1. Canonical Forms and Logic Miniminization
Today:
First Hour: Canonical Forms
Section of Katz’s Textbook
In-class Activity #1
Second Hour: Incomplete Functions, Introduction to Logic Minimization
Section and of Katz’s Textbook
In-class Activity #2

2. Canonical Forms Truth tables uniquely define Boolean functions
There are two standard (canonical) forms for Boolean expressions that derived from the truth table
Represent the function’s 1s (on-set) called Sum of Products (SOP) form
Represent the function’s 0s (off-set) called Product of Sums (POS) form

3. SOP Expressions Sample truth table
A B C F F 1
F = A' B C + A B' C' + A B' C + A B C' + A B C
F' = A' B' C' + A' B' C + A' B C'

4. Notion of a minterm
A minterm is 1 for exactly 1 row of the truth table
A minterm is a product (AND) of all the literals for that row
A truth table function can be represented in terms of its 1s
A truth table function can be represented by summing (ORing) its minterms

5. Shorthand notation for minterms of 3 variables
# A B C minterms
A' B' C' = m0
A' B' C = m1
A' B C' = m2
A' B C = m3
A B' C' = m4
A B' C = m5
A B C' = m6
A B C = m7
Shorthand notation for minterms of 3 variables

6. Shorthand notation for SOP expressions
SOP Shorthand
Shorthand notation for SOP expressions
F(A,B,C) = A' B C + A B' C' + A B' C + A B C' + A B C
=  m(3,4,5,6,7)
F(A,B,C)' = A' B' C' + A' B' C + A' B C'
=  m(0,1,2)

7. POS Expressions F = (A + B +C) (A + B + C') • (A + B' + C)
Sample truth table
A B C F F 1
F = (A + B +C) (A + B + C')
• (A + B' + C)
F' = (A + B' +C') (A' + B + C)
• (A' + B + C') (A' + B' + C)
• (A' + B' + C')

8. Notion of a Maxterm
A Maxterm is 0 for exactly 1 row of the truth table
A Maxterm is the complement of the corresponding minterm of that row
A Maxterm is a sum (OR) of complements of all the row literals
A truth table function can be represented by its 0s; that is, by a product (ANDing) of Maxterms

9. Shorthand notation for Maxterms of 3 variables
# A B C Maxterms
A + B + C = M0
A + B + C' = M1
A + B' + C = M2
A + B' + C' = M3
A' + B + C = M4
A' + B + C' = M5
A' + B' + C = M6
A' + B' + C' = M7
Shorthand notation for Maxterms of 3 variables

10. Shorthand notation for POS expressions
POS Shorthand
Shorthand notation for POS expressions
A B C F F 1
F(A.B.C) = (A + B +C) • (A + B + C') (A + B' + C) =  M(0,1,2)
F(A,B,C)' = (A + B' +C') • (A' + B + C) (A' + B +C') • (A' + B' + C) (A' + B' + C')
=  M(3,4,5,6,7)

11. Canonical Conversions
Complementary relations between minterms & Maxterms
F(A,B,C) =  m(3,4,5,6,7) =  M(0,1,2)
SOP POS
F'(A,B,C) =  m(0,1,2) =  M(3,4,5,6,7)

12. Do Activity #1 Now Reference: Section 2.2.2 of Katz’s Textbook
Minterms: (Shorthand)
m0, m1, … etc
 m(0,1,...)
Minterms: (Canonical)
A’B’C’, A’B'C … etc
A’B’C’ + A’B'C+ … etc
Maxterms: (Shorthand)
M0, M1, … etc
 M(0,1,..)
Maxterms: (Canonical)
(A’+B’+C’), (A’+B’+C), … etc
(A’+B’+C’) (A’+B’+C)  … etc

13. Incomplete Functions Example: Don’t cares
Outputs associated with inputs that can be ignored
These inputs CANNOT happen in a good design
Assign logic values to these outputs to simplify circuits
Example:
Binary Coded Decimal (BCD)

14. BCD Numbers
# A B C D
x A
x B
x C
x D
x E
x F
Decimal
numbers
Not decimal!
Not a valid input!
Cannot happen!

15. Don’t Cares & Canonical Forms
Shorthand Representation
Representation of “D” in BCD
D(A,B,C,D) = m1 + m3 + m5 + m7 + m9
+ d11 + d13 + d15
= [m(1,3,5,7,9) + d(11,13,15) ]
D(A,B,C,D) = M0 • M2 • M4 • M6 • M • D10 • D12 • D14
= [M(0,2,4,6,8) • D(10,12,14) ]

16. Logic Minimization
Minimization is the process of reducing a complex logic expression or equation into a simpler form with fewer terms by removing redundancies and terms having no effect on the output.
This is similar to simplifying and reducing complex algebraic expressions by combining and collecting like terms.
As in algebraic reduction, rules must be followed that guarantee the value of the expression is not changed by the simplification.

17. Used to measure complexity
Variables & Literals
Used to measure complexity
Variable: each input variable or its complement in an expression
Literal: each appearance of a variable or its complement in an expression
Example
Z = A B C + A B + A B C + B C
Variables = A, B, C
Literals = = 10

18. Rationale for Logic Minimization
Reduce complexity of the gate level implementation
• reduce number of literals (gate inputs)
• reduce number of gates
• reduce number of levels of gates
fewer inputs implies faster gates in some technologies
fan-ins (number of gate inputs) are limited in some technologies
fewer levels of gates implies reduced signal propagation delays
minimum delay configuration typically requires more gates
number of gates (or gate packages) influences manufacturing costs

19. Tradeoffs Time and Space Trade-Offs Traditional methods:
reduce delay at expense of adding gates
New methods:
trade off between increased circuit delay and reduced gate count
Power Savings

20. Alternative Gate Realizations
Two-Level Realization
(inverters don't count)
Multi-Level Realization
Advantage: Reduced Gate
Fan-ins
Complex Gate: XOR
Advantage: Fewest Gates

21. Apply the laws and theorems to simplify Boolean equations
Minimization Example
Apply the laws and theorems to simplify Boolean equations
Example: full adder's carry out function
Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin
Simplify by minimizing the number terms
Apply: Laws & Theorems 3 and 9, or 1D, 3, 5 and 8

22. Simplifying Equations
Example: full adder's carry out function
Cout = A' B Cin + A B' Cin + A B Cin' + A B Cin
Alternative
[9]
= A' B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin
= (A' + A) B Cin + A B' Cin + A B Cin' + A B Cin
= (1) B Cin + A B' Cin + A B Cin' + A B Cin
= B Cin + A B' Cin + A B Cin' + A B Cin + A B Cin
= B Cin + A (B' + B) Cin + A B Cin' + A B Cin
= B Cin + A (1) Cin + A B Cin' + A B Cin
= B Cin + A Cin + A B (Cin' + Cin)
= B Cin + A Cin + A B (1)
= B Cin + A Cin + A B
[3]
[8]
[5]
[3]
[1D]
[8]
[5]
[8]
[1D]
[5]
[1D]
NOTE: Minimization is just an exercise in applying laws of Boolean Algebra
learnt earlier!

23. Do Activity #2 Now For Next Class: Due: End of Class Today
RETAIN THE LAST PAGE (#3)!!
For Next Class:
Bring Randy Katz Textbook
Required Reading:
Sec & 2.3 (omit 2.3.6) of Katz
This reading is necessary for getting points in the Studio Activity!