() - , 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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