Nonlinear & Neural Networks LAB.

PART 1. 디지털 회로 설계의 기초

5. State Reduction ( 상태 간소화 )

Nonlinear & Neural Networks LAB.

State Reduction ( 상태 간소화 )

-. 플립플롭은 상태를 표현하는 소자

; 상태의 수를 줄이면 플립플롭의 수도 줄일 수 있다 .

; 상태의 수를 줄이면 회로의 복잡도도 줄 확률이 높다 .

-. 상태 축소의 2 가지 방법

; Row matching methode

; Implication chart

Nonlinear & Neural Networks LAB.

State Diagram :Reset

0/0 1/0

0/0 1/0 0/0 1/0

0/0 1/0 0/0 1/0 0/0 1/0 0/0 1/0

0/0 1/0 0/0 0/01/0 1/0

0/01/0

1/00/1

0/01/0

1/0 0/0 1/00/1

신호 입력 X, 출력 Z4bit 단위로 끊어져 들어오는 입력 1010 or 0110 의 마지막 bit 에서 1 출력 X = 0010 0110 1100 1010 0011 . . .

Z = 0000 0001 0000 0001 0000 . . .

Row Matching Method

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S7 S8 0 0

S4 S9 S10 0 0

S5 S11 S12 0 0

S6 S13 S14 0 0

S7 S0 S0 0 0

S8 S0 S0 0 0

S9 S0 S0 0 0

S10 S0 S0 1 0

S11 S0 S0 0 0

S12 S0 S0 1 0

S13 S0 S0 0 0

S14 S0 S0 0 0

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S7 S8 0 0

S4 S9 S10 0 0

S5 S11 S12 0 0

S6 S13 S14 0 0

S7 S0 S0 0 0

S8 S0 S0 0 0

S9 S0 S0 0 0

S10 S0 S0 1 0

S11 S0 S0 0 0

S12 S0 S0 1 0

S13 S0 S0 0 0

S14 S0 S0 0 0

S’10

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S7 S8 0 0

S4 S9 S’10 0 0

S5 S11 S’10 0 0

S6 S13 S14 0 0

S7 S0 S0 0 0

S8 S0 S0 0 0

S9 S0 S0 0 0

S’10 S0 S0 1 0

S11 S0 S0 0 0

S13 S0 S0 0 0

S14 S0 S0 0 0

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S7 S8 0 0

S4 S9 S’10 0 0

S5 S11 S’10 0 0

S6 S13 S14 0 0

S7 S0 S0 0 0

S8 S0 S0 0 0

S9 S0 S0 0 0

S’10 S0 S0 1 0

S11 S0 S0 0 0

S13 S0 S0 0 0

S14 S0 S0 0 0

S’7

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S’7 S’7 0 0

S4 S’7 S’10 0 0

S5 S’7 S’10 0 0

S6 S’7 S’7 0 0

S’7 S0 S0 0 0

S’10 S0 S0 1 0

S’3

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S’3 S4 0 0

S2 S5 S’3 0 0

S’3 S’7 S’7 0 0

S4 S’7 S’10 0 0

S5 S’7 S’10 0 0

S’7 S0 S0 0 0

S’10 S0 S0 1 0

Nonlinear & Neural Networks LAB.

Transition Table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S’3 S4 0 0

S2 S5 S’3 0 0

S’3 S’7 S’7 0 0

S4 S’7 S’10 0 0

S5 S’7 S’10 0 0

S’7 S0 S0 0 0

S’10 S0 S0 1 0

S’4

Nonlinear & Neural Networks LAB.

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S’3 S’4 0 0

S2 S’4 S’3 0 0

S’3 S’7 S’7 0 0

S’4 S’7 S’10 0 0

S’7 S0 S0 0 0

S’10 S0 S0 1 0

Final Reduced State Transition Table

Nonlinear & Neural Networks LAB.

Reset

S1

S3'

S7'

S2

S4'

S10'

0,1/0

0,1/0

0/0

0/0

1/01/0

1/0

1/0

1/00/1

S0

0/0

0/0

Corresponding State Diagram

Nonlinear & Neural Networks LAB.

Implication Chart Method

신호 입력 X, 출력 Z

serial sequence 010 or 110 이라면 언제든 1 이 출력되는 신호열

State transition table:

PRESENT STATENEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S1 S2 0 0

S1 S3 S4 0 0

S2 S5 S6 0 0

S3 S0 S0 0 0

S4 S0 S0 1 0

S5 S0 S0 0 0

S6 S0 S0 1 0

Nonlinear & Neural Networks LAB.

Implication Chart Method

Enumerate all possible combinations of states taken two at a time

Naive Data Structure:Xij will be the same as XjiAlso, can eliminate the diagonal

Implication Chart

Next StatesUnder allInputCombinations

S0

S1

S2

S3

S4

S5

S6

S0 S1 S2 S3 S4 S5 S6

S1

S2

S3

S4

S5

S6

S0 S1 S2 S3 S4 S5

Nonlinear & Neural Networks LAB.

Implication ChartFilling in the Implication Chart

Entry Xij ?Row is Si, Column is Sj

Si is equivalent to Sj if outputs are the same and next states are equivalent

Xij contains the next states of Si, Sj which must be equivalent if Si and Sj are equivalent

If Si, Sj have different output behavior, then Xij is crossed out

Example: S0 transitions to S1 on 0, S2 on 1; S1 transitions to S3 on 0, S4 on 1;

So square X<0,1> contains entries S1-S3 (transition on zero) S2-S4 (transition on one)

S1-S3S2-S4

S0

S1

Nonlinear & Neural Networks LAB.

Implication Chart Method

Starting Implication Chart

S2 and S4 have differentI/O behavior

This implies thatS1 and S0 cannot

be combined

S1

S2

S3

S4

S5

S6

S0 S1 S2 S3 S4 S5

S1-S3 S2-S4

S1-S5 S2-S6

S3-S5 S4-S6

S1-S0 S2-S0

S3-S0 S4-S0

S5-S0 S6-S0

S1-S0 S2-S0

S3-S0 S4-S0

S5-S0 S6-S0

S0-S0 S0-S0

S0-S0 S0-S0

Nonlinear & Neural Networks LAB.

Implication Chart Method

Results of First Marking Pass

Second Pass Adds No New Information

S3 and S5 are equivalentS4 and S6 are equivalentThis implies that S1 and S2 are too!

Reduced State Transition TableReduced State Transition Table

S0-S0 S0-S0

S3-S5 S4-S6

S0-S0 S0-S0

S1

S2

S3

S4

S5

S6

S0 S1 S2 S3 S4 S5

PRESENT STATE

NEXT STATE OUTPUT

X=0 X=1 X=0 X=1

S0 S’1 S’1 0 0

S’1 S’3 S’4 0 0

S’3 S0 S0 0 0

S’4 S0 S0 1 0

Nonlinear & Neural Networks LAB.

Multiple Input State Diagram Example

State Diagram

Symbolic State Diagram

S0 [1]

S2 [1]

S4 [1]

S1 [0]

S3 [0]

S5 [0]

10

0111

00

00

01

1110

10 01

1100

10

00

01

11

00

1110

01

10

1101

00

PRESENT STATE

NEXT STATE

OUTPUT00 01 10 11

S0 S0 S1 S2 S3 1

S1 S0 S3 S1 S5 0

S2 S1 S3 S2 S4 1

S3 S1 S0 S4 S5 0

S4 S0 S1 S2 S5 1

S5 S1 S4 S0 S5 0

Nonlinear & Neural Networks LAB.

Multiple Input Example

Implication Chart

Minimized State Table

S1

S2

S3

S4

S5

S0-S1 S1-S3 S2-S2 S3-S4

S0-S0 S1-S1 S2-S2 S3-S5

S0

S0-S1 S3-S0 S1-S4 S5-S5

S0-S1 S3-S4 S1-S0 S5-S5

S1

S1-S0 S3-S1 S2-S2S4-S5

S2

S1-S1 S0-S4 S4-S0S5-S5

S3 S4

PRESENT STATE

NEXT STATE

OUTPUT00 01 10 11

S’0 S’0 S1 S2 S’3 1

S1 S’0 S’3 S1 S’3 0

S2 S1 S’3 S2 S’0 1

S’3 S1 S’0 S’0 S’3 0

Nonlinear & Neural Networks LAB.

Implication Chart Method

Does the method solve the problem with the odd parity checker?

Implication ChartImplication Chart

S0 is equivalent to S2since nothing contradicts this assertion!

S 1 S 2

S 0 S 1

S 0 - S 2 S 1 - S 1

Nonlinear & Neural Networks LAB.

Implication Chart Method

The detailed algorithm:

1. Construct implication chart, one square for each combination of states taken two at a time

2. Square labeled Si, Sj, if outputs differ than square gets "X". Otherwise write down implied state pairs for all input combinations

3. Advance through chart top-to-bottom and left-to-right. If square Si, Sj contains next state pair Sm, Sn and that pair labels a square already labeled "X", then Si, Sj is labeled "X".

4. Continue executing Step 3 until no new squares are marked with "X".

5. For each remaining unmarked square Si, Sj, then Si and Sj are equivalent.

Nonlinear & Neural Networks LAB.

(e,g) e’

Row matching methode 을 이용한 상태축소 (State reduction) 예제

PSNS Output

X=0 X=1 X=0 X=1

a a b 0 0

b c d 0 0

c a d 0 0

d e f 0 1

e a f 0 1

f g f 0 1

g a f 0 1

PSNS Output

X=0 X=1 X=0 X=1

a a b 0 0

b c d 0 0

c a d 0 0

d e’ f 0 1

e’ a f 0 1

f e’ f 0 1

Nonlinear & Neural Networks LAB.

Row matching methode 을 이용한 상태축소 (State reduction) 예제

PSNS Output

X=0 X=1 X=0 X=1

a a b 0 0

b c d 0 0

c a d 0 0

d e’ f 0 1

e’ a f 0 1

f e’ f 0 1

(d,f) d’

PSNS Output

X=0 X=1 X=0 X=1

a a b 0 0

b c d’ 0 0

c a d’ 0 0

d’ e’ d’ 0 1

e’ a d’ 0 1

Nonlinear & Neural Networks LAB.

등가 상태도

’

’

Nonlinear & Neural Networks LAB.

Present

State

Next State Present Output

X=0 1 X=0

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 hcfdba and iff

ecdada and iff

hcdbga and iff

Implication chart 을 이용한 상태축소 (State reduction) 예제

Nonlinear & Neural Networks LAB.

Nonlinear & Neural Networks LAB.

Present

State

Next State

Output

X=0 1

a’ a’ c’ 0

b f h 0

c’ c’ a’ 1

f f b 1

g b h 0

h c’ g 1

(a,d) a’

(c,e) c’