1

SOP Standard form

In SOP standard form, every variable in the domain must

appear in each term. This form is useful for constructing

truth tables or for implementing logic in PLDs.

You can expand a nonstandard term to standard form by multiplying the

term by a term consisting of the sum of the missing variable and its

complement.

Convert X = A B + A B C to standard form.

The first term does not include the variable C. Therefore,

multiply it by the (C + C), which = 1:

X = A B (C + C) + A B C

= A B C + A B C + A B C

DCABDCBADCBADCBACDBADCBACDBA

DCABBACBA

2

POS Standard form

In POS standard form, every variable in the domain must

appear in each sum term of the expression.

You can expand a nonstandard POS expression to standard form by

adding the product of the missing variable and its complement and

applying rule 12, which states that (A + B)(A + C) = A + BC.

Convert X = (A + B)(A + B + C) to standard form.

The first sum term does not include the variable C.

Therefore, add C C and expand the result by rule 12.

X = (A + B + C C)(A + B + C)

= (A +B + C )(A + B + C)(A + B + C)

))()()((

))()((

DCBADCBADCBADCBA

DCBADCBCBA

3

Converting SOP–POS

))()(( CBACBACBA

ABCCBABCACBACBA

4

Boolean Expressions and Truth Tables—SOP Form

ABCCBACBAX

A B C X

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1

CBA

CBA

ABCCBACBAX

EX

.

5

Boolean Expressions and Truth Tables—POS Form

))()()()(( CBACBACBACBACBAX

A B C X

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 1

)( CBA

)( CBA

)( CBA

)( CBA

)( CBA

))()((

.

CBACBACBA

EX

6

A 3-variable Karnaugh map showing product terms.

.

7

A 4-variable Karnaugh map.

.

8

Karnaugh Map SOP Minimization

.

.

9

Example

.

ABCCABCBACBA

10

Example

.

DCBADCBADCABABCDDCABDCBACDBA

11

Example

.

CABBAA

12

Example

.

CDBADCBADCBACABBACB

13

1. Group the 1’s into two overlapping

groups as indicated.

2. Read each group by eliminating any

variable that changes across a

boundary.

3. The vertical group is read AC.

K-maps can simplify combinational logic by grouping

cells and eliminating variables that change.

Karnaugh maps

1

1 1

ABC

00

01

11

10

0 1

1

1 1

ABC

00

01

11

10

0 1

Group the 1’s on the map and read the minimum logic.

B changes

across this

boundary

C changes

across this

boundary

4. The horizontal group is read AB.

X = AC +AB

14

A 4-variable map has an adjacent cell on each of its four

boundaries as shown.

AB

AB

AB

AB

CD CD CD CDEach cell is different only by

one variable from an adjacent

cell.

Grouping follows the rules

given in the text.

The following slide shows an

example of reading a four

variable map using binary

numbers for the variables…

Karnaugh maps

15

X

Summary

Karnaugh maps

Group the 1’s on the map and read the minimum logic.

1. Group the 1’s into two separate

groups as indicated.

2. Read each group by eliminating

any variable that changes across a

boundary.

3. The upper (yellow) group is read as

AD. 4. The lower (green) group is read as

AD.

AB

CD

00

01

11

10

00 01 11 10

1 1

1 1

1

1

1

1

AB

CD

00

01

11

10

00 01 11 10

1 1

1 1

1

1

1

1

X = AD +AD

B changes

C changes

B changes

C changes across

outer boundary

16

17

18

Example of mapping directly from a truth table to a Karnaugh map.

.

19

Karnaugh Map POS Minimization

.

20

Example

.

))()()()(( DCBADCBADCBADCBADCBA

21

Example

.

))()()()(( CBACBACBACBACBA

22

))()()()(( DCBADCBADCBADCBADCB

23

Figure 4--41 A 5-variable Karnaugh map.

.

24

Illustration of groupings of 1s in adjacent cells of a 5-variable map.

.

25

Quine-McCluskey (Tabular) Minimization

• Two step process utilizing tabular listings to: – Identify prime implicants (implicant tables)

– Identify minimal PI set (cover tables)

• All work is done in tabular form – Number of variables is not a limitation

– Basis for many computer implementations

– Don’t cares are easily handled

• Proper organization and term identification are key factors for correct results

Quine-McCluskey Minimization (cont.) • Terms are initially listed one per line in

groups

– Each group contains terms with the same number of true and complemented variables

– Terms are listed in numerical order within group

• Terms and implicants are identified using one of three common notations

– full variable form

– cellular form

– 1,0,- form

Notation Forms • Full variable form - variables and

complements in algebraic form – hard to identify when adjacency applies

– very easy to make mistakes

• Cellular form - terms are identified by their decimal index value – Easy to tell when adjacency applies; indexes must

differ by power of two (one bit)

– Implicants identified by term nos. separated by comma; differing bit pos. in () following terms

Notation Forms (cont.)

• 1,0,- form - terms are identified by their binary index value – Easier to translate to/from full variable form

– Easy to identify when adjacency applies, one bit is different

– - shows variable(s) dropped when adjacency is used

• Different forms may be mixed during the minimization