Osnovi Racunarske Tehnike - 7

114455

77

77..11

- - . . , "0" "1". - , . , - , , , .

- - () x1, x2, ... , xn, - (). - , . 0 1, , . . , - - f(x1, x2, ... , xn). - . - , . .

() - . - . - . -, . "0",

114466

"1". .

() - .

, . , - . . 7.1.

- . :

1) . ( ) . . 0 - 0 0,4 V, 1 -

x2

(x , x ,..., x1 2 n)

x x1 2

x1

x1x2

x1x2

x1x2

x1x2

x1x2

x1x2

x1x2

x1x2

x1x2

x1x2

x

Vx x = x +x1 2 1 2 x x = x +x1 2 1 2x x = x x1 2 1 2 x x = x x1 2 1 2V

x x1 2 x x1 2+

x x

7.1

x 1 2/ x x 1 2/ x/x x1 2 x x1 2

x

xn. . .

x1x2

114477

2,4 5,0 V. . - , .

2) . - .

3) . , , . , - .

4) . , - - .

5) . . - .

6) . . , . . - .

7) . - .

8) . . 0 - 70 C, -25 +125 C.

77..22

n m - - v0 v1 v0 v1. {v0, v1}n { v0, v1}m. {0, 1} {v0, v1}, - , . .

114488

, , ( ) (. Switching Theory) (. Logic Design).

- {v0, v1,..., vm-1}n {v0, v1,..., vm-1}m, - {v0, v1,..., vm-1}. - . - (. Multivalued Logic).

, - , :

1) , ,

2) () .

- :

) - ,

) 0 1,

) ,

) , ( ) , -.

. () -. , . , .

114499

- .

i- i- i - . -, i- i-1 . , - . i- i .

77..33

- . ( ) - , . -. , . - , . - . - - .

:

1) ,

2) , ,

3) - .

115500

. . 7.2 . - :

g1 = x1x2; g2 = x1/x3; f = g1 g2 = (x1x2) (x1/x3) = ( 1x x2) ( 1x 3x ).

: f = 1x 2x 3x 1x 2x x3 1x x2 3x 1x x2x3 x1 2x 3x , F(1) = {0,1,2,3,4}.

77..44

: fi(x1,x2,...,xn), (i = 1, 2,..., m), - , , . , . , (- ). , - .

- - . - .

. ,

x1x3

x1x2

v

/

g1

g2

f(x ,x ,x )1 2 3

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

x x x1 2 3 f(x ,x ,x )1 2 311111000

7. 2

115511

, - . , , , .

:

1. - ,

2. . ,

3. - .

() . - . - , -:

1. ( ),

2. ,

3. - .

-- (. 7.3). , () - - -, , (. 7.3).

115522

- .

-, , . - 7.4.

- . - , - .

77..55

- . -

f = x x x x1 1 3 1 2 1 2 3x x x

x1

f

x1

x2x3

V

V

V

x4

7.3 a) b)

f1

f2

x1x3

V

V

Vx2

V

V

x1x2x3

f = x x x2 1 2 1 2 3x x f = xx x x x1 2 2 3 4

x1x2

x3x4

x3x1 x x x1 2 3

x x x x1 2 3 4 x2

{,, } 7.4

,

115533

. - . . - -, , - . -, . , . - ( ), ( ) . -, , . - , .

, , - . - . , - , . - .

n- 1 . 0- . m- m- 2n-m, m{0,1,...,m-1} 0-. 1- , 2- , 3- , . m- n-m -, .

n- , - 1. - .

115544

: . 0 1 .

- 1. , .

. 7.5 1- (), 7.5 7.5 - .

( ) r 2r n-r . r=0 , r = n . r - . , , 7.6, , . r=1 - 4 , r=2 - 4 , r=3 8 . 5 , 7.6, i- (i=1,2,3,4) 000 100, 001 101, 011 111, i- (i=0,1,...,7) 00 10.

x2 x2 x2

x1x1 x1

x3 x3x3

*x x1 2x1 2x

x1 2x

x x1 2

x x2 3

x2 3x

x2 3x

x x2 3

x x

13

x x

13

x x13

x

13x

) f = x x x2 1 3 ) f = x xx2 1 3 7.5

)*

*

115555

. 5 . 7.6.

- 1. - . * - 1 * . - , . . 0 - 1 . .

. 7.6 4 5 -.

f(x1,x2,...,xn) g(x1,x2,...,xn) . g f g(x1,x2,...,xn) f(x1,x2,...,xn) 0 f 0, 1 f 1. g f - 1. - .

. .

g f 1 f, 0 f 0. .

pk - pj pk pj . . pj pj

115566

= xipk, pj 1 - 1 pk, pj pk.

1 . . p f p f f. - , 1 .

- () . , - . : (Quine) - (McClu-skey).

7.6

00 01 11 10 00 01 11 100001

1110

0001

1110

*

*

*

**

1

11

1

1

1

x4 5x

x1 = 0 x = 11

x2 3x

f = x x x x x x1 2 3 5 2 3 5 2 3 5 x x x x)

)

x x1 2

x x3 4

00 01 11 1000011110

1 1

1

1

1 1 1

1

f = x x x x 3 4 1 2 4 1 3 4 x x x xx2 3x

)

x x1 2 3x

f = x x x x x x1 2 3 5 2 3 5 2 3 5 x x x x

1*

*

*

* 1

1

1

1

x x4 5

00011110

1

*

000 001 011 010 110 111 101 100

115577

- :

1) (): pxi p ix = p, 2) : p pxi = p,

: x = x x ... x . -

-. . - x = x x ... x . :

. - . . "" - ( ). - , - . .

. .

"" - .

. 7.1 F(1) = {3,4,5,7,9,11,12,13}.

- : , "*". "*" -

115588

. - "*" "" ( "*" ).

7.1

i i i i 3 1x 2x x3 x4 3,7 1x x3 x4 4,5,12,13 x2 3x 3,7 1x x3 x4 4 1x x2 3x 4x 3,11 2x x3 x4 3,11 2x x3 x4 5 1x x2 3x x4 4,5 1x x2 3x 5,7 1x x2x4 7 1x 2x x3 x4 4,12 x2 3x x4 9,11 x1 x2x4 9 x1 x2 3x x4 5,7 1x x2 x4 9,13 x1 3x x4 11 x1 2x x3 x4 5,13 x2 3x x4 4,5,12,13 x2 3x 12 x1 x2 3x 4x 9,11 x1 x2 x4 13 x1x2 3x x4 9,13 x1 3x x4 12,13 x1x2 3x

-- , -. , - - .

- . , - . - .

, , - . {0,1,x} - () 0 1. "", . - .

(- ) .

115599

1 , 0 - , "" - .

. 7.2 - F(1) = {2,4,5,6,7,9,13}.

7.2 -

i i i i

2 0010 2,6 0x10 4,5,6,7 01xx 2,6 0x10 1x x3 4x 4 0100 4,5 010x 5,13 x101 x2 3x x4 ------------- 4,6 01x0 9,13 1x01 x1 3x x4 5 0101 -------------- 4,5,6,7 01xx 1x x2 6 0110 5,7 01x1 9 1001 5,13 x101 ------------- 6,7 011x 7 0111 9,13 1x01 13 1101

- : 1, "*". "*" - . * - ( * - ).

. 7.3 - - F(1) = {0,5,8,12,15} F(*) = {1,2,3,10,13,14}.

116600

, . . . - , . . .

7.3

-

i i i i

0 0000 0,1 000x 0,1,2,3 00xx 1,5 0x01 1x 3x x4 e --------------- 0,2 00x0 0,2,8,10 x0x0 5,13 x101 x2 3x x4 f 1 *0001 0,8 x000 8,10,12,14 1xx0 0,1,2,3 00xx 1x 2x b 2 *0010 --------------- 12,13,14,15 11xx 0,2,8,10 x0x0 2x 4x d 8 1000 1,3 *00x1 8,10,12,14 1xx0 x1 4x c -------------- 1,5 0x01 12,13,14,15 11xx x1 x2 a 3 *0011 2,3 001x 5 0101 2,10 *x010 10 *1010 8,10 10x0 12 1100 8,12 1x00 -------------- --------------- 13 *1101 5,13 x101 14 *1110 10,14 *1x10 --------------- 12,13 110x 15 1111 12,14 11x0 ----------------- 13,15 11x1 14,15 111x

- 1 . i j i- j- , . "*". .

116611

- , . . - :

1. , . , , .

1 . . * .

2. , . * , -. . * *.

3. ( . ), - * ( - ). 2.

4. 2 3 - . - .

3 , - , -. - (Patrick), :

f(x1,x2,...,xn) - a,b,c,... ,

116622

1 . fp(a,b,c,...) 1 f(x1,x2,...,xn). - *. fp . - , - fp f(x1,x2,...,xn). - - .

. - 7.3 7.4.

7.4

0000 0101 1000 1100 1111 a 11xx * * b 00xx * c 1xx0 * * d x0x0 * * e 0x01 * f x101 * fp = (b v d)(e v f)(c v d)(a v c)a = abce v abcf v acde v acdf v abde v abdf v acde v ade v

adf v abcde v abcdf = abce v abcf v ade v adf. : f = x1 x2 2x 4x 1x 3x x4; f = x1 x2 2x 4x x2 3x x4.

() , , . - -

116633

. , - , - .

F = {f1,f2,...,fm} . - F' F, - F'. F' F - .

F' , F -. , . - .

- - , :

1) - m -, . . fi, (i = 1,2,...,m), i- - * - .

2) * ( ).

3) - * .

4) "" - .

5) .

- . fi *. , .

116644

- , - * *, -. , , - - . -, , .

. 7.5 , 7.6 - - :

7.5

i f1 f2 f3 i f1 f2 f3 i f1 f2 f3 2 0010 * * * 2,3 001x * * * 2,3,6,7 0x1x * * 4 0100 * * 2,6 0x10 * * 2,3,10,11 x01x *

----------------------- 2,10 x010 * 4,5,6,7 01xx * * 3 0011 * * * 4,5 010x * * 4,5,12,13 x10x * 5 0101 * * 4,6 01x0 * * 4,6,12,14 x1x0 * 6 0110 * * 4,12 x100 * * -------------------------------- 9 1001 * * ------------------------ 5,7,13,15 x1x1 * 10 1010 * 3,7 0x11 * * 6,7,14,15 x11x * 12 1100 * * * 3,11 x011 * * 9,11,13,15 1xx1 *

---------------------- 5,7 01x1 * * 12,13,14,15 11xx * * 7 0111 * * 5,13 x101 * 11 1101 * * 6,7 011x * * 13 1101 * * 6,14 x110 * 14 1110 * * 9,11 10x1 * *

----------------------- 9,13 1x01 *

:

2,3,10,11 6,7,14,15

15 1111 * * 10,11 101x *

116655

i B f1 f2 f3 4,5,6,7,12,13,14,15 x1xx *

F1(1) = {2, 3, 9, 11, 12, 13, 14, 15};

F2(1) = {2, 3, 4, 5, 6, 7, 12, 13, 14, 15} ;

F3(1) = {2, 3, 4, 5, 6, 7, 9, 10, 11, 12} .

( 7.6) - , :

- f1: x1x2 1x 2x x3,

- f3: 1x x2, 2x x3 x1 2x x4 .

7.6

f1 f2 f3

2 3

9

11

12

13

14

15

2

3

4

5

6

7

12

13

14

15

2

3

4

5

6

7

9

10

11

12

1100 * * * 001x * * * * * * x100 * * * * x011 * * * * * 10x1 * * * * 0x1x * * * * * * * * x01x * * * * 01xx * * * * * * * * 1xx1 * * * * 11xx * * * * * * * * x1xx * * * * * * * *

x2 3x 4x f3

12. - :

f1(x1,x2,x3,x4) = x1x2 1x 2x x3 x1 2x x4, f2(x1,x2,x3,x4) = x1x2 1x x2 1x 2x x3,

116666

f3(x1,x2,x3,x4) = 1x x2 2x x3 x1 2x x4 x2 3x 4x .

- :

f1(x1,x2,x3,x4) = x1 x2 1x 2x x3 x1 x4, f2(x1,x2,x3,x4) = x2 1x x3, f3(x1,x2,x3,x4) = 1x x2 2x x3 x2 3x x4 x1 2x x4.

. 7.7 , . - , . - , 15 9 , 20 11 .

xx12

xx23

xx12

xx12

xx14

xx12

xx12xx

12

xxx

123

xxx

234

xxx

124

xxx

123

xxx

123

xxx

124

V

V

V

V

V

V

V

V

V

f1

f1

f3

f3

f2 f2

V

V

V

V

V

V

x2

V V

V

V

V

7.7

)

)

116677

77..77

. , - 0. - 1 0 . . 7.8 - .

- - . , f

f.

- 0. x x = 0 -: 0 1 . . . 7.9 - .

7.8

x1

x2

x3

f = ( ) (x x )x x x1 2 2 1 3 f = (x ) (x x )1 3 x x3 2 4))

x x1 2

x x3 4

00 01 11 1000011110

00

00

0 0

116688

77..88

n . , . -. - . - -. :

f(x1,x2,,xn) - : f(0,0,...,0) = 0. - , 00...0, 2n-1 {0,1}, 12

n2 .

f(x1,x2,,xn) - : f(1,1,...,1) = 1. , 11...1, 2n-1 {0,1}, 12

n2 .

7.9

f = xx x3 2 1 f = x x x x x x 1 3 2 4 1 2 3x x1

x2

x3x x1 2

x x3 4

00 01 11 1000011110

111

11

1

116699

f(x1,x2,,xn) fd(x1,x2,,xn) : fd(x1,x2,,xn) = .)x,...,x,x(f n21

, . :

f(x1,x2,,xn) = .)x,...,x,x(f n21

- . () .

-, . , , . - -: - . - n . , 2n-1, -, .2

1n2

yy

{0,1}n. A=(a1,a2, ...,an) B=(b1,b2,..., bn) A B - i bi i=1,2,...,n. A B . A B - .

:

A B f(A) f(B).

117700

- :

f(x1,x2,,xn) = a0 a1x1 a2x2 anxn, aj {0,1}, j = 1,2,,n . j,

n+1, n 2n+1.

.

77..88

. : , .

. f1, f2, ..., fm .

. . - , . - , , . - , , 1.

- . - () - . - , , ,

117711

-, . - . . - .

.

1) (E. Post):

. , -, , - .

6.5 K0, K1, S, M L .

:

: - .

. f1, f2, ..., fm . f* - , f*(0,0...,0) = 1. :

) f*(1,1,...,1) = 1 f* -,

) f*(1,1,...,1) = 0 f* 1. f* , , .

. -. 0 1. 0 1 ,

117722

, : .

: , - . - , . .

2) . f1, f2, ..., fm - . g1, g2, ..., gk. g1, g2, ..., gk f1, f2, ..., fm , . :

f1 = F1(g1,g2,...,gk)

f2 = F2(g1,g2,...,gk)

. . . . . . . . . . . . . .

fm = Fm(g1,g2,...,gk)

.

x = x / x ; x = x x ;

x1 x2 = (x1/x2)/(x1/x2) ; x1 x2 = (x1 x2) (x1 x2) ,

, . .

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Osnovi Racunarske Tehnike - 7