 ?
• Maxterm 6 ?
Index Examples – Four Variables
Index Binary Minterm Maxterm
i Pattern mi Mi
0 0000 abcd a+b+c+d
1 0001 abcd ?
3 0011 ? a+b+c+d
5 0101 abcd a+b+c+d
7 0111 ? a+b+c+d
10 1010 abcd a+b+c+d
13 1101 abcd ?
15 1111 abcd a+b+c+d
Minterm and Maxterm Relationship
▪ Review: DeMorgan's Theorem
x · y = x + y and x + y = x  y
▪ Two-variable example:
M 2 = x + y and m 2 = x·y
Thus M2 is the complement of m2 and vice-versa.
▪ Since DeMorgan's Theorem holds for n variables,
the above holds for terms of n variables
▪ giving:
M i = m i and m i = M i
Thus Mi is the complement of mi.
Observations
▪ We can implement any function by "ORing" the
minterms corresponding to "1" entries in the function
table. These are called the minterms of the function.
▪ We can implement any function by "ANDing" the
maxterms corresponding to "0" entries in the function
table. These are called the maxterms of the function.
▪ This gives us two canonical forms:
• Sum of Minterms (SOM)
• Product of Maxterms (POM)
for stating any Boolean function.
Minterm Function Example
▪ Example: Find F1 = m1 + m4 + m7
▪ F1 = x y z + x y z + x y z
x y z index m1 + m4 + m7 = F1
000 0 0 + 0 + 0 =0
001 1 1 + 0 + 0 =1
010 2 0 + 0 + 0 =0
011 3 0 + 0 + 0 =0
100 4 0 + 1 + 0 =1
101 5 0 + 0 + 0 =0
110 6 0 + 0 + 0 =0
111 7 0 + 0 + 1 =1
Minterm Function Example
▪ F(A, B, C, D, E) = m2 + m9 + m17 + m23
▪ F(A, B, C, D, E) = ?

F(A, B, C, D, E) = A’B’C’DE’ + A’BC’D’E +

AB’C’D’E + AB’CDE
 Maxterm Function Example
▪ Example: Implement F1 in maxterms:
F1 = M0 · M2 · M3 · M5 · M6
F1 = (x + y + z) ·(x + y + z)·(x + y + z )
·(x + y + z )·(x + y + z)
xyz i M0  M2  M 3  M5  M6 = F1
000 0 0  1  1  1  1 =0
001 1 1  1  1  1  1 =1
010 2 1  0  1  1  1 =0
011 3 1  1  0  1  1 =0
100 4 1  1  1  1  1 =1
101 5 1  1  1  0  1 =0
110 6 1  1  1  1  0 =0
111 7 1  1  1  1  1 =1
Maxterm Function Example
▪ F( A, B, C, D) = M 3  M8  M11  M14
▪ F(A, B,C,D) = ?

▪ F = (A + B + C’ + D’) (A’ + B + C + D)
(A’ + B + C’ + D’) (A’ + B’ + C’ + D)
 Canonical Sum of Minterms
▪ Any Boolean function can be expressed as a
Sum of Minterms.
• For the function table, the minterms used are the
terms corresponding to the 1's
• For expressions, expand all terms first to explicitly
list all minterms. Do this by “ANDing” any term
missing a variable v with a term (v + v ).
▪ Example: Implement f = x + x y as a sum of
minterms.
First expand terms: f = x ( y + y ) + x y
Then distribute terms: f = xy + x y + x y
Express as sum of minterms: f = m3 + m2 + m0
Another SOM Example
▪ Example: F = A + B C
▪ Express as SOM:

F = A(B + B’)(C + C’) + (A + A’) B’ C

= ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C
= ABC + ABC’ + AB’C + AB’C’ + A’B’C
= m7 + m6 + m5 + m4 + m1
= m1 + m4 + m5 + m6 + m7
This can be denoted in the formal shorthand:
F( A, B, C) = m(1,4,5,6,7)
Canonical Product of Maxterms
▪ Any Boolean Function can be expressed as a Product of
Maxterms (POM).
• For the function table, the maxterms used are the terms
corresponding to the 0's.
• For an expression, expand all terms first to explicitly list all
maxterms. Do this by first applying the second distributive
law , “ORing” terms missing variable v with a term equal to v v
and then applying the distributive law again.
▪ Example: Convert to product of maxterms:
f ( x, y , z ) = x + x y
Apply the distributive law:
x + x y = (x + x )(x + y ) = 1  (x + y ) = x + y
Add missing variable z:
x + y + z  z = ( x + y + z ) (x + y + z )
Express as POM: f = M2 · M3
 Another POM Example
▪ Convert to Product of Maxterms:
f(A, B, C) = A C + B C + A B
▪ Use x + y z = (x+y)·(x+z) with x = (A C + B C), y = A ,
and z = B to get:
f = (A C + B C + A )(A C + B C + B )
▪ Then use x + x y = x + y to get:
f = ( C + BC + A )(A C + C + B )
and a second time to get:
f = ( C + B + A )(A + C + B )
▪ Rearrange to standard order,
f = ( A + B + C)(A + B + C) to give f = M5 · M2
Function Complements
▪ The complement of a function expressed as a
sum of minterms is constructed by selecting the
minterms missing in the sum-of-minterms
canonical forms.
▪ Alternatively, the complement of a function
expressed by a Sum of Minterms form is simply
the Product of Maxterms with the same indices.
▪ Example: Given F ( x , y , z ) = m ( 1, 3 , 5 , 7 )
F( x, y , z ) = m(0, 2,4,6)
F( x, y , z ) = PM(1, 3,5,7 )
Conversion Between Forms
▪ To convert between sum-of-minterms and product-
of-maxterms form (or vice-versa) we follow these
steps:
• Find the function complement by swapping terms in the
list with terms not in the list.
• Change from products to sums, or vice versa.
▪ Example:Given F as before: F( x, y, z ) = m(1, 3,5,7)
▪ Form the Complement: F( x, y , z ) = m( 0, 2,4,6)
▪ Then use the other form with the same indices – this
forms the complement again, giving the other form
of the original function: F( x, y, z ) = PM(0, 2,4,6)
Standard Forms
▪ Standard Sum-of-Products (SOP) form:
equations are written as an OR of AND terms
▪ Standard Product-of-Sums (POS) form:
equations are written as an AND of OR terms
▪ Examples:
• SOP: A B C + A B C + B
• POS: (A + B) · (A+ B + C )· C
▪ These “mixed” forms are neither SOP nor POS
• (A B + C) (A + C)
• A B C + A C (A + B)
Standard Sum-of-Products (SOP)
▪ A sum of minterms form for n variables
can be written down directly from a truth
table.
• Implementation of this form is a two-level
network of gates such that:
• The first level consists of n-input AND gates,
and
• The second level is a single OR gate (with
fewer than 2n inputs).
▪ This form often can be simplified so that
the corresponding circuit is simpler.
Standard Sum-of-Products (SOP)
▪ A Simplification Example:
▪ F( A, B, C) = m(1,4,5,6,7)
▪ Writing the minterm expression:
F = A B C + A B C + A B C + ABC + ABC
▪ Simplifying:
F=

▪ Simplified F contains 3 literals compared to 15 in

minterm F
Standard Sum-of-Products (SOP)
▪ A Simplification Example:
▪ F( A, B, C) = m(1,4,5,6,7)
▪ Simplifying:
F = A’ B’ C + A (B’ C’ + B C’ + B’ C + B C)
= A’ B’ C + A (B’ + B) (C’ + C)
= A’ B’ C + A.1.1
= A’ B’ C + A
= B’C + A
▪ Simplified F contains 3 literals compared to 15 in
minterm F
AND/OR Two-level Implementation
of SOP Expression
▪ The two implementations for F are shown
below – it is quite apparent which is simpler!
A
B
A
C F
A B
B C
C
A
B F
C
A
B
C
A
B
C
SOP and POS Observations
▪ The previous examples show that:
• Canonical Forms (Sum-of-minterms, Product-of-
Maxterms), or other standard forms (SOP, POS)
differ in complexity
• Boolean algebra can be used to manipulate
equations into simpler forms.
• Simpler equations lead to simpler two-level
implementations
▪ Questions:
• How can we attain a “simplest” expression?
• Is there only one minimum cost circuit?
• The next part will deal with these issues.