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1

,

.

1.1

-

.

. A

aA(a A) a (

) A. A B

A B A B.

B A A, B A = B.

, , ,

A, ,

P(A).

A B A, B A

B,

A B={x |x A x B}

A B ,

A B= {x |x A x B}.

A B=, A B .

A = A A = A.

A, B, C :

- A (B C) =(A B) C,

- A (B C) =(A B) C,

- A (B C) =(A B) (A C),

4

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5

- A (B C) =(A B) (A C).

A\ B B A

A\ B={x |x Ax B}.

A A X X

A=X\A.

A BA B

A B={(a, b)| aA, b B}

. (a, b), (a, b)

A B (a, b) =(a, b) a=a b=b.

n A1, . . . , A n -

n-

A1 . . . An= {(a1, . . . , a n)| a1 A1, . . . , a n An}.

I , .. ,

. i I

Ai, I,

(Ai)iI.

(Ai)i

I

A

-

iIAi=A,

- Ai i I,

- ij = AiAj= i, j I.

A B

f :AB

a A

B f(a). f -

( , 1-1 ) a1, a2 A

a1 a2 = f(a1) f(a2). () bB

a A f(a) = b. f ( )

. A A

B

f(A) ={f(a)| a A}

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6

B B A

f1(B) = {a |aAf(a) B}.

f, g: AB f =g f(a) =g(a)

a A. A B

BA.

IA : A A A

IA(a) =a aA.

f :ABg: BC. aA f(a) B

g(f(a))C. A C

f g g f,

gf :A C

gf(a) =g(f(a)) aA.

f : A B, g : B C h : C D

, h (gf) =(h g) f, f IA=IB f.

1.2

: K K K

() K. k1 k2

((k1, k2)) k1, k2 K.

- k1 k2 =k2 k1 k1, k2 K,

- k1 (k2 k3) =(k1 k2) k3 k1, k2, k3 K.

e K

k e=k=e k kK.

, .

1

.

0 [0,1], ,

, (0,1].

, , : K K K K,

- k1 (k2 k3) =(k1 k2) (k1 k3),

- (k1 k2) k3 =(k1 k3) (k2 k3).

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7

(K, , e) K

e.

(K, , e).

1 :

- (N,+, 0)N ,

- (N, , 1) ,

- (Z,+,0)Z ,

- (Z, ,1) ,

- ([0,1], max,0) max,

,

- ([0,1], min, 1) min, -

,

- (P(A), , A) A,

- (P(A), , ) A,

(AA, , IA) A A

, - .

(K, , e) kn

kK n 0, :

- k0 =e,

- kn+1 =kn k n 0.

, knk=kkn

n 0, .

, (AA, , IA) f(n) fn,

n 0.

(K, , e) (K, , e) ,

h :K K

- h(k1 k2) =h(k1) h(k2) k1, k2 K,

- h(e) =e.

.

, -

, -

.

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8

1 (K, , , e , f ) (K, , e)

, (K, , f) ,

, k e=e k=e kK.

(K, , , e , f )

(K, , f) .

2 :

i) (N,+, ,0,1) ,

ii) (Z,+, , 0,1) ,

iii) ([0,1], max, min,0, 1),

iv) ([0,1], min, max,1, 0),

v) (P(A), , , , A) A,

vi) (P(A), , , A, ) A,

vii) Rmax =(R+ {}, max,+, ,0)R+ -

,

viii) Rmin =(R+ {}, min,+, , 0).

(iii) fuzzy semiring ( -

) (vii) (viii)

max-plusmin-plus. .

(K, , , e , f ) (K, , , e, f) , -

h :K K

- h(k1 k2) =h(k1) h(k2) k1, k2 K,

- h(k1 k2) =h(k1) h(k2) k1, k2 K,

- h(e) =e,

- h(f) =f.

, h

(K, , e) (K, , e), h

(K, , f) (K, , f).

, , -

+, 0

1, (K,+, ,0, 1).

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9

1.3

A B . R A B

A B, R AA R A. (a, b) R, aRb.

R A B S B C,

R S AC

R S={(a, c) A C | bB (a, b) R (b, c) S}.

A, A A= {(a, a)| aA}.

R A

- aRa a A,

- aRb = bRa a, bA,

- aRbbRA = a=b a, bA,

- aRbbRc = aRc a, b, cA.

R A -

, . (a, b) Rab(R).

a A aR

[a]R= {b |b Aab(R)}.

R -

A/R.

.

1 R A, -

A A. ,

A A.

2 A -

A.

2 2

. , A

R, SA, R S .

R A (

) , . -

R ,

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a b aRb. a b b a

a, b A. (A, )

(... ). , (A, )

. , . ... (A, )

A a aA,

(A, ).

3 .

i) (N, ) . (N, )

0.

ii) (Z, ) . (Z, )

.

iii) ([0,1], ) . ([0,1], )

0.

iv) ((0,1], ) .

( ).

v) (RN, ) , -

. f1, f2 RN f1 f2f1(n) f2(n)

n N. (RN, ) .

vi) (P(A), ) A -

. .

(A, ) ... B A, supremum B

sup B A :

- bsup B b B,

- b Abb b B, sup Bb.

sup B B ... (A, ).

supremum ...

([0, 1), ) [0, 1).

(an)n0 ... (A, ),

supremum {an |n 0}( ) supn0 an.

2 ... (A, ) - ,

a0 a1 a2 . . . A supn0 an A.

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11

4 ... -:

- ([0,1], ),

- (N {}, ),

- (P(A), ) A.

(A, ) ... f : AA . a A

f f(a) = a. a A

f fixf aa

a f.

... (A, ) - a0

a1 a2 . . . Afsupn0 an =supn0f(an), f -.

1 f :AA -, aa =

f(a) f(a) a, a A.

a, a A, a a, a = sup{a, a} f

- f(a) = f(sup{a, a}) = sup{f(a), f(a)}

f(a) f(a) f .

Tarski.

1 (Tarski) (A, )- ... f :AA- -

. f fixf =supn0f(n)().

f(n)()

n0

, -

f(n)() f(n+1)() n 0. n.

... (A, ), f(0)() = f() =

f(1)(). f(n)() f(n+1)() f ,

1, f(f(n)()) f(f(n+1)())f(n+1)() f(n+2)(),

. , ... (A, ) -, supn0f(n)()

. f.

f

supn0

f(n)()

= sup

n0

ff(n)()

= supn0

f(n+1)()

= supn0

f(n)()

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12

f -, -

(fn())n0 (fn())n1 A

.

supn0f(n)() f. aA f.

a f() f(a) = a, f(n)() a n 0,

supn0f(n)() a, .

1) A, BX. :

- A=A,- A B=A B,

- A B=A B.

2) A, B, C. :

- A = A= ,

- A (B C) =(A B) (A C),

- A (B C) =(A B) (A C),

- B C = (A B) (A C).

3) , .

4) f :A Bg: B C.

i) f, g 1-1, g f 1-1,

ii) f, g , g f ,

iii) f, g , gf .

5) f :A Bg: B C.

i) g f 1-1, f 1-1,

ii) g f , g ,

iii) g f , f 1-1 g .

6) f : A B. A1, A2 A

B1, B2 B :

i) f1(B1 B2) =f1(B1) f

1(B2),

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13

ii) f1(B1 B2) =f1(B1) f

1(B2),

iii) f(A1 A2) =f(A1) f(A2),

iv) f(A1 A2) f(A1) f(A2).

(iv).

(7)

i) N max

,

ii) N min

,

iii) Z max - ,

iv) Z min -

.

8) (iii)-(viii) 2.

9) - A. ((AA), , , ,A)

.

10) -

.

11) Z \ {0}

R= {(a, b)| a, b Z \ {0} ab > 0}.

R .

12) 1.

13) 2.

14) 2.

15) , -

.

16) 4.

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2

A -

. (a1, . . . , a n) A

( ) A a1 . . . a n.

. .

5 AEN ={a , b , . . . , z }.

-

. babb, zzzzzz, cfg -

. .

.

.

A A,

A ={} {a1 . . . a n | n > 0, a1, . . . , a n A}.

6 A = {0, 1, . . . , 9}. A

. , A =

{0, 1} A

.

w= a1 . . . a n, u = b1 . . . b m A, a1, . . . , a n, b1, . . . , b m A

, w = u, n = m ai = bi

1 i n.

A : w=a1 . . . a n, u =

b1 . . . b m A w u wu=a1 . . . a nb1 . . . b m. -

, ,

w, u, v A w(uv) = (wu)v w = w = w w A.

A

14

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15

.1 card(A) > 1

.

wA n 0 n- wn w-

:

- w0 =

- wn+1 =wnw n 0.

w A n, m 0 (

m)

wnwm =wn+m (wn)m=wnm.

( ) w =

a1 . . . a n A

w

=

an. . . a 1. (wu)=uw w, uA

|w| w ,

w = a1 . . . a n A |w| = n. |wu| = |w| +|u| w, u A

|| = 0.

A

(N, +,0).

L A ( )

A. A={a,b,c}, {a, bac10, bca(ac)30b},

{an | n 0}, {anbn | n 0}, {anbm | n, m 0} -

A.

: L1, L2 A

L1 L2

L1L2 ={w1w2 |w1 L1, w2 L2}.

A={a,b,c}, L1 ={ab2, c10, bca} L2 ={b

2, ac3, }

L1L2 ={ab4, ab2ac3, ab2, c10b2, c10ac3, c10, bcab2,bca2c3, bca}.

,

{} ,

- L1(L2L3) =(L1L2) L3 (1)

- L1{} =L1 ={}L1 (2)

- L1(L2 L3) =L1L2 L1L3 (L1 L2) L3 =L1L3 L2L3 (3)

1 A.

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16

L1, L2, L3 A.

(3). wA

w (L1 L2)L3 w=uv uL1 L2 vL3

w=uv (uL1 uL2) vL3

w=uv uv L1L3 uvL2L3

wL1L3 w L2L3

wL1L3 L2L3.

- L = =L (4).

L A.

(P(A), , ) (1) (2) P(A) {}.

(3), (4) P(A)

.

,

L A n 0 n- (n-) Ln

L :

- L0 ={}

- Ln+1 =LnL n 0.

L L :

L =

n0

Ln

= {} L L2 L3 . . . .

L ={a} aA, {a} ={an |n 0}.

a A {a}

a. , {w} w = a1 . . . a n a1, . . . , a n A w {w} = {a1} . . . {an}.

w u

{w, u}

, (w3u10c bac9b3)

{w3u10c, bac9b3}, .

(

) P(A).

( X) X = LX M L, M A.

F F = FL M. F

F F .

.

.

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17

2 L1, L2, M1, M2 A . L1 M1 L2 M2,

L1L2 M1M2.

w L1L2. w = w1w2 w1 L1 M1 w2 L2 M2,

w M1M2. L1L2 M1M2.

3 L, M A.

(E) X =LX M

LM. L , L,

LM .

f(E) : P(A) P(A)

f(E)(F) =LF M

F P(A). (E)

f(E) . f(E) (

) (E). fixf(E). 4, ... (P(A), ) -,

f(E) -. , F0 F1 F2

. . . P(A). supn0Fn =

n0

Fn.

f(E)(supn0Fn) = f(E)

n0

Fn

= L

n0

Fn M

=

n0

LFn M

=

n0

(LFn M)

= supn0(LFn M)

= supn0f(E)(Fn)

(LFn M)n0

2 .

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18

, 1, fixf(E)

fixf(E) = supn0f(n)

(E)()

= supn0(Ln1M Ln2M . . . M)

=

n0

LnM

=

n0

Ln

M

= LM

f(0)

(E)() = , f

(1)

(E)() = L M = M,

f(2)

(E)() =f(E)(M) =LMM, f

(3)

(E)() =f(E)(LMM) =L(LMM)M) =L

2MLMM,

f(n)

(E)() =Ln1

M . . . M n 1.

(E) LM.

L Y -

. Y = LM. LM (E)

LM Y. .

w Y |w| =n.

Y = LY M

= L(LY M) M

= L2Y LM M

= L2(LY M) LM M

= L3Y L2M LM M

= . . .

= Ln+1Y LnM . . . M.

wLnM . . . M LM.

. w Ln+1Y. w = w1 . . . w n+1u

w1, . . . , w n+1 L uY.

|w| = |w1 . . . w n+1u|

= |w1| + . . . + |wn+1| + |u|

n+ 1 + |u| L,

.

7 A={a, b} X =(ab3b)Xa10.

3 X = (ab3 b)a10.

(ab3 b), .

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X1 =L11X1 L12X2 . . . L1nXn M1

X2 =L21X1 L22X2 . . . L2nXn M2

. . .

Xn =Ln1X1 Ln2X2 . . . LnnXn Mn

Lij, Mi A 1 i, jn.

-

3 , ,

. .

8 A={a,b,c}

X1 = (a2b3 c)X1 a10X2 (1)

X2 = (c4 acb)X1 bX2 c (2).

. 3

X1 =(a2b3 c)(a10X2 ) (1

).

X1 (1) (2)

X2 = (c4 acb)(a2b3 c)(a10X2 ) bX2 c

= (c4 acb)(a2b3 c)a10X2 (c4 acb)(a2b3 c) bX2 c

= ((c4 acb)(a2b3 c)a10 b)X2 (c4 acb)(a2b3 c) c.

X2

3

X2 =((c4 acb)(a2b3 c)a10 b)((c4 acb)(a2b3 c) c).

X2 (1) X1,

X1 =(a2b3 c)(a10((c4 acb)(a2b3 c)a10 b)((c4 acb)(a2b3 c) c) ).

.

;

3 -

.

.

( ) a(ba) =(ab)a.

, , -

.

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20

9 A={a,b,c}

X1 = aX1 bX2 cX3 (1)

X2 = cX1 bX2 (2)

X3 = bX1 aX3 (3).

(2)

X2 =bcX1 (2

)

(3)

X3 =a(bX1 ) (3

).

X2, X3 (2),(3) , (1)

X1 = aX1 bbcX1 ca

(bX1 )

= aX1 bbcX1 ca

bX1 ca

= (a bbc cab)X1 (ca )

X1 =(a bbc cab)(ca ).

X1 (2) (3) X2 X3,

X2 = bc(a bbc cab)(ca )

X3 = ab(a bbc cab)(ca ) a.

.

1) .

2)

.

3) A. ( ) L A

L ={w |w L}.

L, M A

(LM) =ML.

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21

4) A. L A

L2 L. L A,

L L ( )

L.

5) 8

.

6)

X1 = aX1 cX2 bX3

X2 = bX1 cX2 aX3

X3 = cX1 bX2.

7)

X1 = aX1 bX2 cX3

X2 = bX1 cX2 aX3

X3 = cX1 aX2 bX3 .

8) A. P(A) alt : alt(L, M) =

{a1b1a2b2 . . . a nbn | a1a2 . . . a n L, b1b2 . . . b n M, a1, . . . , a n, b1, . . . , b n

A}. A = {a,b,c,d}, alt(L, M) -

.() L={a3, bcab, cda3},M ={a4,cbc,d6}.

()L ={ca3, c10a},M ={a4,cbc,d6}.

() L ={a3, bcab, cda3}, M ={a8, c7, dacbabd}.

9) A w, u A. w u w, u

w u={w1u1 . . . w nun | w=w1w2 . . . w n, u=u1u2, . . . u n,

w1, w2, . . . , w n, u1, u2, . . . , u n A}.

L, M A

L M =

wL,uM

w u.

L

M L M

.

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3

.

3 1 ( )

A =(Q,A,q0, , F )

- Q ,

- A ,

- q0 Q ,

- :Q AQ -

,

- F Q .

: QA Q,

, :

- (q, ) =q,

- (q, wa) =((q, w), a)

q Q, wA, aA.

wA A (q0, w) F.

A (-

) A L(A)( |A|),

L(A) ={w A |(q0, w) F}.

1 -

, finite automaton.

.

22

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23

3 A = (Q,A,q0, , F ) . q Q,w,u A

(q,wu) =((q, w), u).

u. |u| = 0,

u =

(q,w) =(q, w) =((q, w), ).

u A |u| k |u| =k+1.

u=ua u A, aA|u| =k.

(q, wu) = (q, wua)

= ((q,wu), a)

= (((q, w), u), a)

= ((q, w), ua)

= ((q, w), u)

.

A = (Q,A,q0, , F ) :

, ,

a qi qj

(qi, a) =qj.

. w = a1a2 . . . a n

. (q0, w) F q1, . . . , q n qn F (q0, a1) = q1, (q1, a2) =

a2, . . . , (qn1, an) = qn.

q0a1q1

a2q2 . . . q n1

anqn.

. , w = a1 . . . a n

,

(q0, w) = qn F. w A

A A

w. L(A) A . ;

.

10 A = (Q,A,q0, , F ) A = {a, b}, Q =

{q0, q1},F ={q1}

a b

q0 q1 q0q1 q1 q1

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q0 q1a

b a,b

b10a3b, ab1000a40b23, b10000. (q0, b10a3b) = q1,

(q0, ab1000a40b23) = q1

(q0, b10000) = q0. b

10a3b, ab1000a40b23

L(A), b10000 L(A). -

q0,

q1 a.

L(A) =ba(a b).

11 A = (Q,A,q0, , F ) A = {a, b}, Q =

{q0, q1, q2},F ={q2}

a b

q0 q1 q2q1 q0 q1q2 q2 q0

A

q0

q1

q2

a

b

a

b

b

a

(q0, a5b30ab9a2) = q2,

(q0, a2013b3000ab19a2000) = q2,

(q0, a5b30ab8) = q0 a

5b30ab9a2, a2013b3000ab19a2000 L(A),

a5b30ab8 L(A).

.

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25

L(A)

300 (

!) 26 (.. -)...

.

. ,

11.

3 ( A)

A0 = (Q,A,q0, , F ), A1 = (Q,A,q1, , F ) A2 = (Q,A,q2, , F ).

A0 =A. X0 =L(A0),X1 =L(A1)

X2 =

L(A2). A X0. w A A0(q0, w) =q2.

2 :

() w a w= au u A (q0, w) =

(q0, au) = ((q0, a), u) =

(q1, u) = q2.

u A1. ,

u A A1. (q1, u

) =q2(q0, au

) =((q0, a), u) =(q1, u

) =q2 au L(A0).

() w b w = bv v A (q0, w) =

(q0, bv) =((q0, b), v) =

(q2, v) =q2.

v A2. ,

v A A2. (q2, v

) =q2(q0, bv

) =((q0, b), v) =(q2, v

) =q2 bv L(A0).

X0 =aX1 bX2.

A1 w A

, (q1, w) =q2. 2 :

() w a w= au u A (q1, w) =

(q1, au) = ((q1, a), u) =

(q0, u) = q2.

u A0. ,

u

A

A0.

(q0, u

) =q2(q1, au

) =((q1, a), u) =(q0, u

) =q2 au L(A1).

() w b w = bv v A (q1, w) =

(q1, bv) = ((q1, b), v) =

(q1, v) = q2. -

v A1. ,

v A A1.

(q1, v) =q1

(q1, bv) =((q1, b), v

) =(q1, v) =q2

bv L(A1).

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26

X1 =aX0 bX1.

A2.

w A A2 (q2, w) = q2.

2 :

() w a w= au u A (q2, w) =

(q2, au) = ((q2, a), u) =

(q2, u) = q2. -

u A2. ,

u A A2.

(q2, u) = q2

(q2, au) = ((q2, a), u

) = (q2, u) = q2

au L(A2).

() w b w = bv v A (q2, w) =

(q2, bv) = ((q2, b), v) =

(q0, v) = q2.

v A0. , v A A0.

(q0, v) =q2

(q2, bv) =((q2, b), v

) =(q0, v) =q2 bv

L(A2).

(q2, ) =q2

A2.

X2 =aX2 bX0 .

X0 = aX1 bX2

X1 = aX0 bX1

X2 = aX2 bX0 .

X0, A,

. X1 = baX0

X2 =a(bX0 )

X0 = abaX0 ba

(bX0 )

= abaX0 babX0 ba

= (aba bab)X0 ba

X0 =(ab

a ba

b)

ba

.

L(A) =(aba bab)ba.

.

A =(Q,A,q0, , F ) Q={q0, q1, . . . , q n}.

( )

Lij ={a A |(qi, a) =qj} 0 i, jn Xi 0 i n. 2

2 Xi Ai = (Q,A,qi, , F ) Xi =

L(Ai) 0 i n.

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X0 = L00X0 L01X1 . . .L0nXnM0

X1 = L10X0 L11X1 . . .L1nXnM1

. . .

Xn = Ln0X0 Ln1X1 . . .LnnXnMn

Mi = qi F Mi = qi F, 0 i n.

L(A) =X0 A.

1) A =({q0, q1, q2}, {a, b}, q0, , {q0, q2})

a b

q0 q1 q0q1 q2 q1q2 q2 q2

.

2) A =({q0, q1, q2, q3}, {a,b,c}, q0, , {q1, q3})

a b c q0 q2 q1 q0q1 q2 q3 q1q2 q2 q2 q2q3 q1 q3 q0

.

q2;

3) A = {a,b,c,d}.

A

c.

4) A = {a,b,c,d}.

A

a d.

5) A = {a,b,c,d}.

A

c.

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6) A = {a,b,c,d}.

A

c.

7) A = {a,b,c,d}.

A

a

c.

8) A= {0, 1,2, 3,4,5, 6,7,8, 9}. -

.

9) A= {0, 1,2, 3,4,5, 6,7,8, 9}. -

5.

10) A= {0, 1,2, 3,4,5, 6,7,8, 9}. -

3.

11) A={0,1}. -

.

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4

A. L A

A = (Q,A,q0, , F ) L,

L =L(A). Rec(A) -

A.1

Rec(A) boolean , ,

. , Rec(A)

.

-.

4 L, M Rec(A), L M Rec(A).

A = (Q,A,q0, A, F) B = (P,A,p0, B, S) -

L M , L = L(A)

M = L(B). L M.

, -

. w= a1 . . . a n L M

.

q0a1q1

a2 q2 . . . q n1

anqn F

p0a1 p1

a2 p2 . . . q n1

anpn S

A B, .

w ,

w. w -

A B ( )

w

1 Rec recognizable=.

29

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30

,

. -

,

A B . ,

, -

A B, ,

. ,

,

(q0, p0)a1 (q1, p1)

a2 (q2, p2) . . . (qn1, pn1)

an (qn, pn).

. .

C =(QP, A, (q0, p0), C, FS)

C : (Q P) A (Q P)

C((q, p), a) =(A(q, a), B(p, a))

(q, p) Q P, a A.

C((q, p), w) =(A(q, w),

B(p, w))

(q, p) Q P, w A.

w. |w| = 0, w = C((q, p), ) = (q, p) =

(A

(q, ), B

(p, )).

k w A |w| = k+ 1. w = ua u A, a A

|u| =k.

C((q, p), w) = C((q, p), ua)

= C(C((q, p), u), a)

= C((A(q, u),

B(p, u)), a)

= (A(A(q, u), a), B(

B(p, u), a)) C

= (A(q,ua), B(p, ua))

= (

A(q, w),

B(p, w)).

w A.

wL(C) C((q0, p0), w) F S

(A(q0, w), B(p0, w)) F S

A(q0, w) F B(p0, w) S

w L(A) =L w L(B) =M

w L M

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L(C) =L M, L M .

5 L Rec(A), L Rec(A), L = A \ L

L.

A = (Q,A,q0, , F ) 2 L(A) = L.

A = (Q,A,q0, , Q \ F). L = L(A).

, w A

w L(A) (q0, w) Q \ F

(q0, w) F

wL(A)

wL

w L

L(A) =L, L .

6 L, M Rec(A), L M Rec(A).

L M =L M

4 5.

6

4 5 De Morgan.

-

,

( ). 4

5 -

L M 6. , A = (Q,A,q0, A, F)

B =(P,A,p0, B, S) L M -

, L =L(A) M =L(B). 5 L =L(A)

M = L(B) A = (Q,A,q0, A, Q\ F) B = (P,A,p0, B, P\ S).

4, C =(Q P, A, (q0, p0), C, (Q\ F) (P\ S))

C((q, p), a) =(A(q, a), B(p, a))

(q, p) Q P, a A, L M.

5, C =(QP,A, (q0, p0), C, (FP)(QS))

L M ( (F P) (Q S) =(Q P) \ ((Q\ F) (P\ S))).

2 ( - )

.

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32

12 A = {a, b}. L =

{ambm | m 0} .

L Rec(A) A = (Q,A,q0, , F )

.

(q0, a0), (q0, a

1), (q0, a2), . . . .

Q 0 i < j

(q0, ai) =(q0, a

j).

aibj L(A) (q0, aibj) F.

(q0, aibj) =((q0, a

i), bj) =((q0, aj), bj) =(q0, a

jbj) F.

, L .

13 A = {a,b,c,d,e,f,g}.

L ={fedbmgbafmd |m 0} .

L Rec(A) A = (Q,A,q0, , F )

.

(q0, fedb0), (q0, fedb

1), (q0, fedb2), . . . .

Q 0 i < j

(q0, fedbi) =(q0, fedb

j).

fedbigbafjdL(A) (q0, fedbigbafjd) F.

(q0, fedbigbafjd) = ((q0, fedb

i), gbafjd)

= ((q0, fedb

j), gbafjd)

= (q0, fedbjgbafjd) F.

, L .

-,

. Pumping Lemma.

7 L A

n > 0 w L |w| > n, ww= xyzy

xykz L k 0.

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L , A =

(Q,A,q0, , F ) L = L(A). n = card(Q) w =

a1 . . . a m L |w| =m > n. A

q0a1 q1

a2q2 . . . q m1

amqm F.

m > n ()

0 i < j m qi =qj,

q0a1q1 . . . q i1

ai qi

ai+1 qi+1 . . . q j1

ajqi

aj+1 qj+1 . . . q m1

am qm F.

x =a1 . . . a i, y=ai+1 . . . a j, z=aj+1 . . . a m.

w = xyz, i < j y . qi = qj

qi loop y. k= 0,1, 2,3, . . .

q0xqi =qj

zqm F

q0xqi

y

qizqm F

q0xqi

y

qiy

qizqm F

q0xqi

yqi

yqi

yqi

zqm F

. . .

. k 0 xykz L,

.

. 12.

L = {ambm | m 0} Rec(A) n

L 7. w = anbn.

|w| = 2n w= xyz y xykz L k 0. y

.

i) y a, b w -

z. xy2z L n b

n a, .

ii) y b, a w -

x. xy2z L n a

n b, .

iii) y a b, y = ab. 7,

xy2z L, xy2z=xababz, .

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, L -

.

14 A={a}.

L ={ap |p }

.

L Rec({a}) n

L 7. w=ap p p > n.

w= xyzy xykz L k 0. x = a, y= a 0

z = a. k 0 aaka L k 0

+ k+ . + (k = 0)

. k=+ + (+ )+ =(+ 1)(+ )

, . LRec({a}).

1) A, A -

.

2) A =({q0, q1, q2}, {a, b}, q0, A, {q2})

A a b

q0 q1 q2q1 q0 q1q2 q2 q0

B =({p0, p1, p2}, {a, b}, p0, B, {p0, p2})

B a b

p0 p1 p0p1 p2 p1p2 p2 p2

)

, ) ,

) A )

B.

3) A={a, b}.

L1 ={anbm | n m 0} L2 ={a

nbm |m n 0}

.

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35

4) A={a,b,c,d,e,f,g}.

i) L1 ={abcnagebdnf | n 0},

ii) L2 ={d3f2a4bgnad3ebangd2 |n 0},

iii) L3 ={d3a4bgnad3ebangd2dnab| n 0},

iv) L3 ={a10d2f4b2fnad3ebangfd2dnabnd |n 0},

.

5) A={a,b,c,d}.

L ={b3anm

c2b3dnm

|n 0, m > 0}

.

6) A = {a, b}. w A

|w|a ( |w|b) a( b)

w.

L ={w A | |w|a =|w|b}

.

7) , 7, -

13 .

8) A= {a}. L = {a2n

| n 0}

.

9) A= {a}. L = {an2

| n 0}

.

10) A L Rec(A).

n > 0 , |w| > n w L, L

.

11) A L Rec(A).

n > 0 , w L |w|> n, ww=xyz |xy| n, y xykz L k 0.

12) A L Rec(A). L ,

x,y,z A y xykz L

k 0.

13) A,

L A .

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36

14) : n 0

{anbn} , 12, . ,

6, {anbn |n 0} .

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5

-

, -

-.

, -

.

.

.

4 A =(Q,A,q0,

, F)

- Q ,

- A ,

- q0 Q ,

- : Q A Q

,

- F Q .

(q, a) Q A (q, a) .

(q, a) = . , ,

q Q a A,

q.

: Q A Q ( )

. w A A

37

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38

(q0, w) F. () A L(A)(

|A|) ,

L(A)=

{w A

|

(q0, w) F}.

, ,

.

A B , L(A) =L(B).

, -

.

-

.

.

8 A A.

A =(Q,A,q0, , F ) .

A =(Q {q}, A , q 0, , F) q

Q. q Q {q}, a A :

(q, a) =

(q, a) q Q (q, a)

q q Q (q, a) =

q q=q.

w=a1a2 . . . a n L(A).

q0a1q1

a2q2 . . . q n1

anqn F (1)

A w, (q0, w) F. q

(1). (1) A,

w L(A). L(A) L(A). , w L(A),

q0a1q

1

a2q

2. . . q n1

anqn F (2)

A w. A, (2)

q, qn =qF

(2) . (2)

A. w L(A) L(A) L(A). L(A) =L(A), A A .

15 A = ({q0, q1, q2}, A , q 0, , {q2})

a b

q0 q1

q1 q2 q1

q2 q2.

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39

q0

q1

q2

a a

b b

8, A = ({q0, q1, q2, q}, A , q 0, , {q2})

a b

q0 q1 q

q1 q2 q1

q2 q q2

q q q

q0 q1 q2

q

a

b

a

b b

a

a,b

A.

- .

5 - A =

(Q,A,I ,,F)

- Q ,

- A ,

- I Q ,

- : Q A P(Q) -

,

- F Q .

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-

q Q a A

(q, a) .

(q, a) =. , q Q a A,

q.

:QA P(Q),

, :

- (q, ) ={q},

- (q, wa) =

q(q,w)

(q, a)

q Q, w A, a A.

w A

A - q0 I

(q0, w) F . -

A ( ) A

L(A)( |A|),

L(A) ={w A | q0 I (q0, w) F }.

-

, , -

.

.

16 - A =({q0, q1, q2}, {a,b,c},

{q0, q1}, , {q1, q2})

a b c

q0 {q0, q1} {q2}

q1 {q1} {q1, q2}

q2 {q1} {q2}.

X0 = aX0 aX1 bX2

X1 = (a c)X1 cX2

X2 = aX1 cX2 .

( )

L(A) =X0 X1.

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41

q0

q1 q2

a

a b

a,c

c

a

c

-

. ,

-

, .

.

9 - A

A.

A = (Q,A,I ,,F) - .

A

=

(P(Q), A , I ,

, F

)

(P, a) =qP

(q, a)

P P(Q), a A, F = {P P(Q) |

P F}.

(P, w) =qP

(q, w)

P P(Q), w A.

w. , |w| = 0

w = (P, ) = P

qP

(q, ) =

qP

{q} = P.

k w A

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|w| =k+ 1. w=ua u A, a A |u| =k.

(P, w) =

(P, ua)=

((P, u), a)

=

qP

(q, u), a

=

q

qP

(q,u)

(q, a)

=

qP

q(q,u)

(q, a)

= qP

(q, ua)

=

qP

(q, w)

.

w A

w L(A) (I, w) F

qI

(q, w) F

qI

(q, w) F

q0 I (q0, w) F

w L(A)

L(A) = L(A), A

-A.

17 A

- A 16.

, A =(P({q0, q1, q2}), {a,b,c}, {q0, q1},

, F)P({q0, q1, q2}) = {, {q0}, {q1}, {q2}, {q0, q1}, {q0, q2}, {q1, q2}, Q} F=

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43

{{q1}, {q2}, {q1, q2}, {q0, q1}, {q0, q2}, Q}

a b c

{q0} {q0, q1} {q2}

{q1} {q1} {q1, q2}

{q2} {q1} {q2}

{q0, q1} {q0, q1} {q2} {q1, q2}

{q0, q2} {q0, q1} {q2} {q2}

{q1, q2} {q1} {q1, q2}

{q0, q1, q2} {q0, q1} {q2} {q1, q2}.

A

{q2}

{q0}

{q1}{q0, q2}

{q0, q1}

{q0, q1, q2}

{q1, q2}

a,b,c

a

bc

a

b

c

a

b

ca

b,c

a

b

c

a

b

c

a

b

c

- -

, -

. , 9

-

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44

. -

. , 9,

- n-

, 2n . ,

- ,

A = {a,b,c,d}, dac.

.

-

. -

. , -

A = (Q,A,I ,,F)

I =

{qin}, F =

{qt}, qinqt

- qin (q, a), q Q, a A,

- (qt, a) =, a A.

A =(Q,A,qin, , q t).

10 A -

A L( A) =L(A) \ {}.

A = (Q,A,I ,,F) - .

qin, qt Q A = (Q {qin, qt}, A , q in, , q t) q Q

{qin, qt}, a A :

(q, a) =

(q, a) q Q (q, a) F =

(q, a) {qt} q Q (q, a) FqI

(q, a) q=qin

q=qt.

, A .

w=a1a2 . . . a n L(A) \ {}.

q0a1q1

a2q2 . . . q n1

anqn F

A w. ,

qina1q1

a2q2 . . . q n1

anqt

A w . w L(A).

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w L(A). qin qt w .

w=a1a2 . . . a n

qina1q1

a2q2 . . . q n1

anqt

A w. ,

q0 I, qn F

q0a1q1

a2q2 . . . q n1

anqn F

A w, . w L(A)\{},

L(A) =L(A) \ {}.

10,

Rec(A).

11 L, M Rec(A), LM Rec(A).

.

)L M. 10, -

A = (Q,A,qin, A, qt) B = (P,A,pin, B, pt)

L M, . 1 Q P =.

C =((Q P) \ {pin}, A , q in, C, pt) C

r (Q P) \ {pin}, a A :

C(r, a) =

A(r, a) r Q \ {qt}

B(pin, a) r=qtB(r, a) r P\ {pin}.

w= a1a2 . . . a n LM, w = w1w2 w1 = a1 . . . a m L

w2 =am+1 . . . a n M.

qina1q1 . . . q m1

amqt

A w1,

pinam+1 p1 . . . p n(m+1)

anpt

B w2. C,

qina1q1 . . . q m1

amqt

am+1 p1 . . . p n(m+1)

anpt

C w, w L(C),

LM L(C).

1 .

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, w=a1 . . . a n L(C)

qina1r1 . . . r n1

anpt

C w.

A B,

C, 1 m n1

r1, . . . , r m Q rm =qt rm+1, . . . , r n1 P

qina1r1 . . . r m1

amqt

am+1 rm+1 . . . r n1

anpt.

C,

(qt, a),

qina1r1 . . . r m1

amqt

pinam+1 rm+1 . . . r n1

anpt

A B a1 . . . a m am+1 . . . a n, .

a1 . . . a m L am+1 . . . a n M, w= a1 . . . a n LM,

L(C) LM. LM C

LM Rec(A).

) LM. LM =(L\ {})M M

) 10 6.

) L M. LM =L(M\ {}) L ).

) L M. LM =(L\ {})(M\ {}) (L\ {})

(M\ {}) {}, ), 10, 6

{}

( ) .

12 L Rec(A), L Rec(A).

A = (Q,A,qin, , q t) -

L\ {}. A =(Q\ {qt}, A , q in, , qin)

q Q \ {qt}, a A :

(q, a) = (q, a) qt (q, a)

((q, a) \ {qt}) {qin} qt (q, a).

L(A) =L (

), L Rec(A).

2 Rec(A) A

, , , -

.

4, 5, 6, 11 12.

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47

1) A = (Q,A,q0, , F) , , -

. A 5 L(A).

2) - A =({q0, q1, q2}, {a,b,c}, {q0, q2},

, {q0, q2})

a b c

q0 {q1, q2} {q2}

q1 {q1} {q1, q2}

q2 {q1} {q2}.

A, -

.

3) - A =({q0, q1, q2}, {a,b,c}, {q2},

, {q2})

a b c

q0 {q0, q2} {q2}

q1 {q2} {q0, q2}

q2 {q0, q1} {q1}.

A, -

.

4) - A =({q0, q1, q2}, {a,b,c}, {q0, q1, q2},, {q2})

a b c

q0 {q0} {q0} {q2}

q1 {q1}

q2 {q1} {q0} {q2}.

A, -

.

5) A= {a,b,c,d}. -

A

dacA

, aA

d3

a3

c3A

a, ab

2A

cbA

dA

d.

6) Rec(A) -

- .

7) - A = (Q,A,q0, , {q0}).

L(A) .

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48

8) A = (Q,A,q0, A, F) B = (P,A,p0, B, S).

L(A)L(B).

9) A = (Q,A,q0, A, F). L(A).

10) L, M Rec(A), alt(L, M) Rec(A)( alt.

8, 2).

11) L, M Rec(A), L M Rec(A) ( L M

. 9, 2).

12)

2, 3, 4.

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6

. -

,

,

,

. -

, .

.

. -

A =({q0, q1, q2, q3, q4, q5, q6, q7, q8, q9}, {a,b,c}, {q0, q1}, , {q3, q7, q9})

a b c

q0 {q0, q1} {q2} {q2, q7}

q1 {q1, q9} {q4} {q1}

q2 {q4} {q2}

q3 {q3} {q4}

q4 {q3}

q5 {q4, q6} {q6}

q6 {q7} {q6} {q6, q7}

q7 {q8}

q8 {q8} {q8}

q9 {q1}

, ,

-

q5, q6 q8,

A,

A.

49

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50

q0

q1

q2

q3

q4

q9

q6

q7

q5

q8

a

a

b,c

c

a,c

b

a

c

a

ca

bc

a

a,c

a,c

b,c

a

a,c

. A =(Q,A,I,,F) -

. q Q

q0 I w A q (q0, w). q

u A (q, u) F .

. Qac(Qcoac,Qt) ( , )

A.

13 A =(Q,A,I,,F) - .

Qac .

(In)n0 Q

:

I0 =I,

I1 ={q |qQ, q0 I0, a A q(q0, a)} \ I0,

I2 ={q |qQ, q1 I1, a A q(q1, a)} \ (I0 I1),

. . .

In+1 ={q |qQ, qn In, aA q (qn, a)} \ (I0 . . . In),

n 0.

In,n 0, ,

m < card(Q) Im Ii = i > m.

(In)n0

Qac =

0km

Ik.

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51

14 - A =(Q,A,I,,F)

- Aac

.

13,

Qac A. , -

- Aac =(Qac, A , I , ac, Fac),

ac Qac A Fac =F Qac,

A.

.

- Aac

49. I ={q0, q1},

I0 =I ={q0, q1},

I1 ={q2, q4, q7, q9},

I2 ={q3, q8},

I3 =, Ii = i 3.

Qac =I0 I1 I2 ={q0, q1, q2, q3, q4, q7, q8, q9}. -

Aac

q0

q1

q2

q3

q4

q9 q7

q8

a

a

b,c

c

a,c

b

a

c

a

ca

bc

a

a,c

15 A =(Q,A,I,,F) - .

Qcoac .

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52

(Fn)n0 Q

:

F0 =F,

F1 ={p| p Q, p0 F0, aA p0 (p, a)} \ F0,F2 ={p| p Q, p1 F1, aA p1 (p, a)} \ (F0 F1),

. . .

Fn+1 ={p| p Q, pn Fn, a A pn (p, a)} \ (F0 . . . Fn),

n 0.

Fn,n 0, ,

l < card(Q) Fl Fi = i > l.

(Fn)n0

Qcoac =

0kl

Fk.

16 - A =(Q,A,I,,F)

- Acoac

.

15, Qcoac A.

, - Acoac =

(Qcoac, A , I coac, coac, F), coac Qcoac A

Icoac = I Qcoac,

A. .

16 - Acoac -

49. F ={q3, q7, q9},

F0 =F ={q3, q7, q9},

F1 ={q0, q1, q4, q6},

F2 ={q2, q5},

F3 =, Fi = i 3.

Qcoac

= F0 F1 F2 = {q0, q1, q2, q3, q4, q5, q6, q7, q9}. -Acoac 53.

17 A =(Q,A,I,,F) - .

Qt .

-

13

15.

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53

q0

q1

q2

q3

q4

q9

q6

q7

q5

a

a

b,c

c

a,c

b

a

c

a

ca

bc

a

a,c

a,c

b,c

3 - A = (Q,A,I,,F)

- -

.

A Aac

(Aac)coac. 14 16, - (Aac)coac A

.

49, -

3, -

(Aac)coac .

(Aac)coac

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54

q0

q1

q2

q3

q4

q9 q7

a

a

b,c

c

a,c

b

a

c

a

ca

bc

,

, .

A =(Q,A,q0, , F )

A = {a,b,c}, Q = {q0, q1, q2, q3}, F = {q3}

a b c

q0 q1 q2

q1 q3 q2q2 q3 q1q3 q3 q3 q3.

A, 55,

. q1 q2 -

. ,

;

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55

q0

q1

q2

q3

a

b

b

c

b

c a,b,c

A,

q0 p q3a, b a

c

a,b,c

A. ,

,

,

.

.

A = (Q,A,q0, , F )

. Q :

q q - ((q, w) F (q, w) F w A).

Q.

[q] =[q] [(q, w)] =[(q, w)] w A. (6.1)

w A p [(q, w)]. u A

(p, u) F ((q, w), u) F

(q,wu) F

(q, wu) F

((q, w), u) F

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qq.

p (q, w), p [(q, w)] [(q, w)]

[(q, w)]. , [(q, w)] =

[(q, w)]. q Q,

qF [q] F. (6.2)

q F q [q]. (q, w) F (q, w) F

w A, q F q = (q, ) F,

q =(q, ) F. [q] F.

Q/ -

. n 0, Q

n :

qnq - ((q, w) F (q, w) F wA |w| n)

q, q Q.

n 0 q, q Q, qn+1 q= qn q

0 1 2 . . . n0

n =

card(Q/0) card(Q/1) card(Q/2) . . . .

Q , k < card(Q) Q/ i=Q/k ik. = k

q q, q, q Q, (q, w) F (q, w) F

w A |w| < card(Q). Q/

.

A = (Q,A,q0, , F )

, , -

.

18 A = (Q,A,q0, , F ) -

,

.

A = (Q/ , A, [q0], , F),

F ={[q]| q F}

([q], a) =[(q, a)]

qQ, aA. (6.1), .

([q], w) =[(q, w)]

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qQ, wA. w.

|w| = 0 w= ([q], ) = [q] = [(q, )].

w A |w| k w=ua uA, |u| =

ka A.

([q], w) = ([q], ua)

= (([q], u), a)

= ([(q, u)], a)

= [((q, u), a)]

= [(q, w)]

.

A

. A.

, w A

w L(A) ([q0], w) F

[(q0, w)] F

(q0, w) F

w L(A)

(6.2). L(A) = L(A),

.

.

19 A L(A) =

.

A = (Q,A,I,,F) ( 3)

A =(Qt, A , I , , F) .

L(A) = L(A) = Qt =.

Qt Qt = ,

L(A) = .

20 A B L(A) L(B)

.

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L(A) L(B) L(A) L(B) =

L(B) L(B). 5,

L(B) , 4 L(A)

L(B) .

19.

.

4 A B L(A) =L(B) .

L(A) =L(B) (L(A) L(B) L(B) L(A))

20.

A, -

A,

A .

21 A A

L(A) =A .

B =({q0}, A , q 0, , {q0}) (q0, a) =q0 a A. L(B) =A

L(A) =A L(A) =L(B).

,

.

L(A) =A .

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59

1) - A =({q0, q1, q2, q3, q4, q5, q6, q7, q8, q9},

{a,b,c}, {q0, q3}, , {q5, q9})

a b c

q0 {q0, q1} {q2} {q2, q7}

q1 {q1, q9} {q4} {q1}

q2 {q4} {q2}

q3 {q3} {q4}

q4 {q3}

q5 {q4, q6} {q6}

q6 {q7} {q6} {q6, q7}

q7 {q8}

q8 {q8} {q8}

q9 {q1}

A, -

.

2) - A =({q0, q1, q2, q3, q4, q5, q6, q7, q8, q9},

{a,b,c}, {q4}, , {q8})

a b c

q0 {q0, q1} {q2} {q2, q7}

q1 {q1, q9} {q4} {q1}

q2 {q4} {q2}

q3 {q3, q5} {q4}

q4 {q3}

q5 {q4, q6} {q6}

q6 {q7} {q6} {q6, q7}

q7 {q8}

q8 {q8} {q8}q9 {q1}

A, -

.

3) - A =({q0, q1, q2, q3, q4, q5, q6, q7, q8, q9},

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{a,b,c}, {q8}, , {q8})

a b c

q0 {q0, q1} {q2} {q2, q7}q1 {q1, q9} {q4} {q1}

q2 {q4} {q2}

q3 {q3} {q4}

q4 {q3}

q5 {q4, q6} {q6}

q6 {q7} {q6} {q6, q7}

q7 {q8}

q8 {q8} {q8}

q9 {q1}

A, - .

4) A = (Q,A,q0, , F ) A = {a,b,c},

Q = {q0, q1, q2, q3, q4, q5, q6, q7}, F = {q5, q7} -

a b c

q0 q1 q2 q6q1 q3 q1q2 q4 q6 q2q3 q4 q5q4 q3 q5q5 q5

q6 q6 q6q7 q6

, A.

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7

-

Kleene

. ,

.

6 A . L A

A

.

, A={a,b,c,d},

ab2, b(cd3), ab, (((ba)d)c)b db, ((db)ac) c3 abc3

. , Kleene,

{anbn | n 0} .

A 1

Rat(A).

22 Rat(A)

A .

L = {w1, . . . , w n} . L = {w1}

. . . {wn}, {wi}(1 i n)

wi . L, M Rat(A).

L ( M) A - . L M

L M

. L M .

LM, L Rat(A). Rat(A) .

L, A,

.

1 Rat rational= .

61

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62

Rat(A) L. L Rat(A). L

, a1, . . . , a n, A

. a1, . . . , a n

L, L L L, .

Kleene.

23 L Rat(A), L Rec(A).

(. 1, -

4). , 6, 11 12, -

.

22.

24 L Rec(A), LRat(A).

L Rec(A) A =

(Q,A,q0, , F ) . Q= {q0, q1, . . . , q n}. 0 i, j

n k 0 P(k)

ij -

qi, qj

k. L(k)

ij

. -

0 i, j n, k 0 L(k)

ij .

k. k= 0

L(k)

ij = {a |a A (qi, a) = qj}

.

k > 0. () P(k+1)

ij .

(i) qk (). () P(k)

ij .

(ii) qk (). ()

() : qiw1qk

w2qk. . . q k

wm1 qk

wmqj

(1) : qi

w1qk,

(2) : qkw2qk,

. . .

(m1) : qkwm1 qk,

(m) : qkwmqj

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k 1.

L(k+1)ij L(k)ij L(k)ik

L(k)kk

L(k)kj .

L(k+1)

ij .

L(k+1)

ij = L(k)

ij L

(k)

ik

L

(k)

kk

L

(k)

kj .

, L(k)

ij , L(k)

ik , L

(k)

kk, L

(k)

kj

L(k+1)

ij .

F = {qi1 , . . . , q im} 0 i1 < . . . < i m n.

L= L(n+1)

0i1 . . . L(n+1)

0im

q0 F,

L= L(n+1)

0i1 . . . L

(n+1)

0im {}

q0 F. L .

5 (Kleene1956) Rec(A) = Rat(A).

23 24.

1) A a, b A.

A, a3A3b3, a(A)b5, (((A)))

.

2)

a

b

, a

b

c3

, a

b

(ab b5

), (ab)

b2

(a3

), b5

a4

((ab)

)

.

3) {anbn |n 0} .

4) A. Rat(A)

.

5) A. Rat(A)

.

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64

6) A L Rat(A). -

i) L= ;

ii) L= A;

7) A L, M Rat(A). -

L M;

8) A L, M Rat(A). -

L= M;