. , . : , . .

: , [5.1, .30-73], [5.2].

. . .

. :

L1=R1,

. . . (5.1)

Ln=Rn.

Li Ri, i=1, ... n, , , , , ( ), . . - ( ), . , .

, , - , , ( ). . , :

(L1, R1),

. . . (5.2)

(Ln, Rn),

, . , , , , , . .

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

5.2.

(. . 5.1).


/

( )


x1

a[1,1]

a[1,2]

...

a[1,m]

*

x2

a[2,1]

a[2,2]

...

a[2,m]

*

. . .

. . .


xn

a[n,1]

a[n,2]

...

a[n,m]

*

s1

u[1,1]

u[1,2]

...

u[1,m]

u[1,m+1]

s2

u[2,1]

u[2,2]

...

u[2,m]

u[1,m+1]

. . .

. . .

sk

u[k,1]

u[k,2]

...

u[k,m]

u[k,m+1]



. 5.1. .

, (). . , . (), . , , .. ( *), , , (m+1)-, . , .

, , . (k) , - , () . () (.. u[i, j], i=1, ... m+1, j=1, ... k) , ( u[i, j]= '+') ( u[i, j]= '- '). , , , . , , . , , , .

. (T - , T - , T - ). . 0. ( ) , . . 5.2 , . (*) .



Ƹ

Ƹ

T=T

*

*

*

*

*

T=T

*

*

*

*

*

*

T>T

*

*

*

*

*

 

 

 

 

 

*

 

*

 

*

 

*

 

 

 


-

-

-

-

-

-

+

-

-

+

+

-

-

-

-

-

-

-

-

-

+

-

-

-

T:=0

-

+

+

-

+

-

+

-

T:=T+1

+

-

-

+

-

+

-

+

-

-

 

-

 

 

-

 

 

-

 

 

+

 

 

-

 

 

-

 

 

-

 

 

-

+

+

+

-

-

-

-

-

-

-

-

-

-

+

+

+



. 5.2. " ".

5.3.

(5.1):

f1(x1, x2, ... , xk)= E1,

. . . . . . . . . . . . . (5.3)

fn(x1, x2, ... , xk)= En,

, ( )

x1, x2, ..., xk,

, , , , .. . . , xj fi (5.3) yj, cij,

y1, y2, , yk

, .

.

s

E

T

() T E s E s T.

fi(x1, x2, ... , xk)= Ei

x1, x2, ... , xk

fi(T1, T2, ... , Tk) Ei ,

T1, T2, ... , Tk

T1, T2, ... , Tk - ( ) . - .

(5.3) . ()

d1, d2, ... , dk.

, , . .

( ) .

  1. - , ( , ).
  2. , - . ( ) .

(2.1) ,

(2.1.1) , , ,

(2.1.2) , .

(2.2) , , .

fi(y1, y2, , yk),

, y1, y2, , yk ( , ).

, .

F(n)=n!. :

F(0)=1,

F(n)=F(n-1)*n.

F(3) .

1- :

F(0)=1, F(3)=F(2) *3.

2- :

F(0)=1, F(3)=F(2) *3, F(2)=F(1) *2.

3- :

F(0)=1, F(3)=F(2) *3, F(2)=F(1) *2, F(1)=F(0) *1.

4- :

F(0)=1, F(3)=F(2) *3, F(2)=F(1)*2, F(1)=1.

5- :

F(0)=1, F(3)=F(2) *3, F(2)=2, F(1)=1.

6- :

F(0)=1, F(3)=3, F(2)=2, F(1)=1.

F(3) 6- .

5.4.

(5.3), , . ( , ) . . [5.3].

, (5.3) :

X1= phi[1,1] U phi[1,2] U ... U phi[1,k1],

X2= phi[2,1] U phi[2,2] U ... U phi[2,k2],

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.4)

Xn= phi[n,1] U phi[n,2] U ... U phi[n,kn],

i- ki=0

Xi=.

- . , - ( ). A={a1, a2, ... , am} , , (5.4) . X1, X2, ... , Xn , , ( ). phi[i,j], i=1,...,n, j=1,...,kj, :

phi[i,j] (A U {X1, X2, ... , Xn})* .

phi[i,j] , , . . X1, X2, ... , Xn ,

phi= Z1 Z2 ... Zk , Zi (A U {X1, X2, ... , Xn}) i=1, ... , k,

Z1, Z2, ... , Zk , aj {aj}. , phi

{p1 p2 ... pk | pj Zj, j=1, ... , k},

p1 p2 ... pk p1, p2, ... , pk ( ). , (5.4) .

(5.4)

(L1, L2, ... , Ln),

(5.4) X1= L1, X2= L2, ... , Xn= Ln.

(5.4),

X= a X U b X U X U c

A={a, b, c}.

L={ phi c | phi {a, b}*}.

(5.4) . L

L1=L U {phi a | phi {a, b}*}

L2=L U {phi b | phi {a, b}*}.

(5.4) .

(L1, L2, ... , Ln)

(5.4) ,

(L1', L2',..., Ln')

L1 L1', L2 L2', ... , Ln Ln'.

( , ) L.

(5.3) (5.4) . (5.4) . (5.4)

Ti(X1, X2, ... , Xn).

(5.4)

X1=T1(X1, X2, ... , Xn),

X2=T2(X1, X2, ... , Xn),

. . . . . . . . . . . (5.5)

Xn=Tn(X1, X2, ... , Xn).

(L1[0], ... , Ln[0]) = (,,...,).

:

(L1[i],...,Ln[i])= (T1(L1[i-1], ... , Ln[i-1]),

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

Tn(L1[i-1], ... , Ln[i-1])).

,

(L1, ... , Ln)

(5.5), ..

(L1, ... , Ln)= (T1(L1, ... , Ln), ... , Tn(L1, ... , Ln))

. T1, T2, ... , Tn.

:

L[0]= ,

L[1]= {c}, L[2]= {c, ac, bc},

L[3]= {c, ac, bc, aac, abc, bac, bbc},

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

L.

-. -

X::= X

X= X

.

5.5.

(5.1) , / .

(5.1) (S, E) - : S - (5.1) f1, f2, ... , fm ( ) c1,c2, ... , cl, E - , (5.1). , xi, i=1, ... , k, ci, i=1, ... , l, E, - t1, t2, ... , tr, fi, i=1, ... , m, ,

ti1 * ti2 * ... * tik ti0.

, ti=ti', i=1, ... , r, ci=ci', i=1, ... , l. , A S,

A=(t1', ... , tr', f1', ... , fm', c1', ... , cl'),

fi', i=1, ... , m, , fi. , (S, E) ( : ), E, .. E fi', i=1, ... , m, ci', i=1, ... , l, fi ci .

, , , , . . [5.4], . .

((m, d))=m,

((m, d))=d,

()=,

()=,

, , - , - , . D, D1 , , m M, d D, M,

D1, :

: M M,

: M * D M,

: M D1.

, , .

, . : D - , , D1=D U {}, M - , M={d1, d2, ... , dn | di D, i=1, ... , n, n 0}, ={}, - ( ), D. .

( ), " ". .

5.6.

, , , , . , , .

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

- , - .

5

5.1.

function F(x, y: integer): integer;

function G(x, y: integer): integer;

function R(x, y: integer): integer;

:

R(x, y) = x*(y 1),

F(x, y) = R(x + 1, y) - R(x, y - 1),

G(x, y) = F(x, R(x, y)).

G(3, 3).

5.2.

function F(n: integer): integer;

function G(n: integer): integer;

:

F(0)=1,

G(0)=2,

F(n)=G(n-1),

G(n)=F(n-1) + G(n-1).

F(3) G(3).

5.3. E T

{'a', '*', '&', '<', '>'}

E= T U '*' T U E '&' T,

T= 'a' U 'a*' U '<' E '>'

'*a&*a*&a*' ,

'*a&<a&a*>',

'*<*a*&a>&<*a*>*'

E E.

5.4. R .

:

type R= record P1, P2, P3: CHAR end;

function READ(S: R): CHAR; {READ: R CHAR}

function SHIFT(S: R): R; {SHIFT: R R}

function ADD(S: R, C: CHAR): R; {ADD: R * CHAR R}

function REMOVE(S: R): R; {REMOVE: R R}

var X, Y, Z: CHAR;

U: R;

:

SHIFT(ADD(ADD(ADD(U, X), Y), Z)) =

ADD(ADD(ADD(U,Y), Z), X);

REMOVE(U) = SHIFT(ADD(U, '#'));

READ(SHIFT(ADD(U, X))) = X;

:

READ(SHIFT(SHIFT(REMOVE(ADD(ADD(U, 'a'), 'b'))))) =

5.

  1. .. . : . - : ( ), 1987.
  2. Ian Sommerville. Software Engineering. - Addison-Wesley Publishing Company, 1992.
  3. . . , / . - .: , 1982. - . 25-53.
  4. . . / . , . , . . . - .: , 1975. - . 98-197.

, - , . , push .




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