Minimisation

ENEL111

Minimisation

Last Lecture Sum of products Boolean algebra

This Lecture Karnaugh maps Some more examples of algebra and truth tables

Karnaugh Maps

K-Maps are a convenient way to simplify Boolean Expressions.

They can be used for up to 4 or 5 variables. They are a visual representation of a truth table. Expression are most commonly expressed in

sum of products form.

Truth table to K-Map

A B P

0 0 1

0 1 1

1 0 0

1 1 1

B

A 0 1

0 1 1

1 1

minterms are represented by a 1 in the corresponding location in the K map.

The expression is:

A.B + A.B + A.B

K-Maps Adjacent 1’s can be “paired off” Any variable which is both a 1 and a zero in this

pairing can be eliminated Pairs may be adjacent horizontally or vertically

B

A 0 1

0 1 1

1 1

a pair

another pair

B is eliminated, leaving A as the term

A is eliminated, leaving B as the termThe expression

becomes A + B

Returning to our car example Two Variable K-Map

A B C P

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

A.B.C + A.B.C + A.B.C

BC

A 00 01 11 10

0 1

1 1 1

One square filled in for each minterm.Notice the code sequence:

00 01 11 10 – a Gray code.

Grouping the Pairs

BC

A 00 01 11 10

0 1

1 1 1

equates to B.C as A is eliminated.

Here, we can “wrap around” and this pair equates to A.C as B is eliminated.

Our truth table simplifies to

A.C + B.C as before.

Groups of 4

BC

A 00 01 11 10

0 1 1

1 1 1

Groups of 4 in a block can be used to eliminate two variables:

The solution is B because it is a 1 over the whole block

(vertical pairs) = BC + BC = B(C + C) = B.

Karnaugh Maps

Three Variable K-Map

Extreme ends of same row considered adjacent

A BC

00 01 11 10

0

1

A.B.C A.B.C A.B.C A.B.C

A.B.C A.B.C A.B.C A.B.C

0010A.B.C

A.B.C

A.B.C

A.B.C

Karnaugh Maps

Three Variable K-Map example

X A.B.C A.B.C A.B.CA.B.C

A BC

00 01 11 10

0

1

X =

The Block of 4, again

A BC

00 01 11 10

0 1 1

1 1 1

X = C

Returning to our car example, once more Two Variable K-Map

A B C P

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

A.B.C + A.B.C + A.B.C

AB

C 00 01 11 10

0 1 1 1

1

There is more than one way to label the axes of the K-Map, some views lead to groupings which are easier to see.

Don’t Care States Sometimes in a truth table it does not

matter if the output is a zero or a one

Traditionally marked with an x.

We can use these as 1’s if it helps.

AB

C00 01 11 10

0 1 1

1 x x 1

A B C P

0 0 0 0

0 0 1 x

0 1 0 1

0 1 1 x

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

Karnaugh Maps

Four Variable K-Map

Four corners adjacent

AB CD

00 01 11 10

00

01

11

10

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

Karnaugh Maps

Four Variable K-Map example F A.B.C.DA.B.C.D+A.B.C.DA.B.C.DA.B.C.DA.B.C.DA.B.C.D

AB CD

00 01 11 10

00

01

11

10

F =

AB CD

00 01 11 10

00 1 101 1 111

10 1 1

Karnaugh Maps

Four Variable K-Map solution F A.B.C.DA.B.C.D+A.B.C.DA.B.C.DA.B.C.DA.B.C.DA.B.C.D

F = B.D + A.C

1

Product-of-SumsWe have populated the maps with 1’s using sum-of-products extracted from the truth table.

We can equally well work with the 0’s

AB

C00 01 11 10

0 1 1 1

1 1

A B C P

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

AB

C00 01 11 10

0 0

1 0 0 0

P = (A + B).(A + C)

P = A.B + A.C equivalent

Inverted K Maps

In some cases a better simplification can be obtained if the inverse of the output is considered i.e. group the zeros instead of the ones particularly when the number and patterns of zeros is

simpler than the ones

Karnaugh Maps Example: Z5 of the Seven Segment Display

0 0 0 0 1

0 0 0 1 0

0 0 1 1 0

0 1 0 0 0

0 1 0 1 0

0 1 1 0 1

0 1 1 1 0

1 0 0 0 1

X1 X2 X3 X4 Z5

1 0 0 1 0

1 0 1 0 X

1 0 1 1 X

1 1 0 0 X1 1 0 1 X1 1 1 0 X1 1 1 1 X

0

1

2

3

4

5

6

7

8

9

0 0 1 0 1X1X2

X3 X4 00 01 11 10

00

01

11

10

Z5 =

• Better to group 1’s or 0’s?

7-segment display

If there are less 1’s than 0’s it is an easier option:

X

X1X2 X3 X4 00 01 11 10

00 1 0 0 101 0 0 0 111 X X X10 1 0 X X

Changing this to 1 gives us the corner group.

Tutorial - Friday

Print out the CS1 tutorial questions from the website.

Come to see the answers worked through.