MSc Thesis KwasiAduBoahenOpare 20th October 2010

7/31/2019 MSc Thesis KwasiAduBoahenOpare 20th October 2010

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S



(1)

T 4

T 4

T 5

T 5

T 6

T 6

T 7 T 8

T 8 T 9

T 9



T 11

T 11

A (1)

T 2 A

S A ( p)



w







m n



n

m

C in = n (m + 1)

C out = m − 1

C clk = n (m + 1) + m − 1

n

m







AND OR NOT NAND NOR XOR

0 1 w



w

(abc) = (001)

0 w 1

Z 1

0

w

NOR



Q 1 Q 2

Q 3

Q 4 d

Q 1 Q 1

(w1, w2) = (0 , 0), (0, 1), (1, 0), (1, 1) (0, 0) Q 3

Q 4 F 1 (0, 1)

F 0 Q 2

(1, 0) Q 1 F V ss

F 0(1, 1) Q 2 F

0

Q 1

(1, 0) F

F (0, 0)

(0, 0), (1, 0) Q 1 F

NAND d

Q 4 (1, 1)

Q 1 Q 2 Q 4

Q 2

(1, 0)



Q 1 Q 2 Q 4

Q 2 Q 4

AND OR

AND OR

AND OR







(x n − 1, ..., x 2, x1, x0)



( pn − 1, ..., p 2, p1, p0)

2n − 1

(x3, x2, x1, x0)



p

(x3, x2, x1, x0)

( p3, p2, p1, p0) (x3, x2, x1, x0)



x1, x 2, ...x n z1, z2, ...z n

y1, y2, ...y k Y 1, Y 2, ...Y k

F F l Y l yl



D C D S A

B D S

A B

D

A B

D S C



Y 1, Y 2, ...Y k





n

m n (m + 1)

n (m + 1) + m − 1

n

m





B

A B

0

1

1

0

0

1

0

1

1

0

0

1

0

1

1

1

A B

A

B



A

B B

A B

A B

n

m in =

n (m + 1)

n m

m m

in = n (m + 1)

m n



˜A ( p)



r

S

s = m − r



S

A

A( p) p

A( p)

p

k

x

y

A ≡ [T ], m n [t ij ]

[T ] ≡ [T j ], 1 ≤ j ≤ n T j j th

T k ≡ [t ik ], 1 ≤ i ≤ m, j = k

[T j ]T (r )

k ⊂ T k , r = T k



S

T (r )k ⊂ T k T k ≡ [t ik ], 1 + r ≤ i ≤ m, j = k

T k ≡ T (0)k

T (− r )

k ⊂ T k T k ≡ [t ik ], 1 ≤ i ≤ m − r, j = k

S

S =n

j =1

S j , S j skip, S, j th A ( p) .

A( p)

S



•

•

T (r )k T (− r )k+ x

t ik T (r )k , 1 + r ≤ i ≤ m

t ik + x T (− r )

k+ x , 1 ≤ i ≤ m − r,

t ik = t ik + x t ik + x

p = 1 p T pm n

p ≤ n ? A ( p) S A( p) p A ( p)

A

T ( r )k + x + y − 1

T (− r )k + x + y



T p S p

T p S = 0 T p pivotFound = 1

k = p r = 0 x = 0 y = 0T p initExist = 1

k > 1 1 k + y y + 1 k = 1 + yx y = 0

k < n

S A ( p) p



T (r )k T (

− r )k+ x

T (0)k+ x

initExist 1

y r = 0

r < mr = 0

T (r )k+ x + y− 1 T

(− r )k+ x + y

pivotF ound = 0 r

pivotF ound = 1 S T k+ x + y

T k+ x + y

initExist = 0



r = 0 y = 0

r < m x = 1 r = 0

k + x > n(n )

r

T k+ x

pivotF ound = 0 x

T (r )k T (

− r )k+ x

pivotF ound = 0 x

pivotF ound = 1 S T k+ x

T k+ x



initExist = 1

k +1 k + x + y y+1 x > 1 k = k + y+1x = 1 y = 0



s s

s



Q 3 Q 4 Q 7 Q 8

Q 3 Q 4 Q 7 Q 8

A i d

m

n



A

p = 1 k = p k = 1

x y r x = 0 y = 0 r = 6

T 1

S = 0 T 1

x > 1

k y = 0 xk < 12 T k



T 1

r y

r = 0 y = 0 r < 6

T k+ x k + x > nT k+ x T 2

T (r )1 T (

− r )2 r = 0 pivotFound =

0 x 2

T 1 T 3 x

x T k+ x

T 3, T 4 T 9 T 12

pivotF ound = 0 x T 12



(1)

k + x > n T 1

r 1

x 1

T (1)k T (

− 1)k+ x T 2 T 12

r 2 x

1 r = 2 T 6

x = 5 pivotFound 1

S = 4 T 6 T 6

T 2 T 6

T 6 2 k

k = 2 x y x = 1 y = 0



k < n T 2

T 2

T (r )2 T (

− r )2+ x

r = 0 r = 1 T 11 r = 2 x = 9

S = 4 T 11 pivotFound 1

T 11 initExist 1



initExist = 1 y

1 r 0 r < m

T (r )k+ x + y− 1 T (

− r )k+ x + y T (r )

11 T (− r )

12 T 12

T 11 pivotFound = 0 r

1 1

r = 5 pivotFound 1 T 12

S = 1

initExist = 0



T 3 T 12 y + 1 2 T 11

T 12

k 4 x y x = 1 y = 0

T 4

k < n T 4

T 4

T 7 T (r )4

T (− r )

4+ x T 9 r = 2 x = 5

pivotF ound 1 S = 4 T 9

T 5 T 9 T 9

k k = 5 x y x = 1 y = 0



T 4

T 5

k < n T 5

T 5

T (r )5 T (

− r )5+ x

T 12 r = 3 x = 7 pivotFound

1 S = 3 T 12

initExist = 0

T 6 T 12 T 12

k k = 6 x y x = 1y = 0



T 5

T 6

k < n T 6

T 6

T (r )6 T (

− r )6+ x

T 7 r = 3 x = 1 pivotFound

1 S = 3 T 7

initExist = 1



T 6

y 1 r

0 r < m T (r )k+ x + y− 1 T (

− r )k+ x + y

T (r )7 T (

− r )8 T 8 T 7

r = 2 T 8 S = 4

y

2 r 0 T (r )8 T

(− r )9

r = 1 T 9

S = 5 T 9

T 7 T 8



T 8 T 9

x > 1 T 9

k 9 k = k + y +1

x = 1 y = 0

k < n T 9

T 9

T (r )

9 T (− r )

9+ x

r = 0 r = 5 5

1

r = 6 r < m x r 1

0 pivotF ound 1 S = 0

T k+ x T 10 x = 1 x > 1

k k = 10 k = k + y + 1

x y x = 1 y = 0



T 9

k < n T 10

T 10

T (r )10 T

(− r )10+ x T 11

r = 0 x = 1 T 11 S = 6

T 11 initExist = 1

T 11



initExist = 1 y 1 r 0 T (r )11

T (− r )

12 r = 5

T 12 S = 1 T 12

T 11

x > 1 k

k = 12 k = k + y + 1 x = 1 y = 0

k = 12 k < n S

A(1)

p 2 T 2

A

S = 0 T 2

S A (2)

A T 3 T 12



A(1)

S = 35 A(1)

T 3 T 4 T 9 T 12

S

T 2 A

A

A (1)

A(5) S

A(1) p

84

5









T k

T j

T k T j







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MSc Thesis KwasiAduBoahenOpare 20th October 2010