CMPUT 329 - Computer Organization and Architecture II

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CMPUT329 - Fall 2003

Topic C: State Minimization José Nelson Amaral

CMPUT 329 - Computer Organization and Architecture II

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Reading Assignment

Chapter 7, section 7.10.3

CMPUT 329 - Computer Organization and Architecture II

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Equivalent States

State p in N1 is equivalent to a state q in N2 if Z1= Z2 for all input sequences X.

N1

N2

p

qX

Z1

Z2

Given two sequential circuits N1 and N2.

Let Z1 be the output sequence produced from state p in N1with input sequence X.

Let Z2 be the output sequence produced from state q in N2with input sequence X.

CMPUT 329 - Computer Organization and Architecture II

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How to Find Equivalent States?

States that have the same next states and thatproduce the same output are equivalent.

CMPUT 329 - Computer Organization and Architecture II

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Eliminating Redundant States by Row Matching

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F G 0 0 D H I 0 0 E J K 0 0 F L M 0 0 G N P 0 0 H A A 0 0 I A A 0 0 J A A 0 1 K A A 0 0 L A A 0 1 M A A 0 0 N A A 0 0 P A A 0 0

HH

H HH

CMPUT 329 - Computer Organization and Architecture II

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Eliminating Redundant States by Row Matching

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F G 0 0 D H H 0 0 E J H 0 0 F L H 0 0 G H H 0 0 H A A 0 0 J A A 0 1 L A A 0 1

J

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F G 0 0 D H I 0 0 E J K 0 0 F L M 0 0 G N P 0 0 H A A 0 0 I A A 0 0 J A A 0 1 K A A 0 0 L A A 0 1 M A A 0 0 N A A 0 0 P A A 0 0

HH

H HH

CMPUT 329 - Computer Organization and Architecture II

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Eliminating Redundant States by Row Matching

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F G 0 0 D H H 0 0 E J H 0 0 F J H 0 0 G H H 0 0 H A A 0 0 J A A 0 1

D

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F D 0 0 D H H 0 0 E J H 0 0 F J H 0 0 H A A 0 0 J A A 0 1

E

CMPUT 329 - Computer Organization and Architecture II

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Eliminating Redundant States by Row Matching

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C E D 0 0 D H H 0 0 E J H 0 0 H A A 0 0 J A A 0 1

Next State Present Output

Present State

X=0 X=1 X=0 X=1 A B C 0 0 B D E 0 0 C F G 0 0 D H I 0 0 E J K 0 0 F L M 0 0 G N P 0 0 H A A 0 0 I A A 0 0 J A A 0 1 K A A 0 0 L A A 0 1 M A A 0 0 N A A 0 0 P A A 0 0

CMPUT 329 - Computer Organization and Architecture II

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Equivalent States

States that have equivalent next states and thatproduce the same output are also equivalent.

But these equivalences cannot be identifiedby inspection of the state table.

How can we identify these states?

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

a-dc-e

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

c-e

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

a-fe-h

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

a-fe-h

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

d-fc-h

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

a-fe-h

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

a-fe-h

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

b-f

a-be-h

e-fb-d

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-be-h

e-fb-d

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-be-h

e-fb-d

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-be-h

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-be-h

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

c-fa-b

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

c-ed-g

a-gc-fb-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

c-ed-g

a-g

b

c

d

e

f

g

h

a b c d e f g

c-e

b-dc-h

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

c-ed-g

a-g

b

c

d

e

f

g

h

a b c d e f g

c-e

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

a-g

b

c

d

e

f

g

h

a b c d e f g

c-e

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

b

c

d

e

f

g

h

a b c d e f g

c-e

a c

d e

1

0

1

010

0

1

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

b

c

d

e

f

g

h

a b c d e f g

c-e

a

c-e

d

1

0

1

10

1

0

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

b

c

d

e

f

g

h

a b c d e f g

c-e

a-d c-e

10

10

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

b

c

d

e

f

g

h

a b c d e f g

c-e

a

a

CMPUT 329 - Computer Organization and Architecture II

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Implication Table Method

Next State

Present State

X=0 X=1

Present Output

a d c 0 b f h 0 c e d 1 d a e 0 e c a 1 f f b 1 g b h 0 h c g 1

a-d

b

c

d

e

f

g

h

a b c d e f g

c-e

a

ac