# $ "! " # $ % &' #( ) * + & % (, '. / 0, 1 2 *

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1 !"##$ %&'()") *+,-.//.-+-,01,+2,03/3 * ,67.44+/ /

2 # $ "! " # $ % &' #( ) * + & % (, '. / 0, 1 2 *

3

4 ! $ 4 3 & ) 37!!" # $ 56 5 $! 5 5 $ $ % &'()" *+ 3, ,+ 73 * ) $- 3 - ' 3 7,+ *.!/!! ## 3, ('(' 12

5 U " ! # %

6 ( #( % ( $ + S = {2,3,5,7,11,13,17, 19} S = {p p 2 p 20}. a A a A a / A - 3 S 8 / S % ( A + B A B ) % -( A B A B ) A B 2 {2,3} S {1,2,3} S. A A A B A B B A A B A B A B ) % -( A B - + {2,3} S A A + % + ( X = {, {1}, {2}, {1, 2}}. ) A A %!( P(A) )" X = P({1,2}) # 2 A B A P(B) - $$ ) %&'()*++',--./,,* /,'' , 3/(5'016 8&9,5* A B7 n: *+5'9',/0 ;/(-,)9-590 ;91/0,,')9-59,,* 90 *+5'91*< 5621/1660 5'()*++',--./,,* ;91/0,,')9-59++/ 324, 3/(5'0166 A 2 n : 2 A7 3

7 5 %( % ( A B = {x x A x B}, {1,2,3} {1,4} = {1,2,3,4} %( % ( A B = {x x A x B}, {1,2,3} {1,4} = {1} %( % ( {1,2,3} {1,4} = {2,3} A B = {x x A x / B}, $ ) ) A (B C) = (A B) C, A (B C) = (A B) C $$ ) A B = B A, A B = B A A (B C) = (A B) (A C), A (B C) = (A B) (A C). U U A A $ % U ) ( Ā), ) A B = Ā B, A B = Ā B. 1 ) A B = A B. A A = {A1,A 2,...,A n } + n A i = A 1 A 2 A n 9-559/(91-5,/++/ 5621/1660 5'()*++',--./,,* 324, 3/(5'0166 A \ B7 i=1

8 n A i = A 1 A 2 A n. i= A = {A1,A 2,A 3,... } A i = i 1 i=1 A i = {x x A i i = 1,2,... } A i = i 1 i=1 A i = {x x A i i = 1,2,... }. + + I A = {Ai i I} i I i I " $$ ( $ )# ( A i = {x x A i i I} A i = {x x A i i I}. * A B $ a A b B % ( (a,b) * {a,b} = {b,a} a b (a,b) (b,a), A B $ % " ( ) & ) A B = {(a,b) a A b B}. {1,2,3} {1,4} = {(1,1),(1,4),(2,1),(2, 4), (3,1),(3,4)}. $ R A B (a,b) a A b B A B ) R A B. R $ % ( A $ % ( B R A A R A 6,36++',/,1'< 3-11* /;6'01-'1''',/,1' 9'.**0 )6()/,1/112 ;*(' 366('1/++6 )9-5590* (a,b) = {a, {a, b}}7

9 a b a b R R R c d c d, 33) R R R {1, 2} {1} {2}, 35) * S (a,b) R + arb a $ R - b %* a b # (a,b) #( * R A B * A A % ( R B B R[A ] = {b B a A (a,b) R} $ % ( R 1 [B ] = {a A b B (a,b ) R}. R A B $ % ( R 1 B A R 1 = {(b,a) (a,b) R}. R A B S B C $ % ( R S A C ) R S = {(a,c) b B (a,b) R,(b,c) S}. R A B A B $ A B a A % #( b B (a,b) R * A = {a,b,c,d} R A A R = {(a,b),(a,c),(b,d),(c,d), (d, d)}.

10 7, 33 R R R, 35 X = P({1,2}) S = {(A,B) X X A B} f A B a A f b B $ b + a f(a) = b f! f : A B.! ) f : A B g: B C!! (g f)(a) = g(f(a)). f : A B % ( b B a A f[a] = B % ( a A a a f(a) f(a ) a,a A. f b B a A # $ ") &" $$ ( - $ X * X $ $ %, - X Π Π ( * # R A A ) %( ara a A " %( arb bra a,b A " %( arb,brc arc a,b,c A R A A %( %( $ $ $ a A $ $ % R ( R[a] = {x A arx}. - R ) - A = { 3466 } a b + a b + % (

11 $ R A A $ $ $$ $$ a,b A R[a] = R[b] * a b A * R[a] = R[b] R 35 brb + b R[b] = R[a] arb arb R + + * bra R[a] R[b] a ) b ) + + R[b] R[a] R[a] = R[b] * x R[a] R[a] 35 + arx bra R ) + brx x R[b], x R[a] R[a] R[b] $ R A A $ $ $$ R $ $ A $$ arb. R[a] $$ a A A = a A R[a] R[a] R[b] R[a] R[b] = $$ a,b A %( 2 a R[a] %( 2 A = {a} R[a] A. a A a A %( * a,b A R[a] R[b] * + R[a] = R[b] * c R[a] R[b] 35 + arc brc 1 33 R[a] = R[c] = R[b], A Ai i I ) a b a b A i. & " &" $$ (, # ) % R A A a,b A arb,bra a = b

12 * R A A A % %( ( R ) - A A A ) (A, ) (A, ) a,b A a b b a a b $ % -( A % ( $$ % ( % ( % ( (A, ) a A ) %( $ a x a = x x A " %( $ x a a = x x A " %( $ x a x A " %( $ a x x A (A, ) %! ( B A % $$ {a,b} %( 1 N (N, ) %( 2 (Z, ) %( * + X % 35( X 2 ( * m,n N ) m n m n ). 2 (N, ) (P(X), ) * (A, ) ) a b a b a b,a b b a a b a b,a b. $ a A b A $ $ b a ) $ )

13 5 4 8 {1, 2, 3} {1, 2} {1, 3} {2, 3} {1} {2} {3} 1 1 %( a b ({1, 2,..., 5}, ) ({1, 2,...,10}, ) (P({1, 2, 3}), ) %( c A a c b, 3), (A, ) A ) a A + # a ) +, 3 + # ) # ('"& $$" * $ (A, ) P(a) A $ $$ a A $$ [P(x) x a] P(a), P(a) $$ a A * % ( B = {a A P(a) }, A B % ( b B + ) [P(x) x b], % ( + P(b) 2 B = P(a) a A +!!!! '!) $ P(k) $ $$ $

14 4 P(0) $$ k 0 [P(0)&P(1)&... &P(k)] P(k + 1) P(n) $$ n N!!!! '!) $ P(k) $ $$ $ P(0) $$ k 0 P(k) P(k + 1) P(n) $$ n N - 3 ), n N P(n) : ( n) 2 = n 3. %( P(0) : 0 2 = 0 %( $ * k 0 P(k) : ( k) 2 = k 3 + +) ( k + (k + 1)) 2 = (1 + + k) (1 + + k) (k + 1) + (k + 1) 2 = ( k 3 ) k(k + 1) (k + 1) + (k + 1)2 2 = ( k 3 ) + k(k + 1) 2 + (k + 1) 2 = ( k 3 ) + (k + 1) (k + 1) 2 = k 3 + (k + 1) 3. * P(k) P(k + 1) P(k) P(k+1) 1 3 k 0 P(n) n N ( $$ $$##( ( "&## ()( $ # " " $ * %+ ( ' 3 ) *, + )

15 36, 3) $ %( + " %( % + *,# #+ #( $ ) + 3 $ , % # + ( -! +!" # % - -( $ $ {0,1} $ $ $ $ % {... } % ( # 6666# -, '# &, %%,# % ( ε x ) - x = XYZZY = = TKTP = 4 ε = 0, - - ) %(,$1$,,,* ',$1$,,,*" %( x = 00 y = 11 xy = 0011 yx = 1100 " %( x xε = εx = x " ()66('1/+3660,',6+122 /(6, 16(5/6 ;''+9 : 9+/1-,* *-193**1'0 1-+5'1**0 +, ,,/++*',/1,2411//1< )9'++*,/ /' 1-91* 3'05660+*',1* *,1/11* /,'3/(5'5,' +*,5/00*0 ; /0 1*5'* *,.9++',--, 9,9'11* ,,*. /,'1/ /',/0 +*,5/11*--,1/9('*0 5*00*+1* *'*0 5/,5/',/5,' 7

16 33 %( x y z (xy)z = x(yz) " %( x y xy = x + y $ Σ Σ ) - Σ = {0,1} Σ = {ε,0,1,00,01,10,... } A Σ Σ $ $ %! ( * - + Σ + x (x) ) %* (x) {0,1} x Σ ( $ + A = {x Σ (x) = 1} Σ ) % - ( $ $ A ) + * π: Σ {0,1} A π = {x Σ π(x) = 1} π!!!" + $ ) $ ) Σ Γ... % ( - Σ = {0,1} $ % () Σ $ ) a b c % (... Σ = {a1,...,a n } " + Σ = n ) u v w x y % (... ) x y xy ) x %( abc = 3 " %( x = a1...a m y = b1... b n " + xy = m + n ) ε

17 35 n a ) a n " %( a n = aa...a }{{} n kpl %( a i b j c k = i + j + k x k ) x k %( (ab) 2 = abab " %( x k = k x " %( x 0 = ε $ Σ ) Σ {a,b} = {ε,a,b,aa,ab,ba,bb,aaa,aab,... }!! $ # ε % ( 2 u + w = ua * Σ w Σ % ( w R + ) %( " ε R = ε %( w w = ua u Σ a Σ w R = a u R + # ) (011) R = 1 (01) R = 1 (1 0 R ) = 11 (0 ε R ) = 110 ε R = 110 ε = ) * Σ, x,y Σ (xy) R = y R x R y %( y = ε (xε) R = x R = ε R x R %( $ * y y = ua u Σ * a Σ + ) x u (xy) R = (xua) R = a (xu) R R ) = a (u R x R ) = (a u R )xr ) = (ua) R x R ) R = y R x R.

18 3 "&( $ $ ) "&( $ ( #( + X % - ( - f : N X % -( + $ % -( ) X X = {x0,x 1,...,x n 1 } % X ( n X = {x0,x 1,...} % + ( * % ( X # # Σ Σ - f : N Σ * Σ = {a 1,a 2,...,a n }, Σ ) #" a1 < a 2 < < a n Σ $ % ( ) %( 6) %' ε ( 3) %' a1,a 2,...,a n ( 5) " %( $ f ) 0 ε 1 a 1 2 a 2 n a n n + 1 a 1 a 1 n + 2 a 1 a 2 2n a 1 a n 2n + 1 a 2 a 1 3n n 2 + n a 2 a n a n a n n 2 + n + 1 a 1 a 1 a 1 n 2 + n + 2 a 1 a 1 a 2

19 3 % " $ 2" + ( !. # $$ $ $$$ $ $$ $ $$ $$$ ) # - * 1 Σ $ $ $ % " ( Σ! * P(Σ ) = A )! ) Σ A = {A 0,A 1,A 2,... }. * Σ ) x 0,x 1,x 2,...! à ) à = {x i Σ x i / A i }., ) à + A A ) à = Ak k N à x k à x k / A k = Ã. 2 A, A0,A 1,A 2,... # 3 x0,x 1,x 2, i j x i A #) j à A k à A 0 A 1 A 2 A 3 x x x x

20 37 # # & ( & ) ) ' () " $ -! %- * + ( % ( " 1 3 % #( $ 34 * + "! $ - "! + "! + ) {! 1, + " = ) ) 0, *!! ) %!!! " #$ $! " % && ' ( ' )! )! 2 ) confuse(c) halt(c,c) == 1! confuse(c)!! -051'9'./0 *6,, ('11/+2)6 9+/1/11-25,'05/(1*',--./0-95,'< / '9'./0 ( ','*1,', ',1/0 :, +, 7 *(55**0 911*/0./ 1* 19.',1-5,/ '+- 16,36++//0 01 :,1*0.*(.'0 3-5*'0/0 + : 5'/+/0,201*5,' /',*++' 1616< 3-1./ : 01 : + *0 **1'3-,1/0 3-5*',/5,',636(16',',/0 5/,5/',/0 './* '0 + : )61< /,'3/(5'5,' 2/3 +<,*++'*1 16,,6 5621/120,1*0.*(.'0 +**)/00-5,/0 7

21 3

22 $#$$ ( $ $$ $$ $$ $ % ( $ $ % ( - 5 3, + 55 $ # + $ ' # $ + $ - % ( < < < < <, 5 3), 3 > 0, 4

23 3 q q q 0 q f a a q 1 q 2 q1 q2, 55) q 7 q 0 q 1 q 2 q 3 q 4 q 6 q 5, 5 ) " +

24 34 q 0 q 1 error, 5 ) $ 5 + " + %" -#( $ {0,1,...,9} {E,e} $ + $$ % -( +! - 5 ) digit. exp + q 0 q 1 q 7 q 1 q 1 q 2 q 4 q 2 q 3 q 4 q 3 q 3 q 4 q 4 q 6 q 5 q 5 q 5 q 6 q 6 q 6 q 7 q 3 # ) digit. exp + q 0 q 1 q 7 error error error q 1 q 1 q 2 q 4 error error q 6 q 6 error error error error error error error error error error

25 56 d q 0 q 1 q 2 d d, 5 7) - &" ) $ ( $$" ) '"& $ ( " ( ) $ - 5 " + " + )!! ) & ) ( & & #$ " " ( " $! " & ' ) " #$ " % && & )! & ) " $ $$! $ ) ' & ' ) &&! & ) && && & ) & ) $ )!!! $ ) % && && &&! "#"! )!! "#"! ) " $ # + # 1 +

26 53 + * %, d {0,1,...,7} ( " )! ) & ) ( & & ( % && && & & ) & ' ) & & ) $ ) ' & & ) & & ) $ ) & & ) & & ) $ ) $ ) % && #! ) ) % % %!! # # "#"! ) ' ) + % ( % " " ()

27 55! ) ) " & $$ & " & ' ) & ) " " & ) % ( & & ( % && & ' ) && & ' ) & ' ) " " % & & & & ) ) " " % & ) $ ) ' & & & & ) ) " " % & ) $ ) & & & ) " " & ) % & ) $ ) $ ) % && " "!!! ) " " ) %!! # # "#"! ) ' ) %

28 5,241/0*-,** ' 0 ; - 1 0*-,*;66* 9,)*-,25,'554* q 1 q 2 q 0, 5) $ &" ") $ ( $$ ) # "") (& $ ( ) +!. % 5() δ δ + $ $$ q0 + +, + +! + $ $$ + $ $ $ ) $$ % ( = (Q,Σ,δ,q 0,F), Q $ % ( " Σ % -(" δ : Q Σ Q %! (" q 0 Q $$ % (" F Q $$ % ( % 6(/++',/0 *-193**1'0 2.'0 90,/0 +9,)/+3*-<,''(1236, -051'9 δ7./++6 /,'1/121 1'+*5**'91 )* : 1*-+-1 9*1 )--(' 16360,''(1236, -051'90 *',19/,19','* /,'12,1*;9)* 7

29 5-5 +! ) = ({q 0,...,q 7, }, { '...! },δ,q 0, {q 2,q 3,q 6 }), δ 34 " δ(q 0,0) = δ(q 0,1) = = δ(q 0,9) = q 1, δ(q 0,.) = q 7, δ(q 0,E) = δ(q 0,e) =, δ(q 1,.) = q 2, δ(q 1,E) = δ(q 1,e) = q 4, $ $ % ( (q,w) Q Σ " $$ $$ x (q0,x) (q,w) q w + (q,w) (q,w ) (q,w) (q,w ), w = aw % ( a Σ q = δ(q,a) + + (q,w ) $ (q,w) % ( q w = aw a q w (q,w) (q,w ). (q,w) $ (q,w ) (q,w ) (q,w) % ( (q,w) (q,w ), (q0,w 0 ) (q 1,w 1 )... (q n,w n ) n 0 (q,w) = (q 0,w 0 ) (q 1,w 1 ) (q n,w n ) = (q,w ). - n = 0 (q,w) (q,w) (q,w) (q,w) (q,w ). $ % ( x Σ (q 0,x) (q f,ε) qf F; $ % ( x ) ) x ) + x + $ $ % - ( ) L() = {x Σ (q 0,x) (q f,ε) qf F }.

30 57 -! # 5 ) (q 0,0.25E2) (q 1,.25E2) (q 2,25E2) (q 3,5E2) (q 3,E2) (q 4,2) (q 6,ε)., q6 F = {q 2,q 3,q 6 } 0.25E2 L() &" ") $ ( $$ ") ) ( ) $ % ( + + ) = (Q,Σ,δ,q 0,F) +! + ) q Q x Σ δ (q,x) = q Q, (q,x) (q,ε). $ q q $ x Σ q q, δ (q,x) F δ (q,x) F; ) q q ) + k ) q q k $ x Σ x k q k q, δ (q,x) F δ (q,x) F; k ) ) %( %( q 0 q q q " q q q k q k = 0,1,2, k (k + 1) %53(

31 5 /!!!! $ = (Q,Σ,δ,q0,F) ) 3 ' ) + q * ) ) k (k + 1) ) ) ) $ )! )! ) ) ' " * (k + 1) ) % k = 0,1,... ( k ) % %5 3(( $ % ( % ( ) a : 1 2, b 3, 2 4, 2, 3 2, 3, : 4 3, 5, 5 1, 4,

32 5 b a a b a a a b b b b a, 5) b b b a a, 5) 1 % 3 5 ( ) : 1 2, 3 2, 3, : 2 4, 2, : 4 3, 5, 5 1, 4, & " 5 $ 5 3 ) #$ $$ $ $$ $ $ 2 a a b 3,

33 5 * = ( ˆQ,Σ, ˆδ, ˆq 0, F), = ( Q,Σ, δ, q 0, F) ) Q ˆQ $ % ( f : Q ˆQ q Q a Σ f( δ( q,a)) = ˆδ(f( q),a). %55( ) # + Q " - ˆQ ) f q Q x Σ q = δ % + ( q 0,x) ( $ f( q) = ˆδ (ˆq 0,x); # q ) * f q ) ) x,y Σ x y δ ( q0,x) = δ ( q0,y), ˆδ (ˆq 0,x) ˆδ (ˆq 0,y)., ) %5( ) q x = δ (q 0,x) q y = δ (q 0,y) z Σ δ (q x,z) F δ (q y,z) / F % ( 2 xz yz x y 2 ) 2 xz yz f 1 %55( * f q = δ ( q 0,x) f( q) = ˆδ + (ˆq 0,x) ˆδ(f( q),a) = ˆδ(ˆδ (ˆq 0,x),a) = ˆδ (ˆq 0,xa) = f( δ ( q 0,xa)) = f( δ( δ ( q 0,x),a)) = f( δ( q,a)). )* 8&-*-5,/0 3/(5'1660 f,-()/51''',--, < 0''0,/-(** 90 16,16< 59,5* )9, 1'+* ˆq 9'.**0,**-11** 1'+*,1* ˆQ ˆq0 3/(55')909++* x q = δ ( q 0, x) ˆq = f( q)7 %5(

34 54 a a b a q 0 q 1 q 2 q 3 a b, 5 4), ' ""& ) " &" " $ ( $$ ) + + {a,b} aba - 54 q0 $$ a % q0 q1 (! ) # - $ 2 ) + $$ $ % 5 ( -! δ aaba ) (q 0,aaba) (q 0,aba) (q 1,ba) (q 2,a) (q 3,ε). $ + ) (q 0,aaba) (q 0,aba) (q 0,ba) (q 0,a) (q 0,ε), # ) A P(A) )" P(A) = {B B A}. $$ % ( b = (Q,Σ,δ,q 0,F),

35 6 Q $ " Σ " δ : Q Σ P(Q) % ( % ( " $$" q 0 Q F Q $$ - 5 4! ) a b q 0 {q 0,q 1 } {q 0 } q 1 {q 2 } q 2 {q 3 } q 3 {q 3 } {q 3 } δ(q0,a) = {q 0,q 1 } δ(q 1,a) = %- ( ' + ) (q,w) (q,w ) (q,w) (q,w ), w = aw % ( a Σ q δ(q,a) % (" + + (q,w ) $ (q,w) +, + + ) $$ $ A = L() $$ $ $$ $ $$ $$ A = L( ) * A = L() = (Q,Σ,δ,q0,F) ). ) )

36 3 b a a a q 0 a q 0, q 1 b q 0, q 2 a q 0, q 1, q 3 b q 0, q 2, q 3 b a b q 0, q 3, 536) b a b s 0 a s 1 b s 2 a s 3 a, b, 533) b = ( ˆQ,Σ, ˆδ, ˆq 0, F), ˆQ = P(Q) = {S S Q}, ˆq 0 = {q 0 }, F = {S Q S q f F }, ˆδ(S,a) = q S δ(q,a) ) S {q 0,q 1,q 2,q 3 } + +, + {q0 } + 2+ a b a ) q 0 q1 {q0,q 1 } ˆδ({q0 },a) = {q 0,q 1 } + b q 0 % 536( ˆδ({q0 },b) = {q 0 } 2 {q0,q 1 } 2+ a q 0 q 0 q1 " +, q1 a ˆδ({q0,q 1 },a) = {q 0,q 1 } 2+ b & q0 q1 q2 {q0,q 2 } ˆδ({q0,q 1 },b) = {q 0,q 2 } %5(

37 & 5 33 %5( ) L( ) = L() % ( ' x L() (q 0,x) (q f,ε) q f F x L( ) ({q 0 },x) (S,ε) S q f F. c 2 x Σ q Q ) (q 0,x) (q,ε) ({q 0 },x) c (S,ε) q S. x ) %57( x = 0 (q0,ε) (q,ε) " q = q 0 ({q0 },ε) c $ * x = ya " 57 ) + ) y (q 0,x) = (q 0,ya) (q,ε) q Q (q 0,ya) (q,a) (q,a) (q,ε) (S,ε) S = {q 0 } % ( q Q (q 0,y) (q,ε) (q,a) (q,ε) q Q ({q 0 },y) (S,ε) q S q δ(q,a) ({q 0 },y) c ({q 0 },y) c ({q 0 },ya) c (S,ε) q S q δ(q,a) (S,ε) q q S δ(q,a) = ˆδ(S,a) c (S,a) q ˆδ(S,a) = S (S,a) (S,a) (S,ε) q S ({q 0 },ya) c ({q 0 },x) = ({q 0 },ya) c c (S,ε) q S. ε ε ε {aa,ab}

38 a a ε ε, 535), {aa,ab} ε. ε = (Q,Σ,δ,q0,F)! δ δ : Q (Σ {ε}) P(Q). $ ) ε %( w = aw % a Σ ( q δ(q,a) " a (q,w) (q,w ) %( w = w q δ(q,ε) % $ A = L() $$ ε $$ $$ $ ε $$ A = L( ) * = (Q,Σ,δ,q0,F) ε $ # ε # ε. q Q ε $ ε (q) ε (q) = {q Q (q,ε) (q,ε)}, ε (q) q + ) ε ˆδ(q, a) = F = (Q,Σ, ˆδ,q 0, F), q ε (q) δ(q,a); = {q Q ε (q) F } ε 53 ε b

39 a ε ε ε a ε a ε, 53) ε a a a., 53) $ # )) " $ "##"" $ # " " 2 & $$ $ % ( & & + % ( * A B Σ + ) %( A ) B ) % ( A B = {x Σ x A x B}; %( A ) B ) % % ( ( AB = {xy Σ x A, y B}; %( ) A %!( A k ) k 0 { A 0 = {ε}, A k = AA k 1 = {x 1... x k x i A i = 1,...,k} (k 1); %( ) A $ $ % %, ( ( A = k 0A k = {x 1...x k k 0, x i A i = 1,...,k}.

40 7 - A = {aa,b} B = {ab} A B = {aa,b,ab}, AB = {aaab, bab}, BA = {abaa, abb}, A 2 = {aaaa,aab,baa,bb}, A = {ε,aa,b,aaaa,aab,baa,bb,aaaaaa,aaaab,... }. * ε $, {ε} ε ) {ε}, ) = {ε}. % $ Σ $$ $ + ) %( ε Σ ) + " %( a Σ ) + a Σ " %( ) + r s Σ (r s) (rs) r ) + Σ " %( Σ ) +, Σ ) + r L(r) ) %( L( ) = " %( L(ε) = {ε} " %( L(a) = {a} a Σ " %( L((r s)) = L(r) L(s) " %( L((rs)) = L(r)L(s) " %( L(r ) = (L(r)) - {a,b} + ) r 1 = ((ab)b), r 2 = (ab), r 3 = (ab ), r 4 = (a(b (bb))). 1 ) L(r 1 ) = ({a}{b}){b} = {ab}{b} = {abb}; L(r 2 ) = {ab} = {ε,ab,abab,ababab,... } = {(ab) i i 0}; L(r 3 ) = {a}({b}) = {a,ab,abb,abbb,... } = {ab i i 0}; L(r 4 ) = ({a}{b,bb}) = {ab,abb} = {ε,ab,abb,abab,ababb,... } = {x {a,b} x ε x a ) 3 5 a x b }.

41 2 + % ( 1 L(((r s) t)) = L((r (s t))), L(((rs)t)) = L((r(st))), ) r 1 = abb, r 2 = (ab), r 3 = ab, r 4 = (a(b bb)). ) " = (dd.d.dd )(e(+ ε)dd ε) (dd e(+ ε)dd ), d d = ( ' ) e e = ( ). *, $$ % ( +!!"!!!!! $ a b (a b) ba(a b) (a b ) (a b) 1 + ) 2 + ) (0 1) (0 1) 10 (0 1) 1(0 1) 3,/'0 /,'' *-,/5/3-919* rr 3/(5'1660 )9,5-, +2,2/33'0 *++* r + 5'/+16 7 *,1***,1' 5'/+/,16 )9,./11-* A AA 3/(5'1660 *++* )*,*091**0 *0 A + A /0-+ 7 ;(9;/( +9,-(/ 7,'3/(5'0 +*-,/5/ 9'1*',''0,'', 5'()9'11** 324,* (d +.d.d + )(e(+ ε)d + ε) (d + e(+ ε)d )7 +

42 2 + r s $ r = s L(r) = L(s) %, + + ( 2 + r s +, ) r (s t) = (r s) t r(st) = (rs)t r s = s r r(s t) = rs rt (r s)t = rt st r r = r r = r εr = r r = r = ε r r r = (ε r) + +) r = rs t r = ts ε / L(s), r s L(r) L(s) " + - r = s r s s r (a b ) (a b) a b (a b) ba(a b) (a b) (a b) {a,b} (a b) a b + (a b) (a b ) {a,b} a b ba (a b) " + ba(a b) (a b) a b (a b) ba(a b) &" " $ ( $$ $ )) " # " " + ) $ $$ $$$$ $$ * $$ $ $$$$ $$, r ε r L( r ) = L(r) 5 ε

43 r = : r = ε : ε r = a (a Σ) : a r = s t : s ε ε ε ε t r = st : s t r = s : ε ε s ε, 537) 1 r ε r ε

44 4 ε a ε ε ε b b ε b ε ε, 53) 1 r = (a(b bb)) ε ε b a b, 53) 1 r = (a(b bb)) + ε 5 5, 5 37 ε r r - r + r = (a(b bb)) ε 53 $ " ε, 537 +, + " r = (a(b bb)) 53 + ε + $$$$ $$ $ $$ ) $ + ε! '! + - ) 1 Σ Σ + = (Q,Σ,δ,q0,F (! δ δ : Q - Σ P(Q) 6,,6 59,./0 9' 9++*,2216,-93*-11**< / ,,* 7 /,'1/112 3'0'39'01'*+-9('13' 59,5// *'0./1/( : 3'0',1','6 66(/++','6 *-193**11/)*< )* /116 *00/1-++*, ',/++6 5'/+/++6 9' 9++* -,/'1* /('+*','* /;6./1/(3' : 0',1','6 3'0'3'*-193**11/)* 7 b

45 6 ε ε ε, 5 3) 1 r s q i q q j q i rs q j t r s q i q q j q i rt s q j, 534) % - (, δ(q,r) (q,r) Q Σ ) (q,w) (q,w ) q δ(q,r) r - Σ w = zw z L(r) ) $ $$ $ $$ * 2 + ) ) 3 ) q i r q j q i r s q j s, 5 56)

46 3 r r r 1 r 2 r 3 r 1r 2 (r 3 r 4 r 1r 2 ) r 4, 5 5 3) 2 + %( ) 53 %-( & ) * q ) " # ) q ) * qi qj q ) + % qi = q j ( ' q qi q j 534 %( q 534 %( q $ ) %( %(, 5 55 )) ") # " ") &$ ( # $,! + % 3 3( + + # ) % ( + % ( 2 + # 2 % ( $

47 5 aa a a bb b b b b a ab a ba aa b (aa b)(ba) (bb a) ab bb a ba ab ab (aa b)(ba) (bb a) (ab (aa b)(ba) (bb a)), 5 55) 2 +

48 v u, 55) x = uvw A L = {( k ) k k 0}, + 2 #! # $ A $$ $ $$ $ $$ n 1 x A x n x = uvw uv n v 1 uv l w A $$ l = 0,1,2,... * A ) n ) x A x n, ) x % ( ) * x) n ) * ) q ) x ) u ) x ) q v u ) % 55( + x q w x uv n v 1 uv l w A l = 0,1,2,... ' L % ' ( ' a b " L = L = {a k b k k 0}. * L n 1 L ) x = a n b n x = 2n > n 1 x x = uvw uv n v 1 " q w u = a i, v = a j, w = a n (i+j) b n, i n 1, j 1. 6# uv 0 w = a i a n (i+j) b n = a n j b n L 2 L + ' +! +, # ) + A

49 n A + n n - L = L a n b n a a n b n L a n+1 b n / L L + ' - # L + (aa) (bb) L 1 n a n b n L 1 a n+1 b n / L 1 L1 + * % $$ ( - A = {c r a k b k r 1,k 0} {a k b l k,l 0} 1 5 c a ) b ) A $ ) A + " + A ) B " % ( B + " %5 ( A A # + L = {a k b k k 0} %( + + % ( %( + + ) {a,b,c} B + % ( B {a,b} %5( B c + %( + A B = A L(cc a b ) = {c r a k b k r 1,k 0} + %( L = B {a,b} = {a k b k k 0}., L + + A + %5(

50 " (' $ "&## ()( ") ($ )") $$$ % ( % #( + +, %!( $ $ - + %, 7 (,+ % $&. & 1 2 ( # % ( - L = {( k ) k k 0}., + +, S ) %( S (S) %( S ε - ((())) ) S (S) ((S)) (((S))) (((ε))) = ((())). + %( + %(, 7

51 % ( a ) - ) E % #( T % #( F %! #(" E $ $ + ) E T E + T T F T F F a (E). - + % (a + a) a () E T T F F F (E) F (E + T) F (T + T) F (F + T) F (a + T) F (a + F) F (a + a) F (a + a) a. % $ %! ( V " G = (V,Σ,P,S), Σ V % - ( " N = V Σ $ $ % - ( " P N V % ( " $ $ % - ( S N ' (A,ω) P A ω γ V G % ( γ V γ G γ γ = αaβ γ = αωβ % α,β,ω V A N ( G A ω G γ γ γ V % ( γ V G γ G γ %066014)/0 /,'125,/,, ,,6 5621/112 +2,/002,3/(5' A ω1 ω 2... ω k - 5-**3**0 )9-559*,*3**0 6+'5/,23 9+''0 A +''112'6 *',19/,19','*,66014)6 {A ω1, A ω 2,... A ω k }7

52 V ) γ0,γ 1,...,γ n % n 0 ( γ = γ 0 G γ 1 G... G γ n = γ. - n = 0 γ γ γ V G G γ γ $ γ V G %! ( S γ ' ) G G x Σ G ) $ % (, G $ % - ( ) " ) G G L(G) = {x Σ S x}. G. L Σ %!( + - L = {( k ) k k 0} G = ({S,(,)}, {(,)}, {S ε,s (S)},S), L G = (V,Σ,P,E), V = {E,T,F,a,+,,(,)}, Σ = {a,+,,(,)}, P = {E T, E E + T, T F, T T F, F a, F (E)}. L G = (V,Σ,P,E), V = {E,a,+,,(,)}, Σ = {a,+,,(,)}, P = {E E + E, E E E, E a, E (E)}. G L G % 7(!!! $ + "

53 T L(T) ) P B L(B) C L(C) * P ) A + L(B) L(C) P ) A B C # P = P {A B C} L(A) = L(B) L(C) L(B)L(C) P ) A BC # P = P {A BC} L(A) = L(B)L(C), L(B), + A BA ε # % ( A AB ε # % ( L(A) = L(B) = {x 1...x k k 0, x 1,...,x k L(B)},+ + - A BAC ε # P A C B ) L(A) = {x 1...x k y k...y 1 k 0, x 1,...,x k L(B), y 1,...,y k L(C)}., + * & {S asb as Sb ε} + a b %2 {S as T, T bt ε} ( " $# ) )" $ "&# ) $'( $, + ) - ) A,B,C,...,S,T ' ) " a,b,c,...,s,t " " 0,1,...,9 %!)... ( % () X,Y,Z ' ) u,v,w,x,y,z 2 ) α,β,γ,...,ω ' ) A A ω 1, A ω 2,...A ω k A ω 1 ω 2... ω k.

54 4, + ) A 1 ω ω 1k1 A 2 ω ω 2k2 A m ω m1... ω mkm " $ + $ " A1 )) " # " " $ " " # " (' % L L ( ,+,+ $$ $ % ( A ε A ab $$ $ %! ( A ε A Ba # * + + $$ " % $$ $ $$ $ $$ $$$ * + L Σ = (Q,Σ,δ,q0,F) % ( 2 G L(G ) = L() = L, ) + G ), + - Σ Aq ) q ) Aq0 %( ) q F Aq ε " %( ) a q q % q ( δ(q,a) A q aa q, Aq L(A q ) = {x Σ A q G x}. 83,/'0,*++'1**0 9'5/*++/ )* *,/33*++/ +'0/***(',1/0 5'/+'9;;'/0 366('1/+3',,6 324, 3-919* A a 9+/*1 ;(9.-51'91 7 0,/+;;9 19./1*< / **)/00-, /' 3--1* 5'/+'9;;'/0 5-*-,9'3** 7

55 76 a, b b b, 3), x q - x L(A q ) (q,x) (q f,ε) q f F. L(G ) = L(A q0 ) = {x Σ (q 0,x) (q f,ε) q f F } = L() = L. - 3 ) A 1 aa 1 ba 1 ba 2 A 2 ε ba 2. % $$ $ $$ $$$ $ $$ * G = (V,Σ,P,S) L(G) G = (Q,Σ,δ,q S,F) ) $ G G ) ) Q = {q A A V Σ}. G ) G ) + - S qs G ) + G ) Σ G )! δ G ) A ab q A a qb % qb δ(q A,a) ( G ) G ) ε ) F = {q A Q A ε P }., G G

56 73 " " ") # " ('' ") ")) () " $ + + $ % - - ( ) $ + G x * x L(G) # G $ 2 + +!!!!!!! * γ V G = (V,Σ,P,S) S G γ S γ S = γ 0 γ 1 γ n = γ γ ) % ( G ) " n 1 " a + a G % ( ) %( %( %( E E + T T + T F + T a + T a + F a + a E E + T E + F T + F F + F F + a a + a E E + T E + F E + a T + a F + a a + a., ) γ γ %! ( γ γ, lm ) %( % ( γ γ ; rm %( 2 γ lm γ γ rm 2 ) %( %( # - - % ( % ( γ

57 75 E E + T T F F a a, 5) 1 a + a G - 5 ) G = (V,Σ,P,S) +, G % ( ) %( V {ε} % N = V Σ ( + - S " %( A X1,...,X k A X1... X k G ) τ % ( % #( - 5 a + a # S = γ 0 γ 1 γ n = γ ) %( S " n = 0 " %( S X1 X 2...X k k X1,X 2,...,X k " %( Xi Y 1 Y 2... Y l i ) l Y1,Y 2,...,Y l ", τ S γ τ ) γ

58 7 E 1 E 1 E 2 T 2 F 1 a 1 + T 1 F 2 a 2 E 2 + T 1 T 2 F 2 F 1 a 2 E lm lm E + T lm a + T lm T + T F + T lm a + a a + F lm a 1, ) * + τ G x τ ) ) % x #( % ( * % #( S x lm " + & ) % % $ G = (V,Σ,P,S) $ $$ $$ G $ $$ γ G τ γ G τ x S x S x lm rm % * $$ G $$$,+, G + * x L(G) # x!! 1 - a + a a G % ( *,1**--, /' ;6./ 3'/+'*+1*','++/ +5/,5/0/(6','++/- +*-,/)9,.95,'++/* 5*'5'++* +*-,/)9,.95,'++* 90 )6,/002, : ;--< 3-11* /' *,/01* /'56 9'5/** )9,19* /,'3/(5'5,' +*-,/)9,.95,/1 T + F )* F + F,'-0 )9,19/,'3/(5'0 59,.*,,* '' 7

59 7 E E E + E E E a E E E + E a a a, ) 1 a + a a a a,+ G $ % - ( ) G x G ) $ % - (,+ $ $ % - ( - G, G G L = L(G + 1 ) G {a i b j c k i = j j = k}, % $ $ ") )"), % ( G x G ) + - % ( % ( x ) G ) E T + E T E T T a (E). - a a G

60 77 ) E lm lm T + E T E lm lm lm a + T (E) + T a E " " a T + E a a + E " lm a (E) + E " lm a T E a a E " lm a (E) E " lm a T a a *, lm lm lm lm * A A + Aγ!!!!! *, - G 11 %3( + # E % 74( & G ) G ) E TE E +E E ε T a (E)., G ) E a a E lm TE a a! ae a E a TE a ae lm lm 11 %3( - G "! + % # + ( 2,/00/ < 1*' 2+/',/33'0 k < 1-+// )6,/002,;(9,/,,'0 /0-+*00'05'/+',/,16 5-*-5,/,1** + /, 1 : 19 ('-,1, *0< ;(9.- '0- * / :, 1 ;*(,/< '1, k, *,/* '0 /((*0 +**)/3;' 1/,955**,1')6,/00/116'/0 5'/+'9;;'/ * 9*1 0, 7 k : 5'/+'9;'1 + / 1, : 19 : ('-,1, *0< '-,1 ;*(,/- 7 /6'./0 )6,/002,*+-9('13'1 9*1 5-'1/05'0,/0 /((*0 3-15'55**3;'* /116 0''16 /' 56,'1/++6 16,,6 7 lm lm lm a a.

61 7! ) ) ) ) ) ' ) % % ) ) ) % && ) & ) ) % && ) & ) ) % ) % && ) & ) % && ) & ) ) &! ) & ) % (! )

62 7 % & ) ) ) - + ) ) E TE ae a E a TE a (E)E a (TE )E a (ae )E a (a + E)E a (a + TE )E a (a + ae )E a (a + a)e a (a + a).!!!, 11 %3( ) % -( ) A A ε + a b c d ) S Ab Cd A aa ε C cc ε. ) ) ) ) a b, 11 %3( S Ab c d S Cd ) S G =

63 7 (V,Σ,P,S) ).2 (A) = {a Σ A ax x Σ } {ε A ε} = {A ) } {ε A };.*11*0 (A) = {a Σ S αaaβ α,β V } {ε S αa α V } = { A ) G ) } {ε A }. - ) *11*0 (S) = {a,b,c,d},.*11*0 (A) = {a,ε},.*11*0 (C) = {c,ε} (S) = {ε}, (A) = {b}, (C) = {d}. 1.2 ).2 (ε) = {ε};.2 (a) = {a} a Σ;.2 (X1...X k ) =.2 (X1 )....2 (X i ) {ε}, ε.2 (X1 ),...,.2 (X i 1 ),ε /.2 (X i );.2 (X1 )....2 (X k ), ε.2 (Xi ) i = 1,...,k. ).2 (L) =.2 (ω). ω L, 11 %3( ) 11 %3( A A ω 1 A ω2 ω1 ω 2 ).2 ({ω1 }.*11*0 (A)).2 ({ω 2 }.*11*0 (A)) =. 11 %3( % ' () #!! ) " 2 - > 3... ) -* % ) (" " (*(55**0 911*/0 ) '*1,', ,'116',1/0 ;661/3/(55'/0 +',65,' 324, 12,)60 3/(55')9090 ε7

64 74... $ A ) A " A ) A ω1... ω n /!! [a 11,...,a 1m1 ] '!.2 ({ω 1 }.*11*0 (A)) = {a 11,...,a 1m1 } /! A ω1 % ω1 (!)! [a n1,...,a nmn ] '!.2 ({ω n }.*11*0 (A)) = {a n1,...,a nmn } /! A ωn % ωn (!) -* % A (! (!)" # ) (ωi ) (X1...X k ) (X1 );... ; (Xk ), (a) '! -* a % a! ( ; )' " a (B) B, B!!!! 11 %3( # 11 %3(! "!, A αβ 1 αβ 2, α ε, β 1 β 2 11 %3( ++ A ) A αa A β 1 β 2. α αβ1 ) αβ2 ) β1 β2 - G % 77( E T + E T E T * *'1,',''06 ;9'55/-5,/++',/,,* 1*;*-5,/,,*< /116 *'09* α * *3* ;661/) ε7

65 6 + E TE E +E E ε. "!!!!!, $$ A γ A + Aγ, α + β α ) β 11 %3( $ A Aγ A Aβ α, β ε, A αa A βa ε. A A - E E + T E T T + E TE E +TE TE ε. + $ % -! ( A ab 1... B k, k 0, S ε a B1,...,B k S + -!!! $ G = (V,Σ,P,S).2.*11*0 -.2 ) *'1,',''06 1*;*-5,/,,*< /116 (/5-(,'''0/0 )9, '55//++/ A *'09* 3*,.9++'0/0 3-11*,'++9'0 A *,1* /' 9'.* )9,1** 3'1660 ;661/)909*< )*,/ 9'.**0 ;9',1** 5'/+'9;',1* 1-91/11-* 5'/ *3*11* 7

66 3 3 $ a Σ ) A V Σ ).2 (a) := {a},.2 ) (A) := {a Σ A aβ G } {ε A ε ) G }. 5,.2 ) A X1...X k ).2 (A) :=.2 (A) {.2 (Xi ) 1 i k, ε.2 (X j ) j < i} {ε ε.2 (X j ) j = 1,...,k}..*11*0.2 ) 3 $ B V Σ ).*11*0 (B) := {.2 (β) {ε} A αbβ G ) }, + - S ).*11*0 (S) :=.*11*0 (S) {ε}. 5 2.*11*0 ) A αbβ ε.2 (β) ).*11*0 (B) :=.*11*0 (B).*11*0 (A). # # & ( & # " (', + $ % - ( - X # # X # X # X ) % -( X.s X.t, X X ) - $ % (, A X1...X k - $ % ( ! ) A X 1...X k + - A,X1,...,X k -

67 5 I v = 319 U v = 319 U v = 31 U v = 3 D v = 3 D v = 1 D v = , 7) $ , X - X.v X ) " v $ I +U I.v := U.v I U I.v := U.v I U I.v := U.v U D U.v := D.v U UD U 1.v := 10 U 2.v + D.v D 0 D.v := 0 D 1 D.v := 1 D 9 D.v := 9 ' U UD + ) U1 U ) U2 U1 U2 U ), 7 34#, - $- - t % ( A X1...X k + A.t := f(a,x1,...,x k ) t - " - v - % ( $- -

68 I v = 319 U U s = 10 v = 310 s = 1 v = 319 U 3 s = 100 v = 300 D v = 3 D v = 1 D v = 9, ) ' # - s #- v ) $ I +U U.s := 1, I.v := U.v I U U.s := 1, I.v := U.v I U U.s := 1, I.v := U.v U D U.v := (D.v) (U.s) U UD U 2.s := 10 (U 1.s), U 1.v := U 2.v + (D.v) (U 1.s) D 0 D.v := 0 D 1 D.v := 1 D 9 D.v := 9, - 34# $- - + $ " # # (a + b) c ab + c % ( 2 - " X

69 - X. ) X $ E T + E E 1. := (T. ) (E 2. ) ( + ) E T E. := T. T F T T 1. := (F. ) (T 2. ) ( ) T F T. := F. F a F. := a F (E) F. := E., + " - )!!! " $! $$ " ) ) " $ " ) ) # ) % ) ' ) " $ " % ' ) ' & ' ) & ' ) ' ) && " ' &! " & ) ) " &! " ' ) "! & ' " % ' ) ' ) ) "! &! " &'/+'9;'0,**11*3'0/0 : /./++2116',' 1*(55**0 911*/0 6+'55/'./0 E )* T ;(9.-51'9'./0 * :,/33*0 1/5')4'00'0 7 /5')4'01' 9'.**0 5-'1/05'0,', '3;+',''11',/,1' )6,/002,(-1''0/','0 /,'3/(5',16 '+3/0/6++6 1**++* 7

70 7 " $ # # " % ' ) ' & ' ) & ' ) # ' ) " ' & #! " && & ) ) " &! " ' ) "! & ' " % ' ) "! & #! " ' ) ) " $ # " # % && ) & ) " #! & " % && & ) ) " #! &! " &! ) & ) % # (! ) % & ) & ' ) ) ) ) )

71 & " )") ")) ")"" * 11 %3( ) + 11 %3( n % + O(n) ( + ' % ( + + n O(c n ) c 2 + n 2 $ + O(n 3 ) n ",, ε!!! * G = (V,Σ,P,S) + A V Σ % -( A G ε % + $ G $ $ G $ $ ε L(G) * G = (V,Σ,P,S) 2 G ) ) %( & 11 := {A V Σ A ε G ) }; %( & 11 ) & 11 := & 11 {A V Σ A B 1...B k G ) Bi & 11 i = 1,...,k}. G ) A X1...X k A α 1...α k, { Xi, α i = Xi / & 11" X i ε, Xi & 11

72 1 + A ε + - S ε G S S S S ε - ) ε S A B A aba ε ( & 11 = {A,B,S}) B bab ε S A B ε A aba aa ε B bab bb ε S S ε S A B A B aba aa bab bb. "!!! ' A B A B % ( % $ G $ $ G $ * G = (V,Σ,P,S) 2 G ) +# ) %( A V Σ ) F(A) := {B V Σ A B ) G }; %( ) F F(A) := F(A) {F(B) A B ) G }. ) + G ) + A ω B ω G B F(A) - + S S ε S A B A B aba aa bab bb.

73 + ) F(S ) = {S,A,B} F(S) = {A,B} F(A) = F(B) =, + S aba aa bab bb ε S A B aba aa bab bb aba aa bab bb. % S # + % ((!!,+ G = (V,Σ,P,S) $ % "! ( S S ε 1 A BC A a A,B + - C a S % # $ G $ $ $ G * G = (V,Σ,P,S) + - S G ) S G ) S S ' G ) ε + 7 G ) A a A X 1... X k k 2 % S ε S ε ( 1 a Ca C a a, A X 1...X k k 2 A X 1 A 1 A 1 X 2 A 2 A k 2 X k 1 X k, A1,...,A k 2 % A X 1A 1 A 1 X 2A 2 A k 2 X k 1 X k, { X i Xi, = C a, Xi V Σ; Xi = a Σ.)

74 4 - S B C abcd bbb b c S C a S1 1 S1 1 BS2 1 S2 1 CC d S C b S1 2 S1 2 C b C b B b C c C a a C b b C c c C d d.! * G = (V,Σ,P,S) + 1 G ", x L(G) + ) x = ε x L(G) S ε ) G ) x = a 1...a n x ) ) Nik A ) x i k xik = a i...a i+k 1 N ik = {A V Σ A G a i... a i+k 1 }, 1 i i + k 1 n. 2 x L(G) S N1n Nik - A i = 1,...,n A Ni1 G ) A ai * k 2 G " A xik = a i... a i+k 1 xik xij = a i...a i+j 1 x(i+j)(k j) = a i+j...a i+k 1 j = 1,...,k 1 B xij C x(i+j)(k j) G ) A BC Nik )

75 6 i N ik 1 : b 2 : a 3 : a 4 : b 5 : a 1 B A,C A,C B A,C 2 S,A B S,C S,A k 3 B B 4 S,A,C 5 S,A,C, ) ",, S C B C B A A B C b a a b a, ) - ",, Nik $ %( $ i = 1,...,n ) N i1 := {A V Σ A a i G ) }. %( k = 2,...,n k i = 1,...,n k + 1 Nik ) N ik := k 1 j=1 {A V Σ G ) A BC, B N ij C Ni+j,k j }. - ",, " S AB BC A BA a B CC b C AB a

76 3 N ik, 4) ",,,241/0*-,** ' 0 ; - 1 0*-,*;66* 9,)*-,25,'554* q 1 q 2 q 0 δ ;' *-,** & +. 0, 36) ' + x = baaba $ + - S N15 x # + S N15 baaba, ",, Nik # j 1 k # 1 % 4( N ij Nik Ni+j,k j! )($ ( $$ %! ( + % 36( + + ). )

77 5 % %! ( Q $ " Σ " Γ " = (Q,Σ,Γ,δ,q 0,F), δ : Q (Σ {ε}) (Γ {ε}) P(Q (Γ {ε})) % ( " $$" q 0 Q F Q $$ 2! δ(q,σ,γ) = {(q 1,γ 1 ),...,(q k,γ k )} + q σ γ q1,...,q %' k γ1,...,γ k ( + σ = ε " γ = ε % # ( % # ( γ ε γi = ε $ $ (q,w,α) Q Σ " $$ Γ $$ x (q0,x,ε) (q,w,α) + + q w α (q,w,α) (q,w,α ) (q,w,α) (q,w,α ), w = σw α = γβ α = γ β % ( σ, γ, γ 1 (q,γ ) δ(q,σ,γ). (q,w,α) $ (q,w,α ) (q,w,α) (q,w,α ), (q0,w 0,α 0 ) (q 1,w 1,α 1 )... (q n,w n,α n ) n 0 (q,w,α) = (q 0,w 0,α 0 ) (q 1,w 1,α 1 ) ' x Σ (q n,w n,α n ) = (q,w,α ). (q 0,x,ε) (q f,ε,α) qf F α Γ,

78 a, ε/a q 0 a, ε/a q 1 b, A/ε b, A/ε q 3 b, A/ε q 2 b, A/ε, 33), {a k b k k 0} + " $ x ) $ $ ) L() = {x Σ (q 0,x,ε) (q f,ε,α) q f F α Γ } {a k b k k 0} ) = ({q 0,q 1,q 2,q 3 }, {a,b}, {A,A},δ,q 0, {q 0,q 3 }), δ(q 0,a,ε) = {(q 1,A)}, δ(q 1,a,ε) = {(q 1,A)}, δ(q 1,b,A) = {(q 2,ε)}, δ(q 1,b,A) = {(q 3,ε)}, δ(q 2,b,A) = {(q 2,ε)}, δ(q 2,b,A) = {(q 3,ε)}, δ(q,σ,γ) = - + aabb ) (q,σ,γ). (q 0,aabb,ε) (q 1,abb,A) (q 1,bb,AA) (q 2,b,A) (q 3,ε,ε)., q3 F = {q 0,q 3 } aabb L() ' + + ) % 1 $ $$ $$

79 q 0 ε, ε/s# ε, S/aSbS a, a/ε b, b/ε q ε,#/ε ε, S/ε ε, S/bSaS q f, 35), {S asbs bsas ε} α, β/x 1... X k α, β/x k ε, ε/x k 1 ε, ε/x 1, 3) 1 ) + G ) G G ) x G S x ) lm G " %' {S asbs bsas 35 ε} 3 ( ) abab (q 0,abab,ε) (q,abab,s#) (q,abab,asbs#) (q,bab,sbs#) (q,bab,bsasbs#) (q, ab, SaSbS#) (q, ab, asbs#) (q, b, SbS#) (q, b, bs#) (q,ε,s#) (q,ε,#) (q f,ε,ε). ) abab S asbs absasbs abasbs ababs abab.

80 7 ' (q,w,α) (q,w,α ) (q,w,α) (q,w,α ) % 6( - {ww R w {a,b} } % (,+ + + " " ") # " ") &$ ( # $,+ + ) +!, %1 5 ( $ $ uvwxy # % 2 ( $ L $ $$ $ $$ n 1 z L z n z = uvwxy vx 1 vwx n uv i wx i y L $$ i = 0,1,2,... * G = (V,Σ,P,S) " L ) + ) %' ( G h 2 h z L log2 z * $ k = V Σ G n = 2 k+1 - z L z n k + 1 " A k + 2 z 3 z = uvwxy w A ) vwx ) A " S uay uvaxy uvwxy.

81 S S A A A A u v w x y u v v x x y A, 3),+, S uay A vax A w v x # w ) ) S uay uvaxy uv 2 Ax 2 y... uv i Ax i y uv i wx i y. 2 uv i wx i y L i = 0,1,2,..., G " A vax vx 1, A k + 1 vwx 2 k+1 = n - 4 L = {a k b k c k k 0} + * L + " n z = a n b n c n L 1 z v w x z = uvwxy, vx 1, vwx n. ) ) ) vx a b c v 0 wx 0 y = uwy L uwy L

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`^8)#& : a)&& *"##!#""$# (%%#$))=$ _&*$#'$ P0125RZ X+Y+#& λ >* )*==)# P0125RZ b'"$#& P0125R : c *'!#& P0130R (**#:'&'(&&'""="$(#$Z "*% * #**# &=*=#"$ '+:)('#&$#*#)$;.%(%& ('&#"$& $!'#$$#8& * && )$.#&/#& *'&#$!'#8 & : $)) +=!#& =*"#&> *$ #"& '+:)('#&$#_'( )#"(#& Z :'))!'#8 & #)( #"$ * #**# (#$%! +!((#)) *#& '+:)> ('#&$#*#)#))% 6 $'"#& *^(9)^"$#;.#&/#& *'&#8& $ '& "##&%Z $$% :# =*"#&*$ #"> $&" $ *# &##$% '& +)99' *%=$$%% (* &#"& ) "*$$!8& P"'; '+:)('#$!8&R :': *'"*!#"" =)#"#""% $ * "$)#"" ; d&$#$##!#""$#.#&/#& *'& '& *#& %%))#&& $'( $$#Z :')) "=^$& + & "#: & '& $'#"& "&$ & )'9$$'( & 9#$*% $=^& + Z :'$ *'& 9="$== & + 9%%&!%)#$=*"))% )*> ( & : *#:'#$$ ( & (*#& * )) & P*! e;0r; < + & )"" '& #$=#&& )*(**# f g Z : "& *%=$$$=% '" " )'99(**# f g ; a'& 9="$== )*#"" & +! #$"( & &%(% (*#$Z ($$ " # 9="$= *#:'#$$ ( & &##$%; P. *((#& " &'&Q )*(*#& $#) )) *'& # " *#:'#$$ (#$%%& ($ (**#%Z : )'99(**# "##$== $'( $$#""$# $&9%#& "#$% (* *#& *'& *#:'#$$ & + )) )#"%% (**:%;R. * "$)) & &"#&.#&/#& *'&#8& *%=$$%(#"$% _'( )#& *#)$& $&&#"$ (#"&h (=^+((#& *%"#$))%%& (=^" _&*$#'#8& ) "*(#"$ &%#))% $'( $#)) ; a#)$& $&&#"> $ (#"$! $& '& *"" *#&.#&/#& *'&"" * *"# )'99$#) Z +=!%*"=!% qijj : +=)*%%!% qklm ; -&&$& (**#:'&'& $ * "$ (#"*"# " *#:'#$$ & *'&& & + )) "&! "(9 & ) #> $ & P )*(**#% )*&'$$ ( $$ RZ & + 9%% "#:'#$$ & '"'#$$ ( & :'&'& &"#((%#"$% nn

83 n q 1 q 2 q 0 δ a! e;0q.#&/#& *'&; (**#%Z : *'& *%=&&#"$$%%& )*$#) "" q0 ;.%"$% )%+$#& *'& $'#(## "*)#$$ #& "##$=> (%_&*$#'&" '+: ( & Q =+8""% "##$=(%""% " )* & + 9%%& *'+8 )) ')! & (*#& : 9%%$$%% "#$$& $#) &" : )$& (*#& 9"$)) Z (#*% '& "# $#) Z & + 9%%& *'+8 )> ) *#:'#$$$! "# (**#Z : "##$==*^ & + 9%% *%"#$))=& (*#& *'+8 )$ =+8& "*)&! &! "(( ))! # '#* )); '" *'& #* & & 9="%+$== +=!%*"=!%""% )'99$#) "" (**#:'&' *) *'&& $&&#"$ ( & *#)&h :'" " $ " 9="%+$== )'99$#) "" qijj Z qklm IOD NN! "KN@I "LNIINZ (**#:'&' # *) *#)&;.%"(%))#""$#!'#8 & (%%#$))%Q # $$%&'()* $ +,- GABDEHDE TEJ P&/);.#&/ ( Y+#&R '& "#$"#**' = (Q,Σ,Γ,δ,q 0,qijj,qklm), (#""% '& *'&& IDUT JE %%))#&& :'**'h Q '& *'&& K Σ ".IJOOTKITh '& *'&& EOA@O/OOTKITh ')$$ &Z $$% (*#$ h Γ Σ, / Γ '& *'&& KDDBI δ : (Q {qijj,qklm}) (Γ {, }) Q (Γ {, }) {L,R} "LN0AEIDTh "##$=(%_&*$#'&!'#)$! 8#$ &Z $$% P#R :'" b = Z &##& a = h P##R :'" a = Z &##& b = : = R h P###R :'" b = Z &##& a = : = L h '& *'&& OUAIDUOh q 0 Q {qijj,qklm} δ(q,a) = (q,b, ) qijj Q '& "SNK " SN : qklm Q Z qklm qijj Z "UNNSN UT!! AIDUO; : ;<=5:<746>5=6>9 5=7<=5 :>?@>;33: 75 =>8856> ;;=;3 ;>5<>88>>: =5546;9 3==3A B>5<>88>>: ;;=;9 5= : 63 CD5E>>: 6>4C566>7==> F 38F = :>?@>;3363 ;@E7: >=<787: C744>: 7E7=6><>5=5: A

84 n1 ##$=(%_&*$#'&!'& δ(q,a) = (q,b, ) $)*#&$ '&Z $$% '))"" & $#) "" q : )*#"" & & + (*#& P$ # )*> $ # )'99(*#&R a Z *'& "##$== $#) & q Z *#:'#$$ )*( &" 9 #** & (*#& b Z : "##$%% & + 9%%$% =+8& (**#9 #* &! & "&$ & P L [)_$\Z R [#/+$\R; ))#$$: *#:'#$$$!# (**:% : "##$'"&$# '& :'#$$$Z (#*%)# a f g $ # f g Z : "##$=(%_&*$#'&!' '& #& (%%> #$$)(%$^&Z *& q = qijj $ # q = qklm ; '$"" & :'(9 & *(9 & &%#"$% $#)'#"$ *'& 9="%+$== +$#; a'&& IDUOEEJ '& &)#**' (q,u,a,v) Q Γ Z (#""%!'# ')) (Γ {ε}) Γ a = ε Z (#*%)# (=^" u = ε $ # v = ε ;.#) &$& (q,u,a,v) #&$#$##!#&& $)*#&$ '&Z $$% *'& '& $#) "" q Z & : + & & + "#"%)$^ & "#"%)$^ "& & )"$ + & 9%%& + '#* 9%%& )$! "(( 9'))$ )) *%=$$=& 9'))) '& '" u & Z )'99& & + 9%%& '& *'+8 )) '& (**# ; c +8'))#""$# '& a v Z:'" & + 9%% "#: #$" #! & & + & )"" $ # "& *%=$$=& '" & )'9"" ; &"#((%#"""% a = ε $ 9 *""" : $)) &Z $$% *'& [+! #$"\ (*#& f UAIDUOEEJ K g : $'#""" $ 9 *""" (*#& f g ; ". IIJJUUN x = a1 '& &)#**' ;.#) &&$$ a 2... a n (q0,ε,a 1,a 2...a n ) (q,u,a,v) (*#$%%& =)&"% =*"#&*$ #"((#& (q,uav) Z : )*$#) &&$$ "=^$$))% x =*"#&*$ #""$# ) $#'$ Z :'"" $#) && (q,w) T@IOO KATBOOE $#) &$"& (q Z (*#$%%& $! ))#"&,w ) $ 9 & (q 0,x) ; (q,w) (q,w ), : " (%%#$))%%& *'&& "##$=(%_&*$#'& 9'+: )$ "! "$#Q * #*#)) Z q,q Q u,v Z Γ a,b Γ : c Γ {ε} Q :'" Z &##& h δ(q,a) = (q,b,r) (q,uacv) (q,ubcv) :'" δ(q,a) = (q,b,l) Z &##& (q,ucav) :'" δ(q, ) = (q,,r) Z &##& (q,εcv) :'" δ(q, ) = (q,b,r) Z &##& (q,uε) :'" δ(q, ) = (q,b,l) Z &##& (q,ucε) h (q,ucbv) h (q,cv) (q,ubε) h (q,ucb) h :'" Z &##& δ(q, ) = (q,,l) (q,ucε).#) &$$Z :'$* '! $ ('$' $ # (qijj,w) #!%$ :'+8 (#+#&*%%& (+& $#) &$"&; (qklm,w) <%#""% $#) &$#"" *'&! " KN@I "" ;.#) && (q,w) T@IOO IDUOEIJJKJJE Z (*#$%%& (q,w ) (q,uc) ; (q,w) (q,w ), :'" '& ')( "" $#) &&:'&' Z Z Z Z (q0 Z "#$& $$%,w 0 ) (q 1,w 1 )... (q n,w n ) n 0 (q,w) = (q 0,w 0 ) (q 1,w 1 ) (q n,w n ) = (q,w ).

85 7.#&/#& " SNK "" (**#:'&'& x Σ Z :'" (q 0,x) (qijj,w) " UNN x Q&; a'&& IAEEDKIOLO DJUD '&Q :')) *#& w Γ ; :')) *#& L() = {x Σ (q 0,x) (qijj,w) w Γ }. & $%*%% +'( $ Z $$% *'&& # $!#$" 9="%+$=% "=^$$#))%Z :'$* #!%$ *) *#)& L() ; '" *#$&*#& 9="%+$== (=^" * #*#)) +=)%$$%!#))% "=^$$#))%Z " &'$ & $$% '& ITIOOUDEJE : $$% " BOIODKJJ *#)& L() ;!"" 5 $)) & &%*(%%&Z $$% *="=(=" "##$%!'#8 &*' &&$$ *#)# $&&#"$ $'$ )#")).#&/#& *'&)) '& #$$%#& 9%$#!# )#; #$& *#)& [$&&#"$ (#"&\ : [ $* #"(#"&\!%)#&& ' '&.#&/#& *'&#8& $ 9 *""" #! & *"*#&&; "#(*#*"# *#)# {a!'#8 & $&&#"$ P: $* #"$ R.#&/#& *'&)) 2k k 0} (#""% = ({q 0,q 1,qijj,qklm }, {a}, {a},δ,q 0,qijj,qklm), δ(q 0,a) = (q 1,a,R), δ(q 1,a) = (q 0,a,R), δ(q 0, ) = (qijj,,l), δ(q 1, ) = (qklm,,l). a'&& ) "*&$ "#(*#*"# "=^$$))% aaa $& "! "$#Q (q 0,aaa) (q 1,aaa) (q 0,aaa) (q 1,aaaε) (qklm,aaa). a'"* *'& 9="%+$== $#) "" qklm Z 9%%$))%%& $$% ; aaa / L().#&/#& *'&$!'#8 & *%$!#((#& "#$$%% " ( &$ 9 #"#)) *!#'#)) *#& ')# *%=$^""% %%))#"#)) $'( $#)) : 9#&' $'( $#))h *!#'"#$=*"#""% *%=$$=$ (*#&&%$ '& "#$)$= *! "" e; ; "#(*#*"# 8))#"$%Z *#)& {a $&&#"$!.#&/#& *'&$$! "$! 2k k 0} *!#' '& "#$$$= *! "" e;2; c$*#** (9 & "#(**#&% '& *! "" e;e "#$$$= #>=+$=8$$^(%& *#)& {a k b k c k k 0} $&&#"$! P: $* #"! R *'&; a!#'& =*"#&*$ #"$ (#"*"# '& $%""% &'8 $$$ *%=$%&> $^%Z :'&* (* & "##$=(#% +=)*%%!%%& )'99$#) & # (*#$% &%*=!##&; a! $& *'&& #8 '&Z $$% " 9#$%% *#: "=^$$"$% $ 9 (#"$ & a >Z b > : c >(*#"$% ($$ ( )) & =*"# * )) & A Q*"#Z B Q*"# : C Q*"#; c$$$ &!##(#"& 9#&& a Q& #"'*"# *'& $ * "$ Z $$% (=^"*%%& 9#&#% b > $ # c >*#: #(# # ') :%):))%; "#(*#*"# "=^$$"& aabbcc )##$$=!%

86 0 q q q 0 qijj qklm a/b, q q δ(q, a) = (q, b, ) a! e; Q.#&/#& *'&#8& *!#'"#$=*"& (*#&&%$; a/a, R q 0 q 1 /, L a/a, R /, L a! e;2q a#)& {a $&&#"$!.#&/#& *'&; 2k k 0} a/a, R B/B, R b/b, R C/C, R a/a, R b/b, R q 0 q 1 q 2 /, L a/a, R c/c, L q 4 A/A, R q 3 /, L q 5 B/B, R B/B, R C/C, R C/C, L b/b, L B/B, L a/a, L a! e;eq a#)& {a $&&#"$!.#&/#& *'&; k b k c k k 0}

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