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1972

Automatic Design of Switching Networks Darryl Dhein

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AUTm1ATIC DESIGN OF StllTCJIWJ NETHORKS by

Darryl D. Dhein

A Thesis SUbm:l.tted

in Partial }lllfilLm ~nt of the Requira~ents fo~ the HASTE1~

;Dcgree of I

OF SCIEl'tCE

in

Electrical

,

~n3in c Grin3

Approved by:

Madhu .S.Swaminathan hadhu Prof. George A. Brown ---_.------G. 3ro\m

Prof

-.~--:::-----------

Prof .0 George L. Thompson G. ri'~omp son Prof. VV. F. VValker

'i-I . F . ~·ICJ.lker

DEPA?.THE~\T OF COLLEG~

ELECTRICAL

OF

AP?LI~D

:;:~ ;cnt:E£RlI'~G SCI ~ ~C3 T ~ C :i~ ; OLOGY

ROCHESTE i? I i.: S'l'ITUTi:: 02 :·WC:'::':; ST f: C;,~ ~.: ::;~1 YO !~';:( JA!\"UARY 1972

otf ABSTRACT

This

thesis an

selecting

function. mum

cost

FORTRAN

IV

written

amd

gram

was

the

set

of

mechanization

computer

is

the

included

outline

of

further

ploring

as

an

the

part

of

areas

extension

of

of

of

this

on

Boolean a

This

thesis. an

design

of

switching

well

as

the

given.

study

Also which

present

A

work.

pro

overall

net

theory for

included would

was

approach

of

as

mini

function.

simplified

the

the

is based

a

framework

structure

design is

of

implement this

to

the

within

automation

of

algorithm

of

as

for automatically implicants

prime

program

A programing

automation

method

a

optimization

developed

theory for works.

optimum

The of

develops

be

is

an

worth

ex

TABLE OF CONTENTS Page

iii

LIST OF TABLES

CHAPTER 1.

CHAPTER 2.

LIST OF FIGURES

iv

LIST OF

vi

SYMBOLS

INTRODUCTION

2

1.1 Thesis

Definition

2

1.2 Historical Review

4

1.3

7

Scope

of

Thesis

OPTIMUM SELECTION OF PRIME OF BOOLEAN FUNCTIONS

IMPLICANTS

2.1 Optimized Prime Implicant 9

Selection Method

2.2

CHAPTER 3.

Special

Program Features

2.3 Program Description

11

2.4 Program Results

25

FURTHER DEVELOPMENT OF THE AUTOMATIC DESIGN PR03LEM

84

3.1 Program Structure

86

3.2 Development

of

Design Problem

CHAPTER

4.

10

Sample

Computer 102

CONCLUSIONS

110

APPENDIX

I

PROGRAM FLOW CHARTS

111

APPENDIX

II

PROGRAM LISTING

123

REFERENCES

167

BIBLIOGRAPHY

170

ii

LIST OF TABLES

Table

Page

12

1

First Data Card Entries

2

Second

3

Third Data Card Entries

15

4

Additional Data Card 2 Entries

23

5

Input Variables Problem 1

28

6

Table

35

7

of

Differences

Reductions

Level

13

Data Card Entries

Forming

of

Minterms

First

GrouD

of

39

3

8

Essential Prime

9

Machine Types

Implicant

Codes

47 100

Entry Keys

103

10

Example Computer

11

Programing Functions

104

12

Control Functions

104

13

Compute Mode Specification

105

14

Program Mode Arithmetic Operations

107

15

Programing Functions

in

Detailed

108

-LIST

OF FIGURES

Figure

Page

1

Word Format

17

2

Problem 1

27

3

Literal Weighting

31

4

Prime Implicant Development Problem I Level 1

32

Prime Implicant Development Problem 1 Level 2

34

Prime Implicant Development Problem 1 Levels 3, 4/ and 5

41

7

Prime

45

8

Essential Prime Implicants Problem 1

48

9

Problem 1

Solution

50

10

Problem 2

Specification

52

11

Prime

Implicant

12

Prime

Implicant Listing Problem 2

13

Essential Prime

lk

Problem 2

Solution

56

15

Problem 3

Specification

61

16

Prime

Implicant

17

Prime

Implicant Listing Problem 3

18

Essential Prime

19

Problem 3

Solution

66

20

Problem 4

Specification

69

21

Prime

5

6

Specification

Implicant

Listing Problem 1

Development

Elmplicants

Development

Problem

Problem 2

Problem 3

Implicants Problem 3

Implicant Listing Problem 4

iv

2

53 54

55

62

63 65

70

LIST OF FIGURES

(Cont.) Page

Figure

22

Problem 4

23

Problem

24

Prime Prime

5

Solution

71

Specification

72

Implicant

Development

and

Essential

73

Implicants Problem 5

74

25

Problem

26

Problem 6

Entry

and

Result

76

27

Problem 7

Entry

and

Result

78

28

Problem 8

Entry

and

Result

80

29

Automated

Logic

Design Program

88

30

Combinational Logic Circuit

31

Cost

32

Program

33

Main Program. Flow Chart

114

34

DATASN Flow Chart

115

35

SORT

36

PRIMEI Flow Chart

117

37

ESSPI Flow Chart

118

38

FORMPI Flow Chart

119

39

OPTMPI Flow Chart

120

40

CONV1

Flow Chart

121

41

C0NV2

Flow Chart

122

vs

5 Solution

Speed

Decision

Overlay

Data

Structure

Flow Chart

Design

89 95 113

116

LIST OF SYMBOLS

General Usage

Symbol

logical

X

A bar

+

Is used to denote logic Boolean variables)

XY

over

Adjacency

a

of

denotes

variable

addition

negation

(Union

of

lo,|ic variables denotes logical (intersection of Boolean

multiplication

variables)

Program

Special

Subroutines

DATAEN

Data

Entry

Subroutine

SORT

High

Speed

Sorting Subroutine for DATAEN

C0NV11

Format

PRIMEI

Prime

ESSPI

Essential Prime Subroutine

C0NV12

Format

FORMPI

Reformat,

C0NV13

One Word Format

Conversion

Implicant

for SORT

Subroutine

Determination

Implicant

Conversion Weight

Determination

Subroutine and

Subroutine

for ESSPI

Order Prime

Conversion

Implicants

Subroutine

for

FORMPI CONV23

Multiword Format

Conversion

Subroutine

for

FORMPI OPTMPI

Determination

and

Optimization

of

Solution

Subroutine

C0NV1

One

Word Format

Conversion

Subroutine

for

OPTMPI

C0NV2

Multiword Format

Conversion

OPTMPI

vi

Subroutine

for

CHAPTER 1.

INTRODUCTION

1.1 Thesis Definition

This thesis problem

for

of

develops

optimum

minimization

developed with

provision

implicants

and

The weighting based

the

on

considering

the

inclusion

number

function. was

recommended

areas

of

other

The

cost

multiple

the

of

be

areas

functions.

is

cost

a

to

required

program

con

automatic

In Chapter Three

an

to

used

a

of

part

of

prime

output

inputs

to

algorithm

optimization

implicants

computer

the

implicants

prime

weighting.

gate

structured

switching networks.

the

The

of

set

minimum

prime

logic

of

tinuing development in of

a

non-uniform

for the

a

to

approach

functions.

for

this

accomplish

of

switching

used

the

mechanize

selection

of

allows

detailed

a

design

outline

for further development

are

of

pre

sented.

The

model

in this thesis

used

extensively

used

in logic design

logic

with

a

the

types.

in

solution

true

design

time

gates

when

were

and

cost

model

because

restrictions

discrete made

uniform

This

gate

ease

the

model

up

elements

of

is

was

per

and

the

been

has

level AND-OR

two

input

highly

which

one

for

either

developed

of

for

its

it very closely

represented

for

some

time.

This

were

used

for the

individual

diodes).

was

logic

During

the

(i.e.

this

earlier

number

other

shape

if high

state

of

passive

(by

speeds

maintained

with

had to

the

With

levels has been virtually eliminated. at

present

most

a

cases

greater

provide

AND-OR gates. is the

that

variety cost

a

of

simple

a

gate

to minimize

the memory

independent

of

the

However,

gates.

flop including

gates.

minimum

are

simpler

associated

The two still

has

natural

manual

for

a

For this

sometimes

gating

real

design basis.

to

or

price

reason

less

of

design

procedure

the

minimize

circuits

flip-

a

complex

other

comparable

with

designs

non-

of

because

expensive

of

of

being

understand

For the

however,

structure,

same

and

one

of

work

reasons

was

minimum

requirement.

advantage

easiest

then

integrated

level AND-OR gating

the

and

modern

use

be many times

absolute

and

structure

purchased

few individual states

an

built-in gating

some

functions may be

to

the

in

which

logic

to

used

is

there

affects

of

two

of

exclusive

therefore

and

states

gate

with

which

flip flops,

or

memory units,

cost

factor

Another

to

advent

available

over

savings

the

Also,

gates

of

wave

were

limitations

practical

atirer

maintain

standards)

any reliability.

integrated circuits,

inserted

be

circuitry to

prevalent

a

of

composed

amplifiers,

discrete components,

of

every

be

expensive

period,

the

with

it

most

on

a

is best

a

an

to

adapted

are

teaching switching theory.

developed

well

method.

A historical

area

given

are

in the

These

minimization

include

procedures

known

as

the

review

of

the developments

the method

and

fool-proof

relatively

for this model.

procedures

mapping

and

next

there

Additionally,

Quine-McCluskey in this

section.

1.2 Historical Review

The

networks

switching a

was

formal deductive postulate

sets

well

developed

is

article

was

named

it;

one

184S

development has

circuits

paper

set

this of

veloped

and

(6)'

the

been

of

Logic.

in

calculus

Analysis

was

'

of

shown

to

which

alter

was

Post-

'

The

itself

algebra

two

(3 & 4)'

C.

to

Many

Independent

this

of

as

up

of

1853.

A

algebra

E.

papers

on

major

to

Shannon

switching

for his

Relay Switching in 1938. be

propositions

algebra

set

Huntington in

published

published

were

of

"v

who

One

E. V.

to

"Sets

attributed^

which

from the

'

application

development the

1904

in

another

Symbolic

"A

on

Circuits"

of

of

Classes

of

in

work

early

(Boolean Algebra).

George Boole

after

the

of

Algebra

attributed^

published

in

most

have been proposed.

for the Algebra

ulates

the

system

nate

an

for

point

starting

originated

The

derivable which

by

postulates

from

in turn

George

a

was

Boole.

sub

de

a

"The

paper

(7)'

in 1949.

lished for

of

In 1951

chart

simplification

followed by cation

a

a

systematic

algebraic

While the sense

formulated

basis

included

a

as

sixty-

The

subject.

two

level AND-OR

Electrical

come

to

considered

the

level AND-OP.

Algebra, is

given

minimization

be known

The

as

at

(n

the

as

in switching theory

and

ago,

it

almost

every text

forms the basis section

this

of

thesis

Institute

and

to

through

1.

in

of

i2)

approach

equation

was

~

work

on

is

on

his doctoral thesis

This

equation

and

for the

work

the

was

method

which

the

problem

the use

this

of

method

im

an

has

it

Quine-McCluskey method;

simplification

belov;

'

mathe

a

the

a

Massachusetts p.

in

algebra

still

as

earlier

classical

key

and

is

which

method

Quine 's

in 1952

ago

for

point

in June 1956.

on

provement

simplifi

years

years

works

all

Engineering

Technology

was

hundred

over

by E. J. McCluskey

presented

for

Quine

Boolean

of

five

starting

the

This

Method.

by W. V.

presented

for virtually

This

functions.

method

pub

'

postulates

were

Circuits'

was

method

( 8) the Harvard

as

later improved upon.

matical

Boolean

of

published

and

Switching

tabular

or

Boolean functions

of

further

Two Terminal

Synthesis

became known

method

well

ideas

Shannon developed his

Later,

of

is

now

two

Boolean

is based

(1)

XY + XY

Basically terms

all

to

"minterms"

X

=

the method

a

sum

terms

of

then

and

consists

expanding

lowest

their

of

first

of

level

equation

systematically using

1 to

simplify the result. Since these early a

number

of

selection

papers

in

published

J.

F.

and

Luccio 's The

cost

multiple

outputs

methods

solution.

The

thesis

can

be

by

of

cost.

For

a

yielding

to

set

method

small

combination

give

of

size

For

the

papers

1965^15

in

in

J. McCluskey two

also,

&

by l6^

in this thesis

by

relatively or

absolute

all

problems

of

becomes

near

any

optimum

optimum

would

be

for

for this solution

minimum

provided

significant

less

and

straight

developed

solutions

this

including

implicants

prime

algorithm

the

problems

presented

optimum

testing

approach

E.

and

other

these

Luccio,

large problems,

solved

optimization

automatically. all

method

certain

may be

F.

by

the

by

(17)'

different

the

of

developed

Pyne

the

and

the

of

fact that

variable

use

1964

in 1966.

paper

advantages

include the

forward

in

optimizing the

of

^l3 & l4^

1962;

and

one

noted

by I. B.

papers

1961

Gimpel,

As

method.

later

two

subject

implicants

the prime

of

Quine-McCluskey clude

the

on

have been

there

developments,

desirable

size

the

from the

standpoint

required

of

nonessential

to

implicants

prime

factorial type

a

is very

time

computer

to

consider

maximum

of

ten nonessential this

above

is the best analysis

the

are

level AND-OR

minimization

and

later in the currently

well

defined

fore

not

a

as

to

methods

The

by E. W. Veitch

was

published

(19)

These

routines

directly

currently

for

a

the

solution

the

of

similar

For

selects

The final

problem.

use

M. Karnaugh.

algorithm

completion

graphical

also

solution

of

of

extent

printed

of

the user.

by

specified

There

upon

is

program

implicants.

prime

weighting

solution

The

combinations

be considered.

to

combinations

all

the number

as

increased, being

function.

of

written

sizes

rapid

is

in

The increase

time used.

computer

more

solve

method

improved

graphical

methods

tend

visual

insight

and

with

to

applicable

in

automatic

to are

two common

its basic

in

popular

the

form

form by replace

there

solution

by

digital computer,

1.3

Scope

This selection

of

Thesis

thesis of

optimization

mechanization

develops

prime

implicants

algorithm

of

an

the

algorithm

of

is based

simplified

a

on

for the

Boolean a

function.

minimum

function.

optimum

cost

The

The

of

results

of

a

number

of

strong features subject

presents

an

developed

was

program.

Appendix

the

program.

the

areas

other

present

presented

Appendix

of

II

I

for this thesis

provides

a

a

flow

detailed

8

the

This

Chapter Three

recommended

investigation.

provides

giving

approach.

Chapter Four discusses the

derived from the

and

of

discussed,

in Chapter Two.

covered

outline

development.

limitations

and

is

matter

are

problems

sample

for future

conclusions

The

program

as

an

original

chart

of

the

computer

program

listing

of

CHAPTER 2.

The is

OPTIMUM SELECTION OF PRIME IMPLICANTS OF BOOLEAN FUNCTIONS

method

used

This

below.

given

in

selection

of

the

followed

by

a

is

program

used

in solving the AND-OR

problem

with

uniform

the

program

cost

included

are

an

method

additional

essential

used

for

implicants

and

360

computer

visions

are

incorporated for

entry.

Details

of

2.3.

A

in

section

results

given

are

The

Quine-McCluskey

section

method

of

the

selected.

ones

which

are

required

that

contain

set

of

and

sample

are

non

match

job

of

described their

and

2.4. first determined

in Cadwell.

implicants,

because they minterm.

the remaining

use

with

pro-

of

number

problems

Essential

particular

of

naturalness

its

described

prime

are

cost)

are

as

implicants

a

of

implicants

prime

determination

program

number

in

A

and

ease

method

features to

special

conf iguration.

the

for

charts

selection

optimized

RIT

the

logic

Selection Method

is the Quine-McCluskey

algorithm

prime

flow

The

the

of

in Appendix I.

2.1 Optimized Prime Implicant

The

description

combinational

input.

per

implicants

prime

prime

the

prime

are

The

by the (5)'

After

essential

implicants the

only

optimum

implicants

prime

are

ones

(minimum

necessary to

specify the

weighting the

accomplished

by

the

their probability

order

optimum

first

in

the

a

number

The

to

be

correct

of

user

may

considered

in

has

solutions

then

been

achieved

required

as

the

number

of

prime

combination

and

the

a

until

printed

the

an

considered

continues

which

specify

for the

used

is

then

are

probable

solutions

in

being included

of

impli

weighting

implicant ordering.

prime

Special Program Features

There

features

FORTRAN

are

a

octal

TV

which

is

of

variable

byte

is

information as

0

full FORTRAN required

information.

information in octal, up to

mation,

eighteen

in

one

used

Tv

the

This

RIT

the

to

is

storage

plus

of

saves

10

used

some

360

of

for In

computer

store

one

the

and

word.

variables

coding the the

additional

and

state

equivalent

stores

memory

the

computer

logical

employing

program

literals,

word.

on

integer format

the

number

logical data.

of

required

are

for the

By using

coding

Four bytes

1.

or

a

program

saving technique

storage

using

Even in the one

a

in the

programs

four bytes of

incorporated

including

BASIC FORTRAN

of

most

The best

computer.

factor to be

2,2

for

search

solution. cants

The

solution.

defined

user

by

of

in roughly

implicants

prime

is

This

is then selected.

function

required

state

infor

allows

a

higher theoretical limit run.

A description

which

includes the

in

the

of

the

on

above

size

data

program's

is

method

encoding

input

be

to

problems

of

routine

below

given

2.3.1.

section

2.3 Program Description

The

functional

The first

areas.

In this

section

entered

into the

it

is broken down into

program

the

development

In the

program

next

an

final

the

optimum

section.

has

to

to

be

encode

prime

implicant

section

describes

the

section

and

in making

which

of

entry

the method used

and

is presented,

the method used

the

information

basic

computer

is described.

is

number

a

selection

of

the

implicants.

prime

2.3.1 Data Entry

The entry. cards

as

a

The and

first

the

in

described

deck

program

package.

cribed

is

program

order

Next of

cards

of

deck

come

entry

starting

which

the as

is

data

the

are

11

the

computer

provided

cards

follows:

with

which

data system

the user are

des

Table 1 1st

Data Card

Entries

Co lumn

Entry

1

if only one problem is to be run or if this is the last problem. A 1 is entered if another problem is to be run Blank

2-5

Machine Type Enter a 1 in combinational

Specification; column

logic

5 for

a

design

problem.

Note:

blank. columns card

All columns not indicated should be left All entries must be right justified in

indicated.

These

entries.

12

notes

apply to

all

2

Table

2nd Data Card Entries

Column 1-5

Entry No.

literals__used

of

per

(i.e.

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Table 4 Additional Data Card 2 Entries

Column

Entry

16-20

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2 3 4 5 6 7 8 9 10 Weight prime

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21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 66-70

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26

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5.

A

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AND-OR

AUTOMATED LOGIC DESIGN PROGRAM MINIMIZATION BASED ON A UNIFORM CCST

INPUT NG. NC. NG. NC.

SOLUTIONS 2 3C

INPLT

DATA

LITERALS= OUTPUTS=

CF

PER

PRIME 3 15

TC

1 EE

CONSICEREC=

IMPLICANTS 4

12

5 IC

TAKEN IN 6 7 10 IC

25

COMBINATIONS 8 10

VARIABLE 122 1 ICO 13232C0 -33122C0

33131C0

Figure Problem

1

2

Specification

27.

9 10

10 10

CF

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for

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three

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Table 5 1

Problem

Inout Variables

Entry Status

of

Term

Card Column

Term

1

1

y-v-2-:66z5

1

12

Required Term

^1^3-5

1

13

2

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28

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29

literals

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minterm.

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present

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(4)

XJX3X5

(5)

X1X2X3X5

=

XLX2X3X4X5

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x6x5x4

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Literal Weighting

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Figure 3 Literal Weighting

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PRIME

IMPLICANT

LEVEL

1

*

1

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4

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3 5

1 1

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14 24

1 1 1 1 1 1

7 13 15 16 25 26 31 34

1 1 1 1 1 1 1 1

17 27 35

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36

1

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*

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CEVELCPMENT

6

37

Figure Prime

4

Implicant Development Problem 1 Level 1

32

Referring method

of

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literal. be

in

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search

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starts

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expression

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encoded

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33

term

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LFVFI * * * * *

* * * * * * * * * * * * * * * * * * *

2

L 1 1 I 1

1 1 1 4 4 4 4

1 1 11 1 1 14 14 14 24 24 24

* * * * * * * * * * * * * *

16 16 25 25 26 26 31 34 34

1 1 1 1 1 1 1 1 1 1 1 1 1 1

* * * *

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1 1 1 1

Figure Prime

1 ? 4 5

I I I 1 I L L 1L L L 1L ]L 1L 1L 11 1 ]L

6 6 6

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5 2 3 5 1 ? 5 1 2 4 4

5 3 2 5 1 5 2 4 1 4 3 1

2 5 4

2 1

5

Implicant Development Problem 1 Level 2

34

differs by

a

simplified

to

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power

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Figure 4 has

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Table 6 Table

Differences

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Octal Difference

Term

1

5

4

2

11

10

3

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of

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in Table 6

comparing the term

35

the

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in Figure

second

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solution

fact that

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36

derived

from

optimum

removed

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completion

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duced

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appears

nonnegated

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form are

reductions, derivation

reduction

be applicable.

37

terms

indicates

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in the

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power

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Figure 4

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"1

1

tively. is

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for the first group

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manner

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38

and

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of

given

2"

level three

in Table 7

Table 7

Reductions

1st Group Level

5

112

11

113

3

113

11

114

same

Level 3

of

2nd Group Level 2

112

Result

12

1

1

23

2

1

1

24

13

1

1

3

1

1

34

3

14

1

1

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114

5

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1

1

43+

4

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6

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1

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1

4

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24

1

1

4

1

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4

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2

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1

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2

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1

25

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1

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4

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5

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1

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4

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6

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5

4

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It may be the

First Group

Forming

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1

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39

first of

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40

compare

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LEVEL

3

* * * * * * * * *

* * * * * * * *

1 1 I 4 4 4 4 4

4 ...

6 6

* * * * * *

* * * *

* *

LFVFL * * * * * * * *

3. 5 5 5 6

I 1 L 1 1 L 1 ]L I L 1. ]L

j

3 4 4 2 4 4 5 5

5 .4

4 5

5 1L ]I 1L

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7 15 16 25 26 34

1 1 1 1 1 1

1

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4 4

4

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4 4

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LFVEL

Figure 6 Implicant Development Levels 3, 4, and 5 Problem 1 Prime

41

is only

one

tag,

terms

all

For level three there removed

as

which

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a

Y]_

greater

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42

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43

therefore

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3

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terms

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1

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1

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44

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2

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from the computer.

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information

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6

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information for

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4,

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in

PRIME

LFVFL

IMPLICANT

1

*

1

I

*n

4

1

*

3 5

*n

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6

I 1 I

11 14 24

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7 13 15 16 2b 26 31 34

1 1 1 1 1 1 I 1

1

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27 35 36

1 I 1 I

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1

11

I

5

3

11

4

3

2

1

5

4

2

Figure

7

LFVEL

2

LFVFL

3

LEVEL

LFVEL

LISTING

4

5 4

Prime

1

Implicant Listing Problem 1

45

A cant

is

prime

it

when

implicant

is the only

included.

Such

included

as

required

minterm.

cants

each

for its in two

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prime

implicant

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plicant

prime

term

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included

indicate that this one

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implicants prime

no

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computer

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46

not

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included

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essential

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only

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implicant

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essential

prime

cants

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for Problem 1

denoted

Table 8

by

The

implicant.

the

are

is

treated

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essential

1, 2,

3

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shown

as

an

as

impli

The literals

in Figure 8.

shown

numbers

it

one)

in

below.

Table 8 Essential Prime

Implicant

Literal Code

Meaning

1

Literal included in ne gated form (i.e. Xi). Literal included in nonnegated form (i.e. Xi). Literal not included.

2 3 Output

Code

Meaning Not

I

included

output

2 3

only cases

coded

one

Essential

are

order

in the

for the

output

the

essential

"3"

(essential for

had there been

from

The

as

any,

part

of

output.

output

literals

for this

.

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The

Codes

for this

.

format XsX4X3X2X, network

prime

implicants this

would

right

to

left.

listing

of

the

this

of

have

essential

47

There

problem;

for this

output).

been

,

in

all

output

are

Other outputs,

listed

prime

is

in ascending

implicants

is

an

ESSENTIAL

PRIMF

IMPI ICANTS

LITERALS 13332 3 3233 32312

OUTPUTS 3 3 3

Figure 8 Essential Prime Implicants Problem 1

48

listing

optional

For this

entry.

in the of

problem

essential

essential

as

(9)

A

form three

total

form

of

the

quire

an

is

a

the

gate

problem

AND

minimum

as

gate

The

input

essential

cost

2

includes terms

inputs

gate

three

the

is

problem

required

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OR

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an

OR

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course,

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2

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than

case

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essential

49

the

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a

It may be

direct to

wired

term

original

term, does

determined

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thirteen.

single

all

first

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compared

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When the

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fact

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X4X2X1#

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solution

solution.

Problem

Problem

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solution.

states

essential

plus

inputs

program

listing

a

as

term, being

second

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2.4.2

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two

problem

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+

X3

for the third,

input.

required

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+

X5X!

requires

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of

literal form

=

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in

represented

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computer

in Figure 9.

shown

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is the

problem

upon

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all

implicants

prime

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gives

user

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prime

When this occurs, and

the

which

desired

network

implicants.

ALL

TFRMS

ARE

COVFRFO

BY

THE

ESSENTIAL

LITFRALS

OUTPUTS

13332 33233 32312

3 3

3

Figure 9 Problem 1

50

Solution

PRIME

IMPLICANTS

Problem 2

is to find the

AND-OR

optimum

of

mechanization

Equation 10.

(10)

A

The data input

for Problem

development

plicant

in Figures

11

because there

level the

listing shown

is

network

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reduction

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13. The

output.

exclusive

the number of output

essential

prime

solution

of

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weighting

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five

was

Optimum Prime assigned

used.

Implicant

for the

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second

implicant

implicants

provided

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shown

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are

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are

implicants

are

one

1311

(X4X2X1).

Also,

in the term

(LIT

3),

weight

(WT

The

weight

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is

(OUTPUT 2) is prime

is

im

prime

Problem 2

to

solution

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acknowledge

The

past

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illustrated

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It may be

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status

implicant

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prime

For Problem 2 there

given.

of

and

implicant

in Figure

included

ing

no

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shovm

is

The computer,

12 respectively.

and

in Figure 14,

and

is

X]X2X3X4

Entry,

2.4,1,

Section

+

acknowledgement

Data

2.3.1,

for Problem 2

ment

X]X2X3

computer

and

1,

+

XiX2X4

in Section

plained

by

=

given.

implicant. As

In this in

described

Selection, minterm

51

not

a

case

Section

weight

of

a

5),

weight

2.3.3, one

included in the

is

AND-OR

AUTOMATED LOGIC DESIGN PROGRAM MINIMIZATION BASED ON A UNIFORM COST

INPUT NO. NO. NO. NO.

PER

INPUT

DATA

1TTFRALS= OUTPUTS=

SOLUTIONS

4 1 BF

CONSIDFRED= TO 25 PRIMF IMPLICANTS TAKEN IN COMBINATIONS 23456789 10 30 15 12 10 10 10 10 10 10

OF

VARIARLF 723200 121 300 712200

Figure Problem 2

10

Specification

52

OF

PRIME

DEVELOPMENT

IMPLICANT

LEVEL

1

*

0

1

*

*

2 4

I I

*

5

1

*

15

1

0 0

1 1

2 3

4

I

1

5

1

4

-

LEVEL

-

2

Figure 11 Prime

Implicant Development Problem 2

53

PRIME

IMPLICANT

LISTING

LFVFL *

0

*

2 4

*

5 15

LEVEL 0 0

2

4

1

5

4

Figure Prime

Implicant

3

12

Listing

Problem 2

54

ESSENTIAL

PR I ME

I MPL I CANTS

LITFRALS

OUTPUTS

1 131

3 3

3712

Figure 13

Essential Prime

Implicants

Problem 2

55

SOLUTION

PROBLEM

OUTPUT

NO.

PRIME

I

IMPLICANT

LIT 3

1311

ESSENTIAL

NO. NO.

OF OF

WT 5

OUTPUT 2

PRIMF IMPLICANTS 1131 3212

'AND'

Or
X1X2^3X4> X1X2X3X4>

1024

printed

of

Each term

1332

minterms

more

note

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combination

all

of

possibility

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in

term which

consists

minterms.

entry

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last

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X1X2X3X4

and

1X2X3X4

is

1

Problem 7

Problem 7

the

2

inconsistent, the user

X

out

(R^XtfliXl

from Equation

X2X3X4

of

(X2X3X4

listed

minterm

two minterms,

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to

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of

specification

quired

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The term

3212012

term,

second

specification

XLX2X3X4

cation

the

yielded

2).

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The third

the

X1Z2X3X4 is

being

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good

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result

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than nine

Figure 27

of

all

with

combi

provides

an

a

AND-OR

INPUT

AUTOMATED LOGIC DESIGN PROGRAM MINIMIZATION BASED ON A UNIFORM COST

PER

INPUT

DATA

NO.

LITFRALS=

NO.

OUTPUTS=

11 1 NO. SOLUTIONS TO BE CONSIDFRED= NO. OF PRIME IMPLICANTS TAKEN IN 2 3 4 5 6 7 30 15 12 10 10 10 VARI ABLF 33 3 311 1 111 100 333333 3333200 MORE THAN 9 ALL

COMBINATION

25

COMBINATIONS 8 10

LITERALS

910

10

USFD

Figure 27 Problem

7

Entry

78

and

Result

10

OF

example

2.4.8

this

of

than

terms

8

entered,

the

28

provides

an

Problem

limit

8.

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to

amount

for

all

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only

one

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minutes

be

listing

computer

twenty

and

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implicants,

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specified

eight

considerably

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grow

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in the

fifty

seconds. use

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for

of

type

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no

limit

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time

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combined

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course,

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the

containing

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very

to

be

analyzed

expansion

to

the

nonessential

79

By

used.

practical

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practical

a

combinations

of

factorial

number

a

a

seconds,

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for

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shown

user.

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ei^ht

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can

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size.

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larger problems,

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printout

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time used.

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notifies

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thousand min

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printout.

entry

restriction

the memory

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of

job

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than

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size

than

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size

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the

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retical

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Figure

was

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thousand minterms.

a

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Problem 8 Problem

use

type

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limits

implicants

~

""'

AND-OR

AUTOMATED LOGIC DESIGN PROGRAM MINIMIZATION BASED ON A UNIFORM COST

INPUT DATA NO. 1ITFRALS= 11 NO. OUTPUTS^ 1 NO. SOLUTIONS TO PE CONSIOFRED= NO. OF PRIME IMPLICANTS TAKEN IN

PER

25

COMBINATIONS 73456789 10 30 15 17 10 10 10 10 10 10

VARIABLF 3333333331 100 3333333331200 MORE THAN 1000 MINTFRMS

Figure

Problem

Entry

and

80

USED

28 8 Result

INPUT

OF

and

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For the

in

combinations

analysis

basic

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prime

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stopped

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problems

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OPTMPI.

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83

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ient

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at

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program.

be In

the overall and

a

sample

CHAPTER 3.

FURTHER DEVELOPMENT OF THE AUTOMATIC DESIGN PROBLEM

In the to

approach

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total

as

be

similar

interpretation Designs

problems,

voltage,

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of

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