θεωρητικη πληροφορικη
DESCRIPTION
σημειωσεις μαθηματικο απθ θεωρητικη πληροφορικη 1TRANSCRIPT
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,
.
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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- 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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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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(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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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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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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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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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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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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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
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.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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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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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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, 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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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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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 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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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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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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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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28
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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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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14) : n 0
{anbn} , 12, . ,
6, {anbn |n 0} .
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-
, -
-.
, -
.
.
.
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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40
-
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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42
|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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45
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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46
, 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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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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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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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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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;