osnovi racunarske tehnike - 6
DESCRIPTION
OsnoveTRANSCRIPT
-
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).