1. ECE 331 – Digital System Design
Minimizing Boolean Expressions using K-maps,
The Minimal Cover,
and
Incompletely Specified Boolean Functions
(Lecture #6)

2. Minimizing SOP Expressions
ECE Digital System Design

3. 1. A Canonical SOP expression can be derived from a Truth table.
Remember …
1. A Canonical SOP expression can be derived from a Truth table.
2. A shorthand notation can be used to describe a SOP expression.
ECE Digital System Design

4. Minimizing SOP Expressions# A B C F 1 2 3 4 5 6 7
corresponds to the row #s
F = S m(1, 2, 5, 7)
Shorter-hand Notation
F = A'B'C + A'BC' + AB'C + ABC
F = S (m1, m2, m5, m7)
Canonical Sum-of-Products
Shorthand Notation
ECE Digital System Design

5. Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C) = S m(1, 2, 5, 7)
1. Write out the expression in terms of the minterms.
2. Minimize the SOP expression using a K-Map
ECE Digital System Design

6. Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C) = S m(0, 2, 3, 6)
1. Write out the expression in terms of the minterms.
2. Minimize the SOP expression using a K-Map
ECE Digital System Design

7. Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C,D) = S m(0, 4, 8, 10, 11, 12, 13, 15)
1. Write out the expression in terms of the minterms.
2. Minimize the SOP expression using a K-Map
ECE Digital System Design

8. Minimizing POS Expressions
ECE Digital System Design

9. Minimizing POS Expressions# A B C F 1 2 3 4 5 6 7
corresponds to the row #s
F = P M(0, 3, 4, 6)
Shorter-hand Notation
F = (A+B+C)(A+B'+C')(A'+B+C)(A'+B'+C)
F = P (M0, M3, M4, M6)
Canonical Product-of-Sums
Shorthand Notation
ECE Digital System Design

10. Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C) = P M(4, 5, 6)
1. Write out the expression in terms of the minterms.
2. Minimize the POS expression using a K-Map
ECE Digital System Design

11. Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C) = P M(1, 2, 3, 5)
1. Write out the expression in terms of the minterms.
2. Minimize the POS expression using a K-Map
ECE Digital System Design

12. Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C,D) = P M(0, 1, 4, 8, 9, 12, 15)
1. Write out the expression in terms of the minterms.
2. Minimize the POS expression using a K-Map
ECE Digital System Design

13. Selecting a Minimal Cover
ECE Digital System Design

14. ECE 331 - Digital System Design
Definitions
Literal: each appearance of a variable, either uncomplemented or complemented
Implicant: a product term that implies F=1
Prime Implicant: an implicant that cannot be combined into another implicant that has fewer literals
Cannot be further minimized
Essential Prime Implicant: a prime implicant that covers a minterm uniquely.
ECE Digital System Design

15. ECE 331 - Digital System Design
Definitions F
A B C
0 0
0 1
1 1
1 0 1 1
Which are the implicants, prime implicants, and essential prime implicants?
ECE Digital System Design

16. Definition: Implicants
A B C
0 0
0 1
1 1
1 0 1 1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)
ECE Digital System Design

17. Definition: Implicants
A B C
0 0
0 1
1 1
1 0 1 1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)
A'C', A'C, A'B', A'B, BC (pairs of minterms)
ECE Digital System Design

18. Definition: Implicants
A B C
0 0
0 1
1 1
1 0 1 1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)
A'C', A'C, A'B', A'B, BC (pairs of minterms)
A' (quartet of minterms)
ECE Digital System Design

19. Definition: Prime Implicants
A B C
0 0
0 1
1 1
1 0 1 1
Prime Implicants: BC, A'
ECE Digital System Design

20. Definition: Essential Prime Implicants
A B C
0 0
0 1
1 1
1 0 1 1
Essential Prime Implicants: BC, A'
ECE Digital System Design

21. ECE 331 - Digital System Design
Definitions
Minimal Cover (SOP): the sum (ORing) of prime implicants
Solution may not be unique
For SOP, must cover each “1” at least once
A minimal solution is one with the fewest product terms in the SOP expression, and the fewest literals in each product term.
ECE Digital System Design

22. Selecting a Minimal Cover (SOP)
Identify all prime implicants
Select all essential prime implicants
Select prime implicant(s) to cover remaining terms by considering all possibilities
Sometimes selection is obvious
Sometimes “guess” next prime implicant
Continue, perhaps recursively
Try all possible “guesses”
Write minimum Boolean expression
May not be unique
ECE Digital System Design

23. Selecting a Minimal Cover
Example:
Determine the minimal cover for the following K-Map: a b c d 00 01 11 10 1 F
1. Identify Prime Implicants
2. Identify Essential Prime Implicants
3. Determine Minimal Cover
ECE Digital System Design

24. ECE 331 - Digital System Design
Example #1 F a b c d 00 01 11 10 00 1 1 01 1 1 1 11 1 1 1 10 1 1
Prime Implicants: a'b'd, bc', ac, a'c'd, ab, b'cd
Essential Prime Implicants: bc', ac
Minimal Cover (SOP): F = a'b'd + bc' + ac
ECE Digital System Design

25. Selecting a Minimal Cover
Example:
Determine the minimal cover for the following K-Map: y z w x 00 01 11 10 1 F
1. Identify Prime Implicants
2. Identify Essential Prime Implicants
3. Determine Minimal Cover
ECE Digital System Design

26. ECE 331 - Digital System Design
Example #2 y z w x 00 01 11 10 1 F
Prime Implicants: xy'z', w'xy', w'xz, xyz, wxy, wxz'
Essential Prime Implicants: none
Minimal Cover: F = xy'z' + w'xz + wxy
F = w'xy' + xyz + wxz'
ECE Digital System Design

27. Incompletely Specified Functions
ECE Digital System Design

28. Incompletely Specified Functions
Some minterms should (or will) never occur.
For example, Binary Coded Decimal (BCD)
These are considered “don't care” outputs.
When minimizing a SOP expression using a K- Map, treat a “don't care” as a “1” whenever it is beneficial.
When minimizing a POS expression using a K- Map, treat a “don't care” as a “0” whenever it is beneficial.
ECE Digital System Design

29. Incompletely Specified Functions
Exercise:
Derive the minimum SOP expression for the following incompletely specified logic function:
F(A,B,C,D) = S m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)
ECE Digital System Design

30. Incompletely Specified Functions
Exercise:
Derive the minimum POS expression for the following incompletely specified logic function:
F(A,B,C,D) = S m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)
ECE Digital System Design