Osnovi Racunarske Tehnike - 6

112211

66

66..11

f: S L S L . - f(x), x - ( ) S ( , - , ), f(x) L ( -, , ).

, , - .

S - n Si, .

S = S1 S2 Sn = in

1iSX= .

n

,LSX:f in

1i=

f(X), X = (x1,x2,...,xn), f(x1,x2,...,xn), xi, (i = 1, 2,..., n), Si.

112222

,LSX:f min

1i=

m- L. . - .

Si L - , :

,}1m,...,1,0{}1m,...,1,0{X:f in

1i=

mi, (i = 1,2,...,n) n Si L, .

- Si 0, 1,..., mi-1}, i = 1, 2,...,n , L - 0, 1, 2,...,m-1}. , .

,}1,0{}1m,...,1,0{X:f in

1i=

.

Zm 0,1,...,m-1}. - f: 0,1}n Zm. . f: 0,1}n R , R . , , . - - .

, Si L (). ,

112233

,}1m,...,1,0{}1m,...,1,0{:f n

m E = 0,1,...,m-1}.

( -) E . -

,}1,0{}1,0{:f n

B = 0,1}.

, , , . , .

(. Galois function) L Si GF(pk). , f: GF(pk)}n GF(pk).

() (. fuzzy function) - :

f: fn I , I {0,1}.

- , , .

, : - 0,1}n - - . - - .

, - -. ,

112244

, .

. - .

66..22

- . , . , .

() - .

- "=" ( -): (, , - "" "+"), (, , "" "", ) (, , ).

- :

112255

a = a ();

a = b b = ();

a = b b = c a = c ().

, :

1) x y = y x; x y = y x ( ), 2) (x y) = x (y ); (x y) z = x (y z) ( ), 3) x (y ) = (x y) (x z); x (y ) = (x y) (x z) (

),

4) 0 1 - :

0 x = x; 1 x = x ( ).

0 1 0 1. , , . , .

5) :

x x = 1; x x = 0 ( )

, . . 2 , . , - . , .

- , - : , . x = y x, x y, .

112266

- ( : , , ):

) xx = ( , ); ) x x = x; x x = x ( ); ) 1 x = 1; 0 x = 0 ( ); ) x y x y = x; (x y) (x y ) = x ( ); ) x x y = x; x (x y) = x ( ); ) yx = x y ; yx = x y ( ).

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

- B - 0 1. - , 6.1 .

- . .

112277

6.1 , x y x y x y x y x x 0 0 0 0 0 0 0 1

0 1 1 0 1 0 1 0

1 0 1 1 0 0

1 1 1

1 1 1

n - :

f: {0,1}n {0,1} , f: Bn B , B {0,1} .

f(x1,x2,...,xn) f(X), X = (x1,x2,...,xn), xi B, (i = 1,2,...,n ) f(X) B. x1, x2, ..., xn, B.

n- K = (k1,k2,...,kn) (x1,x2,...,xn) B . k1k2...kn, k1, k2, ..., kn - . - , n 2n.

- 0, 1 x1,x2,...,xn - , . - :

1) 0, 1, x1, x2 ,..., xn ,

2) E1 E2

1 2, 1 2 , 21 E,E ,

3) .

112288

- . . , - , - .

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

66..33

, - - , . . - .

2n () . , . , . 6.2 - -.

112299

6.2 x1x2 ... xn-1xn f(x1, x2, ... , xn-1, xn) x1 x2 x3 f(x1, x2, x3)

00...00 f(00...00) 0 0 0 0

00...01 f(00...01) 0 0 1 1

00...10 f(00...10) 0 1 0 1

... ... 0 1 1 0

... ... 1 0 0 0

... ... 1 0 1 1

11...10 f(11...10) 1 1 0 0

11...11 f(11...11) 1 1 1 0

)

)

:

. Fn n - .2F

n2n =

. - 2n . - Fn = 2 x 2 x ... x 2 (2n ) =

n22 .

- . .

. 6.3.

6.3

x y x y x (x y) x x y x y x (x y) x 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1

113300

, k, - () K = k1k2...kn . k . :

==

n1i

ini .2kk

f F(1) - 1 F(0) - 0. . 6.2 1 - F(1) = 1,2,5}.

, . - , , - .

- , .

(())

n- - 2n n- - (), . 0 1 , . 0 1 , - . . 6.1 .

113311

() - . 2n - () . 6.2 . - - , . . , . 5 - . , i- (i=1,2,3,4) 001 101, 011 111.

- 0 1 .

6.1

000 100

111

001

011

101

110010x1

x2

x3

0000 1000

1110

0010

0110

1010

11000100x1

x2

x3

0001 1001

1111

0011

0111

1011

11010101

x4

6.2

x x1 2

x x3 4

00 01 11 1000011110

11

1

11

x x1 2 3x

1

111

111

1

x x4 5

00011110

000 001 011 010 110 111 101 100

113322

66..44

6.4 6.5 . , . . - . ..

6.4

x 01 K0 K1 S M L

f0 00 0 0 + + + f1 01 x x + + + + + f2 10 x x + + f3 11 1 1 + + +

- , - .

,, ,,

- , , :

x 1 = 1; x 0 = x; x x = 1; x x = x; x x ... x = x ; x 1 = x; x 0 = 0; x x = 0; x x = x; x x ... x = x.

:

)x...xx( = x x ... x , )x...xx( = x x ... x .

113333

, , .

x1 x2 = x2 x1; x1 (x2 x3) = (x1 x2) x3; x1 (x2 x3) = x1 x2 x1 x3 .

6.5

x1 x2

1100 1010

K0

K1

S

M

L

f0 0000 0 0 + + + f1 0001 x1 x2 , + + + f2 0010 21 xx + f3 0011 x1 x1 + + + + + f4 0100 12 xx + f5 0101 x2 x2 + + + + + f6 0110 x1 x2 + + f7 0111 x1 x2 , + + + f8 1000 x1 x2 () f9 1001 x1 x2 + +

f10 1010 2x x2 + +

f11 1011 x2 x1 + f12 1100 1x x1 + +

f13 1101 x1 x2 + f14 1110 x1 / x2 () f15 1111 1 0 + + +

, :

x x = 0; x x = 1; x 0 = x; x 1 = x .

113344

. :

x x = 0; x x = x ; 0 x = 1; x 0 = x ;

1 x = x; x 1 = 1.

,

x1 / x2 = x2 / x1; x1 x2 = x2 x1;

x1 / (x2 / x3) (x1 / x2) / x3 ; x1 (x2 x3) (x1 x2) x3 .

:

x / x = x ; x / x = 1 ; x / 0 = 1 ; x / 1 = x ;

x x = x ; x x = 0 ; x 0 = x ; x 1 = 0 .

- . :

x1 x2 = 21 xx ; x1 x2 = 21 xx ; x1 x2 = x1 x2 x1x2 ; x1 x2 = 2121 xxxx ; x1 x2 = 1x x2 ; x2 x1 = 1x 2x ; x1 / x2 = 21 xx x1 x2 = 21 xx .

113355

66..55

f(x1,...,xi ,...,xn) xi, xi f(x1,...,xi ,...,xn) :

f(x1,...,0,...,xn) = f(x1,...,1,...,xn)

xi , f(x1,...,xi ,...,xn) xi.

f(x1,...,xi ,...,xn) - .

f(x1,...,xi ,...,xn) f(x1,...,xi ,...,xn) = f(x1,..., ix ,...,xn).

, - , -. xi n-1 :

Di f(x1,...,xn) = f(x1,...,0,...xn) f(x1,...,1,...,xn).

f(x1,...,1,...,xn) Di f(x1,...,xn) = 0, - 0.

, , , .

Fn n , - . n - :

: An n :

,AA1n

...A2n

nA

1nn

2A 012n1n2

nn

=

113366

0 , - 0 1.

n :

=

=n

0j

2jn .2j

n)1(A

1n

2 , 10 , 218 . 6.4 6.5 - .

66..66

, , . , - , , .

K = k1k2...kn n - k - 6.3 (.101). pk :

.)k,...,k,k(K,0,)k,...,k,kK,1

pn21

n2k

==

( 1

pk . n 1 , K

113377

k, 0 . ( ) :

.0k,x1k,x~,x~...x~x~p

ii

iin21k

==== ix

, K n21 x~...x~x~

1 1. 0 pk 0. pk : xi pk - ki = 0, ki = 1.

sk :

.)k,...,k,k(K,1

,)k,...,k,kK,0s

n21

n2k

==

( 1

sk . n - 0 , K - k, 1 . - ( ) :

.1k,x0k,x~,x~...x~x~s

ii

iin21k

==== ix

sk : - xi sk ki = 1, ki = 0.

2n 2n.

. 6.6 - .

113388

6.6

x1 x2 x3 f(X) p1 p4 p6 s0 s2 s3 s5 s7 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0

p1 = 1x 2x x3 p4 = x1 2x 3x p6 = x1x2 3x

s0 = x1 x2 x3 s2 = x1 2x x3 s3 = x1 2x 3x s5 = 1x x2 3x s7 = 1x 2x 3x

. - , .

. n , - 0, :

f(x1, x2, , xn) = k1)K(fp= ,

k , pk K 1, . , , ().

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

113399

0. - . ( 6.7).

. n , - 1, :

f(x1, x2, , xn) = k0)K(fs= ,

k , sk K 0, . , , ().

. . (. 112).

. n , - 1, :

f(x1, x2, , xn) = k1)K(f

p=

,

k , pk K 1, . , , ().

.

. n :

f(x1,x2,..,xn) = c0 c1x1 cnxn c12x1x2 c1nx1xn cn-1,nxn-1xn c123x1x2x3 c12nx1x2xn

c0, c1, ..., c12...n {0,1}, 2n .

- : ix = xi 1 , , x x = 0.

114400

ci1i2...im ( 6.7). , . ( 6.7).

- , . - (Reed-Muller). , . -, . .

. 6.6 (. 107) , , :

: f(x1,x2,x3) = 1x 2x x3 x1 2x 3x x1x2 3x , : f(x1,x2,x3) = (x1 x2 x3) (x1 2x x3) (x1 2x 3x ) ( 1x x2

3x ) ( 1x 2x 3x ) : f(x1,x2,x3) = 1x 2x x3 x1 2x 3x x1x2 3x , : f(x1,x2,x3) = x1 x3 x2x3 x1x2x3

. . ,nm1,x~...x~x~ im2i1i , ,nm1,x~...x~x~ im2i1i . . . m = 1 , m = n . r = n - m .

( ), () .

114411

(). - .

, . . . , - .

p xi p. p = p (xi ix ) = pxi p ix - xi. , - p, p. - p 1 - xi ix = 1. 2r=2n-m .

: p = p(xi ix ) = pxi p ix , . p.

sk = n21 x~...x~x~ , - xi : s = s xi ix = (s xi)(s ix ) .

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

114422

- , . .

u1 u2 = u1 u2 u1u2 , u1 u2 .

u1 u2 = 1u u2 u1 2u , u1 u2 .

x = x 1, - .

, .

(())

() :

1) ( , - -),

2) () .

F(X) f1(X1), f2(X2),..., fm(Xm), Xj X, (j = 1,2,...,m), (-, , ) . :

114433

F(X) = f(fi1(Xi1),,Fim(Xim)), Xij X .

66..77

, - , . . - , .

xi n n-1 xi , . .

n - :

f(x1,...,xi,...xn) = ix f(x1,...,0,...xn) xi f(x1,...,1,...xn)

(C. Sannon), . - , xi, . xi =0 xi =1. xi =0 :

f(x1,...,xi,...xn) = 0 f(x1,...,0,...xn) 0 f(x1,...,1,...xn) : f(x1,...,0,...xn) = f(x1,...,0,...xn).

xi =1.

n - - (). .

-:

114444

- :

f(x1,...,xi,...xn) = [xi f(x1,...,0,...xn)] [ ix f(x1,...,1,...xn)] .

. n () .

- :

f(x1,...,xi,...xn) = ix f(x1,...,0,...xn) xi f(x1,...,1,...xn).

. n () .

n - :

f(x1,...,xi,...xn) = f(x1,...,0,...xn) xi [f(x1,...,0,...xn) f(x1,...,1,...xn)] .

-. n n .

- :

f(x1,...,xi,...xn) = f(x1,...,0,...xn) xi Di f(x1,...xn) .

Dif(x1,...,xn) xi. ( ) ( 6.6) :

ci1i2im = Di1i2im f(x1,...,xn),

Di1i2im f(x1,...,xn) f xi1, xi2,...,xim. c0 = f(0,0,...,0).

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Osnovi Racunarske Tehnike - 6