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() - , 0 1 , - 0 1.
, , ( ). n () f(x1,,xn).
1. - f(x1,,xn), n , 2n ( , ).
: , .
1. n=1. n=1 , , , 0 1. , 21=2 .
2. , n=k.
3. n=k+1. k+1 . f(x1,,xk+1). : f(x1,,xk,0) f(x1,,xk,1). , , , k . 2k . , f(x1,,xk+1) 2k+2k=2*2k=2k+1 . , n=k+1 , , . , - n.
( ( )
. 1, - 22=4 : 00, 01, 10, 11, 23=8: 000, 001, 010,011, 100, 101, 110, 111.
( ( )
. . : ( ). , - , . .
n- (n=1, 2, 3) ( ) . - . (=((1, ..., (n) (1*20++(n*2n-1, , (0,1,0,1,1) 1*20+1*21+0*22+1*23+0*24=11. , () (). () n- , . , ( ), - ( ). , , , , . , , - .
, .
2. n .
1 ( ).( . 2 =4 . :
xf0(x)f1(x)f2(x)f3(x)
00011
10101
, f0 (x)=0, f3(x)=1, f1(x)=x. f2(x). ( ) x .
f0(,)f1f2f3f4f5f6f7f8f9f10f11f12f13f14f15
000000000011111111
010000111100001111
100011001100110011
110101010101010101
2 =16. :
, , , , . , 1310=11012 f13(x,y), : f13(0,0)=1, f13(0,1)=1, f13(1,0)=0, f13(1,1)=1.
. , : .
1) f0(x,y)=0, f15(x,y)=1. .
2) f1(x,y) : f1(x,y)=x*y=x(y=x&y=xy. f1(x,y) f14(x,y), f14(x,y)=.
3) f7(x,y) (. f7 f8(x,y), ( ) f8(x,y)=x(y=
4) f13(x,y) f13(x,y) = x(y. f12(x,y) f2(x,y) = = .
5) f11(x,y) f11(x,y)=, - f4(x,y)= =
6) f9(x,y) (. - f6(x,y), 2, f6(x,y)=(.
7) f3(x,y)=x, f12(x,y)=
8) f5(x,y)=y, f10(x,y)=
, . , = 28 =256 . 5 . .
. , , - ( - ). , , . , , , .
, . , . .
() , {(, (,( }.
() , 2 {(, (}. , 0 1 , (, ). : , , ; , 2.
( ).
, - , , .
, , .
. , , , , :
1)=x ;
2) x(y=y(x - ';
3) x*y=y*x - ;
4) x((y(z)=(x(y)(z=x(y(z - (;
5) x*(y*z)=(x*y)*z=x*y*z - *;
6) x*(y(z)=x*y(x* z - ;
7) x(y*z =(x(y)(x(z) - ;
8) = *- (;9) = - *;10) = 1, =0
11) x*1 = x, x*0 = 0
12) x(0=x, x(1=1
10-12 ;
13) x = 1
14) x = 0
, . : . , , , . , , , , , -, , , , -, - , .
:
15) A(A*B=A - ';
16) A*(A(B)=A - .
15 :
A(A*B=11A*1(A*B =6A*(1*B)=2A*(B(1)=12A*1=11A
= , .
( 16:
A*(A(B)=12(A(0)*(A(B)=7A(0*B =3A(B*0=11A(0=12A
(, 15 16 - B, 15 B=1, 16 - B=0. , :
17) A(A=A
18) A*A=A
:
19) A(*B=A(B
: A(*B =7(A()*(A(B) =1(A(B)=(A(B)*1=A(B
(
- , , 11, 12, 17, 18 :
1. , 0, .
2. ' 1, 1.
3. , 1, .
4. 0, 0.
5. - , .
6. (8, 9) :
20) = 1 *2 * ... m
21) = 1
EMBED Equation.3 2 m
7. , , , - .
.
22) x(y=y(x, x*y=y*x - 2 ;
23) x((y(z)=(x(y)(z, x*(y*z)=(x*y)*z - 2 ;24) x*(y(z)=x*y(x*z - 2;25) x (0=x; x*0=0; x*1=x - ;26) x (x=0 - 227) x*x= x - .
22-27 .
, . , {, &, (, (}.
, :
29) x(y=x*y(x*y
30) x(y=*y(x*: :
29=25=28=28=24
: .
(x(1)y((x(1)=28=30*(=9,1(v)(yx=1,3,14(x()(0=12(x()=3,6x(=14,12.
, , ( x(1):
(x(1)y( (x(1)=(x(1)(y(1)=
EMBED Equation.3 .
(
8.- =(1xn) , , , 1xn . . 8 . - .
. ' () - , ' () . , ' () , ( ), 0 (1).
: ' : 0, , , ,( , ' : 1, x, , xy, ,. x1(x2(x1, a(b(, , y*y, - ', .
( ' (), , ( ' ).
() ' .
: f(x1, x2, x3) = x1 2 (x2x3 ( x3
( () '.
: f(x1, x2, x3)=(x1(2)((2(3)3
( 3. (), - .
. . , .
1. , , 2.
2. .
3. .
4. 0 ( x(0 =x) ( x(x=x).
:
1) , , 2
2) .
3) .
4) 1 ( x*1=x) ( x*x=x) .
(: :
f(x,y,z,w)=(xy(z)=(xy(z)(()=(xy(z)(x()=xy(xz(xy(z=
= xy(xz(z
:
f(x,y,z,w)=(xy(z)=(xy(z)(()=(xy(z)(()=(xy()(xy(z)(x()= =((y)((x)(z(x)(z(y)(x()=(x()(x(z)(y(z)(x().
( , ={1, , n} . , .
() ' (), .
={x1, , xn} 0 1 ( 2 ( (n, 1 1 ( 2 ( (n, , , =1, 2, ..., n. 4 ( 1). - 1 = 1 ( 2 ( (n , (=((1,,(n) , , 1. , (4), = 1, i= , 0, =. ( (, ( .
(. M={x1, x2, x3}. 1 , , :
x1*2 *x3 ; 1*x2*x3 ; x1*x2*3. , 1 1 (1,0,1), (0,1,1), - (1,1,0).
( ( K, 1, .
5 ( 0). - 0 R= 1 ( 2 ( (n (=((1,,(n) x1, , xn , 0.
, (2), (i i = , 0, = , 1, = .
(. M={x1, x2, x3}. 0 M: 1(x2(3 x1(2(3. 0 : 1(x2(3 (1,0,1), x1(2(3 (0,1,1).
( () , , , 1 .
. M={ x1, x2, x3} x12x3(1x2x3(x1x23 , x12x3 x23 1x2 x3 , .
() , , , 0.
. M={x1, x2, x3}. (x1(2(3)( 1(x2 (3)( 1(2(x3). , (x1
2)(x2
3)( 1(2(x3) , .
( () () f=f(x1xn) -, () ( ) , -, , (), .
f=f (x1,,xn) . fi i- , i 1, Ri - 0, , =. : x1(x2((xn=x; x1*x2**xn=xi. 6 ( ). , :
;
.
). , , ) . , . ) :
, .
, ) - , , ) .
b) ( ).
( .
, - .
.
EMBED Equation.3 ( 1. - :
1) f (x1, , xn) = fi Ki; 2) f (x1, , xn) = (fi (Ri).
, 1) 2) ) b) .
( 2. , () .
, , : =fiKi =Kj. , , , .
(, , , . , - .
:
7. - f(x1,,xn) .
. .
1) 2 . 2) , , ., f(x1,,xn) () . , , . , . , ( 1, ) 1, - 0. ( ), .
. ( 8. , - .
. :
1. () (. 3).
2. , (), (), 1 (0):
1) p ( q), (), i i=, .
2) () p ( g) 11 13 (12 14) : p=(xi(i)=pxi(p (g=g(xi= =(g(xi)(g(i))
(. f(x, y, z) = (x ( z) . ) : f(x, y, z)= (x ( z)=()(x ( z) = ( ( )(x ( z) =
=(
EMBED Equation.3 (
EMBED Equation.3 )(x ( z) (
EMBED Equation.3
EMBED Equation.3 ( Z) =
EMBED Equation.3
EMBED Equation.3 ( (y ( )z =
EMBED Equation.3
EMBED Equation.3 ( y z (
EMBED Equation.3 Z) :
f(x, y, z)= (x ( z)=()(x ( z) = ( ( )(x ( z) ==( ( y)(X ( Z)(X ( y ( Z) = =(( y )( ( )(X ( ( Z )( ( ( Z)(X ( y ( Z )(X v v Z).
( , .
() , , :
1. , 1 (0).
2. 1 (0).
3. 1 (0) ().
. :
yzF(, y, z)
0000
0010
0101
0111
1001
1010
1101
1111
1) :
f(x,y,z)= yz x xy xyz
2) :
f(x,y,z)=(xyz)(xy
)(
y
)
( . , ().
f(x1, . . . ., xn) g(x1,,xn), - ( , g 1, , f 1, , , g f, g(()=1 ( f(()=1.
. , , , , / , .
, , -, , , -, .
, ( f(()=1. , g(x1,,xn), g(()=1
(f) - 1 (f) i .
, , -, , , -, .
, , .
, , .
() , , , ( ) ( ) .
. f (x,y,z) = xy xz yz . ( (). f , , . . f f=(Q , , Q .
, f. r, , f. , r=1 , 1 . , , r Q f f=r(Q. r(Q f, , , (Q.
, , . , , , , - f . , . ( , :
1) , , ;
2) ;
3) .
.
() . , f, , ( 1), 1. 1 (, , - (- *. ( 0, - (- .
1 : - - , (, , ).
. : f=y ( x( xz (yz.
, y, x, xz, yz 1 5- : (0,1,0); (0,1,1); (1,0,0); (1,0,1); (1,1,1). 1 y; yz; x; xz; xyz.
, f :
y
yzx
EMBED Equation.3 xzxyz
y**
x
**
xz**
yz**
. , , *. , - , * , * . , , .
, - - . y x ( y, x
EMBED Equation.3 ). , , . , . , , , * , . : y, yz, x
EMBED Equation.3 , xz.
, , , . , - xyz, xz, yz. : {y, x, xz} {y, x, yz}. : f1=y(x(xz f2=y(x(yz .
(, . , , . , , , .
. , f(x1xn) , (). ( f) :y
yzx
EMBED Equation.3 xzyz
A y**
B x
**
C xz**
D yz**
g =A1((Ai , , . g , . : =A(A(D)B(B(C)(C(D)
, , (). , . , . :
=A(A(D)B(B(C)=(A(AD)(B(BC)(C(D)=AB(C(D)=ABC(ABD.
, , , : ABC ={y, x, xz} ABD={y, x, yz}, : f1=y ( x( xz f2=y ( x( yz. f1 f2 .
( . : Ax(A=A ( , , Ax(A=A (x()=A*1=A)
( Ax A , , Ax A x.
: Ax(A=A(Ax(A
.
f(x1xn), , . ( f0) f0, f1, f2, ..., f , f f+1 . f f+1 (=) : f , n-i, n , , , ; . .
, , , n .
. f, : .
, , 1. :
, , , - . . , ..
( . . , :
(B=AC(B(AB. :
(B= AC((B(AB = AC(B(AB(() =AC(B(AB ( : f , - . f.
. , : f(x,y,z)=x(yz(z. , f , y, ' , ' . . , : f1= x(yz(z(yz. . , f1. , , f1 . , : f2= x (z(yz.
(. - :f=xy(xz(y=xy(xz(y(xy(y(yz = xz(y(xy(y(yz(y = xz(y. , f xz ( y. ( , , . , , , , .
. , .
, , , , . , , .
( , . . f=x(z(z. , f : x(z= x(z(z. f, : f = x(z.
( . . : , , f.
. : f=(x()((z)(x(y()
, : f=xy(
( . . ( ) , . , , ( ), . 2-, 3-, 4- 5- :
?
( ) . (0,0), (0,0,0), ... 12, 123, ... 0; (0,1), (0,0,1), 12, 123, - 1; (1,0), (0,1,0), 12, 123, - 2 . 1, 1. , , 0, , 0.
. , , .
, 1 ( - ), , , . . , 12 2 ( ),
EMBED Equation.3 3. n=2, 3, 4 , . , . ( , ) , . - , . - , . 2, 4, 8, 16 ( ) . , .
n(5 , y, , .
. f(x,y,z) :
?
?
. f(x1,x2,x3,x4) :
?
. . , , , . , . , , =xy(xz , . , , : =(y(z). , , .
, , , (. ). , . , , . , , , . , , . . , , , , . , . .
f :
1. A, B, C, D :
,
,
,
,
,
2. , . {A, C}, {A, D}, {B, D}, {C, D}. {C, D}.
3. , {C, D}.
4. 1, 1, 1 1 . :
,
,
5. , . {A1, C1}.6. 2 , {A1, C1}:
7. , 2 2 . :
(, 14 {&, (} {}, 9 {&, (} {}.
(
. , , , , , , . , .
. {&, (, }.
, :
,
,
.
, .
.
( ) .
, , .
, .
, .
:
x1x2x3f1f2
00011
00100
01001
01101
10000
10111
11010
11110
:
1. :
2. , 1 ( ) .
3. , 1 :
4. , . , :
) 1, , , , , ( , ). 1 , ;
) .
.
1 :
.
1(5=
2(4=
3(4=
5(6=
, (.
5. , 1 , ; ; * :
) 1 ;
) 1;
) * , ( ) ( 1), .
.
:
12121122
**
**
**
**
**
**
, , 1 . , ( :
: . :
.
( . (), - .
9 ( (, &, ). - {(, &, }, .
. .
1. . 2 . , :
, - (, &, , . , .
2. ,
3. +1:
(*)
, , . - (, &, . , (*) . , , , , -
( 10. :
1) {(, &, };
2) {(, &, };
3) {(, };
4) {&, };
5) {(, };
6) {|};
7) {}.
. . , .
, {(, &, }, 2. (29):
.
:
;
;
;
;
.
( . , .
, f(1,...,xn) 0, f(0,...,0)=0.
, f(1,...,xn) 1, f(1,...,1)=1.
f*(x1,...,xn) f(1,...,n) f , f*=f.
{0,1} , : 0(0, 0(1, 1(1. , - (=((1, ..., (n) (=((1, ..., (n), (, (({0,1} (((,, (n((n(f(()(f(().
f(1,...,xn) , , :
ai({0,1}
, 0, 1, , , , 0, 1, S, M, L.
0, 1, S, M, L .
, .
( ), - .
11. P0, P1, S, M, L .
. . , ( ). 0, 1 , , . , (, ). & (. L ( (.
.
f1, f2, f3(P0, f1(0,0)=f2(0,0)=f3(0,0)=0. (0,0): .
P1.
f1, f2, f3(S, (f1((x1,(x2)=f1(x1,x2), (f2((x1,(x2)= f2(x1,x2), (f3((x1,(x2)= f3(x1,x2) g(x1,x2)=f1(f2(x1,x2), f3(x1,x2)).
g*(x1,x2)=(g1((x1,(x2)=(f1(f2((x1,(x2), f3((x1,(x2)) = f1((f2(x1,(x2),(f3((x1,(x2)))= =f1(f2(x1,x2), f3(x1,x2))= g(x1,x2), g(S.
f1, f2, f3(, 1(b1, a2(b2 g(x1,x2)=f1(f2(x1,x2), f3(x1,x2)). g(a1,a2)= f1(f2(a1,a2), f3(a1,a2))( f1(f2(b1,b2), f3(b1,b2))=g (b1,b2). , g(.
f1, f2, f3(L , , f1(x1,x2)=0(11(22, f2(x1,x2)=b0(b11(b22, f3(x1,x2)=c0(c11(c22. g(x1,x2)= f1(f2(x1,x2), f3(x1,x2))=a0(a1f2(x1,x2)(a2f3(x1,x2)= =a0(a1(b0(b11(b22)(a2(c0(c11(c22)==(0(11(22)((a1b1(a2c2)x1((a1b2(a2c2)x2= d0(d11(d22, g(L
(, , , , f(x)=x.
( ( ). F={f1, f2, , fk,} , :
1) , 0;
2) , 1;
3) ;
4) ;
5) .
. F . , 0. , , 0 , , . , F , . , F , 0.
, 1, S, M, L.
. , F f0, f1, f2, f3, f4, f0(0, f1(1, f2(S, f3(M, f4(L. , 1, ..., n ( ) - f(1, ..., n) . f0, f1, f2, f3, f4, , , , {(, } {&, }.
, 1, 2, 3 x1,,xn 0 1.
f0(1, ..., n) g(x1)=f0(x1, , x1) f0 1.
g(0)=1. .
) g(1)=1. g(x1)(1 , f1, 0.
) g(1)=0. g(x1)=(x1.
) f2(S. f2 , (=((1,,(k) (=(((1,,((k) , f2(()=f2((). h(x1) f2 ((x1) , : h(x1)= f2((1(x1),, ((x1)), ((x1)=1, (=0 ((x1)=(1, (=1. :
h(0)=f2((1(0), , ((0))=f2((1,,(k)=f2(((1,,((k)=f2((1(1), , ((1))=h(1),
h(0)=h(1) , , h(x1)(0 h(x1)(1. g(x1)=(x1 .
, f3(x1,,xr)(, . f4 (=((1,,(r) (=((1,, (r) , (((, f3(()>f3((). f3 ( ((((((), ( (k+1 , i- :
(k=((1, , (i-1, 0, (i+1, , (r), (k+1=((1, , (i-1, 1, (i+1, , (r), (((kf3((+1). f3(x1,,xr) j (j (j=1, 2, , i-1, i+1, , r) : q(xi)=f3(((1, , (i-1, xi, (i+1, , (r). : q(0)=1 q(1)=0. q(xi)=(xi, =1, 2, ..., n.
1&2 x1(x2. f4(1,,). . ' , - i xj. , , xi=x1, xj=x2.
f4 :
f4(1,,)=121(3,,)(12(3,,)(23(3,,)(4(3,,).
1 ( 0. 3,, , 1 1, : ((1, 2)=12((1((2((, (, (, ( - 0 1.
:
((1+((b,2+(()=(1(()(2(()(((1(()(((2(()((=12((, (=((((.
( 12, (= 0 =(1((x2, (=1.
, f0(0, f1(1, f2(S, f3(M, f4(L, 1-5 , {(, } {&,}.
( 1. f(x1,,xn) 0 1 (xi f(x1,,xn).
. .
, 0 1. 2 ( ). , , .
. 0 1 , 0 ( 1), , 1 ( 0), ( 0 1). , , .( . , - .
- , . , - . , f, 0, 1, ( f(1,...,1)=1) . , , : , , 1 ( f(1,...,1)=1), ( f(1,...,1)=0).
, . , , : f1=0, f2=1, f3=x1x2, f3=x1(x2(x3.
, f1 , 1, f2 - , 0, f3 - , f4 - ( f1, f2 f3). ,
3. .
_1195890189.unknown
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